Network Graph

Network Graphs and Tellegens Theorem

The concepts of a graph Cut sets and Kirchhoffs current laws Loops and Kirchhoffs voltage laws Tellegens 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 Graph

M

Fig. 1


Special graphs

Fig. 2

The concepts of a graphSubgraphG1 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

32

G1

32

G1

1 4

32

G2

1

2

G3

1 4

32

G4

3

G5

Fig. 3


Associated 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.

kjkv k

jkv


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


Incident matrix

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

=

011

ika

If branch k leaves node i

If branch k enters node i

If branch k is not incident with node i


For the graph of Fig.5 the incident matrix Aa is

=

100110110000011000001101000011

Aa

Cutset and Kirchhoffs 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

Fig. 6 Cutset


A cutset is a set of branches that the removal of these branches causestwo 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


removalConnected Graph

Unconnected Graph

removal

Fig. 8


Fig. 9

Cut set

1

2

34

5

6

7 89

1011

12

13

14

15

1617

18

19

2021

22

2324

2526

27

28

29

(c)Fig. 9


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 tAnd from Fig. 9 (b)

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


Cut sets should be selected such that they are linearly independent.

Cut sets I,II and III are linearly dependentFig. 10


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 Kirchhoffs 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

Fig. 11

Loops and Kirchhoffs voltage laws

I II III

IV

V

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

Fig. 12

Loops and Kirchhoffs voltage laws

For any lumped network , for any of its loop, and at anytime, 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

0)()()()()( 48752 =++ tvtvtvtvtvfor all t

KVL

Tellegens Theorem

Tellegens 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

Tellegens Theorem

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

bvvv ,......, 21 1 2

, ,...... bj j jkv

kjThen

1

0

b

k kk

v j=

=and

1

0b

k kk

v j=

=1

0b

k kk

v j=

=

Tellegens Theorem

ApplicationsTellegens 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( ) ( ) 0

b

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 absorbedby all branches of the network.

For all t

Conservation of Energy

Resistor

Capacitor

Inductor

212 k k

C v

2k kR j For kth resistor

212 k k

L i

For kth capacitor

For kth inductor

Conservation of Complex Power

1

1 02

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 12 2

b

k kk

V J V J=

=

Conservation of Complex Power

1V1J

2V2J

kJ kV

N Linear

time-invariant

RLC Network

The real part and phase of driving point

impedance

1J 1V kVkJ

inZ

1 1 ( )inV J Z j= From Tellegens theorem, and let P = complex power delivered to the one-port by the source

21 1 1

1 1 ( )2 2 in

P 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

21

1 Re[ ( )]2av in

P Z j J=

2

2

1 Re[ ( )]2

b

k kk

Z j J=

=

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

Driving Point Impedance

21

1 ( )2 in

P Z j J=

2

2

1 ( )2

b

m m

kZ j J

=

=

2 2 21 1 1 12 2 2i i k k li k l l

R J j L J Jj C = + +

R L C

2 2 22

1 1 1 122 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

AverageElectric Energy Stored

avPM E

( )2av M EP P j= +

21

1 ( )2 in

P Z j J=From

21

2( )inPZ 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 4 times the difference of EM and EE

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Network Graph