Inkjetprintheadperformance enhancement by feedforward ... · PDF file• DOD electrostatic...

Inkjet printhead performanceenhancement by feedforward inputdesign based on two-port modeling

Inkjet printhead performanceenhancement by feedforward inputdesign based on two-port modeling

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op

maandag 12 februari 2007 om 12.30 uur

door

Matthijs Benno GROOT WASSINK

werktuigkundig ingenieurgeboren te Leiden

Dit proefschrift is goedgekeurd door de promotoren:

Prof. ir. O.H. BosgraProf. dr. ir. D.J. Rixen

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf. ir. O.H. Bosgra Technische Universiteit Delft, promotorProf. dr. ir. D.J. Rixen Technische Universiteit Delft, promotorProf. dr. ir. J. van Eijk Technische Universiteit DelftProf. dr. ir. M. Steinbuch Technische Universiteit EindhovenDr. ir. J.F. Dijksman Philips Applied Technologies EindhovenProf. dr. D. Lohse Technische Universiteit TwenteDr. ir. S.H. Koekebakker Oce-Technologies B.V.Prof. ir. R.H. Munnig Schmidt Technische Universiteit Delft, reservelid

This research is supported by Oce-Technologies B.V. in Venlo, The Netherlands.

The research reported in this thesis is part of the research program of the DutchInstitute of Systems and Control (DISC). The author has successfully completedthe educational program of the graduate school DISC.

ISBN 978-90-9021484-9

Copyright c© 2007 by M.B. Groot Wassink

All rights reserved. No part of the material protected by this copyright notice maybe reproduced or utilized in any form or by any means, electronic or mechanical,including photocopying, recording or by any information storage and retrievalsystem, without the prior permission of the author.

Voorwoord

Degene die het promoveren associeren met vier jaar lang zwoegen achter een com-puter in een hokje op de universiteit, kan ik direct een illusie armer maken: deafgelopen vier jaar hebben mij in ieder geval het tegendeel bewezen. Zo heb ikde enorme vrijheid in het onderzoek, het verdiepen en verbreden van kennis envaardigheden, het samenwerken met Oce en het deelnemen aan internationale con-ferenties ervaren als een combinatie die uniek is bij een eerste ’baan’. Toegegeven,het zwoegen klopt wel af en toe, maar ja, bij welke baan heb je dat nou niet?

Kortom: het is een prachtige tijd geweest. Maar wat het vooral mooi heeftgemaakt is de samenwerking met een (flink) aantal mensen. In dat kader gaat mijngrootste dank uit naar Okko. Hij heeft mij zowel de vrijheid als steun gegevenbij het opzetten en uitvoeren van dit onderzoek: zijn onovertroffen kennis eninzicht is van enorm belang geweest bij de totstandkoming van dit proefschrift.Ook Daniel ben ik veel dank verschuldigd. De inhoudelijke discussies vanuit zijnexpertise heb ik enorm gewaardeerd en hebben het behaalde resultaat aanzienlijkverbeterd.

Daarnaast heb ik het geluk gehad om tijdens de promotie vier goede afstudeerderste hebben kunnen begeleiden: Anton, Niels, Ferry en Pieter. Het pressiemiddel’de exponentiele functie’ heeft zeer zeker effect gehad: jullie resultaten zijn danook terug te vinden in dit boekje, waarvoor dank! Ook dank aan Oce voor hetmogelijk maken van dit onderzoek en de ondersteuning die ik vanuit Venlo hebgekregen: Sjirk, Herman, Rob, Marc en vele anderen dank!

En wat zou promoveren zijn zonder mede-lotgenoten? Met veel plezier denk ikterug aan de vele humorvolle en relativerende gesprekken tijdens de talloze koffie-en lunchpauzes. Dank daarvoor aan alle oud-collega’s van de vroegere vakgroepSysteem- en Regeltechniek en het huidige Delft Center for Systems and Control.Het zijn er teveel om op te noemen... Tot slot: Elske, ouders, familie en vrienden,ook jullie dank voor jullie begrip en luisterende oren de afgelopen jaren!

Matthijs Groot Wassink,Den Haag, December 2006.

i

Contents

Voorwoord i

1 Introduction 1

1.1 Inkjet technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 A historical overview . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 A generic manufacturing technology . . . . . . . . . . . . . 8

1.2 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 An archetypal PIJ printhead . . . . . . . . . . . . . . . . . 9

1.2.2 Limitations of current designs . . . . . . . . . . . . . . . . . 11

1.2.3 Towards a controlled environment . . . . . . . . . . . . . . 16

2 Problem formulation 21

2.1 The research objective . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 A decomposition in research questions . . . . . . . . . . . . . . . . 22

2.3 The structure of this thesis . . . . . . . . . . . . . . . . . . . . . . 24

3 Experimental exploration 25

3.1 Description of the experimental setup . . . . . . . . . . . . . . . . 25

3.1.1 Piezo sensor signal . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2 CCD camera . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.3 Laser-Doppler interferometry . . . . . . . . . . . . . . . . . 31

3.2 Description of the experimental printheads . . . . . . . . . . . . . 33

3.3 Identification method . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Piezo-based experimental identification . . . . . . . . . . . . . . . . 40

3.4.1 With bridge-structure . . . . . . . . . . . . . . . . . . . . . 40

3.4.2 Without bridge-structure . . . . . . . . . . . . . . . . . . . 43

3.5 Laser-vibrometer based experimental identification . . . . . . . . . 45

3.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 47

iii

iv CONTENTS

4 Modeling of the ink channel dynamics 49

4.1 PIJ printhead model survey . . . . . . . . . . . . . . . . . . . . . . 49

4.2 The two-port model . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 The acoustic path . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.2 The fluidic path: the nozzle . . . . . . . . . . . . . . . . . . 60

4.2.3 The fluidic path: drop formation . . . . . . . . . . . . . . . 71

4.2.4 The fluidic path: a review . . . . . . . . . . . . . . . . . . . 78

4.2.5 The actuation path . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 The bilateral coupling . . . . . . . . . . . . . . . . . . . . . . . . . 86


5 Model validation 91

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Piezo-based validation . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Laser-vibrometer based validation . . . . . . . . . . . . . . . . . . 94

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96


6 The control framework 101

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2 The lifted ILC control structure . . . . . . . . . . . . . . . . . . . . 102

6.3 The control goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4 ILC design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4.1 LQ-optimal control . . . . . . . . . . . . . . . . . . . . . . . 112

6.4.2 Constrained ILC . . . . . . . . . . . . . . . . . . . . . . . . 116


7 Application of feedforward control 119

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2 Piezo-based ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2.1 SISO ILC: reducing residual vibrations . . . . . . . . . . . . 123

7.2.2 MIMO ILC: minimizing cross-talk . . . . . . . . . . . . . . 128

7.2.3 Constrained MIMO ILC . . . . . . . . . . . . . . . . . . . . 131

7.3 Laser-vibrometer based ILC . . . . . . . . . . . . . . . . . . . . . . 133

7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136


8 Conclusions and recommendations 143

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

CONTENTS v

A Hamiltonian ILC design 149

Bibliography 153

Glossary of symbols 163

Summary 167

Samenvatting 169

Curriculum Vitae 171

Chapter 1

Introduction

The importance of inkjet technology as key-technology for today’s industry has beenand still is the driving force behind the major improvements that this technologyhas undergone over the last decades. This thesis contributes to that developmentof inkjet technology. As justification of our particular research approach, an in-ventory of the current state of the art of this technology is essential. To thatpurpose, this chapter presents a characterization of inkjet technology. As a re-sult, the limitations of current designs will emerge, based on which several possibleresearch directions are identified.

1.1 Inkjet technology

In this section, a historical overview is presented of inkjet technology. Simul-taneously, the unique capabilities of piezoelectric inkjet technology compared toother forms of inkjet technology are addressed as well. Next, an inventory ofthe applications of piezoelectric inkjet technology is given illustrating its versatilefunctionality.

1.1.1 A historical overview

The rapid development of inkjet technology started off around the late fifties.Since then, literally countless inkjet devices have seen the light of day. In thisoverview, the attention is mainly restricted to the development towards the twomost important inkjet concepts of today, namely piezoelectric and thermal inkjet,see Fig 1.1. At the end of this section, both concepts are discussed vis-a-vis. Fora more extensive overview of the history of inkjet technology, one is referred to[Pon00].

The foundation of inkjet technology is attributed to the Belgian physicist Plateau

1

2 INTRODUCTION 1.1

inkjet technology

continuous

drop on

demand

binary deflection multiple deflection ...

thermal

piezoelectric

...

electrostatic

squeeze bend push shear (DOD)

(CIJ)

(TIJ)

(PIJ)

Figure 1.1: Classification of inkjet technology

and English physicist Lord Rayleigh. Though Plateau was the very first to publishon this field with his article ’On the recent theories of the constitution of jets ofliquid issuing from circular orifices’ in 1856 ([Pla56]), most of the credit belongsto Lord Rayleigh. He published a series of founding papers including ’Instabilityof jets’ in 1878 ([Ray78]), ’On the instability of cylindrical fluid surfaces’ in 1892([Ray92b]), and ’Investigations of capillarity’ in 1899 ([Ray92a]). Still, it tookseveral decades before application of these physical principles took place in work-ing devices. The first pioneering work in that direction was performed in the late1940s by an employee of the Radio Corporation of America (RCA), who inventedthe first drop-on-demand device. By means of a piezoelectric disc, pressure wavescould be generated that caused a spray of ink drops, see Fig. 1.2. However, thisinvention was never developed into a commercial product.

Figure 1.2: The first drop-on-demand inkjet device (US Patent 2,512,743)

The honor of the first commercial inkjet apparatus is considered to go to the

1.1 INKJET TECHNOLOGY 3

Minograf of the Siemens-Elema company released in 1952. Instead of being aninkjet printer, it was merely a voltage recorder quite similar to current seismicapparatus.

The early work of Plateau and Lord Rayleigh and the two jet-writing concepts canbe regarded as first steps towards inkjet printing. The rapid growth of electronicinformation systems in the late sixties induced a renewed scientific interest andstarted research into the two major directions of inkjet technology: continuousinkjet (CIJ) and drop-on-demand (DOD), see Fig. 1.1. During the sixties, progresswas established in three important regions:

• DOD thermal inkjet. With sudden steam printing, a researcher from theSperry Rand Company basically invented thermal inkjet printing, see Fig. 1.3.By boiling aqueous ink at certain time instances, a drop of ink could be gen-erated. The strength of this design clearly was not acknowledged, since thecompany did not elaborate this idea into a commercial product. The ideawas abandoned until the late seventies when Canon and Hewlett Packard(HP) picked it up.

Figure 1.3: Sudden steam printing (US Patent 3,179,042)

• DOD electrostatic pull inkjet. The basic working principle comprises thefollowing. Conductive ink is held in a nozzle by negative pressure. Byapplication of a high voltage pulse to an electrode located outside the nozzle,a charged droplet of ink is pulled out. By application of the appropriatedeflection field, the droplet can be located on the substrate. Companiesdeveloping electrostatic pull inkjet devices were the Casio, Teletype, andPaillard company. With the model 500 Typuter, the Casio company released

4 INTRODUCTION 1.1

in 1971 a printer of this type. The Inktronic Teletype machine in the late1960s was marketed by the Teletype company.

• Continuous inkjet. The major achievement in CIJ was the synchroniza-tion of the jet breakup. By adding periodic (acoustic) actuation, the ran-dom drop formation process becomes synchronized to that period as waspredicted by Lord Rayleigh. Consequently, the resulting droplets can becharged and deflected to the desired position. Main players in the field wereSweet of the Stanford University who came up with the Inkjet Oscillograph.This device was elaborated for use by the Stanford Research Institute (SRI)for inkjet bar coder work for Recognition Equipment Incorporated (REI).The A.B. Dick Company elaborated Sweet’s invention to be used for charac-ter printing. With their Videojet 9600 in 1968, it was the first CIJ printingproduct ever.

Despite these developments in inkjet technology, the products that came to themarket can be characterized as unreliable and having a poor print quality. Inthe seventies, the DOD electrostatic pull principle was abandoned due to poorprinting quality and reliability. The development of DOD thermal principle wasput on hold. Of the principles in development, only CIJ remained and was de-veloped further. In addition, the seventies are marked with the emergence ofthe DOD piezo-electrical inkjet, abbreviated as PIJ, principle. More specifically,these developments comprised the following:

• CIJ with binary drop deflection. This approach is depicted in Fig. 1.4. Thecharged droplets are deflected to the paper or to the gutter where it isrecycled. This track of research and development continued the work thatwas started in the sixties. Main players are the A.B. Dick Company, REI,the Mead Company, and IBM. The A.B. Dick company and REI continuedtheir work in bar code printing. The Mead company introduced DIJITin 1973 used for advertising purposes. The huge research efforts of IBMresulted in one product only, the IBM 6640.

HV

dropgenerator

chargeelectrode

high voltagedeflection plate

gutter

paper

Figure 1.4: CIJ with binary drop deflection


• CIJ with multiple drop deflection. This approach is illustrated in Fig. 1.5.Two companies that were involved in this branch of CIJ were the Sharp andApplicon company. The former released their Jetpoint in 1973, the lattertheir color image printer in 1977.

HV

dropgenerator

chargeelectrode

high voltagedeflection plate

gutter

paper

Figure 1.5: CIJ with multiple drop deflection

• DOD piezo-electrical inkjet. Generally, the basis of piezo-electrical inkjet(PIJ) printers is attributed to three patents. The first one is that of Zoltanof the Clevite company (US Patent 3,683,212), proposing a squeeze mode ofoperation. The second one of Stemme of the Chalmer University (US Patent3,747,120) utilizes the bend mode of piezoelectric operation. Finally, Kyserand Sears of the Silonics company (US Patent 3,946,398) used a diaphragmmode of operation. Common denominator of these three patents is theuse of a piezoelectrical unit to convert a pulse of electrical energy into amechanical pressure to overcome the surface tension forces holding the inkat a nozzle. Drops are only created when an actuation pulse is provided,hence drop-on-demand. Obviously, the main discriminator between thesepatents is the used dominating deformation mode of the piezoelectric mate-rial together with the geometry of the ink channels. The patents of Howkins(US Patent 4,459,601) describing the push mode version and Fischbeck (USPatent 4,584,590) proposing the shear mode, completed the now commonlyadapted categorization of printhead configurations. In general, four types ofPIJ printheads can be distinguished, namely the squeeze, push, bend, andshear mode, see Fig. 1.6.

Major advantages of PIJ over CIJ printers include the fact that there isno need for break-off synchronization, charging electrodes, deflection elec-trodes, guttering and recirculation systems, high pressure ink-supplies andcomplex electronic circuitry. The first piezoelectric DOD inkjet printer toreach the market was in 1977 with the Siemens PT-80. Silonics was thesecond company to introduce a piezoelectric DOD printer, namely the Qui-etype in 1978.

6 INTRODUCTION 1.1

shear

squeeze bend

push

Figure 1.6: Classification of piezoelectrically driven inkjet printheads

All the inkjet printers that had been introduced so far had failed to be com-mercially successful. It proved to be extremely difficult to combine print quality,throughput, cost, and reliability all into one single inkjet printing device with ei-ther CIJ or PIJ. Though CIJ is capable of attaining high throughput, it requiredhigh costs to achieve the required high print quality in addition to reliability.With PIJ, it turned out to be problematic to achieve both excellent print qual-ity and reasonable throughput simultaneously. The realization of high density ofpiezoelectric actuators was difficult. Consequently, it was impossible to miniatur-ize the design to an acceptable format.

The invention of thermal inkjet (TIJ) in the early eighties fundamentally changedinkjet research. By the replacement of the piezoelectric by a thermal transducer,the main bottleneck of PIJ concerning miniaturization was resolved. Not onlythe size of the thermal transducer was favorable being a simple resistor, but alsothe low cost of manufacturing. TIJ can be manufactured using mass-productionbased on IC-manufacturing technology making the cost per nozzle much lowerthan the cost per nozzle of a PIJ printhead. Typically, a TIJ nozzle costs aroundseveral euro-cents whereas a PIJ nozzle cost lies around ten euro-cents. Both thefact that inkjet printers now could be miniaturized and its low cost of manufac-turing made TIJ to the superior inkjet technology at that time. Canon was thefirst company to bring TIJ to the market in 1981. Their lead in the TIJ devel-


opment was translated in a great number of patents, practically giving Canonthe means to control the TIJ market. Of the companies that Canon licensed itspatents to, HP was the only company that could keep up the pace with Canon.The milestones in TIJ printing are so extensive that a list is omitted.

After the introduction and immense success of TIJ, PIJ research efforts werelargely diminished. Only a few companies continued their research into PIJ. In thenineties, only a few companies that conducted research in PIJ were left, amongwhich Spectra, Xaar, Seiko-Epson, Trident, and Lexmark. CIJ-based printersand research practically disappeared, except for some sporadic publications (e.g.[Die98], [Sch99], [Hei00]). An important impulse to PIJ research was provided byongoing developments in the manufacturing of multilayer piezoelectric actuators.One of the major barriers now had been lifted: that of miniaturization. Epson’sadvances in piezoelectric transducer fabrication have allowed it to remain com-petitive.

Despite the eminent success of TIJ printing, there are some fundamental advan-tages of PIJ over TIJ:

• Ink properties. TIJ only works with aqueous inks whereas PIJ can work witha broad latitude of ink properties, including hotmelt ink. This is favorablein two ways. First, certain applications require a special type of materialto be deposited such that PIJ is the only technology capable of doing so.Second, the types of ink that can be used with PIJ results in general in ahigher print quality.

• Durability. PIJ printers have a higher durability than their TIJ equivalents.Typically, a PIJ nozzle is capable of jetting around 10 billion drops perlifetime whereas a TIJ nozzle is only capable of around 200 million droplets.The reason for that is the harm that is posed to the heater element of a TIJprinter. Each time a droplet is jetted, it is heated and cooled quite quicklysuccessively. This affects the life-time considerably.

• Attainable jetting frequency. PIJ printers can achieve higher jetting frequen-cies than TIJ printers.

• Drop-size modulation. Since control of the bubble collapse is not possiblewith TIJ, drop-size modulation is fundamentally not possible with TIJ.With PIJ, the necking of the drop-formation process can be controlled andtherefore gives an opportunity for drop-size modulation. This can be usedto further increase the resolution and thus print-quality.

At present, both TIJ and PIJ printing have evolved into the two most importanttechnologies when it comes to printing. The initial advantages of TIJ over PIJhave been levelled over the years by further development of the PIJ technology.

8 INTRODUCTION 1.1

Also, current applications of inkjet technology simply require the sketched uniquecapabilities of PIJ that the TIJ technology is unable to provide.

1.1.2 A generic manufacturing technology

A fundamental strength of the PIJ technology is its ability to deposit a widevariety of materials on various substrates in certain patterns. Next to this char-acteristic, several additional advantages can be mentioned that apply to inkjettechnology in general. To start with, its on-demand character makes it a veryflexible manufacturing technology. Furthermore, when used for manufacturing,the use of PIJ printing usually reduces the number of manufacturing steps neces-sary. Additionally, due to its additive character, there is a reduction of the use ofpossibly expensive materials or equivalently a reduction in waste as well. Finally,it is a non-contact and non-contaminating process which can be very favorable ina manufacturing process. Altogether, these characteristics make inkjet technologya very versatile manufacturing technology.

The importance of PIJ printing for the industry is best illustrated by the largerange of applications. Due to this wide variety of applications it is practicallyimpossible to present a complete overview. Also, each categorization of the ap-plications remains artificial to some extent. Nevertheless, the following divisionis adopted:

• Graphics. Most likely, inkjet technology is first associated with this field ofapplications. This is hardly surprisingly given the huge amount of (desktop)printers present in offices and the like. Accordingly, the amount of printertypes is also large. A subdivision can be made based on for example thetype of ink used (e.g. aqueous, hotmelt, UV-curable), substrate (e.g. paper,textile, food, canvas), and format (e.g. narrow or wide format printing).Some of these fields are dominated by TIJ printing, others by the PIJ print-ing. In general, PIJ printers are utilized in case the ink cannot be depositedby TIJ printers or the required quality is high.

• Displays. In the display market, PIJ technology is used to manufacture FlatPanel Displays (FPD), Liquid Crystal Displays (LCD), color filters (a partof LCDs), Polymer Light Emitting Diodes (PLED), and flexible displays.The accompanying performance criteria are one of the major driving forcesbehind much research and development efforts concerning PIJ. Examplescan be found in e.g. [Has02; Ben03].

• Electronics. Within this market, PIJ printheads are used to create func-tional electrical traces using conductive fluids on both rigid and flexiblesubstrates. One of the first applications of inkjet technology within thisfield was that for the production of Printed Circuit Boards (PCB). Otherapplications include the fabrication of electric components and circuits such

1.2 SYSTEM DESCRIPTION 9

as Radio Frequency Identification (RFID) tags, wearable electronics, solarcells, fuel cells, and batteries. Challenges for the PIJ technology within thisfield include the spreading of the ink and the required guarantees of conti-nuity of the jetted lines. Examples of the manufacturing of electronics withPIJ technology can be found in e.g. [Hei05; Szc05; Kno05].

• Life science. This market is rapidly expanding with new requirements forprecise dispensing of DNA and protein substances. The high costs of thesefluids make PIJ technology with its precision placement and tight flow con-trol an excellent dispensing tool. Applications include the use for DNA re-search, various medical purposes such as dosing of drugs, and food science.A quite futuristic application is the use of inkjet printing for the fabricationof living tissue. Examples can be found in [Che96; Jam98; Coo01; Rad05].

• Chemical. Within this market, the PIJ technology is mainly used as toolfor research purposes. Again, the unique capacity of the technology fordispensing small doses of liquids specifically makes it useful for this market.Applications include material and substrate development as well as coatingpurposes. Examples can be found in [Oht05; Nak05]

• Optical. Jetting of UV-curable optical polymers is a key technology forthe cost-effective production of micro-lenses. These tiny lenses are usedin devices from fiber optic collimators to medical systems. The ability ofPIJ technology to precisely jet spheres in variable but consistent drop sizesprovide opportunities for the cost reduction of existing optical componentsand innovative new designs, see e.g. [Cox96; Che02; Bie04].

• Three-dimensional mechanical printing. This category claims the PIJ tech-nology as tool for rapid prototyping, small volume production, and the pro-duction of small sensors. Examples can be found in [Wal02; Voi03; Yeo04].

As discussed, performance requirements imposed by various applications are quitestrict. In light of future applications, it is expected that these requirements willbecome even tighter. In combination with current limitations, this motivatesongoing research, as will be discussed in Section 1.2.3.

1.2 System description

1.2.1 An archetypal PIJ printhead

The variety encountered in PIJ printhead design is enormous. Apart from thevariation in actuation principle (see Fig. 1.6), the possibilities in geometry areseemingly endless. Despite the differences between the various designs, somecommon denominators can be distinguished:

10 INTRODUCTION 1.2

1. Basic working principle. Though the operation of a PIJ printhead involvesmany fields of science, a major role is assigned to that of acoustics.

2. The ink channel design. Despite the sketched diversity in printhead designs,four basic components keep returning. These include the channel itself, thenozzle, the ink supply, and the piezo-unit.

3. The operation of printheads. Typically, actuation pulses are manually shapedinput pulses based on physical insight of the design.

The work presented in this thesis focusses on the common principles of PIJ print-heads, among which the ones listed above, yet will be elaborated on one particularPIJ printhead design. Since the fundamental characteristics of this design doesnot differ from most other PIJ printhead designs, the results presented through-out this thesis will be still generally applicable. So to speak, the employed PIJprinthead design is truly an archetypical one.

In this section, a description of the working principle of the used PIJ printheaddesign is given. At the same time, it provides a perfect example of the threesketched characteristics above. Here, the focus lies on the basics rather than thedetails of the design. Those will be discussed in Chapter 3 and 4. Note thatvarious experimental curves shown in the remainder of this chapter have beenmeasured with one of the experimental printheads, see Chapter 3.

x=0 [reservoir]

x=L [nozzle]

t

V puls

ink channel

piezo unit

1

2

3

4

5

Figure 1.7: A schematic side view of an inkjet channel and its working principle

In Fig. 1.7, a schematic side view of a channel of the PIJ printhead subject inthis thesis is depicted. A schematic front view of an array of channels is depictedin Fig. 1.8. As can be seen, all piezo-units are connected to the same substrate.The channel has a length of several millimeters. The reservoir is connected to thechannel as an open end. As explained in [Gro03], the piezo-unit is concurrently


ink channel

ink channel

substrate

piezo unit

piezo unit

Figure 1.8: A schematic view of an a cross-section of a PIJ printhead

used as actuator and sensor. Physically, it senses the force that results from thepressure distribution in the channel acting on the piezo’s surface that bordersthe channel. This force creates a charge on the piezo-unit. Since only changesin charge are measured, in fact the time derivative of the instantaneous presentforce is sensed. Furthermore, since the resulting voltage drop of this current overa resistance is measured, we have that a voltage is the resulting sensor signal.For the trapezoidal pulse used for actuation, a typical sensor signal is depicted inFig. 1.9, p. 13. Typically, around 75 nozzles per inch are integrated in an arraythat forms a printhead.

To fire a droplet, a trapezoidal pulse is provided to the piezo actuator, see Fig. 1.7.Then, ideally, the following occurs, see e.g. [Bog84; Ant02]. To start with, a neg-ative pressure wave is generated in the channel by enlarging the volume in thechannel (step 1). This pressure wave splits up and propagates in both directions(step 2). These pressure waves are reflected at the reservoir that acts as an openend and at the nozzle that acts as a closed end (step 3). Note that the negativepressure wave reflecting at the nozzle causes the meniscus to retract. Next, bydecreasing the channel’s volume to its original value a positive pressure wave issuperimposed on the reflected waves exactly when they are located in the middleof the channel (step 4). Consequently, the wave traveling towards the reservoir iscanceled whereas the wave traveling towards the nozzle is amplified such that itis large enough to result in a droplet (step 5).

Another common denominator is the operation of an PIJ printhead. For mostdesigns, an input wave form is manually shaped based on physical insight in theworking of a printhead. Clearly, for the design presented here, the actuationpulse is tuned to the first eigenfrequency of the ink channel. Additionally, some-what more complex waveforms are designed for purposes like smaller droplets anddamping of the residual vibrations. Details will be discussed in Chapter 4.

1.2.2 Limitations of current designs

The applications discussed in Section 1.1.2 require certain performance criteria tobe met. For a PIJ printhead, an important set of requirements is related to theresulting drop properties, namely:

12 INTRODUCTION 1.2

• Drop-speed. The resulting droplets are required to have a certain speed,typically around several m/s.

• Drop-volume. Depending on the application under consideration, the perfor-mance requirement concerning volume typically varies from 5 to 15 picoliter.Smaller drop-volumes are for example required with the manufacturing ofPolyLEDs. The smallest drop-volumes are around 2 to 3 picoliter. For someapplications, it is required that the drop-size can be varied during opera-tion. For example, for large areas that need to be covered large drops aredesired, whereas for high resolution printing small drops are desirable. Thisis referred to as drop-size modulation.

• Drop-speed and -volume consistency. The variations in drop-volume anddrop-speed between successive drops and between the nozzles must staywithin a certain percentage band, typically ranging from 2 to 15 percent.This is to avoid irregularities in the printed object. In this thesis, onlydrop-to-drop consistency is considered.

• Drop-shape. The drop-shape is influenced negatively by the formation oftails or satellite drops. These are highly undesirable for the quality of theprint. For example, for the production of PolyLEDs, tails or satellites inducecross-contamination.

• Jet straightness. The droplets have to be deposed in a straight line to thesubstrate, typically within 5 to 14 mrad accuracy. Note that as the drop-volume decreases, this requirement becomes even more important.

These requirements are only explicitly concerned with the drop itself. The fol-lowing important two requirements are more related to the jetting process:

• Productivity. The productivity of a PIJ printhead is mainly determined bythe jetting frequency, defined as the number of drops that a channel jetswithin a certain time, and the amount of nozzles per inch (npi-ratio), see[Bru05] for details. Though these two parameters are highly dependent onthe specific design of printhead, typically it is around 10-20 kHz at 50-100npi to guarantee acceptable productivity.

• Stability. Stability of the jetting process is one of the most important perfor-mance requirements for PIJ printheads. In this context, stability is definedas the absence of nozzle failure per a certain amount of jetted drops, e.g.one failure per one million jetted drops.

In addition to these requirements, more general requirements are imposed, in-cluding the lifespan of the printhead (typically more than ten billion actuationsper channel), the materials compatibility (a wide variety of inks must be depos-able), the maintainability, and the cost of production and manufacturability of


the printhead. In this thesis, we restrict ourselves to the requirements posed forthe drop itself plus the two requirements concerning the jetting process itself.

Meeting these performance requirements is severely hampered by the followingoperational issues that are associated with the design and operation of printheadsas discussed in Section 1.2.1. Major issues that are generally encountered are thefollowing:

• Residual vibrations. After a drop has been jetted, the fluid-mechanics withinan ink channel are not at rest immediately: apparently traveling pressurewaves are still present. These are referred to as residual vibrations. InFig. 1.9, the system’s response to a standard actuation pulse is depicted.Also, the time instant of drop-ejection is indicated (around 17 µs in Fig. 1.9).Usually, the fixed actuation pulse is designed under the assumption that achannel is at rest. To guarantee consistent drop properties, one has towait for these residual vibrations to be sufficiently damped out to fulfillthis assumption. Since this takes about 100 to 150 µs, it limits the maxi-mally attainable jetting frequency with all the consequences concerning theproductivity and drop-consistency of a printhead. If the presence of resid-ual vibrations is ignored and the jetting frequency is increased nonetheless,drop-properties start varying. As example, the so called Drop-on-Demand(DOD) speed curve is depicted in Fig. 1.9, showing the dependency of thedrop-speed on the jetting or DOD frequency. As can be seen, considerablespeed fluctuations result.

0 10 20 30 40 50 60 70 80 90 100−2

−1

0

1

2

3

4x 10

−6

Inte

grat

ed s

enso

r si

gnal

[Vs]

Time [µs]2 4 6 8 10 12 14 16 18 20

2

2.5

3

3.5

4

4.5

Dro

plet

spe

ed [m

/s]

DOD frequency [kHz]

Figure 1.9: Residual vibrations (left, measured response (black) and the corre-sponding actuation pulse (gray, scaled)) and its effect on the DOD-speed curve(right)

• Cross-talk. Cross-talk is the phenomenon that one ink channel cannot be ac-tuated without affecting the fluid-mechanics in neighboring channels. Cross-talk occurs in various ways:

14 INTRODUCTION 1.2

1. Electrical cross-talk. This form of cross-talk usually does not play asignificant role. It occurs at the level of electrical circuits that arepresent in any printhead to operate the channels, for example in theform of leakage currents.

2. Acoustic cross-talk. The phenomenon that pressure waves within onechannel influence other channels is called acoustic cross-talk. It canoccur via the ink reservoir. Though it is a more important effect thanelectrical cross-talk, the overall influence can generally be consideredsmall.

3. Structural cross-talk. Structural cross-talk can occur in many ways.For example, as can be seen in Fig. 1.8, all piezo-fingers are connectedto a substrate. As a result, deformation of one piezo-unit induces adeformation of the neighboring units. Another path is via the defor-mation of a channel itself. As a result, the volume of the neighboringchannels changes also which induces pressure waves in those channels.The deformation of the printhead structure can originate from twosources. The first one is the result of a channel being actuated and isreferred to as direct voltage cross-talk. The second one is the resultof the occurring pressure wave that causes deformation of the channeland is called indirect or pressure cross-talk.

0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−6

Time [µs]

Inte

grat

ed s

enso

r si

gnal

[Vs]

−10 −8 −6 −4 −2 0 2 4 6 8 104.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

Channel number [−]

Dro

plet

spe

ed [m

/s]

Figure 1.10: Cross-talk (left, measured response of an actuated channel (gray) anda neighboring channel (black)) and the consequences on the drop-speed (right)

In Fig. 1.10, the effect of cross-talk on the fluid-mechanics of a neighbor-ing channel is shown. Also, its effect on the drop-speed of simultaneousactuation of neighboring channels is depicted. In this figure, the resultingdrop-speed of channel zero is depicted when in turn neighboring channelsare actuated. For example, when the neighboring channel at the right ofchannel zero is actuated, the drop-speed of channel zero drops from 4.9 m/s


to 4.2 m/s. As can be seen, the effect of cross-talk on the drop-speed inparticular is substantial. Though this figure only shows the drop-speed,cross-talk influences other drop-properties as well.

To minimize the effects of cross-talk, a number of measures have been taken.First, operation of ink channels is designed such that two neighboring chan-nels are not actuated simultaneously. However, this limits the possibilitiesconsiderably. Also, ink channels are actuated with a small delay to allowthe worst effects to be damped out. Another measure to minimize the effectof cross-talk involves the printhead design itself. As can be seen in Fig. 1.11,the amount of piezo-units is twice that of the design depicted in Fig. 1.8.The redundant piezo-units B bordering the piezo-unit A form a so calledbridge structure that provide additional stiffness to the design. If piezo-unitA is actuated to jet a droplet, the piezo-units B (short circuited) reduce theeffects of structural cross-talk. However, this reduces the variations in dropspeed only slightly. Furthermore, it is a costly solution, since the numberof required piezo-unit for an array doubles. Also, it limits the attainablenpi-ratio.

ink channel

substrate

piezo unit

B

piezo unit

B

piezo unit

A

Figure 1.11: A schematic view of an a cross-section of a PIJ printhead with abridge structure

• Changing/varying dynamics. There are various phenomena that accountfor changing or varying dynamics. First, some materials suffer from agingand their properties change over time. For example, piezo-material has anotorious reputation when it comes to aging. Second, due to the extremesensitivity of an ink channel’s behavior for small changes in material proper-ties, ink channel dynamics vary even within a range of a couple of channels.Changing or varying dynamics in combination with fixed actuation pulsesaffect the performance negatively. Conventional measures to minimize theseeffects, such as enforcing strict material properties during production, areusually very expensive and boost the cost of production considerably.

• Robustness against disturbances. There are a number of disturbances pos-sibly affecting the performance. To start with, air-bubbles or dirt particlesmay cause a channel stop functioning. Also, the various structural modes

16 INTRODUCTION 1.2

of a PIJ printhead itself influences the performance negatively. Using onefixed actuation pulse simply cannot handle these issues effectively.

These operational issues form boundaries for the attainable performance andhence are a major drive behind the research and development conducted intoinkjet technology. An inventory of solution strategies for these limitations is pre-sented in the following section.

1.2.3 Towards a controlled environment

Applications of inkjet technology as presented in Section 1.1.2 impose tight perfor-mance criteria on the printheads. In the near future, these performance require-ments become tighter. For some of these applications, even today’s performancealready is insufficient. Given these facts, several operational issues have beenidentified in Section 1.2.2 that exactly limit the attainable performance. Theseobservations provide a clear motivation for ongoing research in the field of inkjettechnology.

The objective of this section is to identify suitable research directions that canimprove the performance of PIJ printheads in face of the operational issues. Toobtain such an inventory of possible solution strategies, it is necessary to firstdistance oneself from the specific PIJ printhead and focus on the various disci-plines involved in printhead engineering. In this section, after having obtained anoverview of these disciplines and their individual contributions, the focus againshifts to the printhead design itself and it is discussed how the various disciplinescan offer solution strategies to the issues at hand.

Research and development of the PIJ technology require a wide variety of dis-ciplines to be involved, see Fig. 1.12. After all, due to the complexity of inkjetsystems, it is impossible to attribute all the necessary specialist knowledge toone engineering domain. While restricting to the design and development of aPIJ printhead only, already the following disciplines are typically represented inprinthead engineering:

• Applied physics. The role of applied physics consists mainly of gainingfundamental understanding of the relevant phenomena that form the basisof a PIJ printhead. This is of great importance during practically everyphase of printhead development. Typical examples include studies into thedrop-formation process, the inclusion of air-bubbles, and the assessment ofprint quality.

• Mechatronics. The field of mechatronic engineering can be regarded as thecombination of mechanical, electronic, software, and systems and control en-gineering. To assess the role of mechatronics within printhead engineering,the field is split up according to its origins:


(a) Mechanical engineering. A mechanical engineer applies physical prin-ciples to (re)design a certain device, in this case a PIJ printhead. Theirexpertise mainly aims at the application of the concepts of for example(fluid) dynamics, strength of materials, and applied thermodynamics.

(b) Electrical engineering. Electrical engineering is a discipline that dealswith the application of electricity. Their contribution covers a widerange from the selection of suitable actuators and sensors, designingand testing electrical networks that support the functioning of a PIJprinthead, to the digital signal processing to manipulate the relevantsignals.

(c) Software engineering. This computer science discipline is concernedwith developing large software applications. Their involvement withprinthead engineering usually comes at a later stage, when the print-head is mounted in the complete printing system. Therefore, their roleis somewhat limited during the design of a PIJ printhead itself.

(d) Systems and control. Engineers specialized in systems and control dealwith both the design and operation of a printhead, though main em-phasis is given to the control part. For example, based on knowledgeof the system optimal input pulses can be designed.

Mechatronic engineers form the core of the printhead engineering team.After all, a printhead truly is a mechatronic device. An important remarkconcerns the role of systems and control that is so often associated withmechatronics. Though the systems and control discipline is acknowledged asbeing an important aspect, the application thereof lags considerably behind,especially in the field of printhead engineering.

• Materials engineering. Materials engineering is a multidisciplinary field fo-cusing on functional solids, whether the function served is structural, elec-tronic, thermal, or some combination of these. Their work within the print-head engineering focuses on the choice for materials, keeping an eye onissues such as manufacturability, cost, and function. This discipline playsan important role in printhead engineering, since the consequences of thesechoices have high impact for example on the cost per nozzle.

• Chemical engineering. The involvement of chemical engineering in the de-sign of a PIJ printhead confines itself mainly to the ink. The influence of inkproperties on the functioning of a PIJ printhead however is large. Thoughthe development of ink can be performed quite independently from thatof the printhead, it is important that some critical parameters of ink, e.g.viscosity, are established in mutual consult.

