Download Network Graphs and Tellegen`s Theorem

Network Graphs and Tellegen’s Theorem
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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 systematically
Using graph theories.
Graph consists of nodes and branches connected to form
a circuit.
Fig. 1
Network P
M
Network P
Graph
Graph
The concepts of a graph
Special graphs
1
4
3
2
Isolate node
Self loop
Non plannar
Fig. 2
The concepts of a graph
Subgraph
G1 is a subgraph of G if every node of G1 is the node of G and
every branch of G1 is the branch of G
1
4
3
2
G
1
Fig. 3
1
3
2
3
2
4
G2
G1
1
1
4
3
3
2
2
G3
G4
G5
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
+
jk
+
vk
vk
jk
-
Fig. 4
Graphs with assigned reference direction to all branches are called
oriented graphs.
The concepts of a graph
1
1
2
3
2
5
Fig. 5 Oriented graph
Branch 4 is incident with node 2 and node 3
Branch 4 leaves node 3 and enter node 2
3
4
6
4
The concepts of a graph
Incident matrix
The node-to-branch incident matrix Aa is a rectangular matrix of nt rows
and b columns whose element aik defined by
 1

aik    1
 0

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
 1
 1

Aa   0

 0
 0
1
0
0
0
0
0
1
0
1
1
0
1
0
1
0
1
0
0
1
0
0
0
0

1
 1 
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
Cutset branches
Guassian surface
Fig. 6 Cutset
Cutset and Kirchhoff ’s current law
A cutset is a set of branches that the removal of these branches causes
two separated parts but any one of these branches makes the graph
connected.
An unconnected graph must have at least two separate part.
Fig. 7
Connected Graph
Unconnected Graph
Cutset and Kirchhoff ’s current law
Connected Graph
removal
removal
Unconnected Graph
Fig. 8
Cutset and Kirchhoff ’s current law
1
1
2
6
5
2 7
1
4
1
2
8
3
3
3
(a)
Fig. 9
Cutset 1,2,3
Cutset 1,2,3
(b)
1
3
2
4
19
5
6
7
11
9
15
8
16
10
14
18 21
22
24
23
25
26
29 27
17
28
13
12
Cut set
(c)
Fig. 9
20
Cutset and Kirchhoff ’s current law
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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)
j1 (t )  j2 (t )  j3 (t )  0
for all
t
for all
t
And from Fig. 9 (b)
j1 (t )  j2 (t )  j3 (t )  0
Cutset and Kirchhoff ’s current law
Cut sets should be selected such that they are linearly independent.
III
10
8
9
2
1
3
4
6
I
7
5
II
Fig. 10
Cut sets I,II and III are linearly dependent
Cutset and Kirchhoff ’s current law
Cut set I
j1 (t )  j2 (t )  j3 (t )  j4 (t )  j5 (t )  0
Cut set II
 j4 (t )  j5 (t )  j8 (t )  j10 (t )  0
Cut set III
j1 (t )  j2 (t )  j3 (t )  j8 (t )  j10 (t )  0
KCLcut set III = KCLcut set I + KCLcut set II
Loops and Kirchhoff’s voltage laws
A 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
Not a loop
a loop
Fig. 11
Not a loop
Loops and Kirchhoff’s voltage laws
2
1
2
3
4
2
3
41
4
1
3
5
I
II
III
7
8
6
4
5
9
3
IV
2
1
Cases I,II,III and IV violate the loop
V
10
12
11
Case V is a loop
Fig. 12
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
Write the KVL for the loop shown in Fig 13
1
2
3
4
5
8
7
6
KVL
v2 (t )  v5 (t )  v7 (t )  v8 (t )  v4 (t )  0
9
for all
8
10
Fig. 13
t
Tellegen’s Theorem
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Tellegen’s Theorem is a general network theorem
It is valid for any lump network
For a lumped network whose element assigned by associate reference
direction for branch voltage v k and branch current jk
The product
element k
vk jk
is the power delivered at time
t by the network to the
If all branch voltages and branch currents satisfy KVL and KCL then
b
v
k 1
k jk
0
b
= number of branch
Tellegen’s Theorem
Suppose that vˆ1 , vˆ2 ,...... vˆb and ˆj1 , ˆj2 ,...... ˆjb
ˆjk
voltages and branch currents and if v̂ k and
Then
b

k 1
b

k 1
b
v
vˆk ˆjk  0
vk ˆjk  0
is another sets of branch
satisfy KVL and KCL
k jk
0
k 1
b
and
 vˆ
k jk
k 1
0
Tellegen’s Theorem
Applications
Tellegen’s Theorem implies the law of energy conservation.
b
Since
v
k jk
0
k 1
“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”.
Applications
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Conservation of energy
Conservation of complex power
The real part and phase of driving point
impedance
Driving point impedance
Conservation of Energy
b
 v (t ) j (t )  0
k 1
k
k
For all 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”.
Conservation of Energy
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Resistor
2
k k
R j
For kth resistor
Capacitor
1
2
Ck vk
2
For kth capacitor
Inductor
1 2
Lk ik
2
For kth inductor
Conservation of Complex Power
b
1
Vk J k  0

k 1 2
Vk = Branch Voltage Phasor
J k = Branch Current Phasor
J k = Branch Current Phasor Conjugate
J 2  V2 
J1
V1



J4
V4

J 3  V3 
b
1
1
 V1 J 1   Vk J k
2
k 2 2
V1
V2




J1
J2
N Linear
time-invariant
RLC Network

Jk
Vk

Conservation of Complex Power
The real part and phase of driving point
impedance
J1


V1
Vk

Jk

Z in
Linear timeinvariant RLC
one-port
V1   J1Zin ( j)
From Tellegen’s theorem, and let P = complex power
delivered to the one-port by the source
1
1
2
P   V1 J 1  Z in ( j ) J 1
2
2
b
1
1
2
 Vk J k   Z k ( j ) J k
2
2 k 2
Taking the real part
1
2
Pav  Re[Zin ( j )] J 1
2
b
1
  Re[Z k ( j )] J k
2 k 2
2
All impedances are calculated at the same angular
frequency i.e. the source angular frequency
Driving Point Impedance
1
2
P  Z in ( j ) J 1
2
1 b
2
  Z m ( j ) J m
2 k 2
1
1
1
1
2
2
  Ri J i   j Lk J k  
Jl
2 i
2 k
2 l jCl
R
L
C
2
Exhibiting the real and imaginary part of P
1
1
1
1
2
2
2
P   Ri J i  2 j   Lk J k   2 J l 
2 i
4 l  Cl
4 k

Average
Average
Average
power
Magnetic
Electric
dissipated
Energy
Energy
Stored
Stored
P
av
M
P  Pav  2 j  M  E 
E
From
1
2
P  Z in ( j ) J 1
2
 Z in ( j ) 
2P
J1
2
P  Pav  2 j  M  E 
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