Lecture4.ppt
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EE 529 Circuit and
Systems Analysis
Lecture 4
EASTERN MEDITERRANEAN UNIVERSITY
Matrices of Oriented Graphs
THEOREM: In a graph G let the fundamental
circuit and cut-set matrices with respect to a
tree to be written as
(r )
Bf = B
and
()
U ( )
r
Af = U A r
If the column orderings are identical then
A = B T
Matrices of Oriented Graphs
Consider the following graph
e3
e6
e2 e4 e5 e1
1
0
0
1
1
0
e2
A f 0 1 0 1 0 1 e4
0
0
1
0
1
1
e5
v1
e2
e3
e1
v0
e5
e4
v2
e6
e5
e3
e6
e1
e2 e4
1
1
0
1
0
0
e1
Bf 1 0 1 0 1 0 e3
0
1
1
0
0
1
e6
v3
FUNDAMENTAL POSTULATES
Now, Let G be a connected graph having e
edges and let
xT x1 t , x2 t ,
and
xe t
y T y1 t , y2 t ,
ye t
be two vectors where xi and yi, i=1,...,e,
correspond to the across and through
variables associated with the edge i
respectively.
FUNDAMENTAL POSTULATES
2. POSTULATE Let B be the circuit matrix of the
graph G having e edges then we can write the
following algebraic equation for the across variables
of G
Bx = 0
3. POSTULATE Let A be the cut-set matrix of the
graph G having e edges then we can write the
following algebraic equation for the through
variables of G
Ay = 0
FUNDAMENTAL POSTULATES
2. POSTULATE is called the circuit equations of
electrical system. (is also referred to as Kirchoff’s
Voltage Law)
3. POSTULATE is called the cut-set equations of
electrical system. (is also referred to as Kirchoff’s
Current Law)
Fundamental Circuit & Cut-set Equations
Consider a graph G and a tree T in G. Let the vectors
x and y partitioned as
xT xTb
y T y Tb
xTc
y Tc
where xb (yb) and xc (yc) correspond to the across
(through) variables associated with the branches and
chords of the tree T, respectively.
Then
yb
xb
B U x 0
c
xc Bxb
fundamental circuit
equation
and
U
A = 0
yc
y b = -Ay c
fundamental cut-set
equation
Series & Parallel Edges
Definition: Two edges ei and ek are said to be
connected in series if they have exactly one
common vertex of degree two.
v0
ei
ek
Series & Parallel Edges
Definition: Two edges ei and ek are said to be
connected in parallel if they are incident at
the same pair of vertices vi and vk.
vi
ek
ei
vk
(n+1) edges connected in series
(x1,y1)
(x2,y2)
(x0,y0)
(xn,yn)
1
x0
x
1
x
1 2 0
x3
xn
1 1 1
n
x0 xi
i 1
1
1
1
1
1
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 y0
y
1
0 y2
0
0 y3
1 yn
y0 y1 y2
yn
(n+1) edges connected in parallel
(x0,y0)
1
(x1,y1)
y0
y
1
y
1 2 0
y3
yn
1 1 1
n
y0 yi
i 1
(x2,y2)
1
1
1
1
1
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
x0 x1 x2
(xn,yn)
0 x0
x
1
0 x2
0
0 x3
1 xn
xn
Mathematical Model of a Resistor
A
a
v(t)
i(t)
B
b
v(t ) Ri (t )
Mathematical Model of an
Independent Voltage Source
v(t)
A
a
v(t)
Vs
Vs
i(t)
i(t)
B
b
Mathematical Model of an
Independent Voltage Source
v(t)
a
A
v(t)
Is
i(t)
Is
i(t)
B
b
Circuit Analysis
A-Branch Voltages Method:
Consider the following circuit.