Based on this overview, an inventory of research directions can be drawn up.Not surprisingly, each of these disciplines solves the operational issues from their

18 INTRODUCTION 1.2

particular perspective. A categorization of the various research directions can begiven as follows:

• Mechanical (re)design. This solution approach to the operational issuescomprises a mechanical redesign of the printhead, either by starting fromscratch, applying only minor changes, or anything in between. Some possible(combinations of) directions include:

(a) Geometry. The geometry of an ink channel or an entire printheadinfluences the performance considerably. A few examples thereof arethe following. A reduction of the channel-length induces the creation ofsmaller droplets. The way the ink is supplied to an ink channel largelydetermines the boundary condition of a channel and thus the operationof a printhead. An investigation in the geometry in all its detailstherefore is a suitable research direction in face of the operational issuesencountered.

(b) Actuation. Actuation is one of the key-issues in printhead design. Forexample, the specific implementation of the piezo-electrical actuationnot only determines the amount of cross-talk, but also the controlla-bility and observability of the jetting process. Even the choice for apiezo-electrical actuator could be subject of discussion.

(c) Material. The choice of materials also has its influence on the operationof a printhead. An example is the wetting of the nozzleplate thatmight be solved by using a different type of coating. Also, the cost ofmanufacturing is largely dependent on the choices regarding materialas well.

• Ink properties. Apart from the printhead itself, the ink plays an extremelyimportant role in the jetting process. Rather than focussing on the print-head design itself, the ink is an important research direction as well. Asillustration, recall that the drop-formation is largely dependent on the inkproperties.

• Control. The application of the principles of system- and control to a trulymechatronic device such as the printhead is a promising research direction.As argued, printhead engineering lacks a systems and control up to thispoint while there are a lot of possibilities for this research direction.

The first two research directions characterize current research efforts quite well.These efforts are mainly centered around on the mechanical (re-)design of PIJprintheads. Input for the (re-)design of the printheads originates from amongothers applied physicists. Related, but relatively autonomous tracks comprise thechemical and material engineers performing further research on their particulararea of interest. Apparently, the fact that a printhead is a mechatronic device


does not automatically result in adopting a truly mechatronic approach, i.e. withthe proper attention to systems and control, to solve the performance limiting is-sues. In Fig. 1.12, the sketched characterization of current printhead engineeringis schematically depicted in the figure on the left. Here, the systems and controlapproach plays only a modest role. In our view, however, a more prominent partfor systems and control in general, and the application of control to PIJ print-heads in particular, is indispensable to lift current performance limitations of PIJprintheads. In Fig. 1.12, the importance of systems and control within printheadengineering is depicted in the figure on the right. To get a better understandingfor systems and control as solution strategy for the operational issues, the majorbenefits of this approach are inventoried.

applied physics

mechatronics

materials engineering

chemical engineering

applied physics

mechatronics

materials engineering

chemical engineering

systems &

control

systems &

control

Figure 1.12: Characterization of printhead engineering: current situation (left)and with the proposed direction (right)

Basically, systems and control can play a crucial role in two ways. To start with,its systematic approach to the functioning of complex systems in the aggregateoffers structuring of the research and focus on the major performance determin-ing mechanisms. Second, it provides an additional degree of freedom to enhancethe performance of PIJ printheads by means of control. These added values ofsystems and control within printhead engineering are advantageous for the im-provement of existing printhead designs as well as for the development of newones. For example, the use of control is a very cost-beneficial option to enhancethe performance of existing PIJ printheads without having to perform a redesign.Also, during the design of new printhead, both the systematic approach and theadditional degree of freedom in the form of control provide tools to tune the de-sign such that optimal performance can be achieved.

Having introduced the relevance of systems and control for PIJ printheads in gen-eral, let us elaborate a bit more on the role of control in particular. In most cases,the term control is associated with feedback control. Feedback control aims pri-marily on stabilization and disturbance rejection. However, for a PIJ printhead,being a stable system by its nature, usage of feedback control is not of direct im-

20 INTRODUCTION 1.2

portance. Next to feedback control, feedforward control can be considered. Forsystems that act predictable based on their physical design, feedforward control isa suitable option. A PIJ printhead fulfills this requirement perfectly. Here, feed-forward is considered as tool for the design of actuation signals for PIJ printheads.To the best of our knowledge, the use of feedforward for this purpose has beenvirtually unexplored. The related field of input shaping has been investigated atleast at one occasion, see [Jon97].

In the next chapter, the systems and control approach as solution strategy to liftthe performance limitations of current PIJ printheads is further elaborated to aresearch objective and several research questions.

Chapter 2

Problem formulation

In this chapter, the discussion in the introduction of this thesis is formalized in aresearch objective. This objective is then divided in three main research questions.Finally, the structure of this thesis is outlined.

2.1 The research objective

In the previous chapter, the main performance limiting operational issues thatare commonly encountered in PIJ technology have been discussed. Given thefact that performance criteria for PIJ printhead applications become increasinglytight, these boundaries must be lifted to be able to meet future requirements.Based on an inventory of solution strategies that can resolve these operationalissues, a systems and control approach has been chosen to be explored in thisthesis. To the best of our knowledge, this research direction has been formerlyunprecedented within the printhead engineering community, at least in the openliterature. Therefore, only few work is available that can serve as starting pointfor the research conducted here. In this light, the research objective to fully ex-plore the possibilities of systems and control for PIJ printheads is formulated as:

Develop a unifying modeling and control framework for a PIJ printhead to inves-tigate the possibilities and limitations of current designs in face of the commonlyencountered operational issues.

Let us clarify the various elements present in this objective. To start with, ’aunifying modeling and control framework ’ relates first and foremost to the twobasic ingredients of a systems and control approach, namely modeling and con-trol. To possess unifying properties in light of the research presented here, a modelshould describe the functioning of a printhead on a system level, incorporatingall performance relevant dynamics. The input as well as a firm theoretical back-

21

22 PROBLEM FORMULATION 2.2

ground for these dynamics is often provided by the various disciplines involved inprinthead engineering. The resulting model therefore is able to relate the overallperformance of a PIJ printhead on a system level to the various detail studiesperformed by the various research groups within printhead engineering. Hence,the classification ’unifying’ is adopted. In addition, a solid control frameworkenables the systematic exploration of the to be introduced feedforward controloption together with the obtained insight to come up with practical solutions tothe operational issues at hand. Together, such a unifying modeling and controlframework provide a solid basis to systematically ’investigate the possibilities andlimitations of current designs in face of the commonly encountered operationalissues ’. The word ’possibilities’ reflects the utilization of the resulting frameworkto lift current boundaries posed by the ’commonly encountered operational issues’to enhance the attainable performance of PIJ printheads. At the same time, newboundaries are expected to emerge. These more fundamental ’limitations ’ of cur-rent printhead designs can however offer valuable insight to be used in the designprocess of future PIJ printheads. The generality of the research conducted in thisthesis and the various results is emphasized by the use of the phrases ’current de-signs ’ and ’commonly encountered ’. The results obtained throughout this thesisapply to more PIJ printheads than the ones considered here.

2.2 A decomposition in research questions

In this section, the research objective is decomposed in three main research ques-tions. Together, the solutions to these questions provide an overall solution tothe research objective of this thesis.

Question 1: How should a PIJ printhead be modeled given its intended use forthe proposed systems and control approach?

Basically, this research question is closely related to the suitability of the PIJmodel for the purposes in mind. Within the advocated systems and control ap-proach, the role of the model is versatile. For one, the model should provideinsight in the working of a PIJ printhead, both for the implementation of controland the use for (re-)design purposes. Also, it should facilitate the implementationitself of (feedforward) control. Several additional requirements could be formu-lated. Now, the better the model fulfills these and other requirements, the morebeneficiating the systems and control approach can become.

An important aspect throughout the discussions regarding the modeling (and con-trol) concerns the linearity of the jetting process. Though the jetting of a dropeach time a channel is actuated induces nonlinear behavior, it remains to be seenhow this affects the overall behavior of an ink channel from a systems and controlpoint of view.

2.3 A DECOMPOSITION IN RESEARCH QUESTIONS 23

Question 2: Can we design actuation wave forms which will be implementedas feedforward control such that the performance of current PIJ printheads isimproved?

As discussed previously, the introduction of control provides an additional de-gree of freedom to a PIJ printhead. Without having to perform a redesign ofan existing printhead, its performance can be optimized by a simple tuning of acontroller. For new designs, the performance can be increased by taking the pres-ence of the control into account. Now, for both existing and new PIJ printheaddesigns, the question arises how the incorporation of control can help to overcomethe operational issues and thereby enhancing the performance of PIJ printheads.A related, but certainly equally important question concerns to what extent theattainable performance can be increased.

In this thesis, feedforward is investigated. Given the fact that an PIJ printheadacts predictably based on its physical design and is inherently stable, feedforwardcontrol is the most suitable choice. More specifically, given the highly repetitivecharacter of the jetting process, Iterative Learning Control (ILC) is a logical choiceas control strategy. Though ILC has proven its value for high-precision motionsystems, it has not been used in the field of inkjet technology yet. A systematicexploration of the possibilities of ILC given the operational issues is therefore afitting approach to this research question. Additionally, the generic character ofthe proposed framework renders it generally applicable to a broad range of PIJprintheads.

For the implementation of control, an important issue concerns the choice for thecontrolled and manipulated variable. Two options are considered in this thesis,namely piezo-based and laser-vibrometer based ILC. Though this choice is highlydependent on the particular PIJ design at hand, there is no loss of generality.

Question 3: Can we improve current PIJ printheads such that some basic limi-tations with respect to the attainable performance are lifted?

The utilization of a systems and control approach to the operational issues at handwill lift some of the present boundaries concerning the attainable performance.At the same time, however, several new boundaries will inevitably emerge. Thesource of these boundaries can be attributed to the design itself: they cannot belifted any other way than changing the design itself. Only by the application of theproposed systems and control approach, these new boundaries become apparent.Having identified these fundamental limitations, the question arises how theseboundaries can be dealt with. Several indications will be provided throughoutthis thesis.

24 PROBLEM FORMULATION 2.3

2.3 The structure of this thesis

This thesis is organized as follows. In Chapter 3, the experimental setup andthe various PIJ printheads are introduced. Among other things, the sensor func-tionalities are addressed, the various properties of PIJ printheads are reviewed,and experimental identification is treated. The findings of this experimental ex-ploration form the starting point for both the theoretical and experimental workthat is presented throughout this thesis. By using the results of Chapter 3 asstarting point, the subjects treated in this thesis are directly related to actualverifiable data rather than being somewhat artificial. After having performed ourexperimental exploration, the theoretical modeling of an ink channel is treated inChapter 4. To start with, the need for a new model is thoroughly motivated. Thismodel is constructed as a series of bilaterally coupled multiports and is based onfirst principles only. Special attention is paid to the choices made, e.g. concerningwhite box modeling and the use of a two-port approach. The resulting two-portmodel is validated in Chapter 5. The results are discussed in detail. Also, direc-tions for future research concerning the two-port modeling of an ink channel aregiven. Together, Chapter 4 and 5 provide an answer to research question Q1. Atthe end of Chapter 5, research question Q3 will be addressed based on the resultsobtained so far. Note that at the end of Chapter 7, some of these findings arerevisited to provide conclusive answers to research question Q3. In Chapter 6,the feedforward control framework is introduced. Details concerning this frame-work are treated, such as for example the formulation of a suitable control goal,the ILC controller synthesis, and the incorporation of constraints in the actuationsignal. Next, the implementation of ILC to the experimental setup is addressedin Chapter 7. Both the results of the so-called piezo- and laser-vibrometer basedapproaches are presented. It is shown that the obtained learned actuation pulsesprovide solutions to the two most prominent performance limiting operationalissues: residual vibrations and cross-talk. Consequently, the productivity anddrop-consistency is improved. For several other operational issues, it is indicatedhow they can be solved by the proposed control strategy. Research question Q2

is addressed both Chapter 6 and 7. Research question Q3, that has alreadybeen discussed to some extent in Chapter 5, is revisited based on the results ob-tained in Chapter 7. At the same time, a conclusive answer to this third researchquestion can then be provided. Finally, Chapter 8 presents the conclusions andrecommendations of this research.

Chapter 3

Experimental exploration

In this chapter, the experimental setup used to investigate PIJ printheads is dis-cussed in detail. Special attention is given to the sensor functionalities present.Then, the various PIJ printheads that are used during the research are introduced.The relevant printhead dynamics of these PIJ printheads are identified and the re-sults are presented. Here, these experimental results are not shown for validationor control purposes. For that, one is referred to Chapter 5 and 7. Instead, thedata are used to be able to from this point relate the main topics covered in thisthesis directly to actual verifiable data. In our view, such an approach contributesto the verifiability of our research according to [Buc95].

3.1 Description of the experimental setup

A schematic overview of the experimental setup is depicted in Fig. 3.1. The ex-perimental setup itself is depicted in Fig. 3.2. With this setup, PIJ printheads canbe investigated in various ways. The only actuator is the piezo-unit of the inkjetprinthead. Three sensors are available in this setup. First, the piezo-unit not onlycan be used as actuator but also as sensor. Second, the meniscus (ink-air inter-face in the nozzle) movements can be captured by the laser-vibrometer. Third,properties of the resulting droplet can be monitored by a CCD camera. Thesesensor functionalities will be discussed in detail in the subsequent subsections.

The PIJ printheads under investigation use a hotmelt type of ink that requireheating of the printhead. The required reference temperature is reached by a PIDcontroller (Eurotherm 2408), which measures the printhead’s temperature withthermocouples and controls the input voltages by means of heating elements.Next, to monitor the ink level inside the reservoir, a level sensor is incorporatedin the printhead. Furthermore, a printhead is mounted in vertical direction withthe nozzles faced down, similar to its position in an inkjet printer. To avoid that

25

26 EXPERIMENTAL EXPLORATION 3.1

waveform generator

amplifier switch board

scope

CCD camera + microscope

laser-vibrometer + detector

pc

air pressure unit

temperature control unit

ink level indicator

mirror (45 deg.)

strobe light

actuation signal

piezo sensor signal

image

meniscus velocity printhead

Figure 3.1: A schematic overview of the experimental setup

the ink simply flows out of the nozzles under the influence of gravity, an air pres-sure unit (TS 9150G) makes sure that the pressure in the ink reservoir remainsbelow the ambient pressure.

As depicted in Fig. 3.1, the setup is connected to a personal computer that isequipped with National Instruments IMAQ PCI 1409 and PCI GPIB cards forimage processing and communication, respectively. On the computer, the de-sired actuation signals can be programmed and relevant data can be stored andprocessed. After defining the actuation signal, it is sent to an arbitrary waveformgenerator (Philips PM 5150/Fluke 195). The waveform generator sends the sig-nal to an amplifier unit (Krohn-Hite 7602), which has a certain gain. From theamplifier unit, the signal is fed to a so-called switch-board. The switch-board iscontrolled by the personal computer and determines which channels are providedwith the appropriate actuation signals. For the tracing of both the actuationand various sensor signals, an oscilloscope (Tektronix TDS 420/TDS 3034B) isused. This oscilloscope is connected to the computer and displayed data can bedownloaded to the personal computer.

3.1 DESCRIPTION OF THE EXPERIMENTAL SETUP 27

Figure 3.2: The experimental setup

3.1.1 Piezo sensor signal

The first and most important sensor functionality discussed is that of the piezo-unit. For a detailed discussion on the piezo-unit, one is referred to Chapter 4.Here, the fundamentals are treated, required for the explanation of the simulta-neous use of the piezo-unit as actuator and sensor.

As generally known, a piezo can be used as actuator or sensor, see e.g. [Waa91].For that, one uses the piezo’s indirect (actuator) and direct (sensor) piezo-electriceffect. The former comprises the following. If an electrical potential V is appliedto the piezo-unit, a deformation of the piezo-unit u results. The latter refers tothe following phenomenon. If a force F is applied to a piezo’s surface, an electriccharge q results. Together, this behavior can be described as:

[uq

]

=

[d 1/kC d

] [VF

]

(3.1)

with C the piezo’s capacity, d the piezoelectric charge constant, and k the stiffnessof the piezo. Schematically, (3.1) can be represented as two-port, as depicted inFig. 3.3. The piezo-unit is bilaterally coupled with an impedance Zc representingthe remainder of the ink channel. The use here of the two-port concept antici-pates the derivation of the two-port model of an ink channel to be presented in


Chapter 4. For an introduction of the two-port modeling approach, one is referredto the next chapter.

V

q

u

F + +

+

+

C

d

d

1k

Zc

Figure 3.3: The piezo-block: the ink-channel as impedance

Now, rather than using the piezo-unit as either actuator or sensor, during theresearch presented in this thesis it is used as actuator and sensor simultaneously.This is accomplished as follows. The measured signal q is made up of two con-tributions. The first is that of the applied actuation voltage V via the piezo’scapacity C and is referred to as the direct-path. The second contribution orig-inates from the force F exerted by the ink in the channel via the piezoelectriccharge constant k and is referred to as the indirect-path. Since only this secondcontribution is the required sensor signal, it has to be extracted from the mea-sured signal q. However, the contribution of the direct-path is considerably largerthan that of the indirect-path, being typically 10-20 mA and 50-100 µA, respec-tively. Consequently, it is difficult to measure the sensor signal (indirect-path)simultaneously while using the piezo as actuator. Basically, there are two optionsto do so still:

1. Using software-compensation. Given knowledge of the applied electricalfield V and the availability of an accurate model of the piezo’s capacityC, the contribution of the direct-path can be computed. By subtractingthis contribution from the measured signal q, the required sensor signal canbe established, see e.g. [Dos92; And94]. Note that for the discriminationbetween the direct and indirect-path in our case, a rather accurate modelhas to be available. The model inaccuracies should be at least significantlysmaller than the sensor signal that one is trying to obtain.

2. Using hardware-compensation. Rather than modeling the piezo’s capacityC, an actual piezo is used to predict the contribution of the direct-path. InFig. 3.4 and 3.5, this is schematically depicted. The measured signal q of afull ink channel comprises both the direct- and indirect-path. The measuredsignal q of an empty ink channel only consists of the contribution of the


direct-path. Again, by subtracting both measured signals, the indirect-pathor sensor signal can be obtained.

V

q

u

F

'piezo' 'ink'

direct-path 'piezo'

indirect-path 'ink'

+ +

+

+

+ +

C

d

d

1k Zc

Figure 3.4: Division into a piezo- and ink-block diagram

A drawback of software compensation relates to the required accuracy of thepiezo model. Since modeling of the piezo’s capacity C is extremely difficult givenits nonlinear behavior, this method is hard to implement. On the other hand,hardware compensation requires that both piezo-units are exactly the same. Smalldifferences, e.g. due to drift or production tolerances, are always present. Thisinfluences the accuracy of the resulting sensor signal negatively. Of both methods,hardware compensation is the only feasible method to simultaneously use the piezoas actuator and sensor in case of a PIJ printhead. To minimize the effects of piezocapacity differences, the following measures are taken:

'piezo' 'piezo' 'ink' - = 'ink'

full channel empty channel

+ +

Figure 3.5: The basic principle to obtain the actuation and sensor signal simul-taneously as used in the piezo-sensing device

• Temperature differences. Differences in piezo capacity occur due to temper-ature differences of both piezo-units. By isolating the PIJ printhead thesedifferences are satisfactorily minimized.

• Differences in piezo capacity. Matching the impedance of various piezo-unitsusually results in a satisfactory pair.


• Influence of structural effects on the sensor measurement. Even though theink channel is empty, a small contribution due to the deformation of thestructure may be present in the indirect-path. This effect can be neglectedthough.

For details, one is referred to [Gro03]. The measured frequency response of theelectronic conditioning of the piezo-sensing device, i.e. the subtraction as shownin Fig. 3.5, is depicted in Fig. 3.6. Note that modeling of the piezo-unit itself, i.e.the piezo-block as depicted in Fig. 3.3, is postponed until Section 4.2.5. Appar-ently, as can be seen in Fig. 3.6, the magnitude as well as the phase are distortedfor the low and high frequency range. However, for the frequency range of inter-est, roughly from 20 kHz up to 250 kHz, the resulting sensor signals are minimallyaffected by the piezo-sensing device.

101

102

103

104

105

106

−35

−30

−25

−20

−15

−10

−5

0

5

Mag

nitu

de [d

B]

Frequency [Hz]

101

102

103

104

105

106

−150

−100

−50

0

50

100

Pha

se [d

eg]

Figure 3.6: Measured FR of the piezo-sensing device

Having discussed the technical implementation of the simultaneous use of thepiezo-unit as actuator and sensor, the question arises what the sensor signal rep-resents. Physically, it senses the force that results from the pressure distributionin the channel acting on the piezo’s surface that borders the channel. This forcecreates the discussed electric charge on the piezo-unit (the indirect-path). Sinceonly changes in electric charge are measured, in fact the time derivative of theinstantaneous present force is sensed. Furthermore, since the resulting voltagedrop of this current over a resistance is measured, we have that a voltage is the


resulting sensor signal. A typical sensor signal as result of a standard trapezoidalactuation pulse is depicted in Fig. 3.14, p. 43.

The following remarks are in order. First, the piezo sensor is located in thechannel whereas the droplet formation takes place in the nozzle. Second, due tothe integrating character of the sensor the resulting signal is an average of thepressure that is present in a channel. Finally, since all the piezo’s are connectedto the same substrate, the actuation as well as sensing is influenced by structuralcross-talk. Despite all these facts, the current sensor signal can be regarded asrepresentative for the jetting process.

3.1.2 CCD camera

A second sensor functionality is provided by the Charge-Couple Device (CCD)camera equipped with a microscope, that can observe the generated droplets. Astroboscope provides a short light flash at a defined instant after the droplet isejected and an image is obtained on which the droplet seems to be fixed in the air.A necessary requirement for this approach to succeed is that the repeatability ofthe drop formation is high. Then, since both the time duration and the distancethat the droplet has traveled are known, an estimate of the droplet speed caneasily be obtained. Moreover, it is possible to estimate the volume of the droplet,because the droplet diameter can be determined. Other information which can beobtained concern the droplet’s angle, the formation of satellites and the stability ofthe jetting process. A great advantage of the CCD camera is that direct informa-tion about a droplet is obtained. Unfortunately, this information is only availableat discrete time instants. In case the drop formation is not repeatable, a moreexpensive high-speed camera could be used to obtain the required drop properties.

Note that the resulting droplet properties are the result of image processing. Byaltering some of the parameters of this process, e.g. the threshold used for theblack-white conversion, the outcome may change. This affects the quality of themeasurements negatively.

3.1.3 Laser-Doppler interferometry

The third and last sensor functionality is the laser-vibrometer. The principle ofLaser-Doppler interferometry consists of the splitting of a laser beam in two dif-ferent paths and, finally, combine the beams again. One beam travels over a fixedpath and the path of the other beam is varied. In case a beam is reflected againsta moving object, a Doppler shift takes place. When the object is moving towardsthe beam, the frequency of the signal increases and when the object is movingaway from the beam, the frequency decreases. This way, the combined signal con-tains information about the phase difference and the frequency shift between thetwo signals. This information is measured by a detector. With a Laser-Doppler


interferometer or laser-vibrometer it is possible to measure the velocity of themeniscus inside a nozzle. Here, the meniscus surface is the moving object whichreflects the beam. Unfortunately, this type of measurement can only be appliedto a small range of the dynamics. It is namely not possible to jet during thismeasurement, without taking special measures.

In the experimental setup, the laser-vibrometer is used to measure the meniscusvelocity. It consists of a Polytec OFV-5000 vibrometer controller containing aPolytec VD-02 velocity decoder. Furthermore, a Polytec OFV-512 fiber inter-ferometer and a Polytec OFV-130-3 micro-spot sensor head complete the setup.The resulting laser beam of approximately 3 µm in diameter is aligned via amirror in the center of a nozzle that has a diameter of 32 µm. It is assumedthat a Poiseuille velocity profile occurs in the nozzle during operation, such thatthe laser-vibrometer setup measures the maximum velocity. Due to the use ofa laser-vibrometer via a mirror that is situated directly in front of the nozzleexit, the experiments are restricted to the non-jetting situation. Practically, thismeans that only experiments at a lower voltage can be performed. However, if itis assumed that the ink channel behaves linearly, the resulting learned actuationpulses at a lower voltage can be scaled up to a jetting voltage and implemented.This important linearity assumption will be discussed in detail in the subsequentchapters. The following remarks are in order. First, the impossibility to usemeniscus-based ILC in a jetting situation does not conflict with its intended useas design tool for wave forms. Second, a sensor that is integrated in the printheadas replacement of the laser-vibrometer is currently being developed, see [Gro06a].Then, limitations with respect to the used voltage are removed.

The following remarks are in order with respect to the use of the laser-vibrometer:

• Laser alignment. Due to the reflective property of the nozzleplate, align-ment of the laser beam is quite difficult. Initial alignment is performedbased on a camera image of the laser-spot on the nozzleplate. Since thewetting is clearly visible, the jetting channel can easily be established. Thefinal alignment takes place by observing the resulting sensor signal on thescope. The expected amplitude of the response is known from calibrationexperiments conducted earlier.

• Sensor output. A remaining issue concerns the physical interpretation ofthe resulting sensor signal. If the laser is not aligned in the center of thenozzle, it is not known what velocity is measured. This might still be themaximum component of the meniscus.

• Limited measurement capabilities. To start with, only in a non-jetting situa-tion the measurements can be carried out. Second, a considerable phase-lagis introduced by the velocity decoder of the Polytec equipment. Since thisphase-lag is known, it can be compensated for.

3.2 DESCRIPTION OF THE EXPERIMENTAL PRINTHEADS 33

• Heating of the ink. The heating of the ink by the laser can be neglected dueto the low power intensity of the laser beam.

3.2 Description of the experimental printheads

A schematic representation and nomenclature of the PIJ printheads used in theresearch presented in this thesis are depicted in Fig. 3.7. Specific details concern-ing the geometry and physical properties of these printheads are listed in Table 3.1and 3.2, respectively. All printheads used in this thesis are similar, except for onepoint. This concerns the presence of the so-called bridge structure, see Fig. 3.8.As discussed in Chapter 1, this bridge structure is used for the minimization ofstructural cross-talk effects. Some printheads have the bridge structure (233e02and 293e02) while others have not (DG074). During the discussions throughoutthis thesis, it is clearly indicated which printhead has been used.

channel

connection

nozzle

reservoir

piezo-finger substrate

Figure 3.7: Nomenclature of an ink channel

channel A

channel B

substrate

piezo unit

piezo unit

substrate

piezo unit

B

piezo unit

B

piezo unit

A

channel A

channel B

Figure 3.8: Cross-section of a PIJ printhead without(left) and with (right) bridgestructure

As can be seen in Fig. 3.7 and Table 3.1, the channel and connection have a differ-ent cross-section. Normally, a change in cross-section gives rise to an impedancechange with corresponding transmission and reflection conditions. However, dueto the flexibility of the (actuated) channel wall, the impedances of both the chan-nel and connection match. Hence, effectively, there is no impedance change and


Channel (actuated) length 7.61 mmheight 106 µmwidth 266 µm

Channel (not actuated) length 0.40 mmheight 106 µmwidth 266 µm

Connection length 1.06 mmheight 230 µmwidth 230 µm

Nozzle length 100 µmdiam. (start) 100 µmdiam. (end) 32 µm

Table 3.1: Data of the printhead geometry

Density ρ 1090 kg/m3

Dynamic viscosity µ 0.011 Pa sSurface tension ν 0.028 N/mSpeed of sound c 1250 m/sEffective speed of sound ceff 900 m/s

Table 3.2: Overview of the physical properties of ink

the effect of the changing cross-section can be neglected.

In Table 3.2, a distinction is made between the speed of sound and the effectivespeed of sound. The former applies for the non-actuated parts of the ink channel.The latter is used for the actuated channel. Due to the fluid-structure interaction,the effective speed of sound is lower. By using these different values for variousparts of an ink channel, this effect is accounted for.

Throughout this thesis, it is assumed that all channels are identical. The validityof this assumption as well as the consequences if not, are discussed in Chapter 6and 7. In Fig. 3.9, an overview of the nomenclature of the various transfer func-tions is provided that is adopted in this thesis. The direct transfer functionsare denoted by Ha and Hb, the indirect or cross transfer functions by Hab andHba. The identification is performed using two of the three sensor functionalities.First, the piezo is used as actuator and sensor. This is referred to as piezo-basedidentification. Second, the laser-vibrometer instead of the piezo-unit is used assensor. This is referred to as laser-vibrometer based identification.

The printhead’s main eigenmodes can be determined using modal analysis. Ingeneral, resonance frequencies can be computed according to:


Ha Hb

Hab

Hba

uA uB

yA yB

channelA

channelB

inkchannel

inkchannel

Figure 3.9: Nomenclature of two neighboring channels

fr =ceff

λ(3.2)

with ceff the effective speed of sound and λ the wave length of the appropriatestanding wave in an ink channel. In principle, the ink channel’s basic resonancefrequency is the 1/4 λ mode, given the fact that one open (reservoir) and oneclosed (nozzle) end is present. Note that λ equals in our case Lch + Lco + Ln.However, for frequencies up to approximately 100 kHz, the nozzle acts as anopen rather than a completely closed end. Therefore, for low frequencies the inkchannel acts more as a 1/2 λ resonator, see [Ant02]. This phenomenon can beexplained as follows. Suppose that the nozzle dynamics can be described by anequivalent mass-spring-damper system, where the mass represents the ink in thenozzle. For low frequencies, the mass-spring-damper system oscillates whereasfor high frequencies it does not. Thus, the mass-spring-damper system, i.e. ournozzle, acts as a low-pass filter. This phenomenon is discussed in more detail inSection 4.2.2.

Now, in Table 3.3 and 3.4, the theoretical resonance frequencies of an ink channelare listed, accounting for the occurring switch in resonating behavior at approx-imately 100 kHz. Anticipating on the identification of the frequency responsesin Section 3.4 and 3.5, the corresponding measured resonance frequencies for thepiezo-based (293e02) and laser-vibrometer based (233e01) transfer functions arelisted also. These values have been determined based on Fig. 3.13 and 3.17. Whencomparing the theoretical and measured values of Table 3.3 and 3.4, the followingremarks are noteworthy. In the laser-vibrometer based case, the second modeconsiderably deviates from the theoretical predicted frequency. This will be ad-dressed in Chapter 5. Furthermore, the remaining (small) differences result fromthe particular differences of the 233e01 and 293e02 printhead.


theoretical mode frtheoretical measured mode frmeasured ∆ fr1/2 λ (0.50) 50 kHz 0.48 (≈ 1/2 λ) 48 kHz 2 kHz

2 · 1/2 λ (1.00) 100 kHz 0.91 (≈ 2 · 1/2 λ) 90 kHz 10 kHz5 · 1/4 λ (1.25) 125 kHz 1.19 (≈ 5 · 1/4 λ) 118 kHz 7 kHz7 · 1/4 λ (1.75) 175 kHz 1.83 (≈ 7 · 1/4 λ) 182 kHz 7 kHz

Table 3.3: Overview of the theoretical and measured (293e02, see Fig. 3.13, p. 41)resonance frequencies in the piezo-based approach

theoretical mode frtheoretical measured mode frmeasured ∆ fr1/2 λ (0.50) 50 kHz 0.43 (≈ 1/2 λ) 43 kHz 7 kHz

2 · 1/2 λ (1.00) 100 kHz 0.76 (≈ 3 · 1/4 λ) 76 kHz 24 kHz5 · 1/4 λ (1.25) 125 kHz 1.30 (≈ 5 · 1/4 λ) 129 kHz 4 kHz7 · 1/4 λ (1.75) 175 kHz 1.71 (≈ 7 · 1/4 λ) 170 kHz 5 kHz

Table 3.4: Overview of the theoretical and measured (233e01, see Fig. 3.17, p. 46)resonance frequencies in the laser-vibrometer approach

0 2 4 6 8 10 12−2

−1

0

1

2

3

4

Position [mm]

Sca

led

pres

sure

[Pa]

0 2 4 6 8 10 12−2

−1

0

1

2

3

Position [mm]

Sca

led

pres

sure

[Pa]

0 2 4 6 8 10 12−3

−2

−1

0

1

2

Position [mm]

Sca

led

pres

sure

[Pa]

0 2 4 6 8 10 12−4

−3

−2

−1

0

1

2

Position [mm]

Sca

led

pres

sure

[Pa]

Figure 3.10: Pressure waves in an ink channel at 107 kHz sinusoidal actuation atfour time instances; right traveling wave (gray), left traveling wave (gray dotted),resulting standing wave (black), and piezo-unit actuation (black dotted)


Finally, one last phenomenon is to be addressed. If the piezo-unit is actuatedwith a sinusoid at 107 kHz (or a multiple thereof), the ink in the channel belowthe piezo-unit’s surface oscillates with the same frequency whereas the ink in theremainder of the ink channel, connection and nozzle is almost completely at rest,see Fig. 3.10. Fig. 3.10 is obtained using a finite volume model of the ink channeldynamics, see [Wij04]. Note that this effect is also clearly visible in Fig. 3.17, p. 46.One possible explanation for this phenomenon is the occurrence of destructiveinterference below the piezo-unit’s surface, and comprises the following. Supposethat the piezo-unit can be modeled as a finite set of point sources each emittingtraveling waves in both directions of an ink channel, see Fig. 3.11. If it is assumedthat the piezo-unit deforms uniformly over its length (see Section 4.2.5), thesepoint sources oscillate uniformly for every frequency. If the piezo-unit is actuatedwith a frequency whose wavelength corresponds to the length of the piezo-unit (λ= l), destructive interference occurs. Now, the generated pressure waves for twopoint sources spaced at exactly d = 1/2 λ are illustrated in Fig. 3.11. As can beseen, the waves of this set of sources are amplified below the piezo-units surface,yet are canceled at any other location. Since the piezo-unit can in principle berepresented by an infinite set of point sources spaced at 1/2 λ apart, this effectonly is increased if more point sources are taken into account, see Fig. 3.11. As aresult, the ink below the piezo-unit is oscillating whereas the fluid-mechanics inthe remainder of the ink channel, connection, and nozzle are almost at rest. Thefrequency at which this phenomenon occurs can be computed as:

piezo unit substrate

d

Figure 3.11: Illustration of destructive interference phenomenon

fr =ceff

λ=

900

7.61 10−3 s= 118 kHz (3.3)

This theoretically computed value corresponds nicely to the measured anti-resonanceat 107 kHz, see Fig. 3.17, p. 46.


3.3 Identification method

In this section, the identification method of the Frequency Response (FR) fromthe piezo actuator to either the piezo sensor or laser-vibrometer is discussed. Notethat the construction of the accompanying Frequency Response Functions (FRF)based on these established FRs is discussed in Chapter 5: it additionally requiresthe choice of a model structure and subsequent determination of its parameters.For now, the focus lies on the determination of a non-parametric model to be usedfor validation of the theoretical model to be constructed.

To start with, one has to select a particular type of input signal. The followinginput signals have been considered:

• Sinusoids. Sinusoids are employed as input signals during a (pseudo) sine-sweep identification procedure. A finite number of sinusoidal input signalsare then provided to a system. Important properties of the (pseudo) sine-sweep are the following. First, the energy content is the same for eachfrequency. For systems having a lot of noise, this is very advantageous. Thesignal-to-noise ratio then remains large. Second, the transient effects can beminimized by increasing the time spent per frequency. At the same time,this also relates to a first drawback of the sine-sweep measurement. Sincethe experiments take relatively much time, the effect of drift affects theoutcome. This is particularly true for the piezo, known for its drift. Onemajor cause for piezo drift is formed by the temperature fluctuations ofthe piezo, according to the pyroelectric effect [Waa91]. Another drawbackconcerns the resolution of the sine-sweep. Since only a finite number ofsinusoids are used, some frequencies are not excited at all.

• Step. Identification procedures applying a step response can be performedfast. In face of the piezo drift, this can be very advantageous. However,a number of disadvantages are present. First, a step remains band limitedsuch that high frequencies are often not excited. Second, due to the shortmeasurement, transient effects affect the result. Since lower frequencies areparticularly vulnerable for this effect, the quality of the identification of thelower frequency range is influenced negatively.

• White noise. Another option is the use of white noise for system identifica-tion. One important property of white noise is its flat frequency spectrum:the energy is equally distributed over the frequencies within the bandwidthof the white noise signal. At the same time, this property may cause thesignal-to-noise ratio to deteriorate. This should be taken into account whenapplying white noise as input signal for identification.

For the identification of the inkjet channel FR, the sinusoids were selected as in-put signal. The application of a sinusoids as input for the piezo, superposed on

3.3 IDENTIFICATION METHOD 39

the bias voltage, is schematically illustrated in Figure 4.16. The choice for theamplitude of the sinusoids is discussed in subsequent sections. Next to the choiceof the type of input signal, the selection of the sample frequency is of importance.To avoid aliasing, the signal that is being sampled should not contain frequenciesbeyond the Nyquist-frequency. The Nyquist frequency fN is defined as half thesampling frequency fs. Given the fact that there are no significant inkjet channeldynamics present beyond 4 MHz, the sample frequency of 10 MHz suffices. Formost experiments presented and discussed in this thesis, we are only interestedin frequencies up to 500 kHz. Therefore, a Krohn-Hite 7206 low-pass filter witha cut-off frequency of 500 kHz is employed. Its FR is depicted in Fig. 3.12.

104

105

106

−40

−30

−20

−10

0

10

Mag

nitu

de [d

B]

Frequency [Hz]

104

105

106

−400

−300

−200

−100

0

100

Pha

se [D

eg.]

