The concepts of a graph Cut sets and Kirchhoff’s current

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 Graph

M

Fig. 1

The concepts of a graph

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

32

G1

32

G1

1 4

32

G2

1

2

G3

1 4

32

G4

3

G5

Fig. 3

The concepts of a graph

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.

kj

kv kjkv

The concepts of a graph

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 graph

Incident 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 graph

For 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

Fig. 6 Cutset

Cutset and Kirchhoff’s current law

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

Cutset and Kirchhoff’s current law

removalConnected Graph

Unconnected Graph

removal

Fig. 8

Cutset and Kirchhoff’s current law

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

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 law

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

Cut sets I,II and III are linearly dependent

Fig. 10

Cutset and Kirchhoff’s current law

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

Fig. 11

Loops and Kirchhoff’s 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 Kirchhoff’s 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 =++−− 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 the

element k kv j t

k

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

0

1

=∑=

b

k

kk 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,ˆ21 1 2

ˆ ˆ ˆ, ,...... bj j j

kv̂ ˆkj

Then

1

ˆˆ 0

b

k k

k

v j

=

=∑and

1

ˆ 0

b

k k

k

v j

=

=∑1

ˆ 0

b

k k

k

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”.

0

1

=∑=

b

k

kk 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 k

k

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

2k kC v

2

k kR j For kth resistor

21

2k k

L i

For kth capacitor

For kth inductor

Conservation of Complex Power

1

10

2

b

k k

k

V J=

=∑

kV = Branch Voltage Phasor

kJ = Branch Current Phasor

kJ = Branch Current Phasor Conjugate

1V

2V

3V

2J

1J

4V

3J

4J

1 1

2

1 1

2 2

b

k k

k

V J V J=

− =∑

Conservation of Complex Power

1V1J

2V2J

kJk

V

N Linear

time-invariant

RLC Network

The real part and phase of driving point

impedance

1J1V

kV

kJ

inZ

1 1 ( )in

V J Z jω= −

From Tellegen’s theorem, and let P = complex power

delivered to the one-port by the source

2

1 1 1

1 1( )

2 2inP V J Z j Jω= − =

2

2

1 1( )

2 2

b

k k k k

k

V J Z j Jω=

= = ∑

Taking the real part

2

1

1Re[ ( )]

2av in

P Z j Jω=

2

2

1Re[ ( )]

2

b

k k

k

Z j Jω=

= ∑

All impedances are calculated at the same angular

frequency i.e. the source angular frequency

Driving Point Impedance

2

1

1( )

2inP Z j Jω=

2

2

1( )

2

b

m m

k

Z j Jω=

= ∑

2 2 21 1 1 1

2 2 2i i k k l

i k l l

R J j L J Jj C

ωω

= + +∑ ∑ ∑

R L C

2 2 2

2

1 1 1 12

2 4 4i i k k l

i k l l

P R J j L J JC

ωω

= + −

∑ ∑ ∑

Exhibiting the real and imaginary part of P

Average

power

dissipated

Average

Magnetic

Energy

Stored

Average

Electric

Energy

Storedav

PM

ΕE

Ε

( )2av M E

P P jω= + Ε − Ε

2

1

1( )

2inP Z j Jω=

From

2

1

2( )

in

PZ j

Jω∴ =

( )2av M E

P P jω= + Ε − Ε

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 4ω times the difference of EM and EE