Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR...

53
10/27/2004 07 inductance 1 TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde kringen Mijn naam, als u het weten wilt is i Ik zit in allerlei berekeningen En aan het einde hef ik mij weer op U vraagt niet, denk ik, om verhandelingen Maar wenst dat ik mezelf nu eens ontpop Welnu - ik ben de wortel uit -1 Ik functioneer, ofschoon ik niet besta Denk daar maar eens langdurig over na (Uit: Drs. P en Marjolein Kool, Wis- en natuurlyriek met chemisch supplement, Amsterdam: Nijgh en van Ditmar, 2000) Ch. 5

Transcript of Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR...

Page 1: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 1TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Package Inductance and RLC Circuit Analysis

FACTOR

Jawel, ik sta bekend als fenomeenTenminste onder mathematiciEn in wiskundig aangelegde kringen

Mijn naam, als u het weten wilt is iIk zit in allerlei berekeningenEn aan het einde hef ik mij weer op

U vraagt niet, denk ik, om verhandelingenMaar wenst dat ik mezelf nu eens ontpopWelnu - ik ben de wortel uit -1

Ik functioneer, ofschoon ik niet bestaDenk daar maar eens langdurig over na

(Uit: Drs. P en Marjolein Kool, Wis- en natuurlyriek met chemischsupplement, Amsterdam: Nijgh en van Ditmar, 2000)

Ch. 5

Page 2: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 2TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

New Today

Inductive effects, for switching (transient) behavior

Application of Complex Numbers

Relation between complex exponentials and harmonic(sinusoidal) functions

Damping factor, damped natural frequency, undamped natural frequency

Oscillatory behavior (opslingering)

Concepts of overdamped, critically damped, underdamped and undamped responses

Page 3: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 3TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Package ParasiticsNeed transition from submicron on-chip world to PCB (printed Circuit Board) worldRelatively large ‘package parasitics’Parasitics = unwanted but non-avoidable electrical effects Inductance of package must often be accounted forOn-chip interconnectinductance also becoming important

Bull Microprocessor Chip

Page 4: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 4TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Bondwires

Bondwires: connection from silicon chip to package pins

Page 5: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 5TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Model for Invertor with Inductance

Cin

Rnvin

Rp

5 V

Lground

Lpower

voutCinCin

Rnvin

Rp

5 V

Lground

Lpower

vout

Many interesting properties

Can we determine speedand other properties of such systems?

Addition of L to R and C might result in oscillating behavior!

Need complex numbers to deal with it.

Page 6: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 6TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Inductance (Inductantie)

L

i +

–v

dtdiLv =

C

i +

–v

dtdvCi =

L and C exhibit dual roles of iand vCauses many interesting and useful properties of electrical circuits and circuit analysisWe will apply this right now!

Page 7: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 7TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

DC Equivalent Circuit

DC value of capacitor current:

00 ==⇒= idt

dvcdt

dv cc

DC model for C is open circuit

00 ==⇒= vdtdiL

dtdi LL

DC value of inductor voltage:

DC model for L is

Dual!

short circuit

Page 8: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 8TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

First Order RL Circuits

0=++−dtdiLRiV L

Ls

( ) ( )( )

RVe

RVtiti stt

LR

sLL +⎟⎠⎞

⎜⎝⎛ −=

−− 00

VA

+

-Vs

t=t0

R

+

-L vL(t)

+

-

iL(t)

VA

+

-Vs

t=t0

R

+

-L vL(t)

+

-

iL(t)

1. Use KVL for circuit with t > t0 :

sLL V

Li

LR

dtdi 1

+−=

2. Rearrange:

3. Compare to RC case:

scc V

RCv

RCdtdv 11

+−=

5. Write down result (by inspection):

F(t) = (IV – FV) e-(t-t0)/τ + FV4. Remember general formula:

RC: τ = RC

RL: τ = L/R

F(t): waveform

IV: initial value

FV: final value

Page 9: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 9TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Cascaded Invertors with Inductance

