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
The concepts of a graph
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
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.
kjkv k
jkv
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
=
011
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
=
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
Cutset and Kirchhoffs 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 Kirchhoffs current law
removalConnected Graph
Unconnected Graph
removal
Fig. 8
Cutset and Kirchhoffs 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 Kirchhoffs 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 tAnd from Fig. 9 (b)
1 2 3( ) ( ) ( ) 0j t j t j t+ = for all t
Cutset and Kirchhoffs current law
Cut sets should be selected such that they are linearly independent.
Cut sets I,II and III are linearly dependentFig. 10
Cutset and Kirchhoffs 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 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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