Figure 3.12: Measured FR of the Krohn-Hite 7206 low-pass filter with a cut-offfrequency of 500 kHz

Based on the traced output signals and knowledge of the provided inputs, thefollowing procedure is applied to construct a non parametric model, see [Pei96].For each frequency point, the Fourier components of the input and output aredetermined using a Discrete Fourier Transform. At the same time, possible trendspresent in the data are eliminated in the procedure of [Pei96]. The FR is thenobtained by a simple division per frequency point. Note that though relatedto an Empirical Transfer Function Estimate (ETFE), there are several criticaldifferences. First, the trend is removed in the procedure of [Pei96]. Second, sincethe exact frequency points are known, the computations are more accurate. Thenoise present can be filtered out more accurately. Since the inkjet channel has


rather much noise, this effect is considerable.

3.4 Piezo-based experimental identification

The naming of the experimental identification presented depends on the sensorfunctionality employed. If the piezo is used as sensor, the resulting identificationis referred to as piezo-based. In this section, the results thereof are presented.

Given our interest in an array of channels, two neighboring channels are selectedfor identification. As discussed in Section 3.2, the presence of a bridge structureinfluences the transfer functions considerably due to its effect on the structuralcross-talk. Therefore, the piezo-based identification is carried out for two differentprinthead geometries: with and without the bridge structure.

3.4.1 With bridge-structure

In Fig. 3.13, the direct and cross FR are depicted. During identification, the am-plitude of the sinusoids was chosen such that the inkjet channel was not jetting.Once the nonlinearity as a result of droplet ejection is eliminated, the systembehaves linearly. This has been verified by various superposition experiments. Ofcourse, validity of the resulting model in the jetting situation remains to be seenand is addressed below.

At first glance, the +1 slope in the direct FR seems surprising. As discussed inSection 3.1.1, this is caused by the differentiating character of the piezo as sensor:it senses changes in electric charge (i.e. current) rather than the electric chargeitself. Stated otherwise, the changes in channel pressure are measured instead ofthe pressure itself. From a physical point of view, it makes more sense to controlthe channel pressure rather than the changes thereof. For example, if there is nochange in pressure, the channel is not necessarily in rest. Therefore, an integratoris added. This has several important consequences. One is the importance of thevarious resonance frequencies that are visible in Fig. 3.13. The first resonancefrequency at 45 kHz (corresponding to the theoretically computed one) seems lessimportant compared to the other resonance frequencies due to its limited mag-nitude. However, adding an integrator renders the first resonance frequency themost important one. The apparent physical importance of the first resonancefrequency is confirmed by the actuation pulse that is used, see Chapter 1. Thispulse is namely completely tuned to the first eigenfrequency of the inkjet channel.

As can be seen in Fig. 3.13, the cross FR does not have a large magnitude.Also, the resonance frequencies are barely recognizable. Apparently, the bridge-structure is quite effective in the reduction of the cross-talk. The conclusion thatthe cross-talk is eliminated is not correct, however. Though with limited mag-

3.4 PIEZO-BASED EXPERIMENTAL IDENTIFICATION 41

103

104

105

106

−80

−60

−40

−20

0

Mag

nitu

de [d

B]

Frequency [Hz]

103

104

105

106

−600

−400

−200

0

200

Pha

se [D

eg.]

104

105

−90

−80

−70

−60

−50

−40

−30

−20

Frequency [Hz]

Mag

nitu

de [d

B]

104

105

−700

−600

−500

−400

−300

−200

−100

0

Pha

se [D

eg.]

Figure 3.13: Measured FR from the piezo-actuator to the piezo-sensor; direct(above) and cross (below) (293e02)


nitude, the cross-talk effect is still large enough to affect the droplet propertiesnegatively, as will be discussed in subsequent chapters. In addition, the bridgestructure limits the attainable nozzles per inch and thus productivity.

A substantial phase lag is present in both measured FRs, that is only partiallyresulting from the various (anti-)resonances. Another part originates from variousdevices in the hardware loop (see Fig. 3.1):

• The amplifier. The amplifier for the pulses generated by the waveform gen-erator introduces a phase lag due to its limited bandwidth. For frequenciesbelow its bandwidth, the phase lag can be approximated by a linear phasedelay of 0.08 degrees per kHz, resulting in 80 degrees delay at 1 MHz.

• The waveform generator and scope. The internal clock of the waveformgenerator and that of the scope may cause a delay. Both clocks are samplingat 10 MHz, but are not coupled. In the worst case, this results in a delayof 0.1 µs or a phase delay of 36 degrees at 1 MHz.

• ZOH sampling. The signal sampling in combination with a Zero-Order-Hold (ZOH) also introduces a delay of half a sample interval. For a samplefrequency of 10 MHz this results in 9 degrees delay at 1 MHz.

• The low-pass filter. As discussed, during some experiments, a Krohn-Hitelow-pass filter is used, see Fig. 3.12. This results in approximately 400degrees additional phase lag at 1 MHz. Note that this filter was not appliedduring the identifications as presented in Fig. 3.13.

• The piezo-sensing device. The piezo-sensing device not only introducesphase lag for frequencies beyond approximately 100 kHz, but also phaselead for the low-frequency range, see Fig. 3.6. The phase lag at 1 MHzequals 140 degrees.

• The laser-vibrometer. The phase introduced by the laser-vibrometer can becomputed according to the following formula ([Pol00]):

∆φ(fr) = −100fr

frc− 0.00038fr (3.4)

where fr is the frequency in Hz at which the phase lag is to be computed.frc is the cut-off frequency of the low-pass filter of the laser-vibrometerand equals 1.5 MHz. Though this effect is not relevant for the piezo-basedidentification, it is for the laser-vibrometer based identification. This willbe discussed in the next section.

For the piezo-based FRs, the total phase lag amounts to 274 degrees at 1 MHz ifthe Krohn-Hite low-pass filter is not used.

3.4 PIEZO-BASED EXPERIMENTAL IDENTIFICATION 43

A final remark concerns the following. There is a peculiarity in the amplitude inthe low-frequency range for the direct FR as depicted in Fig. 3.13. One shouldexpect that the magnitude of FRF goes to −∞ for low frequencies. After all,for those frequencies the ink channel has two open ends and the ink can oscillatefreely. In practice, however, the measured FR goes to a certain small, but con-stant, value. It is assumed that this mismatch is caused by electronic conditioningof the piezo-sensing device used during the measurement, see Fig. 3.6.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sen

sor

sign

al [V

]

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time [s]

Sen

sor

sign

al [V

]

Figure 3.14: Measured response in the jetting mode from the piezo input to thepiezo output; direct (left) and cross (right) (293e02)

The measured response of an actuated and neighboring channel to a trapezoidalpulse are depicted in Fig. 3.14. A frequency spectrum of the response of the ac-tuated ink channel as depicted in Fig. 3.14 reveals that the dominating frequencyof the response equals that of the first eigenfrequency of the ink channel. Appar-ently, despite the limited magnitude around 45 kHz, the standard actuation pulseis designed such that this mode is excited the most. Linearity will be discussedin Chapter 5.

3.4.2 Without bridge-structure

In Fig. 3.15, the direct and cross FRs for a PIJ printhead without bridge struc-ture is depicted. Except for several small printhead specific differences, the directFR is similar to that of a printhead with bridge structure. However, this doesnot hold for the measured cross FR. In the absence of a bridge structure, thecross FR is far more evident. Though less smooth, several important resonancefrequencies can be detected. These correspond fairly good to those of the directFR. Note that for the phase, the same arguments hold as the FRs measured witha printhead having a bridge structure.


105

−50

−45

−40

−35

−30

−25

−20

−15

Mag

nitu

de [d

B]

Frequency [Hz]

105

−600

−500

−400

−300

−200

−100

0

Pha

se [D

eg.]

105

−55

−50

−45

−40

−35

−30

−25

−20

Mag

nitu

de [d

B]

Frequency [Hz]

105

−400

−300

−200

−100

0

100

200

Pha

se [D

eg.]

Figure 3.15: Measured FR from the piezo-actuator to the piezo-sensor; direct(above) and cross (below) (DG074)

3.5 LASER-VIBROMETER BASED EXPERIMENTAL IDENTIFICATION 45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time [s]

Sen

sor

sign

al [V

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

Sen

sor

sign

al [V

]

Figure 3.16: Measured response in the jetting mode from the piezo input to thepiezo output; direct (left) and cross (right) (DG074)

In Fig. 3.16, the responses of the actuated as well as a neighboring inkjet channelto a standard trapezoidal actuation pulse are depicted. The response of theactuated channel seems to be in anti-phase with that of a non-actuated channel.This corresponds to the physical behavior of the actuator block. By decreasing thevolume of the actuated channel, the actuators of both neighboring channels arelifted via the substrate. Consequently, the volume of the non-actuated channelsis enlarged, leading to the exact opposite response in pressure than the actuatedchannel.

3.5 Laser-vibrometer based experimental identification

As indicated in the previous section, the naming of the experimental identificationdepends on the sensor functionality used. Here, the laser-vibrometer is used assensor functionality. Correspondingly, the resulting identification is referred to aslaser-vibrometer based. Note that the measurements presented here are obtainedfrom a printhead without a bridge structure.

In Fig. 3.17, the FR from the piezo-actuator to the meniscus velocity is depicted.The following remarks are noteworthy. First, the magnitude of the first resonancefrequency of the direct FR turns out to be dependent on the used excitation volt-age. This effect is not present in the cross FR. To visualize this effect, the directFR has been measured using three different excitation voltages. For the crossFR, one excitation voltage sufficed. This nonlinear behavior can be explainedas follows. Even at these relatively low excitation voltages, the beginning of thedrop formation process can be observed. At this point, the viscous forces becomelarger than the surface tension forces at the free-surface. As a result, the outwardmeniscus velocity detected by the laser-vibrometer is larger than the inward ve-


104

105

106

−70

−60

−50

−40

−30

−20

−10

0

Mag

nitu

de [d

B]

Frequency [Hz]

104

105

106

−1500

−1000

−500

0

Pha

se [D

eg.]

sweep amplitude 1Vsweep amplitude 2.5Vsweep amplitude 4V

104

105

106

−80

−70

−60

−50

−40

−30

−20

Mag

nitu

de [d

B]

Frequentie (Hz)

104

105

106

−5000

−4000

−3000

−2000

−1000

0

Pha

se [D

eg.]

Figure 3.17: Measured FR from the piezo-actuator to the meniscus velocity; direct(above) and cross (below) (233e01)

3.6 CONCLUDING REMARKS 47

locity due to the inertia of the newly formed droplet beginning. These effects areconfirmed by simulations of a finite volume model programmed in Flow3D, see[Wij04]. The larger the excitation voltage, the more distorted the sine-responsebecomes and the smaller the magnitude of the identified FR. This effect onlyoccurs at the first resonance frequency and cannot be detected at higher frequen-cies. Second, considerable phase lag can be seen in Fig. 3.17. This originatesfrom substantial time-delays present in the system as discussed with piezo-basedidentification. In addition, considerable phase lag is introduced by the Polyteclaser-vibrometer.

The measured FR at 2.5 V is selected for use in the sequel of this thesis. Notethat the choice for the 2.5 V FR has been rather arbitrary, the 1.0 V FR couldhave been used equally well instead. However, since the first resonance frequencyis hardly present in the 4.0 V FR, this last FR would not have been a properchoice. In Fig. 3.18, the measured response to a standard trapezoidal actuationpulse at a jetting frequency of 10 kHz at 2.5 V is depicted. In the sequel of thisthesis, further attention is paid to this nonlinearity as well as the limitation ofthe laser-vibrometer setup.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Men

iscu

s ve

loci

ty [m

/s]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−4

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time [s]

Men

iscu

s ve

loci

ty [m

/s]

Figure 3.18: Measured response in the jetting mode from the piezo input to themeniscus velocity; direct (left) and cross (right) (233e01)

3.6 Concluding remarks

In this chapter, a comprehensive experimental exploration of PIJ printheads hasbeen performed. To start with, the experimental setup itself has been introduced.The various sensor functionalities have been discussed, in particular the use of thepiezo-unit as actuator and sensor simultaneously. Next, the PIJ printheads havebeen introduced. The geometry and FRs of various PIJ printheads have beenpresented. Also, the various FRs were clarified using the physical background of


the printheads. With the results discussed in this chapter, a solid basis for theupcoming issues in this thesis has been obtained.

The following two assumptions, introduced in this chapter, are of particular im-portance in the sequel of this thesis:

• Identical channels. The validity of the assumption that all channels areuniform still is an open issue. Validity of this assumption would simplifythe identification and control of an PIJ printhead considerably. In case smalldifferences in channel dynamics turn out to be present, the approach canpossibly be made robust against the corresponding model uncertainties. Inthe next chapters, the uniformness of the ink channels will be investigatedfurther.

• Linearity of the jetting process. Another important assumption concerns thelinearity of the jetting process. Based on the jetting process itself, validity ofthis assumption is certainly not trivial. After all, the jetting of a drop eachtime a channel is actuated induces nonlinear behavior. The actual impactof this effect on the operation of a PIJ printhead from a systems and controlpoint of view is a subject that will be investigated further in the subsequentchapters of this thesis.

In this chapter, it has been assumed that the PIJ printhead behaves lin-early from a systems and control perspective. The identification has beenperformed while keeping the excitation voltages low such that the channelswere not jetting. Now, several linearity related questions emerge. For one,the usefulness of the identified FRs in the jetting case must be reviewed.This is a particular relevant issue for the laser-vibrometer based approachwith the apparent nonlinear behavior with respect to the channel’s firsteigenfrequency. In the remainder of this thesis, these issues are given theappropriate attention.

In the next chapter, modeling of a PIJ printhead is discussed. Having gained in-sight in various physical properties of PIJ printheads in general, and our ’archetypal’experimental printheads in particular, the modeling can start with the appropri-ate prior knowledge. Next to this physical background, a preview was given ofthe adopted approach to the modeling of an ink channel. The use of two-ports,employed here to explain the functioning of the piezo-unit, forms namely a keyfeature of our modeling approach. At the end of the next chapter, the insight inthe working of a PIJ printhead obtained in this chapter will be further extended.

Chapter 4

Modeling of the ink channel

dynamics

This chapter starts with a survey of known mathematical PIJ printhead models.Given our modeling purposes, the need for an alternative model will become appar-ent. In this chapter, therefore, a new theoretical model is derived. To that purpose,an ink channel is divided into a number of functional blocks each representing apart of that channel. During the derivation, all the assumptions necessary arediscussed in detail. It will not only turn out that the unique characteristics of themodel result in a model that breaks the trade-off between accuracy and model com-plexity, but simultaneously form a suitable framework for control and redesign.The derivation of the model is concluded by pointing out future research direc-tions concerning this model. Validation of the resulting so called two-port modelare postponed until the next chapter.

4.1 PIJ printhead model survey

Given the research objectives as formulated in Chapter 2, let us start this chapterby formulating the associated model requirements:

• Accuracy. Though accuracy is an obvious requirement, it still is of impor-tance to state our exact objectives in this matter. As depicted in Fig. 4.1,the resulting model is to be used for the control and redesign purposes asformulated in Chapter 2. The objective with respect to accuracy is there-fore formulated in light of these intended model applications. To start with,the behavior of an ink channel on an input-output level is to be predictedaccurately. Consequently, the model can be used for the application ofcontrol. Also, the major performance determining mechanisms of an ink

49

50 MODELING OF THE INK CHANNEL DYNAMICS 4.1

channel should be predicted accurately. This provides the required insightfor redesign.

• Model complexity. The resulting model’s complexity is to be kept as low aspossible. First, for the use of the model for the redesign purpose in mind,focussing on the performance determining processes in the first place leadsto a simple model that provides the insight needed. Second, linked to themodel complexity is the computational complexity. It is favorable to keepthe computational load as small as possible since it facilitates the use forcontrol.

Modeling

Control (Re-)design

Figure 4.1: Modeling for control and (re-)design

Since modeling usually comprises a trade-off between accuracy and model com-plexity, the requirements posed above form no exception. This is particularly truefor a PIJ printhead. Modeling a PIJ printhead or even one ink channel is consid-ered a complex issue. Gaining insight into the origins of this supposed complexityis of importance when discussing PIJ printhead modeling. This complexity ismainly caused by the following interconnected issues:

• Multiscale and multiphysics modeling. PIJ printhead dynamics cover a widerange in applied mechanics. To start with, the piezo-unit dynamics can bedescribed by the governing equations of solid mechanics. The ink dynamicson the other hand require the relationships of fluid-mechanics. Even withinthe fluid-mechanics, various rather diverse topics are represented in a PIJprinthead. A good example is the droplet formation, where free-surface flowis the center of attention. Another example concerns the simplification of thegoverning equations towards acoustics. Often, this simplification is highlydesirable to reduce the often high computational load. Finally, the modelingof the electrical circuitry has not even been considered yet. Altogether,the described presence of diverse topics in continuum mechanics requiresthe knowledge of all those fields in detail to successful modeling the PIJprinthead. Also, it gives rive to several other difficulties when modeling aPIJ printhead.

• Fluid-structure interaction. Fluid-structure interaction (FSI) occurs whena solid interacts with a fluid. In case of a PIJ printhead, the piezo-unit

4.1 PIJ PRINTHEAD MODEL SURVEY 51

interacts with the ink in a channel. A major cause of the associated difficultyis the moving boundary. Stated otherwise, the domain occupied by themedium is one of the unknowns in the problem. In fact, this is related tothe free-surface problem that occurs during droplet formation. There alsothe boundary is moving and is not known prior to the computation. Anotherassociated difficulty with FSI is the coupling, see below.

• Coupling. More generally speaking, next to the FSI, coupling is anotherissue. Given the usage of several fields in continuum mechanics for variousparts of an ink channel, the coupling of these parts to one model is not trivial.This is especially true in case many parts are present, such as with a PIJprinthead. Additionally, the admissible time step can be severely limiteddue to this method of coupling, leading to large computational times.

• Geometry. The particular geometry of a PIJ printhead often forms an issue.Due to its complexity, e.g. the geometry of a nozzle, it is difficult to generatea proper mesh when using a Computational Fluid Dynamics (CFD) package.

complete ink

channel dynamics

nozzle dynamics

and drop formation

drop formation

only

analytical [Bel98] [Dij84] [Dij99][Ten88a] [Ten88b][Sak00] [Kol02b] [Ber03]

[Fro84] [Shi05] [Mar06][Kol02a]

[Egg95]

numerical FV [Wij04] [Lio02]FE [Wij04] [Bad98; Bad01] [Wil99] [Yeh01]FD [Pan02] [Che99] [Sch86]

[Sch87] [Asa92] [Yu03][Yu05]

combination [Sei04] [Wu05]

Table 4.1: Overview of available piezo-electric ink channel models

Given the model requirements and the complexity of PIJ printhead modeling, thequestion arises whether a suitable model already is available in open literature.As can be seen in Table 4.1, a great number of printhead models can be found.The categorization of these models presented here is based on the fact whetherthe governing equations are solved using analytical or numerical techniques:

1. Numerical. Numerical means not continuous thus discrete. To solve thegoverning equations, being partial differential equations, numerically, onehas to discretize those in place and time. Based on the usual discretizationmethods in place of the common CFD packages, the following subdivisionis adopted:

• Finite volume. The finite volume method discretizes a volume into anumber of cells of arbitrary shape. Subsequently, the governing equa-tions are solved on these discrete control volumes by ensuring conser-


vation of mass, momentum, and energy in the fluids between finitevolumes.

• Finite element. With the finite element method, the governing dif-ferential equations are solved in terms of minimum residuals over anelement. The unknowns inside an element are approximated by shapefunctions which amplitudes are controlled by nodal displacements.

• Finite difference. At each intersection of the lines of the finite dif-ference mesh used, the governing differential equation is replaced bya finite difference approximation. The main disadvantage is that itrequires structured meshes, and coordinate transformations for com-plicated geometries.

For the time discretization, numerous options are available. Examples in-clude the explicit and implicit Euler method, the midpoint rule, and thetrapezoid rule. In explicit time integration schemes, the admissible timestep for the solver during integration is limited. This limit is determinedby the Courant number, defined as the ratio between the time step and thewave propagation time within an element:

C =∆t

∆x/c(4.1)

where ∆x is a characteristic length of the cell and c the speed of sound. Toensure correct computations in explicit methods, the Courant number maynot exceed 1.

For the modeling of free surface, e.g. for the modeling of droplet formation,additional methods are employed within a CFD package. These are used totrack the moving boundary and can be categorized into surface or volumemethods. Examples of the former is the Marker and Cell method (MAC).An example of the latter is the Volume of Fluid (VOF) method.

2. Analytical. In some cases, the governing equations can be solved analytically.Usually, a number of (simplifying) assumptions are then required. Lumpedparameter approaches are considered to fall into this category.

3. Combination. Models that solve the governing equations analytically insome direction and numerically in others belong to this third group.

As can be seen in Table 4.1, the models are further divided according to the partof an ink channel that is modeled. Not all models incorporate the complete inkchannel. Based on this overview, the question arises whether or not these areuseful for the research presented in this thesis.

4.2 THE TWO-PORT MODEL 53

In general, the numerical models of Table 4.1 are very accurate. Rather thanbeing constructed to assess the overall performance of a PIJ printhead, their goalis often to describe certain phenomena. For example, the working of the nozzle isto be predicted. Also, it is employed to establish the cause of arising problems,such as clogging of the nozzle. For these purposes, the level of detail and theaccuracy of the models render them extremely suitable. At the same time, thehigh accuracy is frequently accompanied by large model complexity. Numericalmodels are often programmed in CFD packages using complex meshes and solv-ing techniques to get an answer. This makes it hard to obtain the insight that isrequired for the application of control or redesign.

Though the model complexity of analytical models is usually low, the accuracyis on average less than that of the numerical models. For the intended purposes,the decrease in accuracy is acceptable. However, sometimes over-simplification orunder-simplification is performed such that the resulting model provides too fewor just too much information, respectively. For the models presented in Table 4.1,this is the case. Especially with respect to the insight the models are supposedto provide the models are not adequate.

The models that combine numerical and analytical models usually combine thedrawbacks of both previous mentioned categories. They only take care of a smallreduction in computational time. However, this is not exactly an issue.

Altogether, there still is a need for a model that really breaks the trade-off betweenaccuracy and model complexity. Such a model that is suitable for the control andredesign purposes in mind cannot be found in the open literature. Therefore,a new PIJ printhead model will be derived in the subsequent sections of thischapter. A thorough discussion of the resulting model as well as the validation ofthe derived theoretical model is provided in the following chapter.

4.2 The two-port model

Given the objectives as formulated in the previous section, it is chosen to employthe concept of bilaterally coupled systems (BCS) for the modeling of an ink chan-nel. The notion of BCS in connection with the modeling of dynamical systems hasbeen first introduced in [Pay61]. The related energy port and multiport systemshave been developed in the work of [Ros72] and [Kar77]. In our view, severalproperties of this concept can play a crucial role in achieving these aims. Themost important properties of a BCS are the following. To start with, it enforcesa causal internal model structure for the system to be modeled. Among otherthings, such a structure guarantees the possibility for physical interpretation ofa system at all times. Next, the interaction of a BCS with its surroundings istaken into account explicitly by means of so called impedance and admittance


relationships. These relationships can be viewed as extension of the system’sboundary to represent a part of the behavior of its surroundings. As a result,the role and effect of the boundary conditions, the input and output impedances,that are imposed to the system becomes clear. Considered at a more abstractlevel, the concept of BCS forms the ideal combination of the main focus of asystems and control approach (directing its attention towards the input-outputrelations of a system, the boundary of a system) and an applied physics or fluid-mechanics approach (that is more concerned with the structure itself of a system).

The successful application of this concept to a PIJ printhead is subject of thecurrent chapter. A PIJ printhead typically consists of an array of ink channels.Here, one such an ink channel is considered as the system to be modeled. To obtaina model of a complete printhead, several channel-models can be integrated to forman array of ink channels. For the application of BCS, the overall boundaries of thesystem under investigation are to be selected first. At the nozzle of an ink channel,the exact place and time instant a drop splits from the ink present in the nozzlerepresents the first boundary. The reservoir and the electrical circuitry form theremaining boundary conditions for an ink channel. Second, having establishedthese boundaries, the system itself can be divided into several subsystems thattogether make up a complete ink channel. In Fig. 4.2, a schematic overviewis given of the two-port model of the inkjet channel depicted in Fig. 1.7. Thispartition into subsystems is based on the specific design of the inkjet channel underconsideration. To start with, the segment of the channel that is actuated by thepiezo-actuator is called the channel block. It differs only from the connectionblock by the fact that the latter is not actuated. The reservoir forms the physicalboundary of an inkjet channel and also forms the boundary of the model. As allthese three blocks can be modeled using acoustics, they are referred to as acousticpath. The following two blocks, those of the nozzle and droplet formation, aremodeled using the basic equations in fluid-mechanics. Together, they make upthe fluidic path. The last block is that of the piezo-actuator. In the variousupcoming subsections, each of these functional blocks will be discussed. Finally,the coupling of these subsystems making up the complete two-port system istreated in Section 4.3.

Reservoir

Piezo actuator

Channel Connection Nozzle

Piezo actuator

Drop formation

Figure 4.2: A schematic overview of the two-port model of an inkjet channel


4.2.1 The acoustic path

The acoustic path consists of the channel, connection, and reservoir. The followingassumptions are done:

• To start with, it is assumed that for these blocks a one dimensional approachcan be used. This implies that only plane waves occur during operation ofan inkjet channel. Flow3D simulations confirm the validity of this approach.

• Second, it is assumed that there is no mean flow and that only small per-turbations occur. This is a valid assumption, since the volume that is jettedis so small that this is hardly noticeable as mean flow in the channel.

• The reservoir is assumed to act as open end. In practice, this is almost true.

• Finally, the dissipation also is assumed to be negligible. Dissipation by theink in the channel is determined by the oscillation frequency of the ink itselfas a result of the actuation, see [And84]. These oscillations are relativelysmall compared to the velocity in the nozzle.

The application of BCS to the modeling of fluid transmission lines has been intro-duced in [Bro62] and [Bro65]. Friction can also be accounted for in this approach,see [Bro69a] and [Bro69b]. A recent elaboration of BCS to the modeling of a fluidtransmission line without friction can be found in [Bos02]. In this section, theapproach is extended to account for the presence of an actuator. This translatesinto adjusting the governing equations for a variable cross-section A(x, t).

The modeling of the channel is treated first. To that purpose, we start with theconservation of mass and momentum for a channel with variable cross-sectionA(x, t):

∂A(x, t)ρ(x, t)

∂t+

∂A(x, t)ρ(x, t)v(x, t)

∂x= 0 (4.2)

∂A(x, t)ρ(x, t)v(x, t)

∂t+

∂A(x, t)v2(x, t)ρ(x, t)

∂x+

∂A(x, t)p(x, t)

∂x= 0 (4.3)

Here, v(x, t), p(x, t), A(x, t), and ρ(x, t) are the velocity, pressure, channel cross-section, and density, respectively. (4.2) can be written as:

∂A(x, t)ρ(x, t)

∂t+ A(x, t)ρ(x, t)

∂v(x, t)

∂x+ v(x, t)

∂A(x, t)ρ(x, t)

∂x= 0 (4.4)

and (4.3) as:


v(x, t)∂A(x, t)ρ(x, t)

∂t+ A(x, t)ρ(x, t)

∂v(x, t)

∂t+ 2A(x, t)ρ(x, t)v(x, t)

∂v(x, t)

∂x(4.5)

+ v2(x, t)∂A(x, t)ρ(x, t)

∂x+

∂A(x, t)p(x, t)

∂x= 0

Using the mass balance (4.4), (4.5) can be written as:

A(x, t)ρ(x, t)∂v(x, t)

∂t+ A(x, t)ρ(x, t)v(x, t)

∂v(x, t)

∂x+

∂A(x, t)p(x, t)

∂x= 0 (4.6)

If A(x, t)v(x, t) is replaced by the flow φ(x, t), (4.2) becomes:

∂A(x, t)ρ(x, t)

∂t+

∂ρ(x, t)φ(x, t)

∂x= 0 (4.7)

Elaborating the partial derivatives in (4.7) leads to:

A(x, t)∂ρ(x, t)

∂t+ ρ(x, t)

∂A(x, t)

∂t+ ρ(x, t)

∂φ(x, t)

∂x+ φ(x, t)

∂ρ(x, t)

∂x= 0 (4.8)

Furthermore, it is assumed that the variations in density and pressure underadiabatic conditions are related through:

dp

dρ

∣∣∣∣adiabatic

= c2w → ∂ρ

∂t=

1

c2w

∂p

∂tand

∂ρ

∂x=

1

c2w

∂p

∂x(4.9)

where cw is the wave propagation velocity. (4.8) can be written as:

A(x, t)

c2w

∂p(x, t)

∂t+ ρ(x, t)

∂A(x, t)

∂t+ ρ(x, t)

∂φ(x, t)

∂x+

φ(x, t)

c2w

∂p(x, t)

∂x= 0 (4.10)

or equivalently as:

∂p(x, t)

∂t+

c2wρ(x, t)

A(x, t)

∂A(x, t)

∂t+

c2wρ(x, t)

A(x, t)

∂φ(x, t)

∂x+ v(x, t)

∂p(x, t)

∂x= 0 (4.11)

For the elaboration of (4.6), we make use of the following relations:

A(x, t)∂v(x, t)

∂t=

∂φ(x, t)

∂t− v(x, t)

∂A(x, t)

∂t(4.12)

A(x, t)∂v(x, t)

∂x=

∂φ(x, t)

∂x− v(x, t)

∂A(x, t)

∂x


Using the relations of (4.12), (4.6) can be written as:

ρ(x, t)∂φ(x, t)

∂t− v(x, t)ρ(x, t)

∂A(x, t)

∂t+ ρ(x, t)v(x, t)

∂φ(x, t)

∂x(4.13)

− ρ(x, t)v(x, t)2∂A(x, t)

∂x+

∂A(x, t)p(x, t)

∂x= 0

Elaborating the partial derivatives of (4.13) results in:

ρ(x, t)∂φ(x, t)

∂t− v(x, t)ρ(x, t)

∂A(x, t)

∂t+ ρ(x, t)v(x, t)

∂φ(x, t)

∂x(4.14)

− ρ(x, t)v(x, t)2∂A(x, t)

∂x+ A(x, t)

∂p(x, t)

∂x+ p(x, t)

∂A(x, t)

∂x= 0

or equivalently:

∂φ(x, t)

∂t− v(x, t)

∂A(x, t)

∂t+ v(x, t)

∂φ(x, t)

∂x(4.15)

+

(p(x, t)

ρ(x, t)− v(x, t)2

)∂A(x, t)

∂x+

A(x, t)

ρ(x, t)

∂p(x, t)

∂x= 0

Now, both equations (4.11) and (4.15) are linearized. Suppose that all variablesare derived by a constant plus a small perturbation:

φ(x, t) = φ0 + φ(x, t)

v(x, t) = v0 + v(x, t)

A(x, t) = A0 + A(x, t) (4.16)

ρ(x, t) = ρ0 + ρ(x, t)

p(x, t) = p0 + p(x, t)

Here, it is assumed that p0, φ0, and A0 are not a function of x. If (4.16) is substi-tuted in (4.11) and (4.15) and the higher order terms are dropped, the linearizedequations are obtained. The tilde is omitted for denoting a perturbation, renam-ing v0, A0, and ρ0 in v, A, and ρ, respectively, the set of conservation laws canbe written in vector form as:

∂

∂t

[p(x, t)φ(x, t)

]

+

[

vc2

wρA

Aρ v

]

∂

∂x

[p(x, t)φ(x, t)

]

= (4.17)

[

− c2wρA

v

]

∂

∂tA(x, t) +

[0

v2 − pρ

]∂

∂xA(x, t)


Note that we assume v 6= 0 for now. The eigenvalues of matrix

"v

c2wρ

AAρ

v

#have

the values λ1,2 = v ± cw. Its corresponding right eigenvectors are:

m1 =

[cwρA1

]

m2 =

[− cwρ

A1

]

(4.18)

If we now define the following state transformation:

[z1(x, t)z2(x, t)

]

=[

m1 m2

]−1[

p(x, t)φ(x, t)

]

=

[A

2cwρ12

− A2cwρ

12

][p(x, t)φ(x, t)

]

(4.19)

then (4.17) can be brought to the form:

∂

∂t

[z1(x, t)z2(x, t)

]

+

[v + cw 0

0 v − cw

]∂

∂x

[z1(x, t)z2(x, t)

]

= (4.20)

[v−cw

2v+cw

2

]∂

∂tA(x, t) +

[ρv2−p

2ρρv2−p

2ρ

]

∂

∂xA(x, t)

Note that z1(x, t) and z2(x, t) have the physical dimension of flow. After appli-cation of the Laplace transform while assuming zero initial conditions and somereshuffling we obtain:

∂

∂x

[z1(x, s)z2(x, s)

]

=

[ − scw+v 0

0 scw−v

] [z1(x, s)z2(x, s)

]

(4.21)

+

[−s(cw−v)2(cw+v)−s(cw+v)2(cw−v)

]

A(x, s) +

[ρv2−p

2ρ(cw+v)p−ρv2

2ρ(cw−v)

]

∂

∂xA(x, s)

This renders the partial differential equation to an ordinary one that can be solvedstraightforwardly. Prior to that, the forcing function A(x, s) is defined to be theproduct of A(x) and A(s). A(x) represents the shape of the piezo-actuator whenactuated. It is assumed that the piezo creates a uniform cross-sectional variationK over its complete length. The amplitude of this mode as well as the trajectoryin time is determined by A(s), though being Laplace transformed. The solutionto (4.21) can be computed straightforwardly. If A(x, s) is replaced by KA(s), thefirst ordinary differential equation (ODE) of (4.21) reads as:

∂

∂xz1(x, s) +

s

cw + vz1(x, s) =

−s(cw − v)

2(cw + v)KA(s) (4.22)

Using the solution at x = 0, z1(0, s), as boundary condition, the solution to (4.22)at x = L can be written as:


z1(L, s) = z1(0, s)e−sL

cw+v − KA(s)(cw − v)

2

(

1 − e−sL

cw+v

)

(4.23)

A similar computation reveals the solution for the second ODE of (4.21):

z2(0, s) = z2(L, s)e−sL

cw−v +KA(s)(cw + v)

2

(

1 − e−sL

cw−v

)

(4.24)

The solution to (4.21) can be written in vector form as:

[z1(L, s)z2(0, s)

]

=

[

e−sL

cw+v 0

0 e−sL

cw−v

][z1(0, s)z2(L, s)

]

(4.25)

+

− (cw−v)

2

(

1 − e−sL

cw+v

)

(cw+v)2

(

1 − e−sL

cw−v

)

KA(s)

Now, (4.25) represents a two-port system as depicted as in Fig. 4.3 (block 1).Here, Lch, Lco, Aco, Sp represent the length of the channel, the length of theconnection, its cross-section, and the surface of the piezo bordering the channel,respectively. Note that v = 0 since we assumed that there is no mean flow. Ascan be seen, the solution admits a nice interpretation as travelling waves. Toobtain the original physical states p(x, s) and φ(x, s), the inverse transformationof (4.19) can be applied to the states z1(x, s) and z2(x, s) (block 3).

For the connection, a similar approach can be used, except that the cross-sectionremains constant and can be left out of the mass and momentum equations. Thesolution is depicted in Fig. 4.3 (block 2).

The last subsystem of the acoustic path is the reservoir. For the waves that comefrom the channel, the reservoir acts as open end, p(0, t) = 0, since the reservoircontains a large amount of ink compared to the channel. Using (4.18) and (4.19),this boundary condition can be written as:

[p(0, t)φ(0, t)

]

=

[ cwρA − cwρ

A1 1

] [z1(0, t)z2(0, t)

]

=

[0

φ(0, t)

]

(4.26)

(4.26) can only be satisfied when z1(0, t) = z2(0, t). In Fig. 4.3, this behavior ofthe reservoir is taken into account (block 4). In the actual system, regarding thereservoir as open end is not completely true. The coupling between the channeland the reservoir also takes place via a connection and so a more gradual transitionto an open end is obtained. The error made however is so small that this is allowedwithout introducing a large error.


-

1 2 3 4

+

+

+

+

+

+

+ z1(0, s)

z2(0, s) z2(L, s)

z1(L, s)

A(s)

φ

p

F

1

e−sLchcw+v

e−sLchcw−v

e−sLcocw+v

e−sLcocw−v

Sp Kcw2

(1 − e

−sLchcw )

2cwρAco

−1 −cwρAco

1

Figure 4.3: Block diagram of the acoustic path

4.2.2 The fluidic path: the nozzle

The fluidic path consists of the nozzle and droplet formation. In this section, thefluid-mechanics in the nozzle are modeled. In light of the model requirementsposed in Section 4.1, various options for the modeling of the nozzle dynamics arepresented and discussed.

The starting point for the discussion forms the governing equation: the Navier-Stokes equation. In addition, the fluid dynamics in the nozzle can be consideredincompressible, as proven in [Wij04] and [Mar06]. The trade-off between accu-racy and model complexity boils down to the number of dimensions consideredwhen solving the Navier-Stokes equation for the nozzle at hand. In this section,four options are considered: two one- and two two-dimensional approaches. Thederivation of the first one-dimensional model comprises a simple elaboration ofNewton’s second law. The second one-dimensional approach is based on solvingthe governing equations for a one-dimensional variable control volume, see e.g.[Han67]. The application thereof to the specific nozzle at hand for a Poiseuilleflow profile can be found in [Hei98]. Here, [Hei98] is extended to allow for theconsideration of more complex flow profiles, based on the work of [Mar04]. Thefirst two-dimensional approach is based on the so called stream-function vorticitymethod for solving the governing equations, see e.g. [Poz97]. This approach hasbeen introduced for modeling nozzle dynamics in [Fro84]. For the nozzle geom-etry under investigation, this has been elaborated in [Mar04] and [Mar06]. Thesecond two-dimensional model uses a CFD-package and is discussed in [Wij04].


Both two-dimensional models will only be shortly discussed in this thesis. A three-dimensional model is deliberately not considered here. Such a model is only usefulin the following cases. First, if the nozzle geometry is not axis-symmetric, e.g. incase of a square or elliptic nozzle, a three-dimensional model forms an added value.Second, in case the nozzle geometry can be regarded axis-symmetric, modelingof the inclusion of air-bubbles or dirt particles, requires a full three-dimensionalmodel also. Note that even in the latter case, a two-dimensional model couldsuffice as well provided that several assumptions are made.