2 k
15 V
30 V
4 k
3 k
1 k
20 V
10 mA
Circuit Analysis
A-Branch Voltages Method:
1. Draw the circuit graph
2 k
15 V
30 V
4 k
10 mA
3 k
1 k
20 V
2
a
3
b
There are:
•5 nodes (n)
4
c
1
6
5
d
7
•3 voltage sources (nv)
e
8
•8 edges (e)
•1 current source (ni)
Circuit Analysis
A-Branch Voltages Method:
1. Select a proper tree: (n-1=4 branches)
Place voltage sources in tree
Place current sources in co-tree
Complete the tree from the resistors
2
a
3
b
4
c
1
6
5
d
7
e
8
Circuit Analysis
A-Branch Voltages Method:
2. Write the fundamental cut-set equations for the tree branches which do
not correspond to voltage sources.
i2
i
3
1 1 1 1 1 i5 0
i6
i7
2
a
3
b
4
c
1
6
5
d
7
e
8
Circuit Analysis
A-Branch Voltages Method:
2. Write the currents in terms of voltages using terminal equations.
i2
i3
i5
i6
i7
v2
2k
v3
4k
v5
1k
v6
3k
10mA
2
a
3
b
4
c
1
6
5
d
7
e
8
Circuit Analysis
A-Branch Voltages Method:
2. Substitute the currents into fundamental cut-set equation.
v3 v5 v6
v2
10m
2k 4k 1k 3k
2
a
3
b
4
c
1
6
5
d
7
e
8
3. v3, v5, and v6 must be expressed in terms of branch voltages using
fundamental circuit equations.
Circuit Analysis
A-Branch Voltages Method:
v3 v2 v1 v2 30
v5 v4 v2 1 15 v2 30 v2 15
v6 v2 v1 v8 v2 30 20 v2 50
v
v
v
v
12k 2 3 5 6 10m
2k 4k 1k 3k
6v2 3v3 12v5 4v6 120
6v2 3(v2 30) 12(v2 15) 4(v2 50) 120
6 3 12 4 v2 90 180 200 120
25v2 350
v2
2
a
3
b
4
c
1
6
5
d
7
e
8
350
14 V
25
Find how much power the 10 mA current source delivers to the circuit
Circuit Analysis
A-Branch Voltages Method:
Find how much power the 10 mA current source delivers to the circuit
v7 v8 v1 v2 20 30 14 36V
P10 mA v7i7 36 10m 360 mW
2
a
3
b
4
c
1
6
5
d
7
e
8
Circuit Analysis
Example: Consider the following circuit. Find
ix in the circuit.
20 V
2
i1
4
4
ix
15 V
5
i2
i3
3
10 V
Circuit Analysis
Circuit graph and a proper tree
20 V
2
1
2
3
i1
6
4
4
4
ix
15 V
5
7
8
5
i3
i2
3
ix i5
10 V
Circuit Analysis
Fundamental cut-set equations
1
2
3
i2 i6 i8 i7
i3 i6 i8
6
4
5
7
8
v2
i2
2
v6
i6
4
v8
i8
3
v3
i3
4
v7
i7
5
Circuit Analysis
Fundamental cut-set equations
1
2
3
6
4
5
7
8
v6 v8 v7
v2
2
4 3 5
v3 v6 v8
4 4 3
Circuit Analysis
Fundamental circuit equations
1
2
3
6
4
5
7
8
v6 v3 v1 v2 v3 v2 20
v8 v4 v3 v1 v2 v5 15 v3 20 v2 10 v3 v2 15
v7 v1 v2 v5 20 v2 10 v2 30
Circuit Analysis
v3 v2 20 v3 v2 15 v2 30
v2
60
4
3
5
2
v3 v3 v2 20 v3 v2 15
12
4
4
3
30v2 15v3 15v2 300 20v3 20v2 300 12v2 360
35v3 77v2 960...................(1)
3v3 3v3 3v2 60 4v3 4v2 60
10v3 7v2 120......................(2)
v3= 9.5639V v2=-8.1203 V
Circuit Analysis
v7 v8
v2 30 v3 v2 15
i5 i7 i8
5 3
5
3
i5 3.48A
ix 3.48 A