Network Graphs and Tellegen’s Theorem

Network Graphs and Tellegen’s Theorem The concepts of a graph Cut sets and Kirchhoff’s current laws Loops and Kirchhoff’s voltage laws Tellegen’s Theorem

The concepts of a graph

The analysis of a complex circuit can be perform systematicallyUsing graph theories.

Graph consists of nodes and branches connected to form a circuit.

Network P Graph

Network P Graph

MFig. 1


14

32

Isolate nodeSelf loop Non plannar

Special graphs

Fig. 2

The concepts of a graphSubgraph

G1 is a subgraph of G if every node of G1 is the node of G andevery branch of G1 is the branch of G

1 4

32G

1

32

G1

1 4

32

G2

1

2

G3

1 4

32

G4

3

G5

Fig. 3

The concepts of a graphAssociated reference directions

The kth branch voltage and kth branch current is assigned as reference directions as shown in fig. 4

Fig. 4

Graphs with assigned reference direction to all branches are called oriented graphs.

kj

kv+

-

kjkv

+

-


Fig. 5 Oriented graph

1 2 3

45

1

23

4

6

Branch 4 is incident with node 2 and node 3

Branch 4 leaves node 3 and enter node 2

The concepts of a graphIncident matrix

The node-to-branch incident matrix Aa is a rectangular matrix of nt rowsand b columns whose element aik defined by

0

1

1

ika

If branch k leaves node i

If branch k enters node i

If branch k is not incident with node i

The concepts of a graphFor the graph of Fig.5 the incident matrix Aa is

100110

110000

011000

001101

000011

Aa

Cutset and Kirchhoff’s current law

If a connected graph were to partition the nodes into two set by a closed gussian surface , those branches are cut set and KCL applied to the cutset

Guassian surface

Cutset branches

Fig. 6 Cutset

Cutset and Kirchhoff’s current lawA cutset is a set of branches that the removal of these branches causes two separated parts but any one of these branches makes the graphconnected.

An unconnected graph must have at least two separate part.

Connected Graph Unconnected GraphFig. 7


removal Connected Graph

Unconnected Graph

removal

Fig. 8


1

3

24

1

652

1

3

8

7

Cutset 1,2,3

3

2

1

Cutset 1,2,3

(a)(b)

Fig. 9

Cut set

1

2

34

5

6

7 89

1011

1213

14

15

1617

18

19

2021

22

2324

2526

27

28

29

(c)

Fig. 9

Cutset and Kirchhoff’s current law For any lumped network , for any of its cut sets,

and at any time, the algebraic sum of all branch currents traversing the cut-set branches is zero.

From Fig. 9 (a)

0)()()( 321 tjtjtj for all t

And from Fig. 9 (b)

1 2 3( ) ( ) ( ) 0j t j t j t for all t

Cutset and Kirchhoff’s current lawCut sets should be selected such that they are linearly independent.

9

6

8

10

5

72 3

1 4

III

Cut sets I,II and III are linearly dependent

Fig. 10

III


Cut set I 1 2 3 4 5( ) ( ) ( ) ( ) ( ) 0j t j t j t j t j t

Cut set II

1 2 3 8 10( ) ( ) ( ) ( ) ( ) 0j t j t j t j t j t

4 5 8 10( ) ( ) ( ) ( ) 0j t j t j t j t

Cut set III

KCLcut set III = KCLcut set I + KCLcut set II

Loops and Kirchhoff’s voltage lawsA Loop L is a subgraph having closed path that posses the following

properties: The subgraph is connected Precisely two branches of L are incident with each node

a loopNot a loop Not a loop

Fig. 11

Loops and Kirchhoff’s voltage laws

12 3 4

1 2

3

4

5

I II

1

2

3

4

III

IV1

2

3

4 56

78

9

10

1112

V

Cases I,II,III and IV violate the loop Case V is a loop

Fig. 12

Loops and Kirchhoff’s voltage laws For any lumped network , for any of its loop,

and at any time, the algebraic sum of all branch voltages around the loop is zero.

Example 1

Fig. 13

Write the KVL for the loop shown in Fig 13

2

8

8

3

5

10

69

4

1

7

0)()()()()( 48752 tvtvtvtvtv

for all t

KVL

Tellegen’s Theorem Tellegen’s Theorem is a general network theorem It is valid for any lump network

For a lumped network whose element assigned by associate referencedirection for branch voltage and branch current kv kjThe product is the power delivered at time by the network to theelement

k kv j tk

If all branch voltages and branch currents satisfy KVL and KCL then

01

b

kkk jv b = number of branch

Tellegen’s Theorem

Suppose that and is another sets of branch voltages and branch currents and if and satisfy KVL and KCL

bvvv ˆ,......ˆ,ˆ 21 1 2ˆ ˆ ˆ, ,...... bj j jkv̂ ˆ

kj

Then

1

ˆˆ 0b

k kk

v j

and

1

ˆ 0b

k kk

v j

1

ˆ 0b

k kk

v j

1

0b

k kk

v j

Tellegen’s Theorem

Applications

Tellegen’s Theorem implies the law of energy conservation.

“The sum of power delivered by the independent sources to the network is equal to the sum of the power absorbed by all branches of the network”.

01

b

kkk jvSince

Conservation of energy Conservation of complex power The real part and phase of driving point

impedance Driving point impedance

Applications

Conservation of Energy

1

( ) ( ) 0b

k kk

v t j t

“The sum of power delivered by the independent sources to the network is equal to the sum of the power absorbed by all branches of the network”.

For all t

Conservation of Energy

Resistor

Capacitor

Inductor

21

2 k kC v

2k kR j For kth resistor

21

2 k kL i

For kth capacitor

For kth inductor

Conservation of Complex Power

1

10

2

b

k kk

V J

kV = Branch Voltage Phasor

kJ = Branch Current Phasor

kJ = Branch Current Phasor Conjugate

1V

2V

3V

2J

1J

4V

3J

4J

1 12

1 1

2 2

b

k kk

V J V J

Conservation of Complex Power

1V

1J

2V

2J

kJ kV

N Lineartime-invariantRLC Network

The real part and phase of driving point impedance

1J 1V

kV

kJ

inZLinear time-

invariant RLCone-port

1 1 ( )inV J Z j

From Tellegen’s theorem, and let P = complex power delivered to the one-port by the source

2

1 1 1

1 1( )

2 2 inP V J Z j J

2

2

1 1( )

2 2

b

k k k kk

V J Z j J

Taking the real part

2

1

1Re[ ( )]

2av inP Z j J

2

2

1Re[ ( )]

2

b

k kk

Z j J

All impedances are calculated at the same angularfrequency i.e. the source angular frequency

Driving Point Impedance

2

1

1( )

2 inP Z j J

2

2

1( )

2

b

m mk

Z j J

2 2 21 1 1 1

2 2 2i i k k li k l l

R J j L J Jj C

R L C

2 2 2

2

1 1 1 12

2 4 4i i k k li k l l

P R J j L J JC

Exhibiting the real and imaginary part of P

Average power

dissipated

AverageMagnetic Energy Stored

Average Electric Energy Stored

avPM

E 2av M EP P j

2

1

1( )

2 inP Z j J

From

2

1

2( )in

PZ j

J

2av M EP P j

Driving Point Impedance

Given a linear time-invariant RLC network driven by a sinusoidal current source of 1 A peak amplitude and given that the network is in SS,

The driven point impedance seen by the

source has a real part = twice the average power Pav and an imaginary part that is

4times the difference of EM and EE

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Network Graphs and Tellegen’s Theorem