Kirchhoff Laws

8/12/2019 Kirchhoff Laws

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Kirchhoff Laws

Pekik Argo Dahono

Kirchhoff Voltage LawFor all lumped connectedcircuits, for all choices of datum

node, for all timest, for all pairs ofnodeskandj

For all lumped connectedcircuits, for all closed node

sequences, for all times t, the

algebraic sum of all node-to-nodevoltages around the chosen

closed node sequence is equal tozero.

jkkj eev =


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Kirchhoff Voltage Law

00

524524

5145342312

=++

=++++

vvvvvvvv

D

22552

4224

45445

4334

3223

2112

15115

eeev

eev

eeev

eev

eev

eev

eeev

==

=

==

=

=

=

==

Kirchhoff Current Law

For all lumped circuits, for all gaussiansurfacesS, for all times t, the algebraic sumof all currents leaving the gaussian surfaceSat timet is equal to zero.

For all lumped circuits, for all times t, thealgebraic sum of the currents leaving anynode is equal to zero.


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Kirchhoff Current Law

00

0

11986543

6541

21

=+++

=+++

=+

iiiiiiiiiii

ii

11i

9i

8i

10i

3i1i

2S

12i

6S

4S

7i

4i

3S

1S

2i 5S

+

5i

6i

Three Important Remarks

KVL and KCL are the two fundamentalpostulates of lumped-circuit theory.

KVL and KCL hold irrespective of the nature ofthe elements constituting the circuit. Hence, wemay say that Kirchhoff laws reflect theinterconnection properties of the circuit.

KVL and KCL always lead to homogeneous

linear algebraic equations with constant realcoefficients, 0, 1, and -1, if written in the fashionas we have discussed.


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From Circuits to Graph

A graph is defined by a set of nodes and a set ofbranches.

If each branch is given an orientation, indicatedby an arrow on the branch, we call the graph isdirected, or simply digraph.

5

11

4

2

2

3

4

5

3 4

5

6 7

The Circuit Graph : Digraph

310

49

48

47

426

425

24

33

212

11

ev

ev

ev

ev

eev

eev

ev

ev

eev

ev

=

=

=

=

=

=

=

=

=

=


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Two-terminal and three-terminal elements

)()()( titvtp =

+

i

v

i1

2 2

i

1

1 2

1i 2i

32v

13v

1i 2i

3i

+ 2+

1

3

3

+

21v

Other representations of three-terminal element

+2i

1i+

2v1v

1i

3i

1i

3i

2i

3i

2i

3i

211 2

3 3

1 2 1 2

33


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Multiterminal elements

=

=

1

1

)()()(n

kkk titvtp

Two-Ports

1

1'

2

2'

1 2

2v

2211

22

11

'

'

iviv

ii

ii

=

=

=


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Multiports

+

+

+

1

'1

2

'2

3

'3

3

2

1

1

'1

Hinged Graphs


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Grounded Two-Ports

Cut Sets

Given a connected digraph, a set ofbranches is called cut set iff

a) The removal of all the branches of thecut set results in an unconnecteddigraph.

b) The removal of all but any one branchleaves the digraph connected.


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Cut Sets

i

45 i6

S2

S1

5i

S

i

3

7i1

i3

i4

3

2

2

S1

1

Examples of Cut set


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Cut Set Law

For all lumped circuits, for all time t, thealgebraic sum of the currents associatedwith any cut set is equal to zero.

2 3

5 6i i

iii

i i

4 5

1 2 3

6 7

0321 =++ iii

Independent KCL Equations

1

2 3

4

5

63

4

1 2

0

0

0

0

654

532

431

621

=+

=++

=+

=+

iii

iii

iii

iii


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Independent KCL Equations

( )

( )

( )

( )

=

0

0

0

0

111000

010110

001101

100011

4

3

2

1

6

5

4

3

2

1

i

i

i

i

i

i

amatrix Aincidencethecalledismatrix4x6The

Incidence Matrix

+

=

(i)nodetouchnotdoeskbranchif0

(i)nodeenterskbranchif1-

(i)nodeleaveskbranchif1

ika

0iAa =

If in Aa, the incidence matrix of the connected digraph, we delete the rowCorresponding to the datum node, we obtain the reduced incidence matrix A,

which is of dimension (n-1)b. The corresponding KCL equation read


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Independence Property of KCL Equations

For any connected digraph withnnodes, theKCL equations for any n-1 of these nodesform a set of n-1 linearly independentequations.

