The concepts of a graph Cut sets and Kirchhoff’s current ...
Transcript of The concepts of a graph Cut sets and Kirchhoff’s current ...
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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
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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
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The concepts of a graph
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
32
G1
32
G1
1 4
32
G2
1
2
G3
1 4
32
G4
3
G5
Fig. 3
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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
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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 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
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The concepts of a graph
For 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
Fig. 6 Cutset
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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
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Cutset and Kirchhoff’s current law
removalConnected Graph
Unconnected Graph
removal
Fig. 8
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Cutset and Kirchhoff’s current law
Fig. 9
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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
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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
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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
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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
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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
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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
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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
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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 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
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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,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
=
=∑
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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”.
0
1
=∑=
b
k
kk 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 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
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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
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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
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1V
2V
3V
2J
1J
4V
3J
4J
1 1
2
1 1
2 2
b
k k
k
V J V J=
− =∑
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Conservation of Complex Power
1V1J
2V2J
kJk
V
N Linear
time-invariant
RLC Network
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The real part and phase of driving point
impedance
1J1V
kV
kJ
inZ
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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=
= = ∑
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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
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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
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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
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2
1
1( )
2inP Z j Jω=
From
2
1
2( )
in
PZ j
Jω∴ =
( )2av M E
P P jω= + Ε − Ε
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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 4ω times the difference of EM and EE