ee 529 circuit and systems analysis
DESCRIPTION
EE 529 Circuit and Systems Analysis. Lecture 5. c. a. 1. 2. d. b. Mathematical Model of a Dependent Source. A. Voltage Controlled Voltage Source. c. a. 1. 2. d. b. Mathematical Model of a Dependent Source. B. Current Controlled Voltage Source. c. a. 1. 2. d. b. - PowerPoint PPT PresentationTRANSCRIPT
EASTERN MEDITERRANEAN UNIVERSITY
EE 529 Circuit and Systems Analysis
Lecture 5
Mathematical Model of a Dependent Source
A
B
C
D
V1
+
V1
_
i1 i2
+
V2
_
c
d
2
a
b
1
1 1
2 2
0 0
0
i v
v i
A. Voltage Controlled Voltage Source
Mathematical Model of a Dependent Source
A
B
C
D
+
V1
_
i1 i2
+
V2
_
ri1
c
d
2
a
b
1
1 1
2 2
0 0
0
v i
v ir
B. Current Controlled Voltage Source
Mathematical Model of a Dependent Source
A
B
C
D
+
V1
_
i1 i2
+
V2
_
gV1
c
d
2
a
b
1
1 1
2 2
0 0
0
i v
i vg
A. Voltage Controlled Current Source
Mathematical Model of a Dependent Source
A
B
C
D
+
V1
_
i1 i2
+
V2
_
i1
c
d
2
a
b
1
1 1
2 2
0 0
0
v i
i v
A. Current Controlled Current Source
Circuit Analysis
A-Branch Voltages Method:
Consider the following circuit. Find v and i
1 1
1/2
5 A
5i A
i
+ v -
4v V
Circuit Analysis
•Draw the circuit graph
1 1
1/2
5 A
5i A
i
+ v -a b c
d
4v V
a b b
d
12
3
4
56
77 4
5 2
4
5
v v
i i
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
a b b
d
a b b
d
12
3
4
56
7
Circuit Analysis
• Write the fundamental cut-set equations for the tree branches which do not correspond to voltage sources.
a b b
d
12
3
4
56
7
3 1 2 6i i i i
4 5 6i i i
Circuit Analysis
•Write the currents in terms of voltages using terminal equations.
1 1
1/2
5 A
5i A
i
+ v -a b c
d
4v V
a b b
d
12
3
4
56
7
1
22 2
33 3
5 2 2
4 4
66
5
1
15 5
2
2
i A
vi v
vi v
i i v
i v
vi
Circuit Analysis
• Substitute the currents into fundamental cut-set equation.
• v2, and v6 must be expressed in terms of branch voltages using fundamental circuit equations.
3 1 2 6
63 25
2
i i i i
vv v
4 5 6
64 22 5
2
i i i
vv v
Circuit Analysis
a b b
d
12
3
4
56
7
2 3v v
6 4 3 7 3 45v v v v v v
Circuit Analysis
• Therefore
63 2
3 3 3 4
3 4
3 4
521
5 52
2.5 2.5 5
5 5 10.............(1)
vv v
v v v v
v v
or
v v
4 3 3 4
3 4
4 3
12 5 5
24.5 4.5 0
...............(2)
v v v v
v v
v v
and
Circuit Analysis
• Subst. Eq. (2) into (1) yields
3 3
3
4
5 5( ) 10
1 V
1 V
v v
v
v
4
2 2 3
1 V
1 A
v v
i i v v
Chord Currents Method
Consider the circuit. Find v using chord currents method.
6 k 9 V
18 mA 4 k
12