For reduced incidence matrix :

0Ai =

Independent KVL Equations

If node (4) is used as the reference (datumnode), then

Mev

or

=

=

=

=

+=

=

=

16

35

24

323

312

211

ev

ev

ev

eev

eev

eev


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Independent KVL Equations

The elements of the matrix are

It can be proved that M=AT

Thus, v=ATe

+

=

(i)nodetouchnotdoeskbranchif0

(i)nodeenterskbranchif1-

(i)nodeleaveskbranchif1

kim

Remarks Note that, in the digraph, (a) we choose current

reference directions, (b) we choose a datum node anddefine the reduced incidence matrix A, (c) we write KCLas Ai=0, (d) then we use associated reference directionsto find that KVL reads v=Ate. Thus whenever we invokethis last equation, we automatically use associatedreference directions for the branch voltages. We alsoassume the same datum node is used in writing KCLand KVL.

When we deal with digraphs which are not connected,

we could either use the concept of the hinged graph tomake the digraph connected or treat each separate partindependently. In the latter, each separate part will haveits own incidence matrix and datum node.


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Tellegen Theorem

Background

ips.relationshspecificno

havevoltagesandcurrentsthethatnotedbeshouldIt

thatverifytoeasyisIt

KVL.obeyvoltagesandKCLobeycurrentsthethatNote

hence

Let

hence

Let

=

=

===

===

===

===

6

1

321

654

654

321

0

112

654

413

321

k

kkiv

vvv

vvv

iii

iii


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Tellegen Theorem

( )

( )

( ) ( ) 0(Ai)eiAeieAiv

then

eAvand0AiSince

:Proof

0ivlyequivalentor

thenKVL,satisfyingvoltagesbranchof

setanybevandKCLsatisfying

currentsbranchofsetanybeiLet

TTTTTTT

T

T

====

==

==

=

=

=

b

k

kk

T

b

T

b

iv

vvv

iii

1

21

21

0

,,,

,,,

L

L

Remarks on Tellegen Theorem

The only requirement of Tellegen theoremis that the voltages and currents mustobey KVL and KCL, respectively.

No other assumptions are used. Thus,

vTi=0 vTi=0 vTi=0 vTi=0

Tellegen theorem depicts only theinterconnection properties of the circuit orthe topology of the digraph.


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Tellegen Theorem and Conservation of Energy

Tellegen theorem is much broader thanconservation energy theory because no timerestriction is used.

( )

0)()(0)()(

0)(

)(0)(

)(

0)()(0)()(

)0)

2

1

12

1

1

1

12

1

21

1221

112

==

==

==

==

=

=

==

==

tidt

tdvtidt

tdv

dt

tditv

dt

tditv

titvtitv

ttt

k

k

kk

k

k

b

k

kk

b

k

kk

kkkk

and

and

and

Thus,

eAv(andLet Ai(

b

1k

b

1k

T

The Relation Between Kirchhoff Lawand Tellegen Theorem

If, for all v satisfying KVL, vTi=0 then isatisfies KCL.

If, for all i satisfying KCL, vTi=0 then vsatisfies KVL.


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Example

=

=

=++

=++

N

k

kRkRk

N

kRkRk

RvvRvviv

iviviv

122211

12211

0//

0

=

=

=++

=++

N

k

kRkRk

N

kRkRk

RvvRvviv

iviviv

122211

12211

0//

0

22221

11 5.08.4// vRvvivRvv

N

k

kRkRk ===

22221

11 6// vRvvivRvv

N

k

kRkRk ===

volts4.2

5.08.46

2

22

=

=

v

vv

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Kirchhoff Laws

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