Download EE 529 Circuit and Systems

EE 529 Circuit and
Systems Analysis
Lecture 5
EASTERN MEDITERRANEAN UNIVERSITY
Mathematical Model of a Dependent
Source
A. Voltage Controlled Voltage Source
a
i1
i2
A
C
+
V1
c
+
V1
_
1
V2
2
_
D
B
b
 i1   0 0 v1 
v     0  i 
 2
 2 
d
Mathematical Model of a Dependent
Source
B. Current Controlled Voltage Source
a
i1
i2
A
C
+
V1
c
+
ri1
_
1
V2
2
_
D
B
b
 v1  0 0  i1 
 v    r 0  i 
 2
 2 
d
Mathematical Model of a Dependent
Source
A. Voltage Controlled Current Source
a
i1
i2
A
C
+
V1
c
+
gV1
_
1
V2
2
_
D
B
b
 i1   0 0  v1 
i    g 0 v 
 2
 2 
d
Mathematical Model of a Dependent
Source
A. Current Controlled Current Source
a
i1
c
i2
A
C
+
+
V1
i1
_
1
V2
2
_
D
B
b
v1   0
i   
 2 
0  i1 
0 v2 
d
Circuit Analysis
A-Branch Voltages Method:
Consider the following circuit. Find v and i
5i A
+ v -

1/2 
5A
1
1
i
4v V
Circuit Analysis
•Draw the circuit graph
5i A
a
+ v -
b

c
1/2 
5A
1
4v V
1
i
d
5
a
b
b
4
v7  4v4
i5  5i2
6
3
1
7
2
d
Circuit Analysis
Select a proper tree: (n-1=4 branches)
 Place voltage sources in tree
 Place the voltages controlling the dependent sources in tree
 Place current sources in co-tree
 Complete the tree from the resistors
5
a
b
6
b
4
3
1
7
2
d
Circuit Analysis
•
Write the fundamental cut-set equations for the tree branches which do
not correspond to voltage sources.
i3  i1  i2  i6
5
a
b
6
b
4
3
1
7
2
d
i4  i5  i6
Circuit Analysis
•Write the currents in terms of voltages using terminal equations.
i1  5 A
5i A
a
+ v -
b

c
1/2 
5A
1
4v V
1
i
d
5
a
b
6
b
4
1
2
3
7
d
v2
i2   v2
1
v3
i3   v3
1
i5  5i2  5v2
i4  2v4
v6
i6 
2
Circuit Analysis
•
Substitute the currents into fundamental cut-set equation.
i3  i1  i2  i6
v6
v3  5  v2 
2
i4  i5  i6
v6
2v4  5v2 
2
• v2, and v6 must be expressed in terms of branch voltages using
fundamental circuit equations.
Circuit Analysis
5
b
a
6
b
v2  v3
4
3
7
2
1
d
v6  v4  v3  v7  v3  5v4
Circuit Analysis
•
Therefore
v6
2
1
v3  5  v3   v3  5v4 
2
2.5v3  2.5v4  5
v3  5  v2 
or
5v3  5v4  10.............(1)
and
1
 v3  5v4 
2
4.5v3  4.5v4  0
2v4  5v3 
v4  v3 ...............(2)
Circuit Analysis
•
Subst. Eq. (2) into (1) yields
5v3  5(v3 )  10
v3  1 V
v4  1 V

v  v4  1 V
i  i2  v2  v3  1 A
Chord Currents Method
 Consider the circuit. Find v using chord
currents method.
9V
6 k
12 k
+ v
18 mA
4 k
-
4 k
6 mA
Chord Currents Method
 Draw the circuit graph.
6 k
a
9V
a
5
4
b
12 k
+ v
18 mA
c
-
c
6
2
4 k
4 k
3
b
6 mA
7
1
d
d
 v = v3
Chord Currents Method
 Select a proper tree
6 k
a
9V
a
5
4
b
12 k
+ v
18 mA
c
-
c
6
2
4 k
4 k
3
b
6 mA
7
1
d
d
Chord Currents Method
 Write the fundamental circuit equations for
the chords which are not current sources.
a
5
4
v2  v3  v6
3
b
c
6
2
7
1
d
v4  v3  v5
Chord Currents Method
 Write all resistor voltages in terms of terminal
currents using terminal equations.
a
5
4
v3  12ki3
3
b
c
v4  6ki4
6
2
7
1
d
v5  9 V
v2  4ki2
v6  4ki6
Chord Currents Method
 Substitute the terminal equations to
fundamental circuit equations
a
v2  v3  v6
5
4
4ki2  12ki3  4ki6
3
b
c
6
2
7
1
d
or
i2  3i3  i6
and
v4  v3  v5
6ki4  12ki3  9
or
2i4  4i3  3m
Chord Currents Method
 Write fundamental cut-set equations for i3 and
i6.
a
i3  i1  i2  i4  18m  i2  i4
5
4
3
b
c
6
2
7
1
d
i6  i1  i2  i7  18m  i2  6m  i2  24m
Chord Currents Method
 Substitute the fundamental cut-set equations
to fundamental circuit equations
a
i2  3i3  i6
5
4
3
b
c
i2  3 18m  i2  i4   i2  24m
5i2  3i4  78m...........(1)
6
2
7
1
d
and
2i4  4i3  3m
2i4  4 18m  i2  i4   3m
4i2  6i4  69m.............(2)
Chord Currents Method
 Substitute the fundamental cut-set equations
to fundamental circuit equations
a
i2  3i3  i6
5
4
3
b
c
i2  3 18m  i2  i4   i2  24m
5i2  3i4  78m...........(1)
6
2
7
1
d
and
2i4  4i3  3m
2i4  4 18m  i2  i4   3m
4i2  6i4  69m.............(2)
Chord Currents Method
 Using Eqns.(1) and (2), i2 and i4 can be
calculated as
i2  14.5mA
i4 
11
mA
6
11 

 v  v3  12ki3  12k 18m  14.5m  m   20 V
6 

Chord Currents Method
 Consider the following circuit. Find v0.
a
10 V
i1
0.4v1 A
5
b
20 
c
+
+
v0
v1
_
d
2i1 V
10 
_
e
5A