After the introduction of these four nozzle models, the drop formation is discussedin the next section. At the end of that section, all four models of the completefluidic path are critically evaluated. Based on this evaluation, a decision is maderegarding the nozzle model to be used in the sequel of this chapter.

A 1D nozzle model: an impedance

5

1Z(s)

An

p

φ

v

Figure 4.4: Block diagram of the fluidic path

Our first approach to the modeling of the nozzle dynamics excels in its simplicity.Apart from the fact that a one-dimensional approach is adopted, it is also assumedthat the nozzle is filled with ink at all times. This allows us to model the nozzleas one fixed impedance. The starting point for the derivation is Newton’s secondlaw, which reads for the nozzle, stated in terms of p(s) and v(s), as follows:

p(s)An = ρAnLnsv(s) + 8πµLnv(s) (4.27)

with An, Ln, and µ being the nozzle’s cross-section, length, and viscosity, re-spectively. The viscous friction due to the pressure gradient across the nozzle isaccounted for in the second term, assuming a Poiseuille flow profile, see [Han67].According to the definition, the nozzle impedance can be written as:

Z(s) =p(s)

v(s)=

ρLnAns + 8πµLn

An(4.28)


To compute output φ(s) given the input p(s), we get:

φ(s) = Anv(s) = Anp(s)

Z(s)=

A2n

ρLnAns + 8πµLnp(s) (4.29)

In Fig. 4.4, the fluidic path is depicted (block 5).

Using the parameters listed in Table 3.1 and 3.2, the corresponding frequencyresponse of (4.29) can be obtained, see Fig. 4.5. As discussed in Section 3.2, thenozzle acts as open end for low frequencies and switches to a closed end for higherfrequencies. This behavior corresponds to the frequency response of Fig. 4.5,showing a corner frequency of the nozzle dynamics of approximately 100 kHz ofthe first order system (4.29).

104

105

106

107

−305

−300

−295

−290

−285

−280

−275

−270

Frequency [Hz]

Mag

nitu

de [d

B]

104

105

106

107

−100

−80

−60

−40

−20

0

Pha

se [D

eg.]

Figure 4.5: Theoretical frequency response of the nozzle block from input p tooutput φ

A 1D nozzle model: a deformable control-volume

The Navier-Stokes equation is solved for a deformable control volume representingthe nozzle. With this approach, it is assumed that the flow can be described usingone dimension only. Furthermore, the surface tension at the meniscus is neglected.

The conservation laws for mass and energy in integral form are invoked to describethe flow inside the printhead’s nozzle. Using the integral form implies the use of


deformable control volumes. A deformable CV is a CV that may change in time,which basically means that a certain volume V and its surrounding area A hassome or all boundaries moving at a certain velocity b(t). Suppose that the fluidvelocity itself is denoted by v(t), an observer fixed to the CV sees a relativevelocity vr(t) of the fluid crossing the control surface (CS) defined by:

vr(t) = v(t) − b(t) (4.30)

Given the incompressibility of the nozzle flow, the conservation of mass of thedeformable CV can be written as:

∂

∂t

∫

CV

ρdV −∫

CS

ρ(vr · n)dA = 0 (4.31)

Here, n represents the outward normal at the CS. (4.31) simply states that therate of change of mass within the CV equals the rate of flow of mass into the CVminus the rate of flow out of the CV. The equation of conservation of energy can bederived by forming the scalar product of the local velocity v(t) with the equationof motion. If one assumes that the nozzle operates adiabatically and gravitationalforces are neglected, the equation of mechanical energy of a deformable CV canbe written as (see [Han67]):

∂

∂t

∫

CV

1

2ρ|v|2dV +

∫

CS

1

2ρ|v|2(vr ·n)dA = −

∫

CS

p(n·v)dA+

∫

CV

v ·(∇·σ)dV (4.32)

Here, ρ is the density, p the pressure, and σ the viscous stress tensor. The firstterm of (4.32) at the left indicates the change in kinetic energy of the control vol-ume. The second term represents the fluxes of kinetic energy in and out throughthe (moving) boundaries. The first term on the right side of (4.32) stands forthe power of the surface forces, in this case pressure. The second term on theright accounts for the shear work due to viscous stresses. (4.32) is a scalar equa-tion of energy and uses the three components of the velocity. Using cylindricalcoordinates the velocity equals:

v(t) = vz(r, θ, z, t) (4.33)

In Fig. 4.6, the nomenclature and the geometry of the nozzle is depicted. Tostart with, the velocity profile is regarded 2D axis-symmetric and (4.33) can besimplified to vz(r, z, t). Only in case an air-bubble or dirt particles are present,validity of this assumption is questionable. Note that even in these cases theassumption of axis-symmetry can be used without introducing major inaccuracies.Furthermore, instead of considering vz(r, z, t), an average velocity vz,av(z, t) overthe cross-section is assumed in the one-dimensional approach. Using (4.31), thismeans that the balance equations can be written in terms of volume flow φ(t)only. The volume flow is:


φ(t)

z

z0 zk(t)

p(z0, t)

r

Ln

R(z)

Figure 4.6: Geometry of the nozzle

φ(t) = vz,av(z, t)A(z) = vz,av(z0, t)A(z0) = vz,av(zk, t)A(zk) (4.34)

The average velocity vz,av depends on the actual occurring velocity profile andcan be computed according to:

vz,av(z, t) =1

A(z)

∫

vz(r, z, t)dA (4.35)

However, by considering only an average velocity vz,av(z, t) the computations ofenergies over the cross-section are in error. To compensate for the error, cor-rection factors are used: the kinetic energy and the momentum-flux correctionfactor, α and β, respectively. These factors greatly depend on the actual flow-profile occurring in the nozzle.

For the one-dimensional case, (4.32) can be written as:

∂

∂t

∫

CV

1

2ρv2

zdV +

∫

CS

1

2ρv2

z(vz,rel·ez)dA = −∫

CS

p(ez ·vz)dA+

∫

CV

v·(∇·σ)dV (4.36)

using the fact that the velocity is only in one direction (normal to the surface).The first term of (4.36) can be rewritten using the fact that the nozzle is axis-symmetric:

∂

∂t

∫

CV

1

2ρv2

zdV =∂

∂t

∫

CV

1

2ρvz(r, z, t)2dV = 2π

∂

∂t

∫ zk(t)

z0

∫ R(z)

0

1

2ρvz(r, z, t)2rdrdz

(4.37)


To further simplify (4.37) from the two-dimensional to the one-dimensional case,(4.37) is written in terms of the average velocity and an additional correctionfactor. The following equation then must hold:

2π

∫ R(z)

0

vz(r, z, t)2rdr = βvz,av(z, t)2πR(z)2 (4.38)

since:

2π

∫ R(z)

0

vz,av(z, t)2rdr = 2πvz,av(z, t)2[1

2r2

]R(z)

0

= vz,av(z, t)2πR(z)2 (4.39)

Stated alternatively, (4.38) states that the flux in kinetic energy of a slice dz of theCV computed using the actual flow-profile vz(r, z, t) must equal the outcome whenusing the average velocity vz,av(z, t). Consequently, correction factor β equals:

β(z, t) =2

vz,av(z, t)2R(z)2

∫ R(z)

0

vz(r, z, t)2rdr (4.40)

β is known as the momentum-flux correction factor and is dependent on the ac-tual occurring flow-profile vz(r, z, t).

Similar to these computations, the kinetic correction factor α can be computed.To that purpose, the second term of (4.36) is written as:

∫

CS

1

2ρv2

z(vz,rel · ez)dA = 2π

∫ R(z)

0

1

2ρvz(r, z, t)2(vz,rel(r, z, t) · ez)rdr (4.41)

If (4.41) is written in terms of the average velocity and the kinetic correctionfactor:

2π

∫ R(z)

0

vz(r, z, t)3rdr = αvz,av(z, t)3πR(z)2 (4.42)

(4.42) simply states that the kinetic energy taken at a point over the cross-sectioncomputed using the actual flow-profile must equal that computed with an averagevelocity vz,av(z, t).

The kinetic correction factor α equals:

α(z, t) =2

vz,av(z, t)3R(z)2

∫ R(z)

0

vz(r, z, t)3rdr (4.43)

Now, using the correction factors α and β, (4.32) can be written as:


∂

∂t

Z zk(t)

z0

1

2ρβ(z, t)vz,av(z, t)2πR(z)2dz +

1

2ρ

�α(z, t)vz,av(z, t)2(vz,av,rel(z, t) · ez)πR(z)2

�zk(t)

z0

(4.44)

= −

ZCS

p(ez · vz)dA +

ZCV

v · (∇ · σ)dV

For further elaboration of (4.44), an assumption regarding the occurring flowprofile is required. Given the pulsating nature of the traveling pressure waveswithin an ink channel, a Womersley velocity profile seems a logical choice, see[Hal55]. A characterization of pulsating flow is provided by the Womersley numberWo. Wo is defined as:

Wo =R

2

√ω

ν(4.45)

with R the characteristic diameter of a tube, ω the pulsating frequency of the flow,and ν the kinematic viscosity of the fluid. However, actuation of an ink channelresults in pressure fluctuations that are quite highly irregularly rather than purelysinusoidally. For that reason, a Womersley profile may not be suitable. On theother hand, the conditions for a Poiseuille profile are not fulfilled either. InSection 4.2.4, it is shown that the accuracy of the nozzle model derived here isnot influenced by adopting a Womersley profile instead of a Poiseuille profile.Therefore, for the further elaboration of (4.44) a Poiseuille profile is adopted.After all, this greatly reduces the complexity of the resulting nozzle model. APoiseuille flow profile can be described as:

vz(r, z) =

(

−∂p

∂z

)R(z)2

4µ

(

1 − r2

R(z)2

)

(4.46)

where p is the driving pressure drop and µ the dynamic fluid viscosity. Note thatthis profile is not dependent on time t. According to (4.35), the average velocityvz,av(z) equals:

vz,av(z) =2π

πR(z)2

∫ R(z)

0

vz(r, z)rdr = −∂p

∂z

R(z)2

8µ(4.47)

The momentum-flux correction factor is computed as:

β(z) =8

R(z)2

∫ R(z)

0

(

1− r2

R(z)2

)2

rdr =8

R(z)2

[1

2r2−1

2

r4

R(z)2+

1

6

r6

R(z)4

]R(z)

0

=4

3(4.48)

and the kinetic correction factor as:


α(z) =16π

πR(z)2

∫ R(z)

0

(

1 − r2

R(z)2

)3

rdr (4.49)

=16

R(z)2

[1

2r2 − 3

4

r4

R(z)2+

1

2

r6

R(z)4− 1

8

r8

R(z)6

]R(z)

0

= 2

Given both correction terms, (4.44) will be now elaborated termwise. The sub-script av is omitted.

1. Since

vz(z, t) =φ(t)

A(z)=

φ(t)

πR(z)2(4.50)

the first term at the left side can be written as, using the product rule ofdifferentiation:

∂

∂t

∫ zk(t)

z0

1

2ρβ(z, t)vz(z, t)2πR(z)2dz =

∂

∂t

∫ zk(t)

z0

2

3ρφ(t)2

A(z)dz (4.51)

=4

3ρφ(t)

∂φ(t)

∂t

∫ zk(t)

z0

1

A(z)dz +

2

3ρφ(t)2

∂

∂t

∫ zk(t)

z0

1

A(z)dz

2. The one-dimensional control volume has two areas with a flux, at z0 and atzk(t). Therefore:

1

2ρ

[

α(z, t)vz(z, t)2(vz,rel(z, t) · ez)πR(z)2]zk(t)

z0

= (4.52)

ρvz(z0, t)2

(

(vz(z0, t) − b(z0, t)) · −1

)

A(z0)

+ ρvz(zk, t)2(

(vz(zk, t) − b(zk, t)) · 1)

A(zk)

where b(z, t) represents the velocity of the boundary. The boundary at z0

is not moving, the boundary at zk is moving and equals the velocity of themeniscus vz(zk, t) and the last term drops out of the equation. The equationcan be simplified to:


1

2ρ

[

α(z, t)vz(z, t)2(vz,rel(z, t) · ez)πR(z)2]zk(t)

z0

(4.53)

= ρvz(z0, t)2

(

(vz(z0, t) − 0) · −1

)

A(z0)

= −ρvz(z0, t)3A(z0) = −ρ

φ(t)3

A(z)2

3. The first term on the right also only has to be evaluated at the two areasat the boundaries.

−∫

CS

(p(ez · vz))dA = (4.54)

− p(z0, t)vz(z0, t) · (−1)A(z0) − p(zk, t)vz(zk, t) · (1)A(zk)

If the pressure jump through the meniscus is neglected, the pressure at thenozzle exit equals zero and the above equation reduces to:

−∫

CS

(p(n · vz))dA = p(z0, t)vz(z0, t)A(z0) = p(z0, t)φ(t) (4.55)

with p(z0, t) the pressure at the nozzle entrance.

4. The fourth term of the energy equation is:

∫

CV

v · (∇ · σ)dV (4.56)

It is assumed that the flow in the nozzle is Newtonian. Since the flow isincompressible also, the viscous stress tensor σ in cylindrical coordinatesequals (see [Byr60]):

σ =

σrr σrθ σrz

σθr σθθ σθz

σzr σzθ σzz

(4.57)

with:


σrr = −2µ∂vr

∂r(4.58)

σθθ = −2µ(1

r

∂vθ

∂θ+

vr

r

)(4.59)

σzz = −2µ∂vz

∂z(4.60)

σrθ = σθr = −µ

[

r∂

∂r

(vθ

r

)

+1

r

∂vr

∂θ

]

(4.61)

σθz = σzθ = −µ

[∂vθ

∂z+

1

r

∂vz

∂θ

]

(4.62)

σrz = σzr = −µ

[∂vz

∂r+

∂vr

∂z

]

(4.63)

To elaborate (4.56), we start by obtaining an expression for a Poiseuilleflow profile (4.46) in terms of flow φ(t) rather than pressure drop p. To thatpurpose, the flow φ(t) of Poiseuille flow is computed by integrating the flowprofile over the surface:

φ(t) =

∫ R(z)

0

vz(r, z) 2πr dr = −πR(z)4

8µ

∂p

∂z(4.64)

To write (4.46) in terms of the volume flow, the pressure gradient is ex-pressed in terms of the volume flow:

∂p

∂z= − 8µφ(t)

πR(z)4(4.65)

(4.46) can then be written as:

vz(r, z) =R(z)2

4µ

8µφ(t)

πR(z)4

(

1 − r2

R(z)2

)

=2φ(t)

πR(z)2

(

1 − r2

R(z)2

)

(4.66)

Recall that the velocity profile is assumed to be axis-symmetric. Hence, thevelocity in the θ direction as well as all derivatives with respect to θ are zero.Furthermore, the velocity in the r-direction is assumed zero due to the onedimensionality of the approach. Given these simplifications, the divergenceof the viscous stress tensor (4.57) can be given as:

∇ ·

0 0 −µ∂vz

dr0 0 0

−µ∂vz

∂r 0 −2µ∂vz

∂z

=

−µ ∂2vz

∂r∂z0

−µ 1r

∂∂r

(r ∂vz

∂r

)− 2µ∂2vz

∂2z

(4.67)


Now, (4.56) can be written as:ZCV

v · (∇ · σ)dV =

ZCV

24 00

vz(r, z)

35 ·

264 −µ∂2vz

∂r∂z

0

−µ 1r

∂∂r

�r ∂vz

∂r

�− 2µ∂2vz

∂2z

375 dV (4.68)

= −µ

ZCV

vz(r, z)

r

∂

∂r

�r∂vz

∂r

�dV − 2µ

ZCV

vz(r, z)∂2vz

∂2zdV

The first term in (4.68) can be elaborated using (4.66) as:

−µ

ZCV

vz(r, z)

r

∂

∂r

�r∂vz

∂r

�dV = µ

ZCV

16φ(t)2

π2R(z)6

�1 −

r2

R(z)2

�dV (4.69)

= µ

Z zk(t)

z0

16φ(t)2

π2R(z)6

�1 −

r2

R(z)2

�A(z)dz

= µ

Z zk(t)

z0

16φ(t)2

πR(z)4

�1 −

r2

R(z)2

�dz

The second term in (4.68) cannot be elaborated further if the nozzle geom-etry R(z) is not known:

2µ

∫

CV

vz(r, z)∂2vz

∂2zdV = 2µ

∫

CV

2φ(t)

πR(z)2

(

1 − r2

R(z)2

)∂2vz

∂2zdV (4.70)

= 2µ

∫ zk(t)

z0

2φ(t)

(

1 − r2

R(z)2

)∂2vz

∂2zdz

Taking all four terms together, the energy equation equals in terms of flow φ(t):

4

3ρφ(t)

∂φ(t)

∂t

Z zk(t)

z0

1

A(z)dz +

2

3ρφ(t)2

∂

∂t

Z zk(t)

z0

1

A(z)dz − ρ

φ(t)3

A(z)2= p(z0, t)φ(t)

(4.71)

+ µ

Z zk(t)

z0

16φ(t)2

πR(z)4

�1 −

r2

R(z)2

�dz − 2µ

Z zk(t)

z0

2φ(t)

�1 −

r2

R(z)2

�∂2vz

∂2zdz

Dividing by φ(t) results in:


4

3ρ

∂φ(t)

∂t

Z zk(t)

z0

1

A(z)dz +

2

3ρφ(t)

∂

∂t

Z zk(t)

z0

1

A(z)dz − ρ

φ(t)2

A(z)2= p(z0, t) (4.72)

+ µ

Z zk(t)

z0

16φ(t)

πR(z)4

�1 −

r2

R(z)2

�dz − 4µ

Z zk(t)

z0

�1 −

r2

R(z)2

�∂2vz

∂2zdz

Using a standard ODE solver of Matlab, equation (4.72) can be solved straight-forwardly. The discussion of the resulting model is postponed until after thediscussion of the drop formation in Section 4.2.3.

2D modeling approaches: stream-function vorticity and CFD

Up to this point, two nozzle models have been introduced. Since the evaluation oftheir accuracy is postponed until after the discussion regarding the drop-formationin Section 4.2.3, the resulting accuracy cannot give rise to the exploration of some-what more complex models yet. However, one specific property of both modelscan be studied: their one-dimensional nature. Based on our interest in smallerdroplets, the meniscus shape in two dimensions becomes of importance. There-fore, two two-dimensional nozzle models are investigated.

For these 2D approaches, it is assumed that the nozzle can be regarded as axis-symmetric. The implications and limitations of this assumption is already dis-cussed in the introduction of this section. The first 2D approach is based onsolving the Navier-Stokes equation using stream-function vorticity method, see[Poz97]. Rather than using the velocities and pressure as variables when solvingthe governing equations, the stream-function and vorticity is used. Consequently,the number of variables in the 2D problem can be reduced: from three (two veloc-ity components and pressure term) to two (stream-function and vorticity). Fur-thermore, the surface tension is neglected. The application of the stream-functionvorticity was first proposed in [Fro84]. A detailed derivation of this approach tothe nozzle geometry at hand can be found in [Mar04] and [Mar06].

The second two-dimensional approach is based on the CFD package Flow3D.Using the axis-symmetry of the nozzle, a 2D model is constructed. In contrastto the previous three nozzle models, the surface tension is accounted for. Also,drop formation is directly incorporated in the computations. Details with regardto this model can be found in [Wij04].

4.2.3 The fluidic path: drop formation

Drop formation is a highly complex phenomenon. As a result, fulfilling the modelrequirements as posed in Section 4.1 is far from trivial. With the exception of[Dij84], the complexity of the models of Table 4.1 is too high. Therefore, the


approach to the modeling of the drop formation of [Dij84] is adopted in this the-sis. In [Mar06], this approach is improved by incorporating the effect of friction.The principles and derivation of the accompanying equations are presented inthis section, heavily based on [Mar06]. The suitability of an improved version of[Dij84] for our PIJ printhead model is based on the following two key characteris-tics. First and foremost, by using an energy balance only to determine the courseof the drop formation process and the resulting drop properties such as velocityand volume, computations are kept as simple as possible. Furthermore, ratherthan maintaining a two-sided coupling with the fluid mechanics in the nozzle, aone-sided coupling is adopted. Basically, the drop formation is computed as post-processing step for all nozzle models except the Flow3D model. This latter modelnamely already incorporates drop formation. Validation of this simplification willbe provided in the sequel of this section.

Figure 4.7: A typical sequence of drop formation as computed in Flow3D depictedstarting from t=18 µs in increments of 4 µs ([Mar06])

The drop formation model to be elaborated targets at the following. To startwith, it is to predict whether and if so, at what time instant a drop is formed.Also, the resulting drop velocity and volume are to be predicted. Four stages canbe distinguished in the drop formation process:

1. Start up. During this first stage, a negative pressure wave hits the connection-nozzle interface, causing the free surface to be sucked into the nozzle. Thispressure course is required for the build-up of sufficient energy for the secondstage, see Fig. 1.7.

2. Drop initiation. At this stage, the pressure at the connection-nozzle inter-face becomes positive and the free surface is being pushed out of the nozzle,see Fig. 4.7.

3. Thinning of the tail. The third stage starts when the pressure decreasesagain. The free surface (called ligament from this point) has obtained


enough velocity and inertia to overcome the surface tension and does notreverse its direction. This will in turn cause the ligament to become thinner.

4. Viscous loss in tail resorption. In the last stage, the ligament is broken anda drop is created, traveling with a certain velocity and volume. The fluid inthe ink channel still oscillates slightly as discussed in Section 1.2.2. Theseresidual vibrations are damped out by viscous dissipation. In general, thesemotions are too small to result in an additional drop.

h(r, t)

r

z

CV

Figure 4.8: Definition of the height of the free surface and the control volumeused in the drop formation model

These four steps are discussed in this section. The input for the drop formationmodel forms the meniscus velocity vzk

(r, t). Both the one- and two-dimensionalmodels can provide this input:

• 1D control volume model. The resulting average meniscus velocity vzk,av(t)is transformed back to a Poiseuille profile vzk

(r, t). Note that vr(z, t) is zeroat all times due the one-dimensional character of the model.

• 2D stream-function vorticity. The two-dimensional stream-function vortic-ity model already outputs the required vzk

(r, t). Furthermore, vr(z, t) at andbeyond the nozzle outlet is assumed null during the upcoming derivation.

As a consequence, since beyond the outlet the ink is supposed to form a cylinder,mass conservation implies that ∂vz/∂z is zero beyond the nozzle outlet. Fromthis point, vzk

(r, t) is written as vz . Based on these assumptions, the free surfaceboundary can be written as:


∂h

∂t= vz|h = vz (4.73)

p|h = µ∂vz

∂r+ σ

(1

R1+

1

R2

)

(4.74)

where σ is the surface tension, R1 and R2 the principal radii of the free surface,and h the height of the free surface as depicted in Fig. 4.8. ph is the pressurejust under the free surface. The height of the free surface is a function of r and t.(4.73) implies that the shape of the free surface can be found by time integrationof the velocity at the nozzle outlet. Now, a control volume (CV) is defined whoseboundaries are the surface of the nozzle outlet and the mentioned free surface,see Fig. 4.8. The equation for the mechanical energy of this deformable controlvolume reads as:

∂

∂t

ZCV

1

2ρv2

zdV +

ZCS

1

2ρv2

z(vz,rel · ez)dA (4.75)

= −

ZCS

p(ez · vz)dA − µ

ZCV

�vz

r

∂

∂r

�r∂vz

∂r

��dV − σ

∂S

∂t

where A is the closed surface around the CV. vz,rel is the velocity relative to theCS. vz,rel is assumed to be equal to vz at the nozzle boundary and zero on themeniscus that coincides with the free surface. n is the outward normal on A. Thephysical interpretation of the various terms of (4.75) is similar to that of (4.32).The additional term of (4.75) represents the enlargement of the free surface S.Since the meniscus is a part of the CV, this term must be included in the energybalance. The enlargement S equals:

S = 2π

Z Rn

0

s1 +

�∂h

∂r

�2

rdr − πR2n (4.76)

where Rn is the radius of the nozzle outlet. Hence, it is assumed that the referencefree surface at t = 0 equals the nozzle outlet surface.

To derive the condition for drop formation and the associated one for the drop

initiation, (4.75) is integrated with respect to time from zero to a certain timeinstant τ :

2π

Z Rn

0

1

2ρhv2

z |t=τrdr = (4.77)

2π

Z τ

0

�Z Rn

0

1

2ρv3

zrdr +

Z Rn

0

vzp|outletrdr − µ

Z Rn

0

h

�vz

r

∂

∂r

�r∂vz

∂r

��rdr

�dt − σS(τ )


Here, the fact is used that the relative pressure just outside the free surface iszero. At τ = 0, the CV is assumed to be zero since the free surface is equalto the nozzle outlet. After τ = 0, kinetic and pressure energy is flowing in theCV. Apart from the energy dissipation due to the viscous effects, the energy isstored in the growing CV as kinetic and surface tension energy. The growth rateof the height of the CV is equal to vz at the nozzle outlet. At a certain point intime, the nozzle outlet velocity will again decrease. The velocity vz in the CVwill also decrease as long as the meniscus surface tension is capable to deceleratethe CV. If the speed reduction at the nozzle outlet is too strong the CV cannotbe slowed down to the outlet velocity by the surface tension in the meniscus. Asa result, the CV pushed out of the nozzle will thin out. This time instant t1 canbe determined as follows.

Suppose that at τ = t1 the kinetic energy accumulated in the CV (left hand sideof (4.77)) is exactly equal to the kinetic energy transported in the CV throughthe nozzle minus the loss in viscous effects and in building the free surface energy.In other words, time t1 is reached when:

π

∫ Rn

0

1

2ρhv2

z |t=t1rdr (4.78)

= 2π

∫ t1

0

{∫ Rn

0

1

2ρv3

zrdr − µ

∫ Rn

0

h

[vz

r

∂

∂t

(

r∂vz

∂r

)]

rdr

}

dt − σS(t1)

Comparing (4.77) and (4.78) it is observed that the instant t1 corresponds to themoment when the energy given to the CV through the pressure forces at the nozzleoutlet is zero. After t1 the pressure energy flowing in the CV should become neg-ative in order to satisfy the energy balance. In the actual drop formation process,t1 for which (4.78) is satisfied corresponds to the instant when the outward veloc-ity vz decreases but still is positive. Therefore, in order to have negative energyfrom the pressure component, the pressure should become negative. Physically, itmeans that at time t1 the nozzle outlet velocity, although still positive, has beendecreasing so much that the fluid outside cannot be decelerated fast enough byinternal dissipation and surface tension, and that a negative pressure would berequired to force the drop velocity to follow the reducing outlet velocity. This isnot possible and therefore t1 is the time instant when thinning takes place and thedrop formation starts, see Fig. 4.7. The instant t1 can be determined by monitor-ing the left and right hand side of (4.78) where vz is known from the flow modelin the nozzle, and S and h are computed from (4.73) and (4.76). In Fig. 4.9,the condition is depicted graphically. Here, Ttransported represents the amount ofkinetic energy transported through the CS, Tsurface the surface tension energy,Tviscous the viscous energy, Tnet the nett energy (Ttransported − Tsurface − Tviscous),


and Tinstantaneous the instantaneous kinetic energy present in the CV.

0 2 4 6 8 10 12 14 16 18 20−8

−6

−4

−2

0

2

4

6

8

10x 10

−10

Ene

rgy

[J]

Time [µs]

Ttransported

Tsurface

+ Tviscous

Tnet

Tinstantaneous

equilibrium

Figure 4.9: Graphical representation of (4.75) and (4.78): the various energyterms involved in the drop formation

Computing precisely the thinning of the tail and the full drop formation isvery complex, not to mention computationally intensive. Again, a simple globalbalance is used to estimate the resulting drop velocity and volume at the end ofthe drop formation process. From an energy perspective, it can be assumed thatthe kinetic energy of the CV is mostly converted into energy associated with thedrop (kinetic and surface tension). Part of the CV is returning to the nozzle.Energy lost in viscous dissipation during the thinning of the droplet tail will beconsidered as a correction later.

The instant t2 chosen for considering the drop as formed is taken as the momentjust before the droplet hits the paper. The drop creation is simply being modeledas the creation of a new free surface which, considering the relatively long distancethe drop has to travel before hitting the paper, can safely be assumed to bespherical. Simulations done with Flow3D validates this assumption, see [Wij04].The energy balance between t1 and t2 can be expressed by:


2π

∫ Rn

0

1

2ρ(hv2

z

)|t=t1rdr = E(t2) +

1

2ρVdv

2d + 4πσr2

d (4.79)

where Vd = 4πr3d/3 is the final drop volume, rd is the drop radius, and E(t2)

represents the energy of the part of the CV that was ejected out of the nozzle butwill return to the nozzle once the droplet breaks loose. This part usually is takenas a certain percentage of the total volume pushed out of the nozzle. Typically,around 70 % to 90 % of the ejected volume is assumed to be transformed in thedrop. Using (4.79), the drop velocity then can be computed as:

vd =

√√√√

2π

Vd

∫ Rn

0

(hv2z) |t=t1rdr − 2

ρVd

[

E(t2) + 4πσ

(3

4πVd

)2/3]

(4.80)

The energy E(t2) remaining in the residual volume is composed of its kinetic en-ergy and of the surface tension energy of its free surface generated after the drophas separated from the main flow. The velocity of the residual volume is verysmall for a realistic actuation. Indeed to form stable and repeatable drops theactuation of the printhead has to be such that the nozzle flow returns quickly toa rest once the drop is ejected so that the next actuation cycle can be started.We can therefore neglect the kinetic energy of the residual volume. The surfacetension contribution to E(t2) can be estimated using (4.76) and assuming thatthe free surface shape can be approximated by a quadratic function h(r), the freesurface height at the nozzle edges being zero and the total volume between thefree surface and the nozzle boundary being set to the residual volume.

When 100 % of the flushed volume is transformed in Vd, the free surface of theresidual volume is equal to the outlet surface and E(t2) = 0. If Vd is taken as 90 %of the flushed volume, the surface tension energy of the residual volume was foundto be of the order of 1 % of the energy of the CV, see [Mar06]. Consequently, theenergy E(t2) can be neglected for the model.

One effect that was not accounted for in the energy balance (4.80) is the vis-

cous dissipation related to the transformation of the fluid cylinder assumed atτ = t1 to a spherical drop at τ = t2. From Fig. 4.7 it is observed that shortlyafter the drop formation process has begun most of the fluid volume is concen-trated in the tip. The resorption of the trailing tail takes more time and generatesnon-negligible deceleration of the drop. The velocity from (4.80) is obtained bycomparing the amount of energy at the point in time the drop creation just started(t1) and at a point in time far from the point of drop break-up (t2). To include theeffect of elongation viscosity, a method similar to what was originally proposedin [Dij84] is used. In this simplified drop model it is assumed that just after timet = t1 the drop volume is a cylinder with the volume Vd. The radius R and lengthl of the cylinder are changing between times t1 and t2 but satisfy πR(t)2l(t) = Vd.


Initially at time t1 the radius is equal to the outlet radius of the nozzle Rn.

An increase of l(t) results in a decrease of R(t). The interface of the stretchingjet and the fluid inside the nozzle moves with a velocity equals to vz. Assumingthat the mass of the jet is concentrated in the tip of the jet and that the velocityof the tip is equal to the drop velocity, the dynamic equilibrium between inertiaforces and viscous elongation forces yields:

ρVd∂vd

∂t= −3µ

vd − vz

l(t)πR(t)2 t1 < t < t2 (4.81)

The elongation rate has been assumed uniform over the cylinder and the elonga-tions viscosity has been taken as the Trouton viscosity which is three times theNewtonian viscosity µ. Since Vd = πR(t)2l(t) is known and constant, (4.81) canbe written as:

ρ∂vd

∂t= −3µ

vd − vz

l(t)2t1 < t < t2 (4.82)

The change in length l(t) is related to the velocity difference between the cylinderends and equals:

dl(t)

dt= vd(t) − vz (4.83)

Substituting this relation in (4.82) yields:

ρ∂vd

∂t= −3µ

dl(t)

dt

1

l(t)2t1 < t < t2 (4.84)

and integrating this relation between t1 and t2 results in:

vd(t2) − vd(t1) =3µ

ρ

(1

l(t2)− 1

l(t1)

)

(4.85)

Finally observing that l(t1) is significantly smaller than l(t2) we obtain:

vd(t2) − vd(t1) ≃ −3µ

ρ

1

l(t1)(4.86)

The velocity correction (4.86) must be added to the velocity obtained with (4.80)to obtain the final velocity of the drop.

4.2.4 The fluidic path: a review

As discussed in Section 4.2.2, the nozzle block and the drop formation are linkedby a one-sided coupling only, except for the Flow3D model. To determine thesuitability of the various nozzle models in combination with the drop formationmodel for use within the two-port model, they are to be evaluated with respect


to accuracy and model complexity. For the accuracy, the following procedureis adopted. At this point, it is assumed that the Flow3D model is the mostaccurate model of the complete fluidic path. This is confirmed by numerous ex-periments, see [Wij04]. The remaining three nozzle models in combination withthe drop formation model (hereafter referred to as fluidic path models) are there-fore benchmarked against the Flow3D model.

To facilitate this, the response to a standard trapezoidal actuation pulse of thevarious fluidic path models is computed. Given the nonlinear character of thejetting process, comparison of any other property (such as a Bode diagram) is notrepresentative for a benchmark of these models. The simulation is programmedas follows. The input to the various models is taken as the pressure history atthe nozzle entrance during the actuation with a standard trapezoidal pulse. Thishistory is obtained by tracing the response of the Flow3D model to this pulse atthe appropriate location. It provides a realistic pressure input for the benchmarkwith the two-sided coupling being accounted for. Also, during the simulation, thecondition for the jetting of a drop (4.78) is monitored. If the criterion (4.78) isfulfilled, the resulting drop speed and volume are determined. Also, the statesof the simulation are re-initialized and the simulation of the nozzle response iscontinued. This way, the drop-formation is also accounted for, albeit on a veryrudimentary level. For example, the jetting of a droplet does not take place in-stantaneously, but more over a certain time span.

The described simulation procedure can only be applied if all fluidic models areavailable in a time domain setting. Since the CV and stream-function vortic-ity (SV) models are formulated in the time domain, and the response of the 1Dimpedance model can be easily obtained in the time domain, this is not a problem.

Prior to presenting the results of this simulation, the effect of the flow profile usedin the 1D CV approach is investigated. To that purpose, the response of (4.72) toa standard trapezoidal actuation pulse is computed for a Poiseuille and a Wom-ersley flow profile, see Fig 4.10. For the computation of the Womersley numberaccording to (4.45), the dominating frequency of the pressure input trajectoryhas been used. Not surprisingly, this frequency corresponds to the channel’s firsteigenfrequency. As can be seen in Fig. 4.10, differences are small. Therefore, theuse of a Poiseuille rather than a Womersley flow profile is justified, simplifyingthe computations considerably.

In Fig. 4.11, all four fluidic model responses are depicted. However, the responseof the Flow3D model is only shown for two time-intervals. Tracing the meniscusin Flow3D equals tracing the tip of the drop that is being formed. Only afterthe tail of the drop has been completely detached from the ink in the nozzle, the’true’ meniscus position of the nozzle can be traced again. The accompanyingresulting drop speed and volumes are listed in Table 4.2, as well as an indication


0 10 20 30 40 50 60 70 80 90 100−40

−30

−20

−10

0

10

20

30

Time [µs]

Men

iscu

s [µ

m]

Figure 4.10: Meniscus response to an standard actuation pulse; using a Poiseuilleprofile (gray) or a Womersley profile (Wo=18, black) within the 1D CV approach

vd (m/s) Vd (pL)model

accuracymodel

complexityjetting %

1D impedance 5.13 16.09 + + + + 100 %CV 10.29 13.67 − − 75 %

2D SV 4.07 15.38 ++ −− 95 %Flow3D 4.23 15.30 + + + − − − -

Table 4.2: Evaluation of the fluidic path models


of the relative model accuracy and complexity. Also, the used percentage of inkthat is jetted away is listed for all models except the Flow3D model. In case ofthe Flow3D model, this percentage is not one of the parameters to be specified apriori, since the drop formation process is completely determined by the Flow3Dcomputations. Note that these percentages have been tuned to match the Flow3Ddrop speed and volume.

0 10 20 30 40 50 60 70 80 90 100−50

−40

−30

−20

−10

0

10

20

30

Time [µs]

Men

iscu

s [µ

m]

1D CV1D impedance2D SVFlow3D

Figure 4.11: Meniscus response of the various fluidic path models to a stan-dard trapezoidal actuation pulse; the 1D impedance (black dashed), 1D CV (graydashed), 2D stream-function vorticity (gray), and 2D Flow3D approach (black)

The 1D impedance nozzle model is quite accurate with respect to the Flow3Dresponse. Both the drop speed and volume are predicted reasonably accurately.Also, the meniscus position is predicted quite satisfactorily, except for the slightmismatch in the time instant of jetting. The 1D CV approach, however, is farless accurate: only the predicted drop volume approximates the Flow3D outcome.The meniscus trajectory, including the time of jetting, deviates considerably fromthe Flow3D computations. Finally, the 2D SV approach is the most accurate ofthe three fluidic path models benchmarked against the Flow3D model. Its non-zero value of the meniscus at t = 100 µs is the result of the re-initialization after


having fulfilled the requirement for the jetting of a drop. The volume in the CVin front of the nozzle outlet is jetted away. However, due to the 2D character, thisonly is an approximation. Therefore, too ’few’ is jetted away resulting in an offsetat t = 100 µs. Regarding the model complexity, the 1D impedance model is themost simple, followed by the 1D CV and 2D SV model respectively. An additionaladvantage of the 1D impedance model is its formulation in the frequency domain,which is favorable for its incorporation in the two-port model to be constructed.Note that the model complexity of the 2D SV approach should be weighted againstits ability to compute the meniscus profile. In Fig. 4.12, the meniscus profile ofthe 2D SV model is compared to that of the Flow3D model. As can be seen,these match quite accurately. The occurring inaccuracies can be explained asfollows. To start with, the surface tension is not accounted for in the 2D SVapproach. Additionally, the true location of the free surface is always assumed tobe on the geometrical outlet according to the transpiration approach, see [Mar06].As a result, the computations are slightly in error. Overall, the 1D impedancemodel will be incorporated in the two-port model due to its advantageous trade-offbetween model accuracy and complexity.

a) b) c) d) e)

f) g) h) i) j)

Figure 4.12: Comparison of the shape of the free surface computed with Flow3D(solid) and the stream-function vorticity model (dashed) at certain times; t=6 µs(a), t=8 µs (b), t=10 µs (c),t=12 µs (d),t=14 µs (e),t=16 µs (f),t=17 µs (g),t=18µs (h),t=19 µs (i),t=20 µs (j) ([Mar06])

4.2.5 The actuation path

Since piezoelectric material acts as a two-port system quite naturally, major mod-eling difficulties with respect to the formulation of the actuation path within thetwo-port framework are not to be expected. However, similar to the complexityof modeling the nozzle dynamics, capturing the full three-dimensional (possiblynonlinear) behavior of the piezo can be quite a challenge. In line with the deriva-


tions so far, a simple model of the actuation path is strived for in this section. Inthis light, the following assumptions are done:

• It is assumed that the piezo actuator deforms according to its zeroth ordermode, A(x, s) = KA(s), with K the maximum displacement of the zerothorder mode.

• Second, cross-talk is accounted for by means of the forcing function A(x, s).This effect can be quantified using for example a FEM package.

• Furthermore, it is assumed that there are no significant structural dynamiceffects. This greatly simplifies the modeling of the piezo-unit. For theapplication investigated in this thesis, this assumption does not introducegreat errors.

• Next, the electronic path is assumed to have no significant influence on thebehavior of the actuation path.

• Finally, the approach here is strictly one-dimensional. The so-called bimorfeffect, that occurs due to the fact that the piezo-unit deforms while gluedto the substrate, is neglected, see Fig. 4.13.

substrate

piezo unit

substrate piezo unit

Figure 4.13: Illustration of the bimorf effect of the piezo-unit and substrate: notactuated (left) and actuation (right)

For a detailed introduction into the behavior of piezoelectric material, one is re-ferred to e.g. [Cra87] and [Waa91]. Here, supported by the assumptions listedabove, rather than accounting for the infinite dimensional character of a piezo-unit, a lumped parameter approach is adopted. Let us start with the accompa-nying full description of piezoelectric behavior:

S1

S2

S3

S4

S5

S6

D1

D2

D3

=

sE11 sE

12 sE13 sE

14 sE15 sE

16 d11 d21 d31

sE21 sE

22 sE23 sE

24 sE25 sE

26 d12 d22 d32

sE31 sE

32 sE33 sE

34 sE35 sE

36 d13 d23 d33

sE41 sE

42 sE43 sE

44 sE45 sE

46 d14 d24 d34

sE51 sE

52 sE53 sE

54 sE55 sE

56 d15 d25 d35

sE61 sE

62 sE63 sE

64 sE65 sE

66 d16 d26 d36

d11 d12 d13 d14 d15 d16 ǫT11 ǫT

12 ǫT13

d21 d22 d23 d24 d25 d26 ǫT21 ǫT

22 ǫT23

d31 d32 d33 d34 d35 d36 ǫT31 ǫT

32 ǫT33

T1

T2

T3

T4

T5

T6

E1

E2

E3

(4.87)


Here, ~E and ~T represent the applied electrical field and the stress, respectively. ~Dand ~S stand for the electric displacement and strain, respectively. Furthermore,d is the piezoelectric charge constant, relating either the strain ~S to the appliedelectrical field ~E in the absence of mechanical stress, or the electric displacement~D to the applied stress ~T in a zero electric field. sE is the compliance for a con-stant electrical field ~E. Finally, ǫT is the permittivity under zero stress ~T . InFig. 4.14, the designation of the axes and directions of deformation are depicted.

1

2

3

6

5

4

y

x

z poling axis

Figure 4.14: Designation of the axes and directions of deformation

For the printhead under investigation, the piezoelectric material is used in the so-called d33-mode (plane stress). This implies that for both the actuator and sensor,the electrodes are perpendicular to the poling axis, see Fig. 4.14. Actuation thentakes place by expansion of the piezo-unit along the poling axis. Sensing in thiscase occurs by measuring the stress along that same poling axis. In case of d33-mode and given the fact that a one-dimensional model is aimed at, (4.87) can besimplified to:

(S3

D3

)

=

[s33 d33

d33 ǫ33

](T3

E3

)

(4.88)

Up to this point, piezoelectric material in general was considered. However, whenmodeling a piezoelectric unit with thickness hp and wet surface Sp, the samebehavior can be written in terms of:

• the tensional force applied to the unit along the poling axis F = SpT3;

• the expansion of the unit along the poling axis u = hpS3 assuming constantstrain over the piezo;

• the voltage applied to the electrodes V = hpE3;

• and the charge measured at the electrodes q = SpD3.

(4.88) then becomes:

[uq

]

=

[1/k dd C

] [FV

]

(4.89)


with d, k, and C are the piezoelectric charge coefficient, the stiffness of the piezo,and the piezo capacity, respectively. Here:

k =Sp

hps33(4.90)

d = d33

C =Spǫ33hp

The values of the parameters (4.90) depend on the specific piezo material used.Also, their values are highly influenced by the specific structure that surroundsthe piezo actuator, such as for example the substrate to which the actuator is at-tached. Therefore, the so-called ’effective’ value of these parameters can best bedetermined using a FEM package Ansys or Femlab. Irrespective of the complexityof the actuator, as long as the effective parameters can be computed (4.89) canbe used as two-port model of the actuator path. The resulting block diagram ofthe actuation path is depicted in Fig. 4.15. As can be seen in Fig. 4.15, the piezocapacity is omitted in the two-port model of the actuation path. As discussed indetail in Section 3.1.1, the piezo’s capacity is compensated for in the measurementsetup.

6

+ +

V

F u

q

d

1/k

d

Figure 4.15: Block diagram of the actuation path

One important issue concerning piezo behavior is the following: the tendency ofpiezoelectric material to nonlinear behavior. To that purpose, the behavior ofthe piezo is more closely inspected. The relation between electrical charge (po-larization) and applied electrical field for a piezoelectric material is depicted inFig. 4.16. Apparently, a hysteresis effect is present. If the electrical field is in-creased above a certain value, this will not result in an increase of polarization,since the saturation polarization Ps is reached. Suppose that from that point the


electrical field is decreased, then the polarization decreases too. However, thepolarization will not become zero but assumes a certain remanent polarizationPr. If the electric field is increased in the opposite direction, the polarizationfirst drops to zero and later to −Ps. If the electric field is then increased again,via −Pr the curve goes back to Ps. For the piezo-unit used in this thesis, thenonlinear behavior is avoided. Due to the use of multilayer piezoelectric material,the used voltages can be kept low. Consequently, one remains within the linearoperating range of the piezo-unit, see Fig. 4.16.

E (V/m)

D (C/m 2 )

P s

P r =D r

-P s

-P r = -D r

Figure 4.16: The dielectric hysteresis curve and the linearization around its oper-ating point

Finally, the following remarks are noteworthy. First, the fluid-structure interac-tion is taken into account via the stiffness of the piezo. A displacement of the piezoresults via the ink in a force sensed by the piezo. This force on its turn causes adisplacement of the piezo via the piezo’s stiffness. This way, the fluid-structure isaccounted for, see Fig. 4.15. Secondly, multiple ink channel models can be coupledto form a complete printhead model by adjusting the forcing function A(x, s).

4.3 The bilateral coupling

In the previous subsections, the subsystems that make up the inkjet channel modelhave been discussed. To couple the various subsystems, normally one uses a stag-gered scheme of some kind. For an overview of the use of staggered schemes, seee.g. [Fel01]. For example, a sequential staggered scheme is depicted in Fig. 4.17

4.3 THE BILATERAL COUPLING 87

and comprises the following steps. First, the response of system 1 to a certain in-put is computed. Second, this response is used as input for system 2. Next, afterhaving computed the response of system 2 to this input, system 1 can be providedwith a new input a timestep ∆t later. The timestep used forms a crucial factorin staggered schemes. To start with, for accuracy the Courant number should bechosen with care (see Section 4.1). Also, the timestep should be sufficiently smallto avoid staggering errors. As a result, the computational load is usually veryhigh, especially if more than two systems are to be coupled.

time step length

system 1

system 2

Figure 4.17: Sequential staggered solution procedure

For most PIJ printhead models, a CFD package is used to model the behavior ofthe acoustic and fluidic path and a FEM package for the actuation path. In thispaper, only first principle modeling has been used such that analytical expressionsare available for the formulation of the two-port systems. One major advantageof the presented approach is that the use of staggered scheme can be avoided.Instead, the Redheffer star product can be used, see [Red60; Red62]. Giventwo two-port systems as depicted in Fig. 4.18, the coupled system can then becomputed according to:

v 5

v 6

v 3

v 4

b 1

d 1

a 1

c 1

v 3

v 4

v 1

v 2

b 2

d 2

a 2

c 2

+

+

+

+

+

+

+

+

Figure 4.18: The coupling of two subsystems using Redheffer’s star product�v1

v6

�=

�a2(1 − c1b2)−1a1 c2 + a2c1(1 − b2c1)−1d2

b1 + d1b2(1 − c1b2)−1a1 d1(1 − b2c1)−1d2

��v5

v2

�(4.91)


Also, blocks with more than two input and output signals can be coupled, onlyslightly more complex Redheffer relations are needed. The two-port model of anink channel is constructed by coupling all subsystems and applying the variousboundary conditions. Validation of the resulting model and a discussion of itsproperties is treated in the next chapter. In anticipation thereof, one observationwith respect to the behavior of an ink channel is discussed in this section already.

As a result of the coupling of the various two-port subsystems, the infinite dimen-sional character is converted to a finite dimensional one. Even more specifically,the behavior of the resulting system turns out to be representable by an extremelylow dimensional resonating system, at most 4th order system. If the system isinterpreted as an equivalent mass-spring-damper system, this can be explainedas follows. Apparently, after coupling maximally two masses play a role in thedynamics of an ink channel. Possibly, one originates from the ink in the channel,coupled with that of the connection and the piezo-unit. The other may representthe mass of ink in the nozzle. The coupling provides the necessary elasticity tothe dynamics of both masses. This interpretation of the working of an ink chan-nel will be further discussed and illustrated in the upcoming chapter. Note thatthis behavior as low dimensional resonating system can already be observed inthe results presented in Section 4.2.4. As can be seen in Fig. 4.11, the responseis governed by the first channel resonance frequency only. The second resonancefrequency already seems absent.


In this chapter, the necessity of a new PIJ printhead model has been demon-strated. Given the requirements concerning accuracy and model complexity, amodeling approach based on the notion of bilaterally coupled systems has beenproposed. To that purpose, the ink channel has been divided in several functionalsubsystems. Each of these subsystems have been modeled as two-port systemsthat have been derived using first principle modeling only. As a result, the modelcomplexity could be kept low. Simultaneously, it has been shown that this doesnot necessarily imply poor accuracy, e.g. in case of the nozzle dynamics. Finally,as another major advantage of the chosen modeling strategy, it has been demon-strated that the computational demanding coupling via staggered schemes canbe avoided by the application of Redheffer’s star product. In the next chapter,the resulting two-port model will be validated using the measured frequency re-sponses as presented and discussed in Chapter 3.

Anticipating on the validation and accompanying discussion presented in the nextchapter, the low dimensional resonating character of an ink channel has been in-troduced. It has been argued that due to the coupling of the subsystems andincorporation of the boundaries, the resulting system’s behavior equals that of an


extremely low dimensional resonating system. This observation will be furtherelaborated in the next chapter.

To improve the resulting two-port model, the following research directions canbe explored. First, an upgrade of the current one-sided coupling between thenozzle dynamics and the drop formation to a two-sided one should be investigated.Though the one-sided coupling does not form an obstacle in the use of the two-port model for the (re-)design and control purposes in this thesis, the presence ofa two-sided coupling may be desirable for future investigations. Second, the use ofmore accurate models for the nozzle dynamics requires further research. Now, theexact meniscus profile is not modeled, whereas this is essential for the research intoe.g. the jetting smaller drops. Other effects, such as the presence of air-bubbles,also require the incorporation of more advanced nozzle models. Finally, the piezo-unit should be modeled more accurately. To start with, the bimorf effect shouldbe accounted for. Also, cross-talk effects and hence the coupling with other inkchannel models should be further investigated. Note that the governing equationsof the ink channel model are capable of modeling these effects: it only requires theuse of more complex forcing functions. However, choosing these forcing functionsproperly is an unresolved issue.

Chapter 5

Model validation

This chapter starts with the validation of the theoretically derived two-port modelof Chapter 4 using the measured frequency responses as presented in Chapter 3.Next, several properties of the system are critically reviewed, including the suit-ability of the resulting model in light of the requirements posed in Chapter 4, thelow dimensional approximation of the behavior of an ink channel, and several ofits fundamental limitations.

5.1 Introduction

In this chapter, the theoretically derived two-port model of Chapter 4 is put to thetest by validating it against measured frequency responses (FRs). As presentedand discussed in Chapter 3, two sets of measured FRs are at our disposal: oneobtained via the piezo-unit and the other via the laser-vibrometer. After havingcoupled the acoustic, fluidic, and actuation path using the proposed Redhefferproduct, an analytical expression of the transfer function of the two-port modelbetween certain predefined inputs and outputs becomes available. These inputsand outputs can be selected as desired. Given the measured FRs available, thevoltage sent to the piezo-unit is chosen as input whereas the measured electriccharge and meniscus speed are chosen as outputs. By substitution of an appro-priate frequency vector in the theoretical transfer function, both two-port FRscan be obtained.

To enable a sensible comparison between the measured and theoretical FRs, theymust be adjusted for various measurement devices present in the setup. Thesubsequent two sections will elaborate on this in detail. In addition, the two-port FRs are provided with additional modal damping. To accomplish that, aweighted least square approximation algorithm was applied to the two-port FRs,see [Sch94]. The resulting state space descriptions are in the controllable canoni-

91

92 MODEL VALIDATION 5.2

cal form. These state space descriptions are then each transformed into the realJordan canonical form, see [Hor85]. Next, without altering the natural frequen-cies, the poles can be shifted away from the imaginary axis to change the amountof damping. This way, each of the resonances can be tuned individually. Thenecessity of adding damping to the two-port model is further addressed in thesequel of this chapter.

This chapter is organized as follows. In Section 5.2, the two-port piezo-based FRis validated against the measured FR. The same is done for the laser-vibrometerbased approach in Section 5.3. In Section 5.4, the resulting two-port model iscritically reviewed. Finally, Section 5.5 ends this chapter with some concludingremarks with respect to the modeling of an inkjet channel.

5.2 Piezo-based validation

104

105

106

0

10

20

30

40

Mag

nitu

de [d

B]

frequency [Hz]

104

105

106

−100

−80

−60

−40

−20

0

Pha

se [D

eg.]

Figure 5.1: Measured FR of the Krohn-Hite 7602 amplifier

Prior to the validation of the piezo-based two-port FR, the presence of the fol-lowing measurement equipment is compensated for:

• Piezo amplifier. The measured FR includes the Krohn-Hite 7602 amplifierused for the amplification of the generated pulses by the waveform generator,see Fig. 3.1. This amplifier is not accounted for in the two-port model.Therefore, the theoretical FR is extended with this amplifier, whose FR isshown in Fig. 5.1.

• Low-pass filter. As discussed in Section 3.3, the Krohn-Hite 7206 low-passfilter is used during the various upcoming ILC experiments. Since from this

5.2 PIEZO-BASED VALIDATION 93

point it will be present in the FRs, it is added here to both the theoreticaland measured FRs. The FR of this filter with a cut-off frequency of 500kHz is depicted in Fig. 3.12.

• Differential action. As discussed in detail in Section 3.1.1, since only changesin the electric charge can be measured, basically a differential action isincluded in the measurement loop. Since this is not accounted for in thetwo-port model, a differentiator is to be added to the theoretical FR. Thedifferential action is approximated by an appropriate lead-lag filter.

• Piezo-sensing device. Finally, the measured FR contains the piezo-sensingdevice dynamics, see Fig. 3.6. For comparison, this effect is also accountedfor in the theoretically obtained model.

After having incorporated the adjustments listed above, the theoretical and mea-sured FRs are compared, see Fig. 5.2. In Fig. 5.3, the measured and the simulatedresponse to a standard trapezoidal actuation pulse are shown.

105

106

−50

−45

−40

−35

−30

−25

−20

−15

Frequency [Hz]

Mag

nitu

de [d

B]

105

106

−800

−600

−400

−200

0

200

Pha

se [D

eg.]

Figure 5.2: FR from the piezo actuator to the piezo sensor; measured (293e02,black) and model (gray)

Based on Fig. 5.2, it is concluded that the two-port model matches the measuredFR from the piezo-unit used as actuator to the piezo-unit used as sensor quite


0 50 100 150−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sen

sor

sign

al [V

]

Time [µs]

Figure 5.3: Sensor signal resulting from a standard trapezoidal pulse (black dot-ted, scaled); measured (293e02, black) and model response (gray)

accurately, especially with respect to the location and magnitude of the resonancefrequencies. In Fig. 5.3, it is shown that the measured and simulated response toa standard trapezoidal actuation pulse match accurately as well.

These results are not trivial. During the measurement of the piezo-based FR, theamplitude of the sinusoids has been chosen such that the ink channel was notjetting. In contrast, the measured response results from a jetting ink channel.During the derivation of the two-port model, the nonlinear effect of the jettingof a drop was neglected. Now, whereas a match of the two-port model withthe measured FR might be expected, it certainly is not trivial for the measuredand simulated responses. Since the match is still accurate in the latter case, thesupposedly nonlinear effect of jetting a drop as seen from the piezo is indeednegligible. When Fig. 5.3 is inspected closely, a small increase of the resonancefrequency of the measured response can be observed though. This is due to thedecrease in ink in the nozzle, causing a slight increase of this frequency. Despitethis small mismatch, the behavior of the ink channel can be regarded linear forthe piezo-based case from a control perspective.

5.3 Laser-vibrometer based validation

Similar to the piezo-based approach, the laser-vibrometer based FRs are adjustedwith respect to the following devices:

• Laser-vibrometer. The laser-vibrometer introduces a considerable phase lag

5.3 LASER-VIBROMETER BASED VALIDATION 95

according to (3.4). At 1 MHz, this already amounts to 467 kHz. Prior tothe comparison to the theoretical FR, the measured FR is compensated forthis phase lag.

• Piezo amplifier. Similar to the piezo-based case, the theoretical FR is ad-justed for the presence of the Krohn-Hite 7602 amplifier in the measurementloop.

As mentioned in Section 3.5, the measured FR at 2.5 V is used throughout thisthesis for validation purposes. Recall that the amplitude of the first resonancefrequency is dependent on the used excitation voltage. After having adjustedboth FRs for the various measurement devices, the measured and two-port FRsare compared, see Fig. 5.4.

104

105

−60

−50

−40

−30

−20

−10

0

Frequency [Hz]

Mag

nitu

de [d

B]

104

105

−1000

−800

−600

−400

−200

0

Pha

se [D

eg.]

Figure 5.4: FR from the piezo actuator to the meniscus velocity; measured at 2.5V (233e01, black) and model (gray)

As can be seen in Fig. 5.4, the two-port FR matches the measured FR quitesatisfactorily, except for the first and most important resonance frequency. Thismismatch can be explained as follows. In the laser-vibrometer based case, froma two-port perspective, the nozzle dynamics are measured coupled with a cer-tain output impedance. This output impedance lumps the connection, channel,reservoir, and piezo-unit dynamics into one single output impedance condition.


Apparently, the two-sided coupling between the nozzle dynamics and this out-put impedance is incorrectly accounted for causing the modeling errors as canbe seen in Fig. 5.4. So, either the nozzle dynamics or the output impedance arenot accurately modeled. Validation of the fluidic path in Section 4.2.4, however,has shown that the nozzle dynamics are modeled rather accurate, see Fig. 4.11.The validation has been carried out using an output impedance that has beendetermined by a Flow3D model. By tracing the pressure history at the nozzleentrance, an input signal was obtained that certainly accounts for the two-sidedcoupling properly. In contrast to this procedure, here the output impedance isdetermined by the two-port model itself. Therefore, it seems a valid conclusionthat there are some modeling inaccuracies present in the output impedance caus-ing the encountered mismatch. In the piezo-based case, the piezo-unit is coupledwith an output impedance representing the channel, connection, reservoir, andthe nozzle dynamics. Based on the fact that the results obtained in this case areaccurate, this impedance apparently is more correct.

The impact of the incorrect two-sided coupling seems to be limited to the firstresonance frequency only, see Fig. 5.4. As discussed in Section 3.2, the nozzleacts more as open rather than a closed end for low frequencies. Therefore, thetwo-sided coupling between the nozzle and the remainder of the ink channel playsa more prominent role for these frequencies. For higher frequencies, the nozzledynamic behavior becomes rather autonomous.

As a result of the discussed model inaccuracies, the usability of the two-port modelin the laser-vibrometer based case is rather limited. As discussed previously, theresponse to an actuation pulse is largely determined by the ink channel’s firstresonance frequency. Since this is not modeled correctly, a considerable mismatchresults between the measured and simulated responses to a certain pulse. There-fore, a comparison between the measured and simulated response is not usefuland is omitted.

Based on the encountered dependency of the dynamics on the applied input volt-age, laser-vibrometer based operation of a PIJ printhead cannot be regarded aslinear. The two-port model does not give rise to an adjustment of this statement.Still, since it only involves a soft nonlinearity, the linearity assumption will againbe reviewed when control is applied to an ink channel, see Chapter 7.

5.4 Discussion

In this section, the resulting two-port model and the accompanying system is crit-ically reviewed. First, it is discussed to what extent the resulting model actuallyfulfills the requirements as posed in 4. Second, several important model propertiesare addressed, such as the low dimensional approximation of system’s behavior

5.4 DISCUSSION 97

and the encountered differences in accuracy in the piezo- and laser-vibrometerbased case. Third, the necessity of adding damping is discussed. Finally, severalshortcomings of the system itself are addressed.

In Section 4.1, the objectives for the modeling of an ink channel have been for-mulated as achieving high accuracy while having low model complexity. Thisway, the suitability of the resulting model for control and (re-)design purposesis enforced. Based on the validation presented in the previous two sections, it isconcluded that the resulting overall accuracy is satisfactory, except for the inac-curacy of the first resonance frequency in the laser-vibrometer case. To guaranteesuitability for the application of control in this case, the two-port model shouldbe improved with respect to this transfer function. Due to the sole use of firstprinciples for the modeling of various blocks and their coupling via the Redhefferstar product, the two-port model has a low model complexity. In addition, theemployment of bilaterally coupled systems offers valuable insight in the workingof an ink channel that partly already have been and partly will be discussed in(the sequel of) this thesis. In conclusion, the two-port model meets the prede-fined goals set in the beginning to a large extent. With the work presented inthe previous and current chapter, a satisfying answer can be provided to the firstresearch question as formulated in Chapter 2.

Concerning the resulting two-port model, the following remarks are in order:

• Sensor locations. In the previous section, two sensor locations and accom-panying transfer functions have been investigated. Based on the obtainedresults, it is concluded that the system properties depend on the specificsensor location. In terms of our two-port approach, the output impedanceencountered by the specific subsystem that incorporates the sensor func-tionality clearly is different. This explains the fact that different results areobtained.

A related issue concerning the sensor location is the following. The adoptedinternal structure of the two-port model, based on the physical structureof an ink channel, cannot be validated using only the two measured input-output relations. For that, additional measurements are required that couldnot be performed due to the limitations concerning the printhead structure.For example, measuring the flow or pressure at the channel-connection tran-sition is not possible for the printhead under consideration without destruc-tion of the printhead.

• Low dimensional character of the ink channel system. In Section 4.3, the lowdimensional character of the ink channel’s dynamics has been introduced.As a result of coupling various subsystems and the application of boundaryconditions, the infinite dimensional character is replaces by a finite dimen-


sional one. Based on Fig. 5.3, two dominating resonance frequencies canbe distinguished. One equals the channel’s first eigenfrequency. The otherequals a higher order resonance mode of the system. Given these observa-tions, the system’s behavior can be described by an equivalent 4th ordermass-spring-damper system.

• Damping. Another issue concerning the two-port model is the necessityof the adding of damping as described in Section 5.1. Clearly, there aresome modeling inaccuracies regarding damping. As discussed earlier, themajority of the damping occurs in the nozzle. However, given the accuracyof the 1D impedance model as shown in Section 4.2.4, it is expected that thedamping in the nozzle block is taken into account correctly. In our view, theadditional damping originates from the other boundary: the reservoir. InChapter 4, the reservoir was assumed to act as an open end. In practice, thereservoir is not a genuine open end and, more importantly, it contributes tothe damping. Further research is necessary to verify whether improving thereservoir block with respect to the damping renders the adding of dampingas described in this chapter superfluous.

Based on the derivation of and the investigations into the two-port model, thefollowing properties of the ink channel system have come up:

• Linearity of the jetting process. As discussed above, the jetting process canbe assumed to be linear for the piezo-based case. For the laser-vibrometerbased case, the jetting process cannot be considered linear. This is due tothe earlier established nonlinear behavior, see Section 3.5. However, thisdoes not imply that the linear control techniques are useless, as will bedemonstrated and argued in Chapter 7.

• Limits on the jetting frequency. The currently used standard trapezoidalactuation pulse is completely geared to the channel’s first resonance fre-quency. This is due to several reasons. For one, it is the most energyeffective, see [Wij06], making use of interfering traveling waves. As a result,the actuation voltage can be kept within admissible limits. From our two-port perspective, the use of one of the two dominating resonance frequenciesof the system is a logical choice. Drawback of this approach is the fact thatthe minimum time required for jetting a droplet is fixed by a channel’s firsteigenfrequency, limiting the attainable jetting frequency. Also, the residualvibrations are dominated by the same frequency. Alternatively, if multiplepiezo-units were present, a pulse train could be used that does not requirethe use the channel’s first eigenfrequency. Then, the limits with respect tothe jetting frequency could be lifted. Upon using such a pulse train, theenergy can be added to the pressure wave gradually without the need toexceed the admissible actuation voltage.


• Controllability and observability. An ink channel is a distributed parametersystem. However, after coupling of the various subsystems, the two-portmodel becomes a lumped parameter model. Though in general this simpli-fication is very useful, in some cases it does not suffice. One important casefor example concerns the controllability and observability of a distributedsystem, since it concerns a definite spatial property. Therefore, the notionsof spatial controllability and observability would be more appropriate toconsider here, see e.g. [Jaı88] and [Tzo94]. Given the limitations of the two-port model with respect to this issue, a discussion concerning controllabilityand observability is restrained to the following remarks:

1. Sensor and actuator position. Drop formation takes place in the nozzle.For control purposes, therefore, having a sensor and actuator locatedin the nozzle would be ideal. Both are not present in the current designat that specific location. Note that the laser-vibrometer forms only atemporary solution, since it can only measure in non-jetting situations.This will be discussed in detail in Chapter 7. Therefore, measuringand control of the meniscus is in the hands of the piezo-unit, which isonly an indirect way. Furthermore, as can be seen in Fig. 5.4, severalanti-resonances are present. Apparently, some meniscus trajectoriescannot be generated. This effect can be explained by the occurrenceof destructive interference, see Section 3.2.

2. Length of the piezo-unit. The piezo-unit senses the force that resultsfrom the pressure distribution in the channel acting on the piezo’ssurface, see Section 3.1.1. This force thus represents an average value.Consequently, it is not easily possible to track traveling waves. Also,certain standing waves will be difficult or impossible to measure. Thepiezo-unit acting as actuator also has similar disadvantages due toits length. Most importantly, it is impossible to generate all wavepatterns. As a consequence, one is restricted in the actuation. Forexample, creating a wave front capable of jetting a drop needs to begenerated using interference.

The mentioned issues with respect to controllability and observability canbe resolved by using multiple piezo-units rather than one. The two-portmodel can serve as starting point for further investigations into these issues,for example by coupling several channel blocks, see Fig. 4.3


The following conclusions are drawn concerning the modeling of an ink channel:

• The two-port model fulfills the requirement with respect to the accuracy, ex-cept for the first eigenfrequency of the system in the laser-vibrometer based


case. The cause of this inaccuracies has been attributed to modeling errorsof the ink channel itself. The requirements regarding the model complexityand the related computational load are met as well. Altogether, the two-port model forms a suitable starting point for the control and (re-)designpurposes in mind.

• The behavior of an ink channel can be approximated by a low dimensionalsystem. In our case, a 4th order linear model is capable of describing thesystem dynamics accurately.

• Several (fundamental) limitations of a PIJ system have been identified.First, the dominance of the channel first eigenfrequency limits the attainablejetting frequency. Second, the geometry and location of the sensor hampersthe control of an ink channel.

In the upcoming two chapters, our attention shifts to feedforward control of thea PIJ printhead. The insight in the working of an ink channel obtained in thepreceding chapters forms a valuable tool for the application of control. Severalissues that have been discussed, such as linearity and the fundamental limitationsof current PIJ printhead designs, will be revisited at the end of Chapter 7.

Chapter 6

The control framework

Motivated by the repetitive character of the inkjet printing process, Iterative Learn-ing Control (ILC) is chosen as feedforward control strategy to enable the switchto a controlled environment for PIJ printheads. In preparation for the implemen-tation of ILC presented the next chapter, the ILC framework is introduced in thischapter. After having discussed the adopted lifted ILC control structure, the con-trol goals for the PIJ printhead under investigation are formulated. Finally, ILCcontroller design is discussed. With the theoretical background on ILC presentedin this chapter, and the insight obtained the previous two chapters in the inkjetsystem, an excellent starting point is provided for the successful implementationof ILC.

6.1 Introduction

From a systems and control perspective, virtually all PIJ printheads are uncon-trolled systems. As discussed extensively in Chapter 1 and 2, a switch to a con-trolled environment is investigated in this thesis. The aim is twofold: to push theprinthead performance to its limits in face of the current operational issues, andto simultaneously establish the corresponding fundamental limitations of a cer-tain printhead design. The switch is performed by the application of feedforwardcontrol to the PIJ printhead under investigation. The choice for feedforward con-trol is motivated by the following. Given the fact that a PIJ printhead performsthe same task over and over again, application of feedforward control generallyyields considerably more performance improvement than feedback control. Ad-ditionally, feedback control is not required to stabilize a passive system such asa PIJ printhead. Furthermore, given the small timescales involved in the jettingprocess, feedback control is considered computationally too demanding.

The repetitive character of the jetting process gives also rise to another choice,

101

102 THE CONTROL FRAMEWORK 6.2

namely that for Iterative Learning Control (ILC) as feedforward control strategy.ILC is par excellence suited for systems that have to perform the same task timeand again. It is a control strategy used to iteratively improve the performance ofthese systems by updating the command signal from one experiment to the next.This update is based on measured data from previous trials, hence the term learn-ing. Two remarks are in order. First, only the trial invariant part of the error canbe reduced by ILC. Second, application of ILC requires that the system returnsto the same initial condition in between the consecutive command applications.If this condition is not met, repetitive control should be applied. It is assumedthat the PIJ printhead fulfills this requirements at all times. For an overviewon ILC, one is referred to [Moo93], [Moo98], [Bie98], or [Lon00]. ILC has beensuccessfully applied to a wide variety of applications in many different engineer-ing areas, ranging from its original ([Ari84]) application of robotics (e.g.[Tay04])to servo-mechanical applications (e.g. [Dij04] and [Roo97]) and chemical batchprocessing (e.g. [Lee96]).

Similar to Chapter 3, the application of ILC is performed for both the piezo- andlaser-vibrometer based cases. Though the general ILC approach is the same forboth cases, there are several small differences. This is clearly indicated wheneverappropriate in this chapter. Starting with the lifted control structure, the controlgoals are formulated next. Synthesis of the controller is then discussed. Specialattention is given to the robustness of resulting controller against model inaccu-racies as well as the constraints imposed by the actuator of PIJ printheads. Thischapter ends with some concluding remarks.

6.2 The lifted ILC control structure

integrator

+ +

+ -

γ

H

L

z−1I

yref − d

yk

ek

uk

∆uk

uk+1

Figure 6.1: ILC control structure in the trial domain in the piezo-based case

Of several ILC structures available, the lifted ILC structure ([Pha88]) is adoptedin this thesis, see Fig. 6.1 and 6.2. For a derivation, one is referred to e.g. [Dij04].The choice for the lifted is based on the following two arguments. First, the liftedsystem description accounts for the finite character of the intervals in contrast to

6.2 THE LIFTED ILC CONTROL STRUCTURE 103

descriptions based on infinite time considerations, like for example the standardsetting. In the latter case, the nonzero value of the error used by the ILC at thestart and end of the trajectory causes problems that need to be handled sepa-rately often resulting in rather heuristic approaches. These nonzero values maybe caused by e.g. system noise. Design in the lifted setting has as main advantagethat the solution explicitly takes into account states of the plant at the beginningand end of the trajectory. Second, the lifted setting allows for the use of standardclassical (optimal) control methods for the analysis and design of learning updateschemes, see [Tou01]. A final, more application related, advantage of the liftedsetting are its numerically favorable properties. This will be further discussed inSection 6.4.

+ +

+ -

γ

H

L

z−1I

yref − d

yk

ek

uk

∆uk

uk+1

Figure 6.2: ILC control structure in the trial domain in the laser-vibrometer case

The lifted ILC structure for the piezo-based and laser-vibrometer based case aredepicted in Fig. 6.1 and 6.2, respectively. The mapping H is the impulse responsematrix of the plant having a state space representation (A, B, C), for an LTI sys-tem a lower triangular Toeplitz matrix. The learning matrix, that still has to bedesigned, is represented by L and may be non-causal and time-varying. z−1 is onetrial delay operator and can be seen as memory block. The trial length N equals1000 given a sample rate of 10 MHz and the DOD frequency of 10 kHz. Signaluk is a vector containing the system’s inputs or states of the ILC system. Sig-nal yk is the system output, yref the reference trajectory, and d the disturbance.Throughout this thesis, the effect of the noise d is assumed to be negligible. Theeffect of noise is discussed in e.g. [Nor01]. ek is the error output. The updateof the system’s input is ∆uk and uk+1 is the input for the next trial k + 1. Atthe k-th trial, signal uk is provided to the system, resulting in the (integrated)output yk. The output yk is then subtracted from the reference yref to obtain theerror ek. Based on this error, the learning controller computes the adjustmentsto the input ∆uk that, added to the previous input, forms the input for the nexttrail uk+1. Apparently, the ILC controller functions as a feedback controller inthe trial domain.


In case of a MIMO system, the above control structures are the same. In caseof two channels, the signals in (6.1) have dimension 2N × 1. H has dimension2N × 2N . The various signals and impulse response matrix are then structuredas follows:

yk =

yAk (0)

yBk (0)

yAk (1)

yBk (1)...

yAk (N − 1)

yBk (N − 1)

uk =

uAk (0)

uBk (0)

uAk (1)

uBk (1)...

uAk (N − 1)

uBk (N − 1)

ek =

eAk (0)

eBk (0)

eAk (1)

eBk (1)...

eAk (N − 1)

eBk (N − 1)

(6.1)

and:

H =

hA(0) hBA(0) 0 0 . . . 0 0hAB(0) hB(0) 0 0 . . . 0 0

hA(1) hBA(1) hA(0) hBA(0) . . ....

...

hAB(1) hB(1) hBA(0) hB(0) . . ....

......

.... . .

. . .. . .

......

hA(N − 1) hBA(N − 1) . . . . . . . . . hA(0) hBA(0)hAB(N − 1) hB(N − 1) . . . . . . . . . hAB(0) hB(0)

(6.2)

For a larger array of channels, (6.1) and (6.2) are adjusted according to the dis-played structure.

In the piezo-based case, the measured sensor signal represents the derivative ofthe pressure in the ink channel, see e.g. Section 5.2. Bringing the derivativeof the channel pressure to zero, however, does not imply that the channel is atrest. Therefore, the measured output is numerically integrated as can be seen inFig. 6.1. Control then is focussed on the channel pressure itself. In Section 6.4,it is shown that adding an integrator also is numerically advantageous. In thelaser-vibrometer case, the integrator is omitted, see Fig. 6.2. As a result, themeniscus speed is controlled rather than its position. Since drop formation ishighly dependent on the meniscus velocity rather than on its position, addingan integrator is not necessary. For a study into refill as well as stability, theavailability of the meniscus position becomes important. In that case, adding anintegrator can be considered. However, integration requires the availability of acorrect initial state, which in case of the meniscus is not trivial as apposed to thechannel’s initial state. After all, the channel pressure can be measured at anytime instant whereas the meniscus position cannot be determined at all. Finally,

6.2 THE LIFTED ILC CONTROL STRUCTURE 105

a discussion on the effect of the scaling γ is postponed until Section 6.4.

observation

actuation

0

1

0

1

N

N1

Figure 6.3: Illustration of the actuation and observation time windows

For some applications, the actuation and observation time intervals are not equalto the complete trial length N . Though in case of a PIJ printhead the observationwindow does cover the the complete trial length, the actuation is to be restrictedto a limited time window. This is depicted in Fig. 6.3. Restriction of the actu-ation is necessary to enable the increase of the jetting frequency. After all, thehigher this frequency, the shorter the available actuation time interval. To enablethe restriction of the actuation and observation windows, the lifted ILC controlapproach can be adjusted according to the following two methods.

The first approach adjusts to the impulse response matrix H . To that purpose,(6.2) is structured as follows:

yk(0)...

yk(N1)yk(N1 + 1)

...yk(N − 1)

=

[H11 0H21 H22

]

uk(0)...

uk(N1)uk(N1 + 1)

...uk(N − 1)

(6.3)

where N1 is the time instant for the actuation to stop. In our case, the trackingbehavior of the complete trial is important, yk, but the actuation, uk, is restrictedto a certain time period. Therefore, (6.3) can be reduced to:


yk(0)...

yk(N1)yk(N1 + 1)

...yk(N − 1)

=

[H11

H21

]

︸︷︷︸

H∗

uk(0)...

uk(N1)

(6.4)

This adjusted H is now used during the design of the learning filter L, see Sec-tion 6.4. The incorporation of actuation and observation intervals in the designof the learning filter L as demonstrated can only be facilitated by the lifted ILCsetting.

integrator

+ +

+ -

γ

H

L

z−1I

yref − d

yk

ek

uk

∆uk

uk+1

Wi

Wo

Figure 6.4: ILC control structure with weightings in the trial domain in the piezo-based case

A second approach expands the ILC control structure with weighting filters, seeFig. 6.4. Wi and Wo serve as weighting on the inputs and outputs of the system,respectively. If these weightings are taken diagonal, they act as time-weights onthe signals. A very small weight on certain parts of the input signal ensures thatthe ILC controller does not generate control signals in that range. Similarly, avery small weight on the output ensures that the ILC algorithm does not try toreduce the errors in that range. To illustrate the choice of the weighting filters,consider the following choice:

Wi =

[IN1 00 0N−N1

]

Wo =

[IN1 00 IN−N1

]

(6.5)

Using these filters, the same objectives are strived for as in the first time windowsapproach:

H∗ =

[IN1 00 IN−N1

] [H11 0H21 H22

] [IN1 00 0N−N1

]

=

[H11 0H21 0

]

(6.6)

6.3 THE CONTROL GOALS 107

The range for the choice of the weighting filter is seemingly endless. Consequently,compared to the approach of time windows, the use of weighting filters offers moreflexibility. For example, allowing the ILC algorithm to generate slightly more er-ror in a time interval where it is of less importance, generally yields better overallperformance.

A final remark concerning the restriction of the actuation window is the following.The restriction is limited, e.g. due to avoiding too high actuation voltages. As aresult, the increase of the jetting frequency is bounded also. Measures to overcomethis include the following. For example, linearity of the jetting process can beassumed and the ILC actuation pulses can be superimposed. Alternatively, ILCactuation pulses can be learned for a sequence of drops. The former solution isadopted in this thesis.

6.3 The control goals

In Section 1.2.2, the performance requirements of a PIJ printhead as well as thecorresponding limitations have been discussed in detail. In this thesis, the focuslies on improving the performance with respect to the following two requirements:

• Enhancing the productivity. The productivity of a PIJ printhead is mainlydetermined by the jetting frequency and the amount of nozzles per inch, seee.g. [Bru05]. As discussed in Chapter 1.2.2, the attainable jetting frequencyis limited by the residual vibrations. The amount of nozzles per inch, alsoreferred to as npi-ratio, is limited by the measure to minimize the effect ofcross-talk, see Section 1.2.2 also. Therefore, to improve the productivity ofa PIJ printhead, the residual vibrations and cross-talk effects are to be min-imized. Changing/varying dynamics and robustness against disturbancesdo not affect the productivity directly.

• Improving the drop-consistency. Apart from the specific requirements withrespect to drop properties such as speed, volume, shape, and straightness,consistency of these properties is the most important property of all. Meet-ing current consistency requirements limits the operation of PIJ printheads.For example, jetting at 10 or 20 kHz yields inadmissible variations in e.g.drop-speed and -volume. Actuation of a random combination of neighbor-ing channels generally yields too large variations in drop properties as well.Again, the residual vibrations and cross-talk are the major performance lim-iting phenomena when considering consistency, see Section 1.2.2. The otherperformance limiting phenomena affect the drop-consistency much less.

Other requirements as formulated in Section 1.2.2, such as stability and drop-speed and -size, are not considered in this thesis. However, this does not imply


that ILC cannot be used to improve the performance concerning these require-ments. At the end of this section, it is shortly indicated how ILC can be employedto improve the performance concerning these issues as well.

Apparently, both improving the productivity and consistency demand for theminimization of the residual vibrations and cross-talk. In the sequel of this section,therefore, the attention is shifted from the control goals as formulated above tothe minimization of the residual vibrations and cross-talk. To measure the effectof ILC with respect to these two issues, the following performance indicators areused:

• IAE of the resulting error signal. The error signal indicates to what extentthe reference trajectory is attained. The error can be expressed in terms ofthe Integrated Absolute Error (IAE):

IAE =

N∑

i=0

|ek(i)| (6.7)

Given an appropriate choice for the reference trajectory, attaining this tra-jectory implies that the residual vibrations and cross-talk are effectivelyminimized. Therefore, the IAE serves as indicator for the performance.

• DOD-speed and -volume curves. A Drop-On-Demand (DOD) curve showsthe relation between drop-speed or -volume and the used jetting frequency.For example, a DOD-speed curve is depicted in Fig. 1.9. Elimination of theresidual vibrations leads to an improvement of the DOD curve, as discussedin Section 1.2.2. Therefore, the DOD-speed and volume curves can be usedas performance indicators for the minimization of residual vibrations. Ide-ally, a DOD curve is a horizontal line.

• Cross-talk curve. The influence of cross-talk on the performance of a PIJprinthead is assessed in a cross-talk curve, see e.g. Fig. 1.10. It depictsthe resulting drop-speed of one particular channel when in turn neighbor-ing channels are actuated simultaneously. If the cross-talk is eliminatedcompletely, the cross-talk curve is a horizontal line. For an array of twochannels, the cross-talk curve reduces to a table.

The link between the formulated objectives and the adopted control frameworkis formed by the reference trajectories. More specifically, these are to be con-structed such that minimization of the residual vibrations and cross-talk is en-forced. The observation that drop properties are completely determined by themeniscus trajectory forms the starting point in the formulation of suitable ref-erence trajectories. Note that this observation implicitly has served as basis forthe derivation of the governing equations for the drop formation in Section 4.2.3,


where the meniscus speed serves as input for the computations. This observationgreatly facilitates the implementation of ILC. Instead of formulating the controlobjectives in terms of drop properties such as drop-speed or -volume, informationthat is available only at certain discrete time instances, a continuous objective cannow be adopted. Still, the relationship between the resulting drop properties andthe meniscus trajectory is far from trivial and cannot be characterized straight-forwardly. For example, various meniscus trajectories may result in similar dropproperties whereas some drop properties may not be realizable for any menis-cus trajectory. Consequently, choosing a suitable meniscus reference trajectoryremains a non-trivial matter. Another complicating matter concerns the usageof the laser-vibrometer, the most sensible sensor when aiming at realization of acertain meniscus trajectory. As discussed in Section 3.1.3, there are a numberof practical disadvantages associated with the use of the laser-vibrometer as sen-sor. Alternatively, the piezo-unit can be chosen as sensor functionality. Then, achannel pressure trajectory can be used for the control purposes in mind. How-ever, given the fact that the pressure trajectory is only an indirect measure ofthe realized meniscus trajectory, specification of a proper reference trajectory forthe channel pressure might be even more difficult. Nevertheless, both options areconsidered and are used in the sequel of this thesis.

Theoretically, the following procedure is to be utilized to construct a suitable refer-ence trajectory. Based on the required drop properties, a corresponding meniscustrajectory can be computed using the relations derived in Section 4.2.3 in generaland the inverse of (4.78) in particular. Basically, this amounts to computing theinverse of the drop formation model. Once this trajectory has been computed,measures to counteract the residual vibrations and cross-talk can be incorporated.If desired, the corresponding pressure trajectory can be obtained using both thepiezo-based and the laser-vibrometer based transfer functions, see [Bos05]. Basedon these TFs, the TF between the channel pressure and the meniscus velocitycan be computed, see Fig. 6.5. Using the inverse of this computed TF, the corre-sponding channel pressure trajectory can be computed given a certain meniscustrajectory.

From a practical point of view, the above procedure is rather complex (computingthe inverse of the drop formation model) and sensitive to modeling errors (com-puting the corresponding channel pressure trajectory). An alternative simple yeteffective approach is the following. The starting point is a measured meniscusvelocity or channel pressure response to a standard trapezoidal actuation pulse,see Fig. 6.6. Suppose that the corresponding drop properties are according to thespecifications. The trajectories are then adjusted as follows:

• Eliminating residual vibrations. The measured trajectories are supposed toconsist of two parts. During the first part, up to the point where condition(4.78) is fulfilled, is left unchanged. In this way, the drop formation is


replacemen

V

V

vmeniscus

vmeniscus

channelpressure

TFp2vTFV 2p

TFV 2v

Figure 6.5: Schematic overview of the TFs playing a role during computation ofa channel pressure reference trajectory

left undistorted. During the second part, the response is governed by theresidual vibrations. By forcing the meniscus velocity or the channel pressureto a rest, this operational issue can be eliminated.

• Eliminating cross-talk. If the responses are measured while only one inkchannel is actuated, the measured trajectories are cross-talk free. Sub-sequently, when these references are used during the ILC computations,cross-talk is effectively eliminated.

There are two important constraints for the construction of the reference trajec-tories. First, to ensure the refill of the nozzle the fluid-dynamics are not broughtto a rest immediately after the ejection of the drop. Details with respect to refillcan be found in e.g. [Yan04]. Second, the fluid-dynamics are brought to a restsomewhat gradually to avoid too high actuation voltages. The sketched approachis illustrated in Fig. 6.6.

In the next chapter, it is shown that this somewhat pragmatic approach to theconstruction of reference trajectories is in fact a very successful one. Still, it isemphasized that this is just one possible choice for the reference trajectory. Acomplete analysis based on the theoretical approach would provide valuable in-sight in the limitations of current printhead designs, e.g. with respect to the dropproperties that are feasible.

The question arises whether the piezo- or laser-vibrometer based approach is moresuitable for the realization of the control goals. Intuitively, one could argue thatthe adoption of the laser-vibrometer based approach leads to better results. Even-tually, the meniscus determines the performance for a large part. The ink channelpressure remains an indirect indicator. Additionally, the limited (spatial) control-lability might prevent the realization of some meniscus movements using solelythe piezo-unit, see Section 5.4. A comparison of both approaches provides insight


0 20 40 60 80 100−2

−1

0

1

2

3

4x 10

−6

Inte

grat

ed s

enso

r si

gnal

[Vs]

Time [µs]0 20 40 60 80 100

−0.5

0

0.5

1

Time [µs]

Men

iscu

s ve

loci

ty [m

/s]

Figure 6.6: Measured sensor signals (black) and reference signals (gray dotted)for the piezo- (left) and laser-vibrometer case (right)

concerning the limitations of certain PIJ printhead designs.

In addition to the control goals formulated above, the following two operationalissues can be handled by ILC also by adopting the right reference trajectories:

1. Drop-speed and -volume (modulation). Drop properties such as speed andvolume can be adjusted by changing the trajectory. This can possibly bedone during operation, enabling drop speed or size modulation. The effec-tiveness of these measures can be established by measuring the resultingdrop-speed and -volume using the CCD camera.

2. Stability. Stability of the jetting process is among other things dependenton the retraction of the meniscus. By adjusting the reference trajectories inthis respect, this could be realized easily. The larger the retraction, the morethe possibility that instabilities occur. Mainly connected with robustnessalso, especially dirt particles and air-bubbles. Therefore, ILC can even beinvoked to improve stability. Stability can be checked by bitmap tests.

Note that drop-shape and straightness have not been discussed here. Thoughboth might very well be controllable with the ILC approach discussed in thisthesis, the 1D approach adopted throughout this work restricts our scope to thedrop properties considered so far. If the approach is extended to 2D, these issuescould be resolved also.


6.4 ILC design

In this section, the design of ILC controllers in the lifted setting is discussed. Forthe synthesis of ILC controllers various approaches can be adopted. Here, LQ-optimal ILC design is treated. It is shown that this method can be used for thedesign of both SISO and MIMO ILC controllers. Also, special attention is paid toissues such as robustness of the resulting controller and limiting the observationand/or actuation interval. Implementing the resulting ILC controllers usuallyresults in rather complex ILC actuation pulses. Since the Application SpecificIntegrated Circuits (ASIC) can only handle actuation pulses that are limited incomplexity, an adjusted ILC algorithm is proposed. This so called constrainedILC constructs actuation pulses that are composed of a predefined number ofpiece-wise affine functions.

Note that throughout this section it is assumed that the structure of the varioussignals as in (6.1) and (6.2) is adopted consequently.

6.4.1 LQ-optimal control

In this section, LQ-optimal ILC design is discussed. The derivations presented inthis section are based on the work of [Tou01] and [Dij04]. Starting point formsthe following ILC system description:

uk+1 = uk + ∆uk

ek = −yk + yref = −Huk + yref (6.8)

∆uk = Lek

with u0 = 0. Recall that the noise d is neglected, as assumed in Section 6.2. Letus first verify whether the conditions for the existence of a solution of the optimalLQ-problem are fulfilled. For that, the system must be both stabilizable and de-tectable. If a system is not stabilizable, then obviously it cannot be stabilized. Ifa system is not detectable, there exist state feedback controllers that do not sta-bilize the system but hide the instabilities from the output. Stability then cannotbe guaranteed. A sufficient condition for stabilizability is that the system is con-trollable. A sufficient condition for detectability is that the system is observable.Given the presence of a bank of integrators in the control structure, controllabilityis automatically fulfilled. In contrast, observability is not a trivial matter. In casethe output matrix H is singular or nearly singular this criterion is not fulfilled.This might occur if the underlying plant contains time-delays or non-minimumphase zeros. To resolve this, define the singular value decomposition of H as:

H = UΣV T =(U1 U2

)(

Σ1 00 Σ2

)(V T

1

V T2

)

(6.9)

6.4 ILC DESIGN 113

where U and V are unitary orthogonal matrices and Σ is a diagonal matrix withthe singular values on the main diagonal ordered from large to small. Furthermore,V V T = V T V = UUT = UT U = I. If Σ2 contains singular values from Σ that are(nearly) zero, H can be approximated as:

H ≈ U1Σ1VT1 (6.10)

Note that U1 and V T1 will in general not be square. Incorporating V1 and V T

1 intothe control structure, see Fig. 6.7, renders (6.8):

uk+1 = uk + ∆uk

ek = −yk + yref = −HV1uk + yref (6.11)

∆uk = V T1 Lek = L∗ek

integrator

+ +

+ -

γ

H

L∗

z−1I

yref − d

yk

ek

uk

∆uk

uk+1

V1

Figure 6.7: Adjusted ILC control structure in the piezo-based

The conditions for the existence of an LQ-optimal control solution now have beenfulfilled for the system (6.11). The design of the ILC controller is formulated interms of the following optimal control problem:

J =

N∑

k=1

yTk Qyk + ∆uT

k R∆uk

=

N∑

k=1

uTk V T

1 HT QHV1uk + ∆uTk R∆uk (6.12)

For an array of n channels, the summation is extended to nN . Furthermore,weighting matrices R and Q must be positive-definite. R has to be positive-definite to prevent infinite input amplitudes. If Q is not positive-definite thenthere may be unstable closed-loop modes that have no effect on the performanceindex. Choosing Q = I and R = βI, results in:


V T1 HT IHV1 = Σ1U

T1 U1Σ1 = Σ2

1 (6.13)

and (6.12) reduces to:

J =

N∑

k=1

uTk Σ2

1uk + β∆uTk ∆uk (6.14)

The solution to the discrete LQ-optimal control problem (6.14) is:

∆uk = −(βI + X)−1Xuk (6.15)

with X the stabilizing solution of the Discrete Algebraic Riccati Equation (DARE):

−X(βI + X)−1X + Σ21 = 0 (6.16)

Since Σ1 is diagonal, the solution X to the Riccati equation (6.16) will be diagonalas well. With σi and xi denoting the i-th elements of Σ1 and X , respectively, thesolution is:

xi =1

2σ2

i

(

1 +

√

1 +4β

σ2i

)

(6.17)

The feedback interconnection matrix L∗ becomes:

L∗ = (βI + X)−1XΣ−11 UT

1 (6.18)

The resulting closed loop system can be analyzed by its closed loop system matrixI − L∗HV1 and equals:

I − (βI + X)−1XΣ−11 UT

1 HV1 = I − (βI + X)−1X = βI(βI + X)−1 (6.19)

having closed loop poles:

λi =β

β + xi=

β

β + 12σ2

i

(

1 +√

1 + 4βσ2

i

) (6.20)

Thus:

λi ≈β

σ2i

≈ 0 for σ2i ≫ β

≈ β

β + σi

√β

≈√

β√β + σi

≈ 1 for σ2i ≪ β (6.21)

For large singular values, LQ-optimal control approximately provides dead-beatperformance with poles in the origin. For small singular values, the gain in the

6.4 ILC DESIGN 115

feedback loop is almost zero. Apparently, β can be viewed as tuning parameterdetermining which dynamics are taken into account in the ILC algorithm. Toclarify this, let us zoom in on the relation between the dynamics in terms of theFR and the singular values. There exists a fundamental difference between theFR and the singular value characterization of a system. The FR is the result ofFourier transforms of -in principle- an infinite time signals. The SVD descriptionof a system is based on a finite time impulse response. The question remainsto what extent a FR describes the system’s behavior for a finite time, as is thecase in the lifted ILC framework. However, if the resulting singular values areordered according to their frequency content, the FR is approximately obtained,see Fig. 6.8. For a more detailed treatment, one is referred to [Dij04]. In con-clusion, with β the relevant system dynamics can be selected, even though beingbased on the singular values.

200 400 600 800 1000−60

−50

−40

−30

−20

20*l

og10

(σi)

[dB

]

Element number i10

010

110

210

3−60

−50

−40

−30

−20

Mag

nitu

de [d

B]

Frequency [kHz]

200 400 600 800 1000−180

−170

−160

−150

−140

−130

20*l

og10

(σi)

[dB

]

100

101

102

103

−180

−170

−160

−150

−140

−130

Mag

nitu

de [d

B]

Figure 6.8: Piezo-based FR and the corresponding SVD; without (above) andwith (below) integrator

By choosing a large β, only the most dominant system dynamics are used inthe ILC algorithm. Focussing on these dynamics only renders the ILC controllerrobust against model inaccuracies. Therefore, the β parameter also is a tuningparameter to enhance the robustness of the resulting ILC controller.


Based on Fig. 6.8, another advantage of the added integrator in the piezo-basedcase becomes apparent. Due to the +1 slope of the FR in combination with acertain β, the wrong dynamics would be taken into account. Rather than select-ing the channel first eigenfrequency around 40 kHz, higher frequent dynamics areselected. By adding an integrator, this problem can be resolved. The first eigen-frequency then correspond to the larger singular values. Apart from a physicalnecessity of the added integrator, it is favorable from a numerical point of view aswell. Note that in the laser-vibrometer based case, the integrator is not requiredfrom a numerical perspective.

Note that to facilitate the computations, the solution (6.16) can be approximatedby:

xi =1

2σ2

i

(

1 +

√

1 +4β

σ2i

)

≈ σ2i + β (6.22)

This approximation holds as long as:

4β

σ2i

≪ 1 (6.23)

In Fig. 6.1 and 6.2 and various other control structures, a scalar learning gainγ is visible. A gain of γ < 1 can be used to increase the robustness of the ILCcontroller against model uncertainties by shifting the closed-loop closer to 1. Notethat this does not affect the final attainable error, see [Dij04].

The LQ-optimal ILC design presented here has several drawbacks. Most impor-tantly, the associated computations become increasingly difficult if not impossiblefor long reference trajectories, mainly due to numerical issues. Therefore, in Ap-pendix A, an alternative ILC design procedure is discussed that can handle theselong trajectories: the Hamiltonian based ILC design.

6.4.2 Constrained ILC

For the implementation of an actuation pulse on a PIJ printhead, use is madeof an Application Specific Integrated Circuit (ASIC). In contrast to a Field Pro-grammable Gate Array (FPGA), an ASIC is capable of handling the high voltageactuation pulses required for PIJ printheads. Unfortunately, an ASIC can han-dle signals that consist of a limited number of piece-wise affine functions. SinceILC pulses usually contain high frequency components, they fail to meet the re-quirements for implementation on an ASIC. Though choosing a suitable β solvesthis issue to a certain extent, the complexity of the resulting ILC pulse simplycannot be reduced sufficiently. In this section, therefore, another simple yet ef-fective modification of the ILC algorithm is discussed that allows for the design

6.4 ILC DESIGN 117

of ILC actuation pulses that fulfill the requirements for ASIC implementation:constrained ILC.

For the design of simplified actuation pulses within the ILC framework severalstrategies varying in complexity can be followed. To start with, given the numberof switching instances a non-linear optimization problem can be formulated thatdetermines the switching instances in time and amplitude, e.g. see [Hat04]. In-terpolation between those points then gives the actuation pulse. However, formu-lation within an ILC framework is not trivial and the computational complexitymakes it unsuitable for implementation on a PIJ printhead. Second, by utiliz-ing a certain set of basis functions the non-linear optimization problem can betransformed into a linear optimization problem within the ILC framework, seee.g. [Pha96; Gor97]. However, since a high number of basis functions is usuallyneeded to obtain reasonable performance, quite complex actuation pulses resultthat still are infeasible for ASIC implementation.

integrator

+ +

+ -

least squares approximation

H

L

z−1I

yref − d

yk

ek

uk

∆uk

uk+1

Figure 6.9: Constrained ILC control structure in the trial domain in the piezo-based case

Alternatively, rather than using a high number of basis functions for the construc-tion of a simplified actuation pulse, an optimized basis is adopted that is basedon known limitations concerning the implementation on an ASIC and physicalinsight in the working of a PIJ printhead. This is accomplished by the follow-ing adjustment of the ILC algorithm, see Fig. 6.9. The resulting ILC controllercomputes, based on the resulting error signal ek, an update ∆uk of the actuationsignal uk. The actuation signal (and the update accordingly), is to be transformedinto a simplified signal. Given a certain number of switching instances tsw thatare fixed in time and determined a priori, a nonlinear least-squares algorithm([Mar63; Lev44]) is used to approximate the update ∆uk with function F (tsw , p):

minp

|∆uk − F (tsw, p)| (6.24)

where p is the amplitude of the approximation function at tsw. The switching


instances tsw are chosen such that the first eigenmode of the ink channel can beeffectively damped by the ILC algorithm. As discussed in Section 3.2 and Chap-ter 5, this eigenmode dominates the response and hence forms a suitable choice.If the actuation is changed such that other modes become dominant, the switch-ing instances should be adjusted accordingly. Typically, around twelve switchinginstances are chosen. Note that omitting this projection step, the unconstrainedlifted ILC framework is obtained.


In this chapter, the theoretical background for the implementation of ILC on aPIJ printhead has been presented. In the next, both piezo- and laser-vibrometerbased ILC is implemented on the experimental setup. This setup is not equippedwith an ASIC such that there are no limitations with respect to the ILC actuationpulses. Nevertheless, in preparation of the implementation of ILC to a commercialPIJ printhead, constrained ILC is implemented as well for the piezo-based case.The performance is benchmarked against the performance of the unconstrainedILC algorithms. At the end of the next chapter, the in Chapter 5 discussedfundamental limitations are revisited.

Chapter 7

Application of feedforward control

This chapter demonstrates the use of lifted ILC to improve the printhead’s perfor-mance. To that purpose, both piezo- and laser-vibrometer based ILC are appliedto various PIJ printheads to reduce the residual vibrations and cross-talk. Nextto the realization of a performance improvement, more fundamental limitationsof current printhead designs become apparent. After having presented the experi-mental results, these results and their implications are discussed in detail. As itturns out, several findings will confirm the suppositions stated in earlier chaptersconcerning the printhead’s limitations.

7.1 Introduction

For the implementation of ILC on the various PIJ printheads, use is made of themeasured FRs rather than the theoretically obtained FRs. Though the theoreti-cally obtained piezo-based FR is sufficiently accurate for the frequency range ofinterest, the laser-vibrometer based FR is not (as argued in Section 5.3). To adopta similar approach to the implementation of ILC throughout this chapter, onlymeasured FRs are used as starting point. In addition, to enhance the generalapplicability of our proposed ILC approach, the employment of measured FRsnot only guarantees the usage of the most accurate system descriptions available,but also lifts the necessity to model a printhead theoretically. Still, an example ofthe successful utilization of the theoretically obtained FR in the piezo-based ILCapproach can be found in [Gro05b]. All the same, based on the measured FRs aspresented in Chapter 3, transfer functions are fitted using weighted Output-Error(OE) least-squares approximations, see [Sch94]. For the piezo-based approach,the 293e02 and DG074 PIJ printheads are used for the SISO and MIMO case, re-spectively. The measured FR from the piezo actuator to the piezo sensor and theaccompanying fitted transfer function is depicted in Fig. 7.1 and Fig. 7.2, respec-tively. To assess the quality of both models, it has been validated using measured

119

120 APPLICATION OF FEEDFORWARD CONTROL 7.1

sensor signals, see Fig. 7.3 and Fig. 7.4. These sensor signals are the result ofactuating a channel with a standard trapezoidal pulse at a jetting frequency of10 kHz. Based on Fig. 7.3 and Fig. 7.4, we conclude that the piezo-based dynam-ics are modeled satisfactorily. Note that the sensor signal of the non-actuatedink channel in Fig. 7.4 (cross) oscillates in anti-phase to the sensor signal of theactuated ink channel in Fig. 7.4 (direct). This corresponds to the fact that a de-crease of one channel induces an increase of its neighboring channels and providesa physical explanation of the obtained sensor signals. For the laser-vibrometerbased approach, the 233e01 printhead is used. The corresponding FR and TFfrom the piezo actuator to the meniscus velocity is displayed in Fig. 7.5. The ac-companying measured and simulated response to a standard trapezoidal actuationpulse are depicted in Fig. 7.6. Note that the differences between the measuredand simulated responses shown in this section can be handled by ILC.

100

101

102

103

−60

−50

−40

−30

−20

−10

Mag

nitu

de [d

B]

Frequency [kHz]

100

101

102

103

−1000

−800

−600

−400

−200

0

200

Pha

se [D

eg.]

Figure 7.1: Frequency response of the 293e02 from the piezo actuator to the piezosensor; measured (black dotted) and model (gray)

The piezo-based MIMO case will be elaborated for an array of two channels. Asdiscussed in Section 3.2, it is assumed that all ink channels are identical. Conse-quently, the MIMO case is simplified. Rather than having to take four transferfunctions into account (Ha, Hb, Hab, and Hba), now two suffice (Ha = Hb andHab = Hba). The validity of this assumption and the possible consequences forthe attainable performance are subject of discussion in subsequent sections of thischapter. The laser-vibrometer based MIMO case is not investigated here due to

7.1 INTRODUCTION 121

105

−50

−45

−40

−35

−30

−25

−20

−15

Mag

nitu

de [d

B]

Frequency [Hz]

105

−600

−500

−400

−300

−200

−100

0

Pha

se [D

eg.]

105

−55

−50

−45

−40

−35

−30

−25

−20

Mag

nitu

de [d

B]

Frequency [Hz]

105

−400

−300

−200

−100

0

100

200

Pha

se [D

eg.]

Figure 7.2: Frequency response of the DG074 from the piezo actuator to the piezosensor, direct (left HA and HB) and cross (right HAB and HBA); measured (blackdotted) and model (gray)

the availability of only one laser-vibrometer.

0 10 20 30 40 50 60 70 80 90 100−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [µs]

Sen

sor

sign

al [V

]

Figure 7.3: Response of the 293e02 to a standard trapezoidal actuation pulse;measured (black) and simulated (gray)

This chapter is organized as follows. To start with, the piezo-based ILC approachis elaborated in Section 7.2. The SISO case serves as demonstration of the use ofILC for the reduction of residual vibrations. Next, the MIMO case is employedto show the minimization of cross-talk effects. Then, the same MIMO setting isadopted for the implementation of the constrained ILC framework. During eachof the treated cases, the performance measures discussed in Section 6.3 are used.In Section 7.3, the laser-vibrometer based ILC approach is discussed. In corre-


spondence with earlier discussions, the experiments are conducted using 2.5 V asactuation voltage. For the implementation, the resulting learned ILC pulses arescaled to the appropriate jetting voltage. After having presented the experimen-tal results, the obtained results and their implications are discussed in detail inSection 7.4. This chapter ends with conclusions regarding the implementation ofILC to PIJ printheads.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [s]

Sen

sor

sign

al [V

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−4

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

Sen

sor

sign

al [V

]

Figure 7.4: Response of the DG074 to a standard trapezoidal actuation pulse;direct (left) and cross (right), measured (black) and simulated (gray)

104

105

106

−80

−60

−40

−20

0

mag

nitu

de [d

B]

104

105

106

−2000

−1500

−1000

−500

0

phas

e [D

eg.]

frequency (Hz)

Figure 7.5: Frequency response of the 233e01 from the piezo actuator to themeniscus velocity at 2.5 V; measured (black) and model (gray)

7.2 PIEZO-BASED ILC 123

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10−4

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [µs]

Men

iscu

s ve

loci

ty [m

/s]

Figure 7.6: Response of the 233e01 to a standard trapezoidal actuation pulse at2.5 V; measured (black) and simulated (gray)

7.2 Piezo-based ILC

7.2.1 SISO ILC: reducing residual vibrations

In this section, ILC is applied to one ink channel of the 293e02 PIJ printhead toreduce the residual vibrations. The used control structure is depicted in Fig. 6.1.The reference trajectory is constructed according to the procedure discussed inSection 6.3. Starting with an integrated sensor signal of a PIJ printhead jettingat 10 kHz resulting from a standard trapezoidal actuation pulse, the first partup to the firing of a drop at 30 µs is copied. After that, the residual vibrationsare eliminated by forcing the reference trajectory to zero, see Fig. 7.7. Note thatthe damping is not enforced too quickly after 30 µs to avoid too high actuationvoltages and to ensure the refill of the nozzle.

The controller synthesis is performed based on the identified transfer function asdepicted in Fig. 7.1 plus an added integrator. The presence of an integrator hasbeen motivated extensively throughout this thesis, e.g. see Section 6.2 and 6.4.β has been chosen such that the printhead dynamics up to approximately 250kHz are taken into account. Beyond 250 kHz, there are no relevant printheaddynamics. γ is chosen as 0.25. Recall that this only affects the convergence speedonly. Furthermore, the length of the reference trajectory allows for the use of theLQ-optimal approach for the design of the ILC controller, see Section 6.4.

For a PIJ printhead, attaining the reference trajectory is of importance during


0 10 20 30 40 50 60 70 80 90 100−2

−1

0

1

2

3

4x 10

−6

Inte

grat

ed s

enso

r si

gnal

[Vs]

Time [µs]

Figure 7.7: Integrated sensor signal; without ILC (black), with ILC (gray), andchosen reference trajectory (black dotted)

0 10 20 30 40 50 60 70 80 90 100−10

−5

0

5

10

15

20

25

30

35

Inpu

t [V

]

Time [µs]

Figure 7.8: Actuation pulse; standard trapezoidal (black dotted) and resultingILC pulse (gray)


the complete duration of one jetting cycle. For a jetting frequency of 10 kHz,this amounts to 100 µs. In contrast, the actuation is restricted to a certain timeinterval to be able to increase the jetting frequency without the immediate ne-cessity of overlapping actuation pulses. In this case, the actuation is limited tothe first 60 µs. Consequently, the jetting frequency can be increased to 16.6 kHzwithout overlapping actuation signals. Note that the adopted restriction of 60µs is not the absolute minimum length of the actuation window. Nevertheless, afurther decrease of this window deteriorates the attainable performance consider-ably. For jetting frequencies beyond the 16.6 kHz, the superposition principle forlinear systems is used. Having assumed linearity of the jetting process at least inthe piezo-based case, this is a valid approach.

The sensor signal resulting from a standard trapezoidal and the learned ILC pulseare shown in Fig. 7.7. The accompanying actuation pulses are shown in Fig. 7.8.Based on Fig. 7.7, the conclusion is drawn that the reference trajectory is at-tained satisfactorily. Since the first part of reference trajectory up to the firing ofa drop is the same as realized by the standard trapezoidal pulse, it is not surpris-ing that the learned ILC pulse resembles the standard trapezoidal pulse for thefirst part. After that, the ILC controller adjusts the actuation pulse such thatthe fluid-mechanics follow the desired trajectory in presence of the restriction ofthe actuation interval. In Fig. 7.8, it can be seen that the ILC actuation pulsecounteracts the pressure oscillation. The peaks just before 60 µs originate fromthe fact that the ILC controller cannot actuate beyond 60 µs while it is requiredthat the channel is in rest after 60 µs nonetheless. If desired, these peaks can besuppressed by additional weightings.

2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5x 10

−4

IAE

Iteration number

Figure 7.9: Integrated absolute error of the error signal against the trial number


101

102

0

0.2

0.4

0.6

0.8

1

1.2x 10

−6

CP

SD

[V]

Frequency [kHz]

Figure 7.10: Cumulative power spectrum of the error signal; standard trapezoidal(black) and ILC pulse (gray)

The IAE criterion for the discussed ILC experiment is depicted in Fig. 7.9. Thoughconvergence occurs monotonously here, in general this is not the case. Especiallyduring the early stages of learning the IAE might temporarily deteriorate com-pared to a previous trial. Though this might affect the drop properties duringoperation negatively, usually only a few iterations or equivalently a couple of mi-croseconds are involved. In Fig. 7.10, the cumulative power spectrum (CPS) ofthe error of the first and last trial is depicted. Based on Fig. 7.10, it is concludedthat the largest error reduction takes place around the channel’s first resonancefrequency at 45 kHz. This is in correspondence with our observation that thechannel’s response and thus the residual vibrations are governed by the this firstresonance frequency.

Finally, the DOD-speed curve is obtained to assess the effect of minimization ofthe residual vibrations on the attainable jetting frequency and hence productivity.In Fig. 7.11, the DOD-speed curve is depicted for the standard trapezoidal andthe ILC learned actuation pulse. Note that for frequencies beyond 16.6 kHz, theILC actuation pulses are superposed as discussed above.

The location of the local minima and maxima of the DOD-speed curve in Fig. 7.11for the standard trapezoidal actuation pulse can be linked to the occurring residualvibrations as follows. In Fig. 7.7, these residual vibrations are depicted. Typi-cally, it takes approximately 150 µs for these residual vibrations to be completelydamp out. If the jetting frequency increases, the time between two successivepulses decreases. Therefore, at a certain jetting frequency, the channel is notrest anymore if the consecutive actuation pulse is given. Assuming linearity, the


5 10 15 20 252

2.5

3

3.5

4

4.5

5

Dro

plet

spe

ed [m

/s]

DOD frequency [kHz]

Figure 7.11: DOD (drop-on-demand) curve; standard trapezoidal (black) and ILCpulse (gray)

response of an ink channel can be obtained by superposing two responses as de-picted in Fig. 7.7 at the appropriate time instant. For example, at 11.8 kHz and12.8 kHz, or a time between two actuation pulses of 85 µs and 78 µs equivalently,the overlapping responses amplify and attenuate each other, respectively. Con-sequently, a local maximum and minimum results. A similar reasoning holds forthe local maximum at 15.8 kHz (63 µs) and minimum at 18.2 kHz (55 µs) andthe subsequent minima and maxima. Additionally, structural modes of the PIJprinthead itself influence the course of the DOD-curves also.

Since residual vibrations are minimized with the ILC actuation pulse, the phe-nomenon of attenuating or amplification is eliminated, theoretically at least upto a jetting frequency of 16.6 kHz. Based on Fig. 7.7, it is concluded that thisis the case. For jetting frequencies beyond 16.6 kHz, the ILC actuation pulsestill outperforms the standard actuation pulse. Typically, 15 % deviations from anominal drop-speed are allowed given the desired print quality. Given a nominaldrop-speed of 3.5 m/s, the lower and upper bound on the drop-speed are 3.0 and4.0 m/s, respectively. These boundaries are indicated in Fig. 7.11. As can beseen, the ILC learned actuation pulse reduces the speed variations such that thejetting frequency can be increased up to 25 kHz.

Finally, both DOD curves show a positive linear trend for frequencies up to ap-proximately 15 kHz. This trend is caused by the wetting of the nozzleplate, see


[Nag06]. Wetting is the phenomenon that the nozzleplate is covered with a thinlayer of ink. Among other things, it slows down the resulting drop and deteri-orates the jet straightness. Since the wetting decreases with an increase of thejetting frequency, the positive trend can be explained. The only effective measureto counteract this phenomenon aims at developing a non-wetting nozzleplate.

7.2.2 MIMO ILC: minimizing cross-talk

0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

sens

or s

igna

l A [V

]

Time [µs]0 10 20 30 40 50 60 70 80 90 100

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

sens

or s

igna

l B [V

]

Time [µs]

Figure 7.12: Integrated sensor signal of channel A (left) and channel B (right);without ILC (black), with ILC (gray), and chosen reference trajectory (blackdotted)

In this section, MIMO ILC is applied to an array of two ink channels, A and B,to simultaneously minimize the effect of cross-talk and residual vibrations. Thesame control structure as in the SISO case can be adopted here, albeit with anadjustment to the structure of the various signals and matrices. The referencetrajectories for both channels are constructed as follows. Starting point forms theresponse of each channel to a standard trapezoidal pulse without the neighboringchannel jetting. This guarantees the absence of cross-talk. From this point, theconstruction is equal to that in the SISO case. In Fig. 7.12, the resulting referencesignals are depicted in case both channels are to be jetting.

Despite the fact that the impulse response matrix is doubled in size compared tothe SISO case, the LQ-optimal ILC design approach can still be used. The β andγ values are the same as those in the SISO piezo-based case. Given our focus onthe minimization of cross-talk, the limitations concerning the actuation intervalis omitted. Starting point for the ILC synthesis form the transfer functions asdepicted in Fig. 7.2.

The resulting sensor signals from the standard trapezoidal and learned ILC ac-tuation pulses are shown in Fig. 7.12. The accompanying actuation pulses are


0 10 20 30 40 50 60 70 80 90 100−10

−5

0

5

10

15

20

25

30

35

40

Inpu

t A [V

]

Time [µs]

Figure 7.13: Actuation pulse; standard trapezoidal (black dotted), the resultingILC pulse for channel A (black) and channel B (gray)

depicted in Fig. 7.13. In Fig. 7.12, small differences between both reference tra-jectories are visible. Apparently, ink channel A and B are not completely identi-cal as assumed. Consequently, the learned ILC pulses for channel A and B differslightly also. This is not bothersome, since both reference trajectories are attainedsatisfactorily and the required performance level is met. As discussed previously,the first part of reference trajectories up to the firing of a drop is the same asrealized by the standard trapezoidal pulse. As a result, the learned ILC pulsesresemble the standard trapezoidal pulse for the first part, though there are someconsiderable deviations. They can be accounted for by the fact that the ILC con-troller is counter-acting the cross-talk effects. After the jetting of the drops, theILC controller adjusts the actuation pulses such that the fluid-mechanics followthe desired trajectory for the damping of the residual vibrations.

The convergence of channel A and B in terms of the IAE is depicted in Fig. 7.14.The CPS of the resulting error signals of the standard and learned ILC actua-tion pulses are depicted in Fig. 7.15. As can be seen in Fig. 7.14, convergence isachieved in approximately 20 iterations. An error reduction of a factor of 3.4 and2.6 is achieved for channel A and B, respectively. Based on Fig. 7.15, it concludedthat the largest error reduction takes place around the first resonance frequency at45 kHz. This is similar to our findings in the SISO case. The differences betweenchannel A and B in both Fig. 7.14 and 7.15 can be attributed to the differencesbetween the channels.

In Table 7.1, the cross-talk curve for channel A and B is listed. The resulting


5 10 15 20 25

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−4

IAE

Iteration number

Figure 7.14: Integrated absolute error of the error signal against the trial number;channel A (black) and channel B (gray)

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−6

CP

S [V

]

Frequency [kHz]

CPS standard ACPS ILC ACPS standard BCPS ILC B

Figure 7.15: Cumulative power spectrum of the error signal of channel A and B;standard trapezoidal (black) and ILC pulse (gray)


drop-speed if only one ink channel is actuated is listed first. The drop-speed ofthis ink channel if a neighboring channel is actuated simultaneously is listed in thefollowing columns using the standard trapezoidal and ILC learned actuation pulse.The deviations in drop-speed in both cases are a measure of the drop-consistency.In case of the standard trapezoidal actuation pulse, the drop-speed consistencyis considerably less than in case an ILC approach is adopted. A similar result isobtained for the drop-volume consistency. In conclusion, MIMO ILC can be usedto minimize the effect of cross-talk and consequently improve the drop-consistency.

standard standard ILC standard ILC variationsingle double double variation variation reduction

channel A 3.96 m/s 3.28 m/s 4.20 m/s 17.2 % 6.1 % 64.7 %channel B 3.37 m/s 2.58 m/s 3.56 m/s 23.4 % 5.7 % 75.9 %

Table 7.1: Comparison of drop-speed with and without ILC

Up to this point, it has been assumed that the ink channels of a PIJ printhead areidentical. Based on the results presented in this section, this assumption has beenproven to be not valid. To investigate the influence of these differences on theattainable performance, the same ILC experiments have been conducted using allfour transfer functions Ha, Hb, Hab, and Hba. The results show that although theconvergence rate is slightly higher, the performance is the same. However, theexperiments conducted in this section have been performed on two neighboringchannels. Channels that are located further apart from each other may show moredifferences in channel dynamics and hence performance. Altogether, for the arrayof channels considered in this section, the assumption still holds.

7.2.3 Constrained MIMO ILC

The final piezo-based ILC experiment discussed in this section is the implemen-tation of constrained ILC. To establish the effect of the imposed constraints, thesame experiments are conducted in the unconstrained setting. As argued in Sec-tion 6.4, ASIC limitations require the use of piece-wise affine actuation signalsonly. To that purpose, the ILC algorithm has been adjusted such that only ILCpulses are learned that fulfill this requirement. In Fig. 6.9, the adopted controlstructure is depicted. For the implementation of constrained MIMO ILC, the ref-erence trajectories are chosen such that channel A is jetting and channel B is atrest. The residual vibrations present in channel A are damped. The correspond-ing reference trajectories are constructed similarly to the previous two cases andare depicted in Fig. 7.16.

The same ILC controller as used for the unconstrained MIMO ILC case has been


0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

sens

or s

igna

l A [V

]

Time [µs]0 10 20 30 40 50 60 70 80 90 100

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

sens

or s

igna

l B [V

]

Time [µs]

Figure 7.16: Integrated sensor signal of channel A (left) and B (right); withoutILC (black), with constrained ILC (gray), with ILC (gray dotted), and chosenreference trajectory (black dotted)

used for the experiments shown in this section. For the constrained case, thealgorithm is adjusted as discussed in Section 6.4. Furthermore, the number ofswitching instances is chosen as 11. The location of these instances is determinedbased on physical insight: the switching instances are tuned to the channel’s firstresonance frequency, that dominates the residual vibrations.

0 10 20 30 40 50 60 70 80 90 100−5

0

5

10

15

20

25

30

35

Inpu

t A [V

]

Time [µs]0 10 20 30 40 50 60 70 80 90 100

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Inpu

t B [V

]

Time [µs]

Figure 7.17: Actuation pulse for channel A (left) and B (right); standard trape-zoidal (black), the resulting constrained ILC pulse (gray), and the ILC pulse (graydotted)

The resulting sensor signals from the unconstrained and constrained ILC actu-ation pulses are shown in Fig. 7.16. The accompanying actuation pulses aredepicted in Fig. 7.17 for both channel A and B. The small nonzero values of

7.3 LASER-VIBROMETER BASED ILC 133

the actuation pulses at 100 µs do not influence the performance negatively. Thesenonzero values can be avoided by the use of additional constraints. It is concludedthat the reference trajectories are attained satisfactorily. Given the fact that theresulting sensor signals are quite similar, it is concluded that constrained ILC iscapable of attaining similar performance as its unconstrained version. Apparently,the actuation signal can be simplified considerably to meet the requirements forthe implementation on an ASIC without sacrificing too much performance. Thecorresponding CPS for channel A and B are shown in Fig. 7.18.

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−6

CP

S A

[V]

Frequency [kHz]10

110

20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−7

CP

S B

[V]

Frequency [kHz]

Figure 7.18: Cumulative power spectrum of the error signal of channel A (left)and B (right); standard trapezoidal (black), constrained ILC pulse (gray), andILC pulse (gray dotted)

7.3 Laser-vibrometer based ILC

In this section, laser-vibrometer based SISO ILC is implemented on the 233e01printhead. As discussed previously, the major limitation of the accompanyingexperiments is the restriction to non-jetting regimes only. The measurement con-figuration does not allow the jetting of a drop. This would cause the measurementto stop. Therefore, all experiments are carried out using 2.5 V. To obtain theDOD-speed and -volume curves, the resulting ILC actuation pulse is scaled to ajetting voltage. Note that various consequences of this restriction is discussed indetail in the next section.

The adopted control structure is shown in Fig. 6.2. In Fig. 7.19, the used referencetrajectory at 2.5 V is depicted. The construction of this trajectory is performedthe same way as in the piezo-based ILC case. The ILC controller has been de-signed using the LQ-design approach based on with the fitted transfer function


0 10 20 30 40 50 60 70 80 90 100−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time [µs]

Men

iscu

s ve

loci

ty [m

/s]

Figure 7.19: Meniscus velocity at 2.5 V without ILC (black), with ILC (gray),and chosen reference trajectory (black dotted)

at 2.5 V, see Fig. 7.5. Finally, note that the actuation window is limited to 50 µs.

0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time [µs]

Inpu

t [V

]

Figure 7.20: Actuation pulse without ILC (black dotted), with ILC (gray)

The resulting sensor signal from the standard trapezoidal and learned ILC actu-ation pulse at 2.5 V are shown in Fig. 7.19. The accompanying actuation pulsesare depicted in Fig. 7.20. Similar to the previous cases, the first part of reference

7.3 LASER-VIBROMETER BASED ILC 135

trajectory up to the firing of a drop is roughly the same as realized by the stan-dard trapezoidal pulse. Therefore, it is not surprisingly that the learned ILC pulseresembles the standard trapezoidal pulse for the first part. The higher voltageduring the first part result mainly from the differences as discussed at Fig. 3.18.The convergence is depicted in Fig. 7.21. The CPS of the resulting error signal ofthe standard and learned ILC actuation pulse is depicted in Fig. 7.22. In Fig. 7.21,it can be seen that convergence is achieved in approximately 10 iterations. A re-duction of a factor of 5.2 is achieved. In Fig. 7.22, it can be seen that the largesterror reduction takes place at the first resonance frequency around 45 kHz as inthe previous cases.

0 5 10 15 20 2510

20

30

40

50

60

70

Iteration number [−]

IAE

Figure 7.21: Integrated absolute error of the error signal against the trial number

For the measurement of the DOD-speed and -volume curves, the laser-vibrometerconfiguration has been taken away. The learned actuation pulse at 2.5 V has beenscaled up to 30 V and implemented on the experimental setup. Using this scaledpulse, the DOD-speed and DOD-volume curves have been measured, see Fig. 7.23.Based on Fig. 7.23, it is concluded that the variations are reduced considerably.A number of remarks are noteworthy. First, the drop-speed is considerably highercompared to the DOD-speed curve measured with the 293e02 printhead. This iscaused by the differences in both printheads. Second, the DOD curves measuredwith the learned ILC pulse show a certain offset compared to the DOD curvesobtained with the standard trapezoidal actuation pulse. The main reason for thisoffset lies in the used reference trajectory. In contrast to the reference trajectoriesused previously, the damping of the residual vibrations is imposed approximately10 µs earlier, see Fig. 7.19. Since the drop-formation process is still ongoing at 25µs, see Fig. 4.7, the drop is so to speak hold back. More specifically, the tail of thedrop that is still connected to the drop is decelerated and slows down the drop


itself as well as reduces its volume. Third, the linear trend visible in Fig. 7.11is almost not present here. The direct control of the meniscus itself rather thanthe related channel pressure provides a better mean to control the wetting of thenozzleplate. This can be explained as follows. The meniscus trajectory is nowconfined such there hardly is any overfill. In the piezo-based case, the meniscusposition simply cannot be controlled so directly. Apparently, in the latter casethere is (more) overfill and thus wetting.

101

102

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

CP

SD

[V]

Frequency [kHz]

Figure 7.22: Cumulative power spectrum of the error signal; standard trapezoidal(black) and ILC pulse (gray)

Based on the results presented in this section, it is concluded that the nonlin-earities can be handled by the proposed ILC approach. Despite the undeniablepresence of these nonlinearities and the limitations of the measurement setup,the performance can still be improved considerably by the application of laser-vibrometer based ILC. Although for the application of ILC considered here thesystem was assumed to behave linear, the validity of the linearity assumptionneeds additional research. To further enhance the performance of this particularILC approach, current research strive for the integration of a sensor in the nozzleto replace the laser-vibrometer sensor. Details can be found in [Gro06a].

7.4 Discussion

In this section, the ILC approaches presented in the preceding sections are re-viewed. To start with, the experimental results are evaluated in light of theformulated control objectives. Next, several issues concerning the implementa-tion of ILC to PIJ printheads are considered. Then, various subjects for furtherresearch regarding the current application of ILC are brought up. Finally, some

7.4 DISCUSSION 137

8 10 12 14 16 18 20 22 24 26 28 304

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Jet frequency [kHz]

Dro

p sp

eed

[m/s

]

8 10 12 14 16 18 20 22 24 26 28 3026

28

30

32

34

36

38

40

42

Jet frequency [kHz]

Dro

p vo

lum

e [p

l]

Figure 7.23: DOD (drop-on-demand) speed- (left) and volume (right) curve ofthe 233e01; standard trapezoidal (black) and scaled ILC pulse (gray)

fundamental limitations of the current PIJ printhead design are discussed.

To assess the performance of ILC, the experimental results are evaluated basedon the following ILC objectives as formulated in Section 6.3:

• Enhancing the productivity. In the previous two sections, it has been demon-strated that the productivity of a PIJ printhead can be improved in twoways. First, due to the active damping of the residual vibrations, the jettingfrequency can be increased up to approximately 25 kHz. Second, minimiza-tion of cross-talk by means of ILC renders the use of the so called bridgestructure obsolete. Consequently, the npi-ratio can at least be doubled.Both the use of higher jetting frequencies and the increase of npi enhancesthe productivity considerably.

• Improving the drop-consistency. It has been shown that application ofMIMO ILC can improve drop-consistency. More specifically, drop-speedvariations have been reduced from 20.3 % to 5.9 % on average.

Apart from the actual realized performance improvements, it is demonstratedthat ILC is a suitable control strategy to overcome the current boundaries withrespect to at least productivity and drop-consistency. Furthermore, it is expectedthat performance with respect to these two formulated control objectives can beincreased even further. In the sequel of this section, various research directionsare pointed out to further enhance the performance. First, however, the followingILC-printhead related issues are considered:

• Piezo- versus laser-vibrometer based ILC. From a performance point of view,the question arises whether piezo- or laser-vibrometer based ILC is more fa-vorable. As argued in Section 6.3, eventually, the drop-formation is the


most important performance determining process of a PIJ printhead. If themeniscus would be completely observable and controllable using the piezo-unit, using the piezo- or laser-vibrometer based ILC approach would notmake any difference. However, as discussed in previous chapters, this isnot the case. Therefore, one might be inclined to attribute one’s prefer-ence to the laser-vibrometer based approach. On the other hand, based onthe experimental results presented in this chapter, the conclusion must bedrawn that both approaches achieve almost similar performance, neglect-ing the small differences in the various printheads for convenience. How-ever, whereas the piezo-based approach is quite straightforward, the laser-vibrometer based approach certainly is not. This can be attributed mainlyto the limitations imposed by the laser-vibrometer as sensor. It is there-fore expected that the attainable performance in the laser-vibrometer basedapproach can be further increased if the sensor functionality is improved.Altogether, further research in the controllability and observability of themeniscus and the development of more suitable sensor functionality in thenozzle is needed to decide in this issue.

• Linearity of the jetting process. In Section 7.2 and 7.3, linearity of thejetting process has been assumed. The obtained experimental results did notprovide reasons to question the validity of this assumption. Whereas this isnot surprising for the piezo-based case, validity is certainly not trivial for thelaser-vibrometer based case. After all, in Chapter 5, nonlinear behavior hasbeen demonstrated in the laser-vibrometer based case. Irrespective of whichILC approach is adopted, it remains to be seen to what extent linearitycan be assumed. For example, the use of superposition of ILC actuationpulses to obtain the DOD curve at high frequencies or to decouple a (large)array of ink channels may lead to new insights with respect to the validityof the linearity assumption. More specifically, if superposition gives riseto actuation signals of high actuation voltages, linearity may be lost. Inconclusion, for the experiments conducted thus far, linearity of the jettingprocess has been a valid assumption. However, further research is requiredto reach a final conclusion.

• Uniformness of ink channels. In this thesis, it is assumed that all ink chan-nels are identical. The validity of this assumption can be questioned basedon the observation that two neighboring channels already show differencesin channel dynamics, albeit small. Measurement of the frequency responseof ink channels further apart show that the variations in channel dynamicscan indeed become considerable. Nevertheless, the main channel resonancefrequency usually only differs a few kHz. At this point, insufficient exper-iments have been conducted to support a finite conclusion regarding theuniformness of ink channels. Further research is required. It is expectedthat the assumption regarding channel uniformness can be used at least in

7.4 DISCUSSION 139

one particular case. If the β parameter is chosen such that the ILC algo-rithm only takes the channel’s first eigenfrequency into account, the use ofthis assumption is justified.

• Robustness of the ILC approach. In case of a PIJ printhead, there are twocases of robustness to be considered:

1. Robustness against model uncertainty. Model uncertainty can originatefrom for example model mismatches (e.g. due to wrongfully assumingchannel uniformness) or aging. As discussed in Section 6.4, the robust-ness against model uncertainty of the ILC approach can be increasedby the parameter β. Given the experiments presented in this chapter,it is concluded that the robustness against model uncertainty of thecurrent ILC approaches is sufficient.

2. Robustness against disturbances. Disturbances that can occur includedirt-particles entering the nozzle and air-bubbles sucked in the nozzle.To improve the robustness against disturbances the reference trajec-tories can be used. For example, by limiting the retraction of themeniscus in the nozzle, the chance of sucking in air-bubbles is reduced.Still, it is not trivial how to improve the robustness in these cases andrequires further research.

Various subjects for future research have been discussed above. In addition, thefollowing more general topics are of interest:

• Further optimization of the ILC approaches. The ILC approaches as em-ployed in this thesis can be improved with respect to several issues. Firstand foremost, the design of the reference trajectories can be improved. Thetrajectories used to obtain the results in this chapter are based on the re-sponse to a standard actuation pulse. As indicated Section 6.3, there aremany alternatives to be investigated for various purposes. Next, only theuse of actuation windows has been investigated. As discussed in Section 6.2,weight filters can be employed to the same purpose offering more freedomin the design of the ILC pulses. Additional research is required to establishthe possible advantages of this approach. Third, the actuator and sensorfunctionality is to be improved, especially the sensor used for the meniscus.Finally, the constrained MIMO ILC approach can be developed further.Though the fixation a priori of the number and location of the switchinginstances renders the corresponding optimization problem linear, it also pos-sibly limits the performance of the constrained ILC algorithm. Extendingthe algorithm to allow the algorithm itself to determine the number andlocation of the switching points can improve the constrained ILC approachconsiderably.


• Application of ILC to improve stability and enable DSM. As discussed inSection 6.3, it is expected that ILC can be used to improve the jet stabilityand enable the use of DSM. Research in both subjects is required.

• Development of a decoupling strategy for large arrays of ink channels. Ap-plication of ILC to a large array of ink channels calls for the developmentof a decoupling strategy. Given the fact that an inkjet typically consists ofan array of 100 to 300 ink channels and that depending on the data to beprinted many different actuation schemes are used, it is simply not realisticto learn for every possible occurring situation. One option is to assume lin-earity and superpose the various actuation signals. However, as mentionedabove, validity of the linearity assumption remains to be seen in case toohigh actuation voltages are present. There are many alternatives, e.g. theuse of a (static) decoupling matrix. This requires additional research.

Finally, some of the fundamental limitations discussed in Chapter 5 are revisitedgiven the experimental results obtained in this chapter:

• The channel’s first eigenfrequency. As discussed previously, the dynamicbehavior of an ink channel is dominated by its first eigenfrequency. Undercertain conditions, the most energy-efficient actuation pulses are tuned tothat particular frequency. The PIJ printheads considered in this thesis formno exception to both observations. The minimum required time for one jet-ting cycle is thus also determined. For example, if the first eigenfrequencyis 45 kHz, a minimum jetting cycle of a multiple of 22 µs (typically two tothree) is required to jet a drop and damp the residual vibrations without toohigh actuation voltages. This roughly corresponds to the results obtainedin this section. As a result, the attainable jetting frequency is limited, evenwhen ILC is applied. This also has been demonstrated in this chapter. TheDOD curves can be improved up to a certain jetting frequency. Beyond thatfrequency, the DOD curve based on an ILC curve deteriorates also.

There are a few solutions possible. First, rather than designing an actu-ation pulse for one drop only, pulses can be designed for multiple drops.This requires research in reference trajectories, see the discussion above.Second, the design can be adjusted to facilitate higher jetting frequencies.For example, the channel’s length can be decreased such that the its firsteigenfrequency is decreased. Another adjustment to the printhead’s designconcerns the piezo-unit as actuator. A division of the piezo-unit in multiplepiezo-units allows for the use of a completely different actuation strategy ofan ink channel. For example, a drop could be extruded out of a channel. Asa result, one avoids the use of the channel’s eigenfrequency thereby liftingthe corresponding constraint in attainable jetting frequency.


• Spatial controllability and observability of the jetting process. As discussedin Chapter 5, the spatial controllability and observability of the PIJ print-heads investigated in this thesis is limited. Without proper actuation andsensing functionality, the attainable performance is limited. For example, ifthe piezo-sensor indicates that the ink channel is at rest, there still may betraveling pressure waves present. Various experiments with shorter piezo-units confirm this observation. To enhance both the controllability andobservability, the following adjustments to the PIJ printhead design is sug-gested. First, the piezo-unit is to be divided in multiple piezo-units. Second,a sensor in the nozzle is to be incorporated.


Based on the experimental results presented in this chapter, the following mainconclusions are drawn:

• The suitability of ILC as control strategy to enhance a PIJ printhead’sperformance in face of commonly encountered operational issues has beendemonstrated. Minimization of the residual vibrations and cross-talk hasbeen proven to be very profitable in terms of the productivity and drop-consistency of a PIJ printhead. For the further exploration of ILC manyresearch directions have been pointed out.

• The operation of a PIJ printhead can be regarded as linear for the exper-iments conducted in this thesis. Further research is necessary to establishthe validity of this assumption in face of various other experiments to beconducted in the near future.

• The two major limitations of current PIJ printheads relate to the first eigen-frequency of a particular design and the limited spatial controllability andobservability of the jetting process. To overcome both issues, a re-designof PIJ printheads with respect to the actuator and sensor functionality isof crucial importance. For the design at hand, a division of the currentpiezo-unit in multiple piezo-units as well as the development of a sensingdevice in the nozzle is advised.

Chapter 8

Conclusions and recommendations

In the beginning of this thesis, the importance of inkjet technology was sketched.The commonly encountered limitations of PIJ printheads were discussed and asolution strategy was pointed out (Chapter 1) that led to the research objectivespecified into three research questions (Chapter 2). In this chapter, the researchis concluded and the recommendations are presented.

8.1 Conclusions

The conclusions presented in this section are categorized according to the threeresearch questions as formulated in Chapter 2.

Question 1: How should a PIJ printhead be modeled given its intended use forthe proposed systems and control approach?

Based on the research objective of this thesis, several requirements for the mod-eling of a PIJ printhead have been formulated. First and foremost, a suitablemodel is to provide insight in the working of a PIJ printhead. Second, the modelcomplexity is to be kept as low as possible while maintaining the model accurateenough for the use for control and (re-)design. Current available models fail tosatisfy these requirements simultaneously and are therefore not completely suitedfor the purposes in mind.

The key to the successful modeling of an ink channel forms the use of bilaterallycoupled systems (BCS). For one, this concept not only fixes an appropriate inter-nal model structure, but also provides an explicit role for the surroundings actingon a system. To apply this concept to an ink channel, the following - generallyapplicable - approach to the modeling of PIJ printheads has been followed. An inkchannel is divided in several functional blocks. Main guidelines for the division

143

144 CONCLUSIONS AND RECOMMENDATIONS 8.1

chosen is the geometry of an ink channel. Subsequently, the dynamic behavior ofthese blocks is modeled using first principles only. During the various derivations,it is assumed that the jetting process behaves linearly. For each block, the result-ing dynamical equations are transformed into the two-port formulation as partof BCS. Finally, the coupling of all the blocks is performed by the applicationof Redheffer’s star product. A model of an array of channels can be obtainedby the coupling of an arbitrary number of ink channel models. In that case, thecoupling as well as the accompanying cross-talk effects are facilitated by meansof the actuating function of the actuator.

The resulting so called two-port model fulfills the requirements for the modelingof a PIJ printhead as formulated a priori to a large extent. To start with, due tothe chosen modeling strategy, the resulting model has relatively low complexity.Also, experimental validation shows that the resulting model is accurate for thepiezo-based case. In the laser-vibrometer based approach, however, the two-portmodel is to be improved with respect to the first and most important resonancefrequency. Third, the two-port model provides insight in the working of an inkchannel as well from a systems and control perspective. The obtained results pre-sented in this thesis show the two-port suitability for the control and (re-)designpurposes in mind.

Based on the resulting two-port model, one important observation concerning theink channel dynamics is the following. Apparently, the dynamic behavior of anink channel can be represented by an extremely low dimensional system, in ourcase a 4th order. Further research is required to further explore this possibility.

To further improve the two-port model, several research directions have been indi-cated. These include the use of more complex nozzle models, the upgrading of theone-sided coupling between the nozzle and the drop formation to a two-sided one,the further development of the piezo-unit modeling, and the adding of dampingto the reservoir block.

Question 2: Can we design actuation wave forms which will be implementedas feedforward control such that the performance of current PIJ printheads isimproved?

The three most prominent performance criteria for a PIJ printhead are its produc-tivity, drop-consistency, and stability. The focus of the research presented in thisthesis lies on the former two. The attainable performance with respect to thesetwo issues is limited by two commonly encountered operational issues: residual vi-brations and cross-talk. In this thesis, it has been demonstrated that feedforwardcontrol, more specifically ILC, is a suitable control strategy to overcome these twoissues and hence increase the performance of PIJ printheads considerably beyondcurrent limits.

8.1 CONCLUSIONS 145

Given the available sensor functionalities, two ILC approaches have been inves-tigated: piezo- and laser-vibrometer based ILC. Also, the limitations for the im-plementation of ILC as posed by the ASIC have been resolved. Based on ourexploration of the possibilities of feedforward control of a PIJ printhead, the fol-lowing conclusions are drawn:

• Productivity. The productivity of an individual ink channel as well as a com-plete array can be increased in two ways by the implementation of ILC. Tostart with, due to the active damping of the residual vibrations, the attain-able jetting frequency can be increased with a factor 2.5 for the printheadsunder consideration. Given an admissible deviation in drop-speed of ± 0.5m/s from a nominal value, the jetting frequency can be increased from 10up to 25 kHz. A second effect results from the minimization of cross-talk.Since the bridge-structure becomes redundant, the npi-ratio can at least bedoubled. As a result, one has more nozzles per inch available for jetting.Both effects contribute to an enhancement of the productivity.

• Drop-consistency. The drop-consistency can be considerably improved bythe application of ILC. It has been demonstrated that the variations in drop-speed can be reduced from 20.3 % to 5.9 % on average for the printheadsunder investigation.

Although it has not been experimentally demonstrated in this thesis, it is expectedthat feedforward control can also improve a PIJ printhead’s performance withrespect to the following issues:

• Stability of the jetting process. Feedforward control can be applied to designactuation pulses such that the stability of the jetting process is improved.Stability is closely related to the meniscus retraction. For example, it hasbeen argued that limiting this retraction reduces the risk of entrapping anair-bubble leading to nozzle failure.

• Drop-speed and -volume (modulation). The ILC control structure uses acertain reference trajectory to learn an actuation pulse that results in adrop of some predefined properties. By switching between various referencetrajectories, drop properties such as speed and volume can be varied duringoperation.

Given the exploratory character of the research presented in this thesis, it isexpected that the performance can be further increased with respect to the pro-ductivity and drop-consistency of a PIJ printhead. Several recommendations areprovided in the next section. Stability and the on-demand realization of certaindrop properties by means of feedforward control is to be investigated.


Two important assumptions have been used throughout this thesis and in partic-ular during the application of control, namely linearity of the jetting process andthe uniformness of the ink channels. From a control perspective, the former hasproven its validity during the implementation of ILC, at least in the consideredcases. Further research is required to establish to what extent this assumptionremains valid. It is noted that the implications of the validity of the linearityassumption are eminent: it facilitates the application of a systems and controlapproach and the accompanying range of (optimization) tools considerably. Va-lidity of the latter assumption has not been conclusively determined in this thesis.In view of the intended extension of the ILC framework to a complete printhead,it certainly deserves further research. For an array of two channels consideredhere, the assumption was valid.

Since the proposed ILC feedforward control strategy is generally applicable, theresults and conclusions presented in this thesis are not limited to the specificprinthead design that has been investigated.

Question 3: Can we improve current PIJ printheads such that some basic limi-tations with respect to the attainable performance are lifted?

During the derivation of the two-port model and the implementation of ILC andthe accompanying discussions, several new limitations of PIJ printheads havebecome apparent. The following more fundamental limitations have emerged:

• Observability of the jetting process. The observability of the jetting processis limited by the current sensor functionality (piezo-unit) in two ways. Tostart with, the placing of the sensor functionality is not optimal. As arguedin this thesis, the preferred sensor location is in the nozzle where the actualdrop-formation takes place. Adding or relocating sensor functionality wouldenhance the observability of the process. Second, the resolution of the piezoused as sensor is limited. Several smaller wave-forms are not sensed due tothe fact that the nett contribution in pressure distribution over the piezo’ssurface is zero. Incorporating multiple smaller piezo-sensors would improvethe observability of the jetting process.

• Controllability of the jetting process. The controllability of the jetting processis limited by the current actuator (piezo-unit) in three ways. First, thelength of the piezo-unit is too long to be able to generate several pressurewave patterns. Ultimo, this limits the performance of a PIJ printhead. Thelength now equals the length of the ink channel. Incorporating multiplesmaller piezo-actuators can lift this limitation regarding the controllability.Second, the controllability of the meniscus movements using the piezo-unitas actuator is limited. Some movements simply cannot be generated in thenozzle. Consequently, some drop properties cannot be formed. Third, with

8.2 RECOMMENDATIONS 147

the current printhead design it is very difficult to simultaneously damp boththe ink channel and nozzle. Again, this affects the performance negatively,in particular the drop-consistency. Incorporating an additional actuator inthe nozzle can lift this limitation.

• Dominancy of the first eigenfrequency. Considering the piezo-unit’s con-straints with respect to the admissible actuation voltage and utilizing theenergetically most favorable actuation mode, the actuation is tuned on theink channel’s first eigenfrequency. Consequently, the residual vibrations arethen dominated by the same frequency. It is demonstrated that the damp-ing is limited by this frequency also. Altogether, the attainable jettingfrequency of a printhead therefore is limited depending on the used eigen-frequency during actuation. To overcome this boundary, the design is to beadjusted such that the eigenfrequency is increased (shorter ink channels) ormultiple piezo-units are incorporated rather than just one.

8.2 Recommendations

The recommendations for further research are formulated as follows:

• Further development of the two-port model. The accuracy of the result-ing two-port can be improved with respect to the following issues. First,the extension of one ink channel to an array of multiple channels (a PIJprinthead) is to be performed. This requires the proper incorporation ofcross-talk effects. Since the derived equations are in principle capable ofhandling these effects, only the exact determination of cross-talk is to befurther investigated. Second, to enable future investigations into formationof smaller drops, the incorporation of more complex nozzle models is to beconsidered. A related issue concerns the coupling of the nozzle dynamicswith the drop formation. At present, these two are linked by a one-sidedrather than a two-sided coupling is adopted. The quality of the two-portmodel would be enhanced if the correct two-sided coupling were incorpo-rated. Third, the boundary condition that represents the reservoir is to beimproved with respect to the damping it introduces. The current necessityto add damping then would become superfluous. Further development ofthe two-port model with respect to these points will improve the insightthat is obtained by the systems and control approach to the modeling of aPIJ printhead.

• Further exploration of control (ILC). The application of feedforward con-trol to a PIJ printhead is to be further explored. First, ILC can be em-ployed to improve the PIJ printhead performance with respect to severaluninvestigated issues such as stability. Second, the robustness against vary-ing or changing dynamics and disturbances of the ILC approach requires


research. Third, the application of ILC to enhance the productivity anddrop-consistency is in need of further optimization. In our view, the at-tained performance with respect to these two issues can be even furtherincreased. The key to all these issues, irrespective whether it deals withfurther extension or improvement of the ILC approach, lies in the design ofthe reference trajectories. It is expected that tuning of these trajectoriesbased on physical insight is highly profitable. Finally, the application of ILCto an array consisting of more than two ink channels is to be investigated.

• Employment of the insight obtained from the derivation of the two-port modeland the application of control for PIJ printhead (re-)design. The insight thatresults from the derivation of the two-port model and the application of con-trol is to be used for the (re-)design of PIJ printheads, e.g. the establishedfundamental limitations. Also, a design can be evaluated from a systemsand control perspective using the two-port model approach. This way, theperformance of a design can be optimized a priori.

Appendix A

Hamiltonian ILC design

In this appendix, an alternative for the LQ-optimal ILC design approach as pre-sented in Section 6.4 is discussed: the Hamiltonian based ILC design. An im-portant motivation for considering an alternative is found in the length of thereference trajectory. The longer the trajectory, the numerically more difficult theLQ-optimal computations become. The Hamiltonian approach offers a numeri-cally attractive alternative.

Two major differences between the Hamiltonian and the LQ-optimal ILC designapproach can be distinguished. To start with, an alternative method is used toselect the observable part of the impulse response matrix H , thereby avoidingthe singular value decomposition. Second, the computations for the update law,requiring the inverse of a matrix of size H , are handled differently.

Rather than using the singular value decomposition, it is possible to select theobservable part of H using a non-square identity matrix I = [I 0]T of size N ×(N − m), where m is the number of (nearly) zero singular values. Note that thismatrix replaces V1 in the sense that it removes the last columns of H . Define thenew output and feedback matrix as:

H = HI

L = IT L (A.1)

respectively. Using H, the measurement horizon is m samples longer than thecontrol signal. The optimal control problem (6.12) with Q = I and R = βIbecomes:

149

150 APPENDIX A

J =

N∑

k=1

yTk Qyk + ∆uT

k R∆uk

=

N∑

k=1

uTk HT Huk + β∆uT

k ∆uk (A.2)

The corresponding Riccati equation then equals:

−X(βI + X)−1X + HT H = 0 (A.3)

The solution of (A.3) can be approximated by:

X = HT H + βI (A.4)

Substitution of this approximate solution in the Riccati equation shows that itis a solution of an optimal control problem with a slightly different weightingQ. Since it still is a solution of an optimal control problem (Q > 0), it gives astable solution and hence a convergent ILC. The feedback interconnection matrixL equals:

L = X−1HT = (HT H + βI)−1HT (A.5)

and the corresponding update law:

∆uk = (HT H + βI)−1HT ek (A.6)

For long trajectories, matrices H , L, and X can become very large. To avoidnumerically intensive computations, e.g. to compute the inverse of (A.4), it ispossible to obtain the responses for any length of the trajectory with a simulationof a Hamiltonian system, hence the naming Hamiltonian based ILC design. Thissystem comprises two linked difference equations, one with a forward (causal) re-cursion and one with a backward (anti-causal) recursion, based on a state spacemodel of the process and the weighting parameter β. It is assumed that the sys-tem has not a relative degree of zero and thus has a throughput matrix D = 0.

Our aim is to obtain a realization of (A.5). A block diagram of the update law(A.6) is depicted in Fig. A.1. Suppose that H has a state-space representation(A, B, C) with zero state initial condition at k = 0, simulated with a forwardrecursion:

xk+1 = Axk + B∆uk

yk = Cxk (A.7)

HAMILTONIAN ILC DESIGN 151

+

+ H

∆uk

H

− 1β

vk

ek

yk

Figure A.1: Graphical representation of the update law

x and k denote the system’s state and time, respectively. HT has a state spacerepresentation (AT , BT , CT ) with zero state initial condition k = N−1, simulatedwith a backward recursion:

qk−1 = AT qk + CT (yk + ek)

∆uk = − 1

βBT qk (A.8)

Combining the linked state space descriptions (A.7) and (A.8) leads to the fol-lowing system:

xk+1 = Axk − 1

βBBT qk

qk−1 = AT qk + CT (Cxk + ek) (A.9)

∆uk = − 1

βBT qk

Suppose that A is invertible, then:

xk+1 = Axk − 1

βBBT (A−T qk−1 − A−T CT Cxk − A−T CT ek)

qk = A−T qk−1 − A−T CT (Cxk + ek) (A.10)

∆uk = − 1

βBT (A−T qk−1 − A−T CT Cxk − A−T CT ek)

or equivalently in state space:

152 APPENDIX A

(qk

xk+1

)

=

(A−T −A−T CT C

− 1β BBT A−T A + 1

β BBT A−T CT C

)(qk−1

xk

)

+

( −A−T CT

1β BBT A−T CT

)

ek (A.11)

∆uk =( − 1

β BT A−T 1β BT A−T CT C

)(

qk−1

xk

)

+1

βBT A−T CT ek

(A.11) contains a Hamiltonian matrix as system matrix. This matrix has n sta-ble and n anti-stable eigenvalues. There exists a similarity transformation thatseparates the stable and anti-stable part:

(I 0−S −I

)(A−T −A−T CT C

− 1β BBT A−T A + 1

β BBT A−T CT C

)(I 0−S −I

)

=

(

A−T

A−T CT C0 A

)

(A.12)

with S = ST the stabilizing solution of the DARE:

S = ASAT − ASCT (I + CSCT )−1CSAT +1

βBBT (A.13)

and AT

the stable closed-loop matrix:

AT

= AT − CT (I + CSCT )−1CSAT (A.14)

The corresponding transformed variables are:

(qk−1

wk

)

=

(I 0−S −I

)(qk−1

xk

)

=

(qk−1

−xk − Sqk−1

)

(A.15)

such that (A.11) becomes:

(qk

wk+1

)

=

(

A−T

A−T CT C0 A

)(qk−1

wk

)

+

( −A−T CT

(S − 1β BBT )A−T CT

)

ek

∆uk =( − 1

β BT A−T (CT CS + I) − 1β BT A−T CT C

)(

qk−1

wk

)

+1

βBT A−T CT ek (A.16)

with initial and terminal conditions w0 = −Sq−1 and qN−1 = 0, respectively. Theresult is the sequence ∆uk which is the new input to the memory block.

Bibliography

[And84] B. A. Anderson. Computational fluid mechanics and heat transfer.McGraw-Hill, New York, 1984.

[And94] E. H. Anderson and N. W. Hagood. Simultaneous piezoelectric sens-ing/actuation: analysis and application to controlled structures. Jour-nal of Sound and Vibration, vol. 174(5):pp. 617–639, 1994.

[Ant02] B. V. Antohe and D. B. Wallace. Acoustic phenomena in a demandmode piezoelectric inkjet printer. Journal of Image Science and Tech-nology, vol. 46(5):pp. 409–414, 2002.