Rn

Rp

5V

Lground

Lpower

CinRn

Rp

5V

Lground

Lpower

Cin

Rint

+

-

Lpower

Package and power supply

5V

Driving gateRp

Rn

Lground

Cint

Interconnect Load gate

Cin

Page 10: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 10TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Simplify Circuit

Rint

+

-

Lpower

Package and power supply

5V

Driving gateRp

Rn

Lground

Cint

Interconnect Load gate

Cin

+

-Vs C

Rt=0t=0

iL(t)

L vc(t)Prototypical RLC series circuit

Principal subject of this chapter

Page 11: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 11TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

RLC Circuit Differential Equations

Lc i

dtdvC =

KCL @ vc

+

-Vs C

Rt=0t=0

iL(t)

L vc(t)+ vL -+ vR –

+ vC

iC=iLR L

0=+++− CL

Ls vdtdiLRiV

KVL around loop0=+++− CLRs vvvV

State variables: vC, iL

Note: KVL for inductor voltage is dual from KCL for capacitor current

Page 12: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 12TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

RLC Circuit Differential Equations

Lc i

dtdvC =

KCL & KVL

0=+++− CL

Ls vdtdiLRiV

+

-Vs C

Rt=0t=0

iL(t)

L vc(t)

⎥⎥⎦

⎢⎢⎣

⎡+⎥

⎤⎢⎣

⎥⎥⎥

⎢⎢⎢

−−=

⎥⎥⎥

⎢⎢⎢

LV

iv

LR

L

C

dtdidt

dv

sL

CL

C 01

10

Lc i

Cdtdv 1

=

LVi

LRv

Ldtdi s

LCL +−−=

1

Rewrite

Page 13: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 13TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

⎥⎥⎦

⎢⎢⎣

⎡+⎥

⎤⎢⎣

⎥⎥⎥

⎢⎢⎢

−−=

⎥⎥⎥

⎢⎢⎢

LV

iv

LR

L

C

dtdidt

dv

sL

CL

C 01

10

Two coupled first order D.E.’s

Next: find vc(t) and iL(t)Satisfy D.E.’sSatisfy initial conditions vc(0), iL(0)

Procedure:Assume general solutionDetermine steady state solutionsFind natural frequencies (s1, s2)Determine coefficients of trans. part

General method identical to RC and RL case

General Procedure

Page 14: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 14TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Assume General Solution( ) ( ) C

tstsCCC VeVeVVtvtv t ++=+= 21 21

( ) ( ) Ltsts

LtLL IeIeIItiti ++=+= 2211

+

-Vin C

Rt=0

iL(t)

L vc(t)

Determine Steady State Solutions

VC= vc(∞) =

IL= iL(∞) =

Vin

0

Page 15: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 15TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Find Natural Frequencies

⎥⎥⎦

⎢⎢⎣

⎡+⎥

⎤⎢⎣

⎥⎥⎥

⎢⎢⎢

−−=

⎥⎥⎥

⎢⎢⎢

LV

iv

LR

L

C

dtdidt

dv

sL

CL

C 01

100.

01

1=

⎥⎥⎥

⎢⎢⎢

+

LRs

L

Cs

det1.

011=⎟

⎠⎞

⎜⎝⎛ −

−⎟⎠⎞

⎜⎝⎛ +

Cx

LLRss2.

012 =++LC

sLRs3. Characteristic Equation of series

RLC circuit

Page 16: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 16TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

D - discriminant

Sign of Discriminant012 =++

LCs

LRs

02 =++ cbsas

aacbbs

242 −±−

=⇒

Note: c≠C

LCc

LRba 1;;1 ===

Will be explained shortly

undampedRe(si) = 0R = 04underdampeds1 and s2∈ CD < 03critically dampeds1 = s2D = 02overdampedlike RC and RLD > 01

Four Cases depending on sign of discriminant

Next: Study cases 1, 2, 3 and 4

LCLR

LRs 1

22

2−⎟

⎠⎞

⎜⎝⎛±

−=⇒

Page 17: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 17TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Damping

Many systems vibrate/oscillate/cycle(more or less) repetitive behaviormechanical systems, electrical, economic, …

Damping is the mechanism that tries to suppress oscillationsand force the system into stable steady-state

‘schokdempers’ – of a car‘dempers’ – of Erasmus bridge Bridges can oscillate!

