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![Page 1: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/1.jpg)
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
![Page 2: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/2.jpg)
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
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The concepts of a graph
14
32
Isolate nodeSelf loop Non plannar
Special graphs
Fig. 2
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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
![Page 5: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/5.jpg)
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
+
-
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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
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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
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The concepts of a graphFor the graph of Fig.5 the incident matrix Aa is
100110
110000
011000
001101
000011
Aa
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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
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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
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Cutset and Kirchhoff’s current law
removal Connected Graph
Unconnected Graph
removal
Fig. 8
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Cutset and Kirchhoff’s current law
1
3
24
1
652
1
3
8
7
Cutset 1,2,3
3
2
1
Cutset 1,2,3
(a)(b)
Fig. 9
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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
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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
![Page 15: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/15.jpg)
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
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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
![Page 17: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/17.jpg)
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
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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
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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
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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
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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
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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
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Conservation of energy Conservation of complex power The real part and phase of driving point
impedance Driving point impedance
Applications
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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
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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
![Page 26: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/26.jpg)
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
![Page 27: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/27.jpg)
1V
2V
3V
2J
1J
4V
3J
4J
1 12
1 1
2 2
b
k kk
V J V J
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Conservation of Complex Power
1V
1J
2V
2J
kJ kV
N Lineartime-invariantRLC Network
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The real part and phase of driving point impedance
1J 1V
kV
kJ
inZLinear time-
invariant RLCone-port
![Page 30: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/30.jpg)
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
![Page 31: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/31.jpg)
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
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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
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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
![Page 34: 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.](https://reader033.fdocuments.us/reader033/viewer/2022061616/56649ed45503460f94be48c5/html5/thumbnails/34.jpg)
2
1
1( )
2 inP Z j J
From
2
1
2( )in
PZ j
J
2av M EP P j
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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