[Ari84] S. Arimoto, S. Kawamura, and F. Miyazaki. Bettering operation ofrobots by learning. Journal of Robotic Systems, vol. 1(2):pp. 123–140,1984.

[Asa92] A. Asai. Three-dimensional calculation of bubble growth and dropejection in a bubble jet printer. Journal of Fluids Engineering, vol.114(4):pp. 638–641, 1992.

[Bad98] A. Badea, C. Carasso, and G. Panasenko. A model of a homogenizedcavity corresponding to a multinozzle droplet generator for continuousinkjet printers. Numerical Methods for Partial Differential Equations,vol. 14(6), 1998.

[Bad01] A. Badea, A. Capatina, and C. Carasso. Optimization of a multinozzleinkjet printer. Computer methods in applied mechanics and engineering,vol. 190(18-19), 2001.

[Bel98] W. M. Beltman. Viscothermal wave propagation including acousto-elastic interaction. Ph.D. thesis, University of Twente, 1998.

[Ben03] R. Bennett. Precision industrial ink-jet printing technology for full colorPLED display and TFT-LCD manufacturing. In Proc. of the 3rd Inter-national Display Manufacturing Conference. Japan, 2003.

153

154 BIBLIOGRAPHY

[Ber03] S. S. Berger and G. W. Recktenwald. Development of an improvedmodel for piezo-electric driven inkjets. In Proc. IS&T NIP 19, pp. 323–327. New Orleans, USA, 2003.

[Bie98] Z. Bien and J. Zu. Iterative learning control: analysis, design, integra-tion and applications. Kluwer Academic Publishers, Boston, 1998.

[Bie04] A. Bietsch, J. Zhang, M. Hegner, H. P. Lang, and C. Gerber. Rapidfunctionalization of cantilever array sensors by inkjet printing. Nan-otechnology, vol. 15(8):pp. 873–880, 2004.

[Bog84] D. B. Bogy and F. E. Talke. Experimental and theoretical study ofwave propagation phenomena in drop-on-demand inkjet devices. IBMJournal of Research and Developement, vol. 28(3):pp. 314–321, 1984.

[Bos02] O. H. Bosgra. Bilaterally coupled systems. Tech. rep., Delft Universityof Technology, 2002.

[Bos05] N. J. M. Bosch, M. B. Groot Wassink, and O. H. Bosgra. Iterative learn-ing control on an inkjet printhead. Tech. Rep. DCT 2005.80, EindhovenUniversity of Technology, 2005.

[Bro62] F. T. Brown. The transient response of fluid lines. Trans ASME/Journalof Basic Engineering, vol. 84-D:pp. 547–553, 1962.

[Bro65] F. T. Brown and S. E. Nelson. Step responses of liquid lines withfrequency dependent effects of viscosity. Trans ASME/Journal of BasicEngineering, vol. 87:pp. 504–510, 1965.

[Bro69a] F. T. Brown. A quasi-method of characteristics with application tofluid lines with frequency dependent wall shear. Trans ASME/Journalof Basic Engineering, vol. 91-D:pp. 217–227, 1969.

[Bro69b] F. T. Brown, D. L. Margolis, and R. P. Shah. Small-amplitude frequencybehavior of fluid lines with turbulent flow. Trans ASME/Journal ofBasic Engineering, vol. 91:pp. 678–693, 1969.

[Bru05] D. Bruijnen, R. v. d. Molengraft, A. Draad, and T. Heeren. Productivityanalysis of a scanning inkjet printer. In Proc. IS&T NIP 21, pp. 129–132. Baltimore, USA, 2005.

[Buc95] J. B. Buckheit and D. L. Donoho. Wavelab and reproducible research.In A. Antoniadis and G. Oppenheim, editors, Wavelets and Statistics,pp. 55–81. Springer Verlag, New York, 1995.

[Byr60] R. Byron, W. E. Stewart, and E. N. Lightfoot. Transport phenomena.Wiley, New York, 1960.

BIBLIOGRAPHY 155

[Che96] M. Chee, R. Yang, E. Hubbell, A. Berno, X. C. Huang, D. Stern, J. Win-kler, D. J. Lockhart, M. S. Morris, and S. P. A. Fodor. Accessing geneticinformation with high-density DNA arrays. Science, vol. 274:pp. 610–614, 1996.

[Che99] P. H. Chen, H. Y. Peng, H. Y. Liu, S. L. Chang, T. I. Wu, and C. H.Cheng. Pressure response and droplet ejection of a piezoelectric inkjetprinthead. International Journal of Mechanical Sciences, vol. 41:pp.235–248, 1999.

[Che02] T. Chen, W. R. Cox, D. Lenhard, and D. J. Hayes. Microjet printingof high precision microlens array for packaging of fiber-optic compo-nents. Proc SPIE Optoelectronic Interconnects, Integrated Circuits, andPackaging, vol. 4652:pp. 136–141, 2002.

[Coo01] P. W. Cooley, D. B. Wallace, and B. V. Antohe. Application of inkjetprinting technology to biomems and microfluidic systems. In Proc. SPIEConference on Microfluidics and BioMEMS, vol. 4560, pp. 177–188. SanFrancisco, USA, 2001.

[Cox96] W. R. Cox, T. Chen, D. Ussery, D. J. Hayes, J. A. Tatum, and D. L.MacFarlane. Microjetted lenslet triplet fibers. Optics Communications,vol. 123:pp. 492–496, 1996.

[Cra87] E. F. Crawley and J. de Louis. Use of piezoelectric actuators as elementsof intelligent structures. AIAA journal, vol. 25(10):pp. 1373–1385, 1987.

[Die98] T. Diepold, E. Obermeier, and A. Berchtold. A micromachined contin-uous ink jet print head for high-resolution printing. Journal of Micro-mechanics and Microengineering, vol. 8:pp. 144–147, 1998.

[Dij84] J. F. Dijksman. Hydrodynamics of small tubular pumps. Journal FluidMechanics, vol. 139:pp. 173–191, 1984.

[Dij99] J. F. Dijksman. Hydro-acoustics of piezoelectrically driven inkjet print-heads. Flow, Turbulence and Combustion, vol. 61:pp. 211–237, 1999.

[Dij04] B. G. Dijkstra. Iterative Learning Control with applications to a wafer-stage. Ph.D. thesis, Delft University of Technoloy, 2004.

[Dos92] J. J. Dosch, D. J. Inman, and E. Garcia. A self-sensing piezoelectricactuator for collocated control. Journal of Intelligent Material Systemsand Structures, vol. 3:pp. 166–185, 1992.

[Egg95] J. Eggers. Theory of drop formation. Physics of Fluids, vol. 7(5):pp.941–953, 1995.

156 BIBLIOGRAPHY

[Fel01] C. A. Felippa, K. C. Park, and C. Farhat. Partitioned analysis of coupledsystems. Computer methods in applied mechanics and engineering, vol.190, issues 24-25:pp. 3247–3270, 2001.

[Fro84] J. E. Fromm. Numerical calculation of the fluid dynamics of drop-on-demand jets. IBM Journ of Research and Development, vol. 28:pp.322–333, 1984.

[Gor97] D. Gorinevsky, D. E. Torfs, and A. A. Goldenberg. Learning approxi-mation of feedforward control dependence on the task parameters withapplication to direct-drive manipulator tracking. IEEE Transactions onRobotics and Automation, vol. 13(4):pp. 567–581, 1997.

[Gro03] M. A. Groninger. A method of controlling an inkjet printhead, an inkjetprinthead suitable for use of said method, and an inkjet printer com-prising said printhead. European Patent 1 378 360 A1, 2003.

[Gro05a] M. B. Groot Wassink, N. J. M. Bosch, O. H. Bosgra, and S. H. Koeke-bakker. Enabling higher jetting frequencies for an inkjet printhead usingiterative learning control. In Proc. IEEE Conf. on Control Applications,pp. 791–796. Toronto, Canada, 2005.

[Gro05b] M. B. Groot Wassink, O. H. Bosgra, D. J. Rixen, and S. H. Koeke-bakker. Modeling of an inkjet printhead for iterative learning controlusing bilaterally coupled multiports. In Conf. on Decision and Controland European Control Conference, pp. 4766–4772. Sevilla, Spain, 2005.

[Gro05c] M. B. Groot Wassink, O. H. Bosgra, and M. Slot. Enabling higher jet-ting frequencies for an inkjet printhead using iterative learning control.In Proc. IS&T NIP 21, pp. 273–277. Baltimore, USA, 2005.

[Gro06a] M. B. Groot Wassink and O. H. Bosgra. Inkjet printer and method forcontrolling this printer. In European Patent 1 690 686 A1. 2006.

[Gro06b] M. B. Groot Wassink, O. H. Bosgra, and S. H. Koekebakker. Enhancingthe functional performance of inkjet printheads by exploiting model-based iterative learning control. Mechatronics, p. submitted, 2006.

[Gro06c] M. B. Groot Wassink, O. H. Bosgra, and S. H. Koekebakker. Minimiz-ing cross-talk for an inkjet printhead using MIMO ILC. In AmericanControl Conference, pp. 964–969. Minneapolis, USA, 2006.

[Gro06d] M. B. Groot Wassink, O. H. Bosgra, S. H. Koekebakker, and M. Slot.Minimizing residual vibrations and cross-talk for inkjet printheads usingILC designed simplified actuation pulses. In Proc. IS&T NIP 22, pp.69–74. Denver, USA, 2006.

BIBLIOGRAPHY 157

[Gro06e] M. B. Groot Wassink, F. Zollner, O. H. Bosgra, and S. H. Koekebakker.Improving the drop-consistency of an inkjet printhead using meniscus-based iterative learning control. In Conference on Control Applications,pp. 2830–2835. Munich, Germany, 2006.

[Gro07] M. B. Groot Wassink, O. H. Bosgra, and S. H. Koekebakker. Enhancinginkjet printhead performance by mimo iterative learning control usingimplementation based basis functions. In American Control Conference,p. submitted. New York, USA, 2007.

[Gun01] S. Gunnarsson and M. Norrlof. On the design of ilc algorithms usingoptimization. Automatica, vol. 37:pp. 2011–2016, 2001.

[Hal55] J. F. Hale, D. A. McDonald, and J. R. Womersley. Velocity profiles ofoscillating arterial flow, with some calculations of viscous drag and thereynolds number. Journal of Physiology, vol. 128(3):pp. 629–640, 1955.

[Han67] A. G. Hansen. Fluid Mechanics. John Wiley, New York, 1967.

[Has02] E. I. Haskal, M. Buechel, and J. F. Dijksman. Ink jet printing of passive-matrix polymer light emitting displays. In Proc. of International Sym-posium of Society for Information Display, vol. 33, pp. 776–779. 2002.

[Hat04] V. Hatzikos, J. Hatonen, and D. H. Owens. Genetic algorithms innorm-optimal and non-linear iterative learning control. Int J Control,vol. 77(2):pp. 188–197, 2004.

[Hei98] J. van de Heijden, H. Reinten, M. Schuwer, and M. E. H. van Dongen.Pompwerking bij een oscillerende vloeistofstroming in een vernauwendebuis. Tech. Rep. R-1472-A, Eindhoven University of Technology, 1998.

[Hei00] J. Heilmann and U. Lindqvist. Effect of dropsize on the print quality incontinuous inkjet printing. Journal of Imaging Science and Technology,vol. 44(6):pp. 491–494, 2000.

[Hei05] J. Heilmann. The applications of inkjet for the printing of PDLC displaycells. In Proc. IS&T NIP 21, pp. 48–51. Baltimore, USA, 2005.

[Hor85] R. A. Horn and C. R. Johnson. Matrix analysis. Camebridge UniversityPress, Camebridge, 1985.

[Jaı88] A. E. Jaı and A. J. Pritchard. Sensors and Controls in the Analysis ofDistributed Systems. Ellis Horwood Ltd, Chichester, 1988.

[Jam98] P. James and R. Papen. Nanolitre dispensing: a new innovation inrobotic liquid handling. Drug Discovery Today, vol. 3(9):pp. 429–439,1998.

158 BIBLIOGRAPHY

[Jon97] M. de Jong, P. J. M. van den Hof, and R. A. de Callafon. Design ofan actuation pulse for an inkjet printhead. Tech. Rep. A-767, DelftUniversity of Technology, 1997.

[Kar77] D. Karnopp. Power and energy in linearized physical systems. Journalof the Franklin Institute, vol. 303:pp. 85–98, 1977.

[Kno05] A. Knobloch. Printed RFID labels based on polymer electronics. InProc. of Digital Fabrication Conference, pp. 121–123. Baltimore, USA,2005.

[Kol02a] P. Koltay, C. Moosmann, C. Litterst, W. Streule, B. Birkenmeier, andR. Zengerle. Modeling free jet ejection on a system level - an approachfor microfluidics. In Proceedings of International Conference on Mod-eling and Simulation of Microsystems, vol. 1, pp. 112–115. San Juan,Puerto Rico, 2002.

[Kol02b] P. Koltay, C. Moosmann, C. Litterst, W. Streule, and R. Zengerle.Simulation of a micro dispenser using lumped models. In Proceedings ofInternational Conference on Modeling and Simulation of Microsystems,vol. 1, pp. 170–173. San Juan, Puerto Rico, 2002.

[Lee96] K. S. Lee, S. H. Bang, S. Yi, J. S. Son, and S. Yoon. Iterative learningcontrol of heat-up phase for a batch polymerization reactor. Journal ofProcess Control, vol. 6(4):pp. 255–262, 1996.

[Lev44] K. Levenberg. A method for the solution of certain problems in leastsquares. Quarterly Applied Mathematics II, pp. 164–168, 1944.

[Lio02] T. M. Liou, K. C. Shih, S. W. Chau, and S. C. Chen. Three-dimensional simulations of the droplet formation during the inkjet print-ing process. International Communications in Heat and Mass Transfer,vol. 29(8):pp. 1109–1118, 2002.

[Lon00] R. W. Longman. Iterative learning control and repetitive control forengineering practice. International Journal of Control, vol. 73:pp. 930–954, 2000.

[Mar63] D. Marquardt. An algorithm for least-squares estimation of nonlinearparameters. SIAM Journal Applied Mathematics, vol. 11:pp. 431–441,1963.

[Mar04] A. M. Markesteijn, M. B. Groot Wassink, and S. H. Koekebakker. Mod-elling of incompressible flow in a printhead nozzle. Tech. rep., DelftUniversity of Technology, 2004.

[Mar06] A. M. Markesteijn and D. J. Rixen. A fast computational model ofnozzle flow and droplet formation in printheads. Physics of Fluids, p.submitted, 2006.

BIBLIOGRAPHY 159

[Moo93] K. L. Moore. Itertive Learning Control for Deterministic Systems.Springer-Verlag, 1993.

[Moo98] K. L. Moore. Iterative learning control: an expository overview. Ap-plied and Computational Controls, Signal Processing, and Circuits,vol. 1(1):pp. 151–214, 1998.

[Nag06] P. M. Nagelmaeker, M. B. Groot Wassink, and O. H. Bosgra. Improvingthe drop consistency of an inkjet printhead by iterative learning control.Tech. Rep. DCT 2006.108, Eindhoven University of Technology, 2006.

[Nak05] T. Nakanishi, I. Ohtsu, M. Furuta, E. Ando, and O. Nishimura. DirectMS/MS analysis of proteins blotted on membranes by a matrix-assistedlaser desorption/ionization-quadrupole ion trap-time-of-flight tandemmass spectrometer. Journal of Proteome Research, vol. 4:pp. 743–747,2005.

[Nor01] M. Norrlof and S. Gunnarsson. Disturbance aspects of iterativelearning control. Engineering Applications of Artificial Intelligence,vol. 14(1):pp. 87–94, 2001.

[Oht05] I. Ohtsu, T. Nakanisi, M. Furuta, E. Ando, and O. Nishimura. Directmatrix-assisted laser desorption/ionization time-of-flight mass spectro-metric identification of proteins on membrane detected by western blot-ting and lectin blotting. Journal of Proteome Research, vol. 4:pp. 1391–1396, 2005.

[Pan02] F. Pan, J. Kubby, and J. Chen. Numerical simulation of fluid-structureinteraction in a MEMS diaphragm drop ejector. Journal of Microme-chanics and Microengineering, vol. 12:pp. 70–76, 2002.

[Pay61] H. M. Paynter. Analysis and Design of Engineering Systems. The MITPress, Cambridge, MA, 1961.

[Pei96] L. Peirlinckx, P. Guillaume, and R. Pintelon. Accurate and fast esti-mation of the fourier coefficients of perodic signals disturbed by trends.IEE Transactions on Instrumentation and Measurement, vol. 45:pp. 5–11, 1996.

[Pha88] M. Q. Phan and R. W. Longman. A mathematical theory of learningcontrol of linear discrete multivariable systems. In Proc. of AIAA/AASAstrodynamics Specialist Conference, pp. 740–746. Minneapolis, USA,1988.

[Pha96] M. Q. Phan and J. A. Frueh. Learning control for trajectory trackingusing basis functions. In Conf. on Decision and Control, pp. 2490–2492.Kobe, Japan, 1996.

160 BIBLIOGRAPHY

[Pla56] J. A. F. Plateau. On the recent theories of the consitution of jets ofliquid issuing from circular orifices. Philosophical Magazine, vol. 12:pp.286–297, 1856.

[Pol00] Polytec. User Manual OFV 5000. Polytec GmbH, Waldbronn, Ger-many, 2000.

[Pon00] S. F. Pond. Inkjet technology and product development strategies. TorreyPines Research, Carlsbad, 2000.

[Poz97] C. Pozrikidis. Introduction to theoretical and computational fluid dy-namics. Oxford University Press, New York, 1997.

[Rad05] D. Radulescu and et al. 3d printing of biological materials for drug deliv-ery and tissue engineering applications. In Proc. of Digital FabricationConference, pp. 96–99. Baltimore, USA, 2005.

[Ray78] J. W. S. Rayleigh. On the instability of jets. Proc of the London math-ematical society, vol. 10:pp. 4–13, 1878.

[Ray92a] J. W. S. Rayleigh. On the instability of a cylinder of viscous liquid undercapillary force. Philosophical Magazine, vol. 34:pp. 145– 154, 1892.

[Ray92b] J. W. S. Rayleigh. On the instability of cylindrical fluid surfaces. Philo-sophical Magazine, vol. 34(207):pp. 177–180, 1892.

[Red60] R. M. Redheffer. On a certain linear fractional transformation. Journalof Mathematics and Physics, vol. 39:pp. 269–286, 1960.

[Red62] R. M. Redheffer. On the relation of transmission-line theory to scatter-ing and transfer. Journal of Mathematics and Physics, vol. 41:pp. 1–41,1962.

[Roo97] D. de Roover. Motion control of a wafer stage. A design approach forspeeding up IC production. Ph.D. thesis, Delft University of Technoloy,1997.

[Ros72] R. C. Rosenberg. Multiport models in mechanics. Trans of the ASME,Journal of Dynamic System Measurement and Control, vol. 94(3):pp.206–212, 1972.

[Sak00] S. Sakai. Dynamics of piezoelectric inkjet printing systems. In IS&TNIP 16, pp. 15–20. Vancouver, Canada, 2000.

[Sch86] T. W. Schield, D. B. Bogy, and F. E. Talke. A numerical comparison ofone-dimensional fluid jet models applied to drop-on-demand printing.Journal of Computational Physics, vol. 67(2):pp. 327–347, 1986.

BIBLIOGRAPHY 161

[Sch87] T. W. Schield, D. B. Bogy, and F. E. Talke. Drop formation by DODinkjet nozzles - a comparison of experiment and numerical simulation.IBM Journal of Research and Development, vol. 31(1):pp. 96–110, 1987.

[Sch94] R. Schrama. Approximate identification and control design with applica-tion to a mechanical system. Ph.D. thesis, Delft University of Technoloy,1994.

[Sch99] J. M. Schneider. Continuous inkjet technology. Recent progress in inkjettechnologies, vol. II:pp. 246–251, 1999.

[Sei04] H. Seitz and J. Heinzl. Modelling of a microfluidic device with piezo-electric actuators. Journal of Micromechanics and Microengineering,vol. 14:pp. 1140–1147, 2004.

[Shi05] D. Y. Shin, P. Grassia, and B. Derby. Oscillatory incompressible fluidflow in a tapered tube with a free surface in an inkjet printhead. Journalof Fluids Engineering, vol. 127(1):pp. 98–109, 2005.

[Szc05] J. Szczech, J. Zhang, K. Kalyan, P. Brazis, and D. Gamota. Printedelectronic using traditional graphic arts printing technologies. In Proc.of Digital Fabrication Conference, pp. 213–214. Baltimore, USA, 2005.

[Tay04] A. Tayebi. Adaptive iterative learning control for robot manipulators.Automatica, vol. 40:pp. 1195–1203, 2004.

[Ten88a] K. F. Teng. A mathematical model of impulse jet mechanism. Mathe-matical and Computer Modeling, vol. 11:pp. 751–753, 1988.

[Ten88b] K. F. Teng and R. W. Vest. Mathematical models of inkjet printingin thick-film hybrid microelectronics. Applied Mathematical Modeling,vol. 12(2):pp. 182–188, 1988.

[Tou01] R. Tousain, E. van der Meche, and O. H. Bosgra. Design strategiesfor iterative learning control based on optimal control. In Proc. IEEEConference on Decision and Control, pp. 4463–4468. Orlando, USA,2001.

[Tzo94] H. S. Tzou and H. Q. Fu. A study of segmentation of distributed piezo-electric sensors and actuators part i: theoretical analysis. Journal ofSound and Vibration, vol. 172(2), 1994.

[Voi03] W. Voit, W. Zapka, L. Belova, and K. V. Rao. Application of inkjettechnology for the deposition of magnetic nanoparticles to form micron-scale structures. IEE Proc Science Measurement and Technology, vol.150(5):pp. 252–256, 2003.

162 BIBLIOGRAPHY

[Waa91] J. W. Waanders. Piezoelectric Ceramics. Properties and applications.Morgan Electroceramics, Eindhoven, 1991.

[Wal02] D. B. Wallace, W. R. Cox, and D. J. Hayes. Direct write using inkjettechniques. In A. Pique and D. B. Chrisey, editors, Direct-Write tech-nologies for rapid prototyping applications - sensors, electronics, andintegrated power sources, pp. 177–227. Academic Press, San Diego, 2002.

[Wij04] H. Wijshoff. Free surface flow and acousto-elastic interaction in piezoinkjet. In Proc. NSTI Nanotechnology Conference and Trade Show,vol. 2, pp. 215–218. Boston, USA, 2004.

[Wij06] H. Wijshoff. Manipulating drop formation in piezo acoustic inkjet. InProc. IS&T NIP 22, p. accepted. Denver, USA, 2006.

[Wil99] E. D. Wilkes, S. D. Philips, and O. A. Basaran. Computational andexperimental analysis of dynamics of drop formation. Physics of Fluids,vol. 11(12):pp. 3577–3598, 1999.

[Wu05] H. C. Wu, H. J. Lin, and W. S. Hwang. A numerical study of the effectof operating parameters on drop formation in a squeeze mode inkjetdevice. Modeling and Simulation in Materials Science and Engineering,vol. 13(1):pp. 17–34, 2005.

[Yan04] K. S. Yang, I. Y. Chen, and C. C. Wang. A numerical study of inkjetrefilling. Proceedings of the Institution of Mechanical Engineers C, vol.218(12), 2004.

[Yeh01] J. T. Yeh. A VOF-FEM and coupled inkjet simulation. In Proc. ofASME Fluids Engineering Division Summer Meeting, pp. 1–5. New Or-leans, USA, 2001.

[Yeo04] W. Y. Yeong, C. K. Chua, K. F. Leong, and M. Chandrasekaran. 3Dinkjet printing for rapid prototyping: development of a physical model.In Proc. of International Conference on Materials Processing for Prop-erties and Performance, pp. 379–385. Singapore, 2004.

[Yu03] J. D. Yu and S. Sakai. Piezo inkjet simulations using the finite differencelevel set method and equivalent circuit. In IS&T NIP 19, pp. 319–322.New Orleans, USA, 2003.

[Yu05] J. D. Yu, S. Sakai, and J. Sethian. A coupled quadrilateral grid levelset projection method applied to ink jet simulation. Journal of Com-putational Physics, vol. 206(1), 2005.

Glossary of symbols

Symbols

Ach Cross-sectional area of the ink-channelAco Cross-sectional area of the connectionb Boundary velocityc Speed of soundceff Effective speed of soundcw Wave propagation velocityC Piezo capacityd Piezo-electrical charge constantek Error signal at trial kez Outward normal in the positive z-directionf Sample frequencyh Height of the free surfacehp Piezo thicknessH Discrete time Hankel matrixk Piezo stiffnessK Maximum displacement of the piezo’s zeroth order model Length of the ink cylinderL Learning filterLch Length of the ink-channelLco Length of the connectionLn Length of the nozzlen Outward normalN Trial lengthp PressurePr Remanent polarizationPs Saturation polarizationq Electric chargeQ Weighting on ILC error/inputr Nozzle radiusR Weighting on ILC control effort

163

164 GLOSSARY OF SYMBOLS

sE Compliance for a constant electrical field ES Surface areat Timetsw Switching instantu Displacementuk Input signal at trial kU Matrix with singular output vectorsv Velocityvav Average velocityvd Drop velocityvr Relative velocityvz Meniscus velocity in the z-directionV Matrix with singular input vectorsVd Drop volumeWi, Wo Input and output weighting filtersxi Diagonal entries of matrix XX Stabilizing solution of a DAREyk Output signal at trial kyref Reference trajectoryz−1 Discrete time delay operatorz1, z2 FlowZ Impedanceα Kinetic energy correction factorβ Momentum-flux correction factor, or ILC tuning parameterγ Scalar learning gain∆uk Update of the input signal at trial k∆φ Phase lag of the laser-vibrometerǫT Permittivity under constant stress Tλ Wave lengthλi Closed-loop poles in the trial domainµ Dynamic viscosityν Surface tensionρ Densityσi i-th singular valueσ Viscous stress tensorΣ Diagonal matrix with singular values on the main diagonalφ Flowω Angular frequency

Abbreviations

ASIC Application Specific Integrated Circuit

GLOSSARY OF SYMBOLS 165

BCS Bilaterally Coupled SystemCCD Charge Coupled DeviceCFD Computational Fluid DynamicsCIJ Continuous InkjetCPS Cumulative Power SpectrumCS Control SurfaceCV Control VolumeDARE Discrete time Algebraic Riccati EquationDOD Drop-on-DemandDSM Drop Size ModulationETFE Empirical Transfer Function EstimateFEM Finite Element MethodFPD Flat Panel DisplayFPGA Field Programmable Gate ArrayFR Frequency ResponseFRD Frequency Response DataFRF Frequency Response FunctionFSI Fluid Structure InteractionHP Hewlett PackardIAE Integrated Absolute ErrorIBM International Business Machines corporationILC Iterative Learning ControlLCD Liquid Cristal DisplayLTI Linear Time InvariantLQ Linear QuadraticMAC Marker And CellMIMO Multiple-Input Multiple-OutputNPI Nozzles Per InchOE Output ErrorPCB Printed Circuit BoardPID Proportional, Integrating, and Differentiation feedback controlPIJ Piezo-electrical InkjetPLED Polymer Light Emitting DiodeRCA Radio Corporation of AmericaREI Recognition Equipment InstituteRFID Radio Frequency IdentificationSISO Single-Input Single-OutputSRI Stanford Research InstituteSV Stream-function VorticitySVD Singular Value DecompositionTF Transfer FunctionTIJ Thermal InkjetVOF Volume Of Fluid

Summary

Inkjet printhead performance enhancement by feedforward input de-

sign based on two-port modeling

Inkjet technology is an important key-technology from an industrial point of view.Its ability to deposit various types of material on a substrate in certain patternsmakes it a very versatile technology. Not surprisingly, the variety of applica-tions is very wide, ranging from standard document printing to the fabrication offlat panel displays. Applications of inkjet technology are often accompanied withtight performance criteria. Usually, these include specifications concerning severaldrop-properties, such as speed and volume, and the consistency of those proper-ties. Also, requirements for the jetting process itself are frequently imposed, e.g.with respect to the productivity and stability. Whereas current performance cri-teria are quite stringent already, they are expected to become even tighter in thenear future.

A typical design of a piezo-electrical inkjet (PIJ) printhead comprises a large arrayof piezo-actuated channels. The shape of the corresponding actuation pulses isdetermined by manually tuning based on physical insight such that the requesteddrop-on-demand results. However, this approach in combination with printheaddesigns has become mature and its possibilities have been exhausted, especially inface of some operational issues that are generally encountered: residual vibrationsand cross-talk. The former issue relates to the fact that the ink in a channel isusually not at rest immediately after drop ejection. On average, it takes approxi-mately 100 µs for the pressure waves to be damped such that a next drop can befired. Cross-talk refers to the fact that if one channel is actuated, the fluid me-chanics in neighboring channels are also actuated. This results in different dropproperties if neighboring channels are actuated simultaneously or shortly afterone another. Altogether, both phenomena limit the productivity as well as thedrop-consistency, and hence the performance, of PIJ printheads considerably.

In this thesis, a systems and control approach to the functioning of PIJ print-heads is proposed to break current boundaries. The aim is threefold. First, such

167

168 SUMMARY

an approach to the modeling of an inkjet printhead provides good insight in itsworking that can be used for the control and redesign purposes in mind. Second,application of feedforward control is a very cost-effective method to improve a PIJprinthead’s performance, e.g. concerning its productivity and drop-consistency.Finally, the proposed approach helps identifying several more fundamental limi-tations of PIJ printheads that can be taken into account in future designs.

The key in the modeling of an ink channel from a systems and control perspectiveis to view the system as a series of bilaterally coupled subsystems. Additionally,to keep the model complexity low, the dynamics of each of these subsystems aremodeled using first principles only. This is also achieved by coupling the vari-ous blocks by using the Redheffer star product rather than staggered schemes.Despite the low model complexity of the resulting so called two-port model, itis still accurate enough to serve as starting point for the intended control andredesign purposes. For one, the two-port model provides sufficient physical in-sight in the jetting process to facilitate the implementation of feedforward control.

Given the repetitive character of the jetting process, the Iterative Learning Con-trol (ILC) framework is used as feedforward control strategy. In this framework,reference trajectory design plays a crucial role in achieving the control objec-tives, i.e. the minimization of residual vibrations and cross-talk. The chosen ILCframework also enables a reduction of the time required for actuation while stillattaining the formulated control objectives. Furthermore, a modified ILC algo-rithm is presented that allows for the design of piece-wise affine actuation pulses.This is necessary to overcome the limitations posed by the electronics of a PIJprinthead, that can only handle extremely simplified actuation pulses. Finally,ILC is implemented on various PIJ printheads using either the pressure in anink channel or the meniscus velocity as sensor signal. The experimental resultsdemonstrate that by meeting the control objectives a considerable improvementof the performance with respect to the drop-consistency and the productivity canbe achieved.

Upon using the systems and control approach for PIJ printheads, several morefundamental limitations of the design emerge, e.g. concerning the maximally at-tainable jetting frequency and the spatial observability and controllability. At thesame time, based on the insight obtained several adjustments to the design areproposed to overcome even those.

M.B. Groot Wassink

Samenvatting

Prestatieverbetering van inkjet printkoppen door het via feedfoward

technieken ontwerpen van aanstuursignalen op basis van tweepoort

modellering

Inktjet technologie vormt een belangrijke sleuteltechnologie voor de industrie.De mogelijkheid om verschillende soorten materiaal op een substraat te kunnenprinten in zekere patronen maakt de technologie tot een zeer breed inzetbare. Hetmag daarom geen verassing heten dat het spectrum aan toepassingen zeer breedis, varierend van het printen van documenten tot de fabricage van zogenaamdeplatte beeldschermen. Doorgaans gelden er voor de toepassingen van inktjet tech-nologie strikte prestatie-eisen. Zo is het gebruikelijk dat er eisen worden gesteldaan diverse druppeleigenschappen, zoals snelheid en volume, evenals de consis-tentheid daarin. Daarnaast worden vaak eisen gesteld aan het jet-proces zelf,zoals bijvoorbeeld betreffende de productiviteit en de stabiliteit. Ondanks hetfeit dat de huidige prestatie-eisen al vrij hoog liggen, wordt verwacht dat dezesteeds strenger worden in de nabije toekomst.

Een typisch ontwerp van een piezo-electrische inktjet (PIJ) printkop omvat eenaanzienlijk aantal piezo-geactueerde kanalen naast elkaar. De bijbehorende actu-atie pulsen worden vastgesteld door handmatig tunen op basis van fysisch inzichtzodat de gewenste druppel resulteert. Echter, deze aanpak in combinatie metverschillende printkop ontwerpen is uitontwikkeld en de mogelijkheden die hetbiedt zijn uitgeput, zeker gezien enkele veel voorkomende operationele proble-men: residuale trillingen en overspraak. Het eerstgenoemde probleem betreft hetverschijnsel dat de inkt in een kanaal niet direct in rust is nadat er een druppel isgejet. Het duurt gemiddeld gezien ongeveer 100 µs voordat de drukgolven zodanigzijn uitgedempt dat een volgende druppel kan worden gejet. Overspraak is de be-naming voor het verschijnsel dat een bepaald kanaal niet geactueerd kan wordenzonder dat de buurkanalen dit ook worden. Een gevolg is dat druppeleigenschap-pen varieren wanneer buurkanalen gelijktijdig of kort na elkaar worden geactueerd.Al met al, beide verschijnselen beperken de productiviteit en de druppelconsis-tentie, en daarmee dus ook de prestatie van PIJ printkoppen aanzienlijk.

169

170 SAMENVATTING

In dit proefschrift wordt een systeem en regelaanpak voor het functioneren vanPIJ printkoppen voorgesteld om de huidige grenzen te doorbreken. Het doel daar-van is driedelig. Ten eerste biedt een dergelijke aanpak voor het modelleren vaneen inkjet printkop goed inzicht in de werking dat weer gebruikt kan worden voorregel en herontwerp doeleinden. Ten tweede vormt feedforward regelen een primakosten-effectieve manier om de prestatie van PIJ printkoppen te verbeteren, bi-jvoorbeeld als het gaat om de productiviteit en de druppelconsistentie. Tot slothelpt deze aanpak om de meer fundamentele beperkingen van PIJ printkoppenaan het licht te brengen. Kennis op dat vlak kan weer gebruikt kunnen wordenbij toekomstige ontwerpen.

De sleutel tot succes bij het modelleren van een inkt kanaal vanuit een systeem enregelperspectief is door het systeem als een serieschakeling van tweezijdig gekop-pelde systemen te beschouwen. Om de complexiteit van het model laag te houden,wordt vervolgens de dynamica van elk van deze subsystemen gemodelleerd dooruitsluitend gebruik te maken van first principles. Dit wordt ook bereikt door dediverse blokken te koppelen met behulp van het Redheffer star product in plaatsvan staggered schemes. Ondanks de lage complexiteit van het resulterende zoge-naamde tweepoort model, is het nog steeds nauwkeurig genoeg om als startpuntte dienen voor de beoogde regel en herontwerp doeleinden. Zo biedt het tweep-oort model voldoende fysisch inzicht in het jet-proces om de implementatie vande feedforward regeling te vergemakkelen. Gegeven het repeterende karakter vanhet jet-proces wordt het Iterative Learning Control (ILC) raamwerk gebruikt alsfeedforward regelstrategie. In dit raamwerk speelt referentie trajectorie ontwerpeen cruciale rol in het bereiken van de regeldoelen, namelijk het minimaliserenvan residuale trillingen en overspraak. Verder maakt het gekozen raamwerk hetmogelijk om de benodigde tijd voor actueren te reduceren zonder dat dit het be-halen van de geformuleerde regeldoelen aantast. Vervolgens wordt een aanpastILC algoritme geıntroduceerd waarmee stuksgewijs-affine actuatie signalen kun-nen worden ontworpen. Dit is noodzakelijk om goed om te kunnen gaan met debeperkingen van de electronica van een PIJ printkop, die slechts extreem vereen-voudigde actuatie pulsen aankan. Tot slot wordt ILC toegepast op verschillendePIJ printkoppen waarbij als sensor signaal ofwel gebruik wordt gemaakt van dedruk in een kanaal of de meniscus snelheid. De experimentele resultaten latenzien dat door de regeldoelen te behalen de productiviteit en de druppelconsis-tentie aanzienlijk verhoogd kunnen worden. Door gebruik te maken van een sys-teem en regelaanpak voor PIJ printkoppen komen een aantal meer fundamentelebeperkingen van de ontwerpen aan het licht, zoals bijvoorbeeld de maximaal haal-bare jet-frequentie en de ruimtelijke regelen waarneembaarheid. Tegelijkertijdworden op grond van het verkregen inzicht verschillende aanpassingen aan hetontwerp voorgesteld waarmee deze overwonnen kunnen worden.

M.B. Groot Wassink

Curriculum Vitae

January 15, 1978 Born in Leiden, The Netherlands

1990 - 1996 VWO (pre-university education), Stedelijk Gymnasium, Lei-den, The Netherlands

1996 - 2002 MSc student Mechanical Engineering at Delft University ofTechnology, Delft, The Netherlands, with a specialization inSystems and Control. Graduated cum laude with a MSc the-sis on Linear Parameter Varying control for a wafer stage, forwhich research was conducted at Philips Center for IndustrialTechnology, Eindhoven, The Netherlands

2002 - 2006 PhD student Mechanical Engineering, Systems and Con-trol group, at Delft University of Technology, Delft, TheNetherlands. This PhD research was sponsored by Oce-Technologies, Venlo, The Netherlands

2006 - present Project manager at Boer & Croon Young Executives, Ams-terdam, The Netherlands

171

Inkjetprintheadperformance enhancement by feedforward ... · PDF file• DOD electrostatic...

Documents

Transcript of Inkjetprintheadperformance enhancement by feedforward ... · PDF file• DOD electrostatic...