Tacoma Narrows Bridge, November 7, 1940

Overdamped – critically damped – underdamped – undampedWhat does it mean?

Page 18: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 18TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Tacoma Narrows Bridge

Page 19: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 19TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Tacoma Narrows Bridge

Page 20: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 20TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

D>0

Case 1: Overdamped Series RLC Response

LCLR

LRs 1

22

21 −⎟

⎠⎞

⎜⎝⎛+−=

LCLR

LRs 1

22

22 −⎟

⎠⎞

⎜⎝⎛−−=

No extra difficulties compared to RC or RL case

LCLR

LRs 1

22

2−⎟

⎠⎞

⎜⎝⎛±

−=

Strictly decaying transient solution

10/9/2004 07 inductance 19TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Solve for s

⎥⎥⎦

⎢⎢⎣

⎡+⎥

⎤⎢⎣

⎥⎥⎥

⎢⎢⎢

−−=

⎥⎥⎥

⎢⎢⎢

LV

iv

LR

L

C

dtdidt

dv

sL

CL

C 01

100.

01

1=

⎥⎥⎥

⎢⎢⎢

+

LRs

L

Cs

det1.

011 =⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ +

Cx

LLRss2.

012 =++LC

sLRs3. Characteristic Equation of series

RLC circuit

Two real natural frequencies

CLR

LCLR 2012

2>⇔>−⎟

⎠⎞

⎜⎝⎛

Now consider case of

§5.8.1

Page 21: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 21TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Case 2: Critically Damped Response

Houston, we have a problem…

α−≡−==L

Rss221

α: damping factor

Then, V1 and V2 are not independent because they refer to same exponential term. Actually, V1+V2 can be replaced by V’.

Remember: for a second-order system, we actually need two independent exponential solutions and two constants to adjust for two initial conditions.

Similarly for iC(t)

Because s1 = s2 = – α we would have( ) ( ) C

tC

ttC VeVVVeVeVtv ++=++= −−− ααα

2121

CLR

LCLR 2012

2=⇔>=⎟

⎠⎞

⎜⎝⎛

Now consider case of

§5.8.2

D=0LCL

RLRs 1

22

2−⎟

⎠⎞

⎜⎝⎛±

−=

Page 22: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 22TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Second Independent Solution to D.E.

tt eedtd αα α −− −=

=− ttedtd α

vc(t) = V1e-αt + V2 t e-αt + Vc iL(t) = I1e-αt + I2 t e-αt + ILModified solutions:

Product rule

dtduv

dtdvuvu

dtd

+=⋅

Derivative (almost) proportional to function

Modified solution compared to RC or RL caseOvershoot in transient solution

D = 0 ⇒ 1 natural frequency

More formal derivation in D.E. course

tttt etetdtdee

dtdt αααα α −−−− +−=+

Page 23: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 23TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Critically Damped Waveform

vc(t) = V1e-αt + V2 t e-αt + Vc

V1 -5

V2 10

Vc 5

α 1

012345678

0 1 2 3 4 5

Page 24: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 24TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Houston, we have another problem…

Case 3: Underdamped Response

We need square-root of negative number…

Solution: complex numbers

FACTOR

Jawel, ik sta bekend als fenomeenTenminste onder mathematiciEn in wiskundig aangelegde kringen

Mijn naam, als u het weten wilt is iIk zit in allerlei berekeningenEn aan het einde hef ik mij weer op

U vraagt niet, denk ik, om verhandelingenMaar wenst dat ik mezelf nu eens ontpopWelnu - ik ben de wortel uit -1

Ik functioneer, ofschoon ik niet bestaDenk daar maar eens langdurig over na

(Uit: Drs. P en Marjolein Kool, Wis- en natuurlyriek met chemischsupplement, Amsterdam: Nijgh en van Ditmar, 2000)

CLR

LCLR 2012

2<⇔><⎟

⎠⎞

⎜⎝⎛

Now consider case ofD<0

LCLR

LRs 1

22

2−⎟

⎠⎞

⎜⎝⎛±

−=

See TIO ‘Complexe Rekenwijze’

Page 25: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 25TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Imaginary Numbers

We need square-root of negative number…

We offer number of which the square is -1

Conventionally denoted as i: i 2 = -1

In EE, i is reserved for current, use j instead (j2 = -1)

xjxjxjx ===− 22Hence:

Now, we have real numbers, e.g. 1, 2, 1.34132, …

And imaginary numbers, e.g. 3j

Their combination is called a complex number

Page 26: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 26TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Complex NumbersComplex number is sum of real and imaginary number,

Addition keeps real and imaginary parts separate:

(1+3j) + (1+3j) = 1+1+j(3+3) = 2+6j

Multiplication uses j2 = -1:(1+3j) x (1+3j) = 1 + 3j + 3j + 9j2 = -8+6j

All normal rules for addition, subtraction, multiplication, division, but real and imaginary parts are kept separate, except for j2, which becomes -1

e.g. 1+3j

Page 27: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 27TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Complex Plane and Polar Form

z = a + bjbjb = Im(z)

a = Re(z)real axis

imaginary axis

r = |z|ϕ = arg(z)

z=a+bj z = r(cosϕ + jsinϕ) = r·ejϕ

ejϕ = cosϕ + jsinϕ

|z1z2| = |z1||z2| arg(z1z2) = arg(z1) + arg(z2) (mod 2π)Multiplication often easier using polar co-ordinates:

Eulers identity will be important

Page 28: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 28TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

LRs2

)Re( =≡α Damping factor

20

2 ωαα −±−=s

α

Underdamped Response

20ω

LC1

0 ≡ω Undamped natural frequency

LCLR

LRs 1

22

2−⎟

⎠⎞

⎜⎝⎛±−=

Would-be natural frequency when R=0

D<0LCL

RLRs 1

22

2−⎟

⎠⎞

⎜⎝⎛±

−=

CLR

LCLR 2012

2<⇔><⎟

⎠⎞

⎜⎝⎛

Now consider case of

§5.8.3

Page 29: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 29TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Damped natural frequency220)Im( α−ω=≡ω sd

2dω

22 ωαα −±−=s

< 0

( )( )2201 αωα −−±−=

220

2 αωα −±−= j 220 αωα −±−= j

> 0

Complex Natural Frequencies

220 αωα −±−= j

djs ωα +−=1 djs ωα −−=2

Main result: two complex natural frequencies

21 ss =

Page 30: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 30TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Complex Arithmetic for Underdamped Response

s1 and s2 two complex natural frequenciesAlso need complex coefficients in linear combination of both independent solutions for vc(t) and iL(t)Result: Familiar general solution, but with complex arithmetic

( ) Ctsts

C Veetv +Φ+Φ= 21 21

( ) Ltsts

L Ieeti +Γ+Γ= 21 21

complexare,, iiis ΓΦ

But we can’t have complex voltages and currents, can’t we?

Page 31: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 31TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

( ) Ctsts

C Veetv +Φ+Φ= 2211

( ) Ltsts

L Ieeti +Γ+Γ= 2211

Ltdjttdjt Iee +Γ+Γ= −−+− ωαωα

21

( ) ( )( ) Lddddt Itjttjte +−Γ++Γ= − ωωωωα sincossincos 21

( ) ( )( ) Lddt Itjte +Γ−Γ+Γ+Γ= − ωωα sincos 2121

Apply Euler’s Identity

Dilemma can be solved by exploiting Euler’s Identity

ejϕ = cosϕ + jsinϕ

Similar for vC(t)

djs ωα +−=1

djs ωα −−=2

Page 32: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 32TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

( ) ( )( ) Lddt

L Itjteti +Γ−Γ+Γ+Γ= − ωωα sincos)( 2121

Looks complicated complexBut physical signals are simply realSolution: Γ1 and Γ2 can not be chosen independentlyIm(Γ1 + Γ2 ) = 0 Im( j (Γ1 - Γ2 )) = 0

Γ1 and Γ2 must be complex conjugates:

Re(Γ1) = Re(Γ2) and Im(Γ1) = - Im(Γ2)

Γ1 = Γ2

Real or Complex?

Like s1 and s2

Then: Γ1 + Γ2 = 2a and j(Γ1 – Γ2) = 2b

Let: Γ1 = a - jb and Γ2 = a + jb

Clearly Real!Finally, let: I1 = 2a and I2 = 2b

( ) Lddt

L ItItIeti ++=⇒ − ωωα sincos)( 21 Similar for vC(See book)

Page 33: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 33TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

12 Γ=+=Γ jba

If we wouldn’t know that Γ1 and Γ2 are complex conjugates, we can ‘discover’ it as follows:

Γ1 and Γ2 are Complex Conjugates

jba −=Γ1 jdc −=Γ2

)(21 dbjca +−+=Γ+Γ

bd −=⇔=Γ+Γ 0)Im( 21

)(21 dbjca −−−=Γ−Γ

( ) caca =⇔=−=Γ−Γ 0Re 21

Page 34: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 34TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Underdamped Response

( ) ( ) Cddt

C VtsinVtcosVetv ++= − ωωα21

( ) ( ) Lddt

L ItsinItcosIeti ++= − ωωα21

Exponentially damped

Sinusoidal Waveform

Steady State

Vi and Ii again follow from initial conditionand derivatives at t=0 (our method) (or substitution of above general solution in original DE (book))

Damped natural frequencyDamping factor

Page 35: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 35TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Underdamped Waveforms

-6

-4

-2

0

2

4

6

8

0 1 2 3 4 5

( ) ( ) Cddt

C VtsinVtcosVetv ++= − ωωα21

αωdV1V2Vc

αωdV1V2Vc

0.52

-505

0.52

-505

Page 36: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 36TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Case 4: Undamped Series RLC Response

01 ωj

LCs ±=−±=

( ) LL ItsinItcosIti ++= 0201 ωω

When R = 0 ⇒ α = 0, ωd=ω0

Oscillating behavior, without damping, oscillation continues forever

Perpetuum Mobile?

LCLR

LRs 1

22

2−±−=

LR2

−≡α LC1

0 ≡ω

( ) CC VtsinVtcosVtv ++= 0201 ωω

( ) ( ) Cddt

C VtsinVtcosVetv ++= − ωωα21

α = 0, ωd=ω0

§5.8.4

Page 37: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 37TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

The natural frequencies can be shown in complex plane

Im

Re

Overdamped

Im

Re

Underdamped

Im

Re

UndampedCritically Damped

Im

Re(2x)

Review of Natural Frequencies

Page 38: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 38TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Digital System Switching Speed

§5.9

Compare switching speed with and without supply line inductance

Completely worked example

But also see other examples in book

Fully worked example 5.7

Page 39: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 39TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Digital System Switching Speed

Determine high-to-low waveform here

Assume initially high for long time

Compare RC and RLC case

Rout = 100Ω

Cin = 1pFL = 10nH

Vs = 5V

Page 40: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 40TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Compare RC and RLC Case

Without L

t=0 1pF

100Ω v ?With L

1pFt=0

100Ω v ? 10nH

Rout = 100Ω

Cin = 1pFL = 10nH

Vs = 5V

Page 41: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 41TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Without L

τ =

v(0) =

v(∞) =

v(t) = FV + (IV – FV)e-t/τ

te10105 −=

1pFt=0

100Ω

5

0

100Ω x 1pF = 0.1 nS

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.20 0.40 0.60 0.80 1.00

Page 42: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 42TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Strategy: Solving Second Order RLC circuits

1. Write down differential equations (determine state vars)2. Write down characteristic equation and solve for s3. Assume general solution, depending on si, as in scheme

below (A, B and C can be current or voltage or …)

4. Solve for steady state solutions, and constants (use I.C. and time derivatives at t=0)

5. Check solution!

CBeAe tsts ++ 21Overdamped: two different, real s

CBteAe tt ++ −− ααCritically damped:s1 = s2 = –α

( ) ( )( ) CtcosBtcosAe ddt ++− ωωα

Underdamped: s1 and s2 complex conjugate (undamped when α = 0)

( ) ( )sImsRe d == ωα

CBeAe tsts ++ 21Overdamped: two different, real s

CBteAe tt ++ −− ααCritically damped:s1 = s2 = –α

( ) ( )( ) CtcosBtcosAe ddt ++− ωωα

Underdamped: s1 and s2 complex conjugate (undamped when α = 0)

( ) ( )sImsRe d == ωα

Page 43: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 43TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Determine (Complex) Natural Frequencies

t=0 1pF

100Ω vC ?10nH

iL+

-

LCLR

LRs 1

22

2−⎟

⎠⎞

⎜⎝⎛±−=

⎥⎥⎦

⎢⎢⎣

⎡+⎥

⎤⎢⎣

⎥⎥⎥

⎢⎢⎢

−−=

⎥⎥⎥

⎢⎢⎢

LV

iv

LR

L

C

dtdidt

dv

sL

CL

C 01

10

20

2

99 101

1020100

1020100

−−− −⎟⎟⎠

⎞⎜⎜⎝

×±

×−=

j99 1066.8105 ×±×−= Underdamped !

Page 44: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 44TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

α = 5 Damping factor

Write Assumed Solution

js 99 1066.8105 ×±×−= Unit: 1/s

js 66.85 ±−= Unit: 1/ns

Change seconds into nano seconds (for convenience)

( ) ( ) ( )( ) Cddt

C VtsinVtcosVetv ++= − ωωα21

( ) ( ) ( )( ) Lddt

L ItsinItcosIeti ++= − ωωα21

ωd = 8.66 Damped Natural Frequency

Page 45: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 45TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Steady State and Initial Values

vC(∞) = 0 iL(∞) = 0

vC(0) = iL(0) =5 0

t=0 1pF

100Ω vC ?10nH

iL+

-

Page 46: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 46TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Determine Constants

( ) ( ) ( )( ) Cddt

C VtsinVtcosVetv ++= − ωωα21

( ) ( ) ( )( ) Lddt

L ItsinItcosIeti ++= − ωωα21

Next: Determine constants (V1, V2, I1, I2)From initial conditions

From derivatives at t=0

(book uses substitution in D.E.)

VC=vC(∞) =0 IL=iL(∞) =0

First: steady-state values

Page 47: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 47TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Use Initial Condition(and Steady-State Value)

( ) ( ) ( )( ) Cddt

C VtsinVtcosVetv +ω+ω= α−21

( ) ( ) ( )( ) CC VsinVcosVev ++== 0050 210

( ) 121 0011 VVV =+⋅+⋅= ⇒ V1 = 5

( ) 121 0011 III =+⋅+⋅= ⇒ I1 = 0

( ) ( ) ( )( )tsinVtcosetv ddt

C ω+ω= α−25

( ) ( )tIeti dt

L ω= α− sin2

3.

( ) ( ) ( )( ) Lddt

L ItsinItcosIeti +ω+ω= α−21

( ) ( ) ( )( ) LL IsinIcosIei ++== 0000 210

2.

1.

Page 48: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 48TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Analysis Tools Reminder

Combined: ( ) ( ) ( )tetetedtd ttt ωα−ωω=ω α−α−α− sincossin3.

Chain Rule: dxdu

dudy

dxdy

⋅=

( ) ( ) ( )xdxdusin

dudxsin

dxd ωω ⋅=

( ) xxu ⋅= ω

( ) ω⋅= ucos( )xcos ωω=

2.

Example:

AnalysisProduct Rule: dtduv

dtdvuvu

dtd

+=⋅

Example: tsinedtd tα− tt e

dtdtsintsin

dtde αα −− +=

tsinetcose tt αα α −− −=

1.

Review!Determine ( )te

dtd t ωα− cos

Review!Determine ( )te

dtd t ωα− cos

Page 49: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 49TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Use Time-Derivative at t=0 for Assumed Solution

( ) ( ) ( )( )tVtetvdtd

ddt

C ω+ωα−= α− sincos5 2

( ) ( )( )tVte ddddt ωω+ωω−+ α− cossin5 2

( ) ( ) ( )( )tsinVtcosetv ddt

C ω+ω= α−25

( ) dt

C Vtvdtd

ω+α−==

20

5Hence:

cos(0) = 1 sin(0) = 0 e0 = 1Note:

cos(0) = 1 sin(0) = 0 e0 = 1Note:Use:

Page 50: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 50TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

( ) 0)0(1)0(00

++×==

LCt

C iC

vtvdtd

⎥⎥⎦

⎢⎢⎣

⎡+⎥

⎤⎢⎣

⎥⎥⎥

⎢⎢⎢

−−=

⎥⎥⎥

⎢⎢⎢

LV

iv

LR

L

C

dtdidt

dv

sL

CL

C 01

10

Use Time-Derivative at t=0 for D.E.

Known!

00150 =×+×=C

( ) 00

==t

C tvdtd

Page 51: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 51TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

( ) ( ) ( )( )t.sin.t.cosetv tC 66889266855 += −

Final result (t in ns):

Combine Time-Derivative Info at t=0

( ) dt

C Vtvdtd

ω+α−==

20

5

( ) 00

==t

C tvdtd

α=ω 52 dV

89.22 =V

( )t.sine.i tL 66805770 5−−=

Use the preceding procedure for iLUse the preceding procedure for i L

α = 5ωd = 8.66earlier:

Page 52: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 52TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

( ) ( ) ( )( )t.sin.t.cosetv tC 66889266855 += −

( )t.sine.i tL 66805770 5−−=

Final result (t in ns):

Inverter Pair Waveforms

t=0 1pF

100Ω v ?

t=0 1pF

100Ω v ?

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.20 0.40 0.60 0.80 1.00

vC(t)

when L = 010nH

Page 53: Package Inductance and RLC Circuit Analysis · Package Inductance and RLC Circuit Analysis FACTOR Jawel, ik sta bekend als fenomeen Tenminste onder mathematici En in wiskundig aangelegde

10/27/2004 07 inductance 53TUE/EE 5DD17 netwerk analyse 04/05 - © NvdM

Summary: Solving Second Order RLC circuits

1. Write down differential equations (determine state vars)2. Write down characteristic equation and solve for s3. Assume general solution, depending on si, as in scheme

below (A, B and C can be current or voltage or …)

4. Solve for steady state solutions, and constants (use I.C. and time derivatives at t=0)

5. Check solution!

CBeAe tsts ++ 21Overdamped: two different, real s

CBteAe tt ++ −− ααCritically damped:s1 = s2 = –α

( ) ( )( ) CtcosBtcosAe ddt ++− ωωα

Underdamped: s1 and s2 complex conjugate (undamped when α = 0)

( ) ( )sImsRe d == ωα

CBeAe tsts ++ 21Overdamped: two different, real s

CBteAe tt ++ −− ααCritically damped:s1 = s2 = –α

( ) ( )( ) CtcosBtcosAe ddt ++− ωωα

Underdamped: s1 and s2 complex conjugate (undamped when α = 0)

( ) ( )sImsRe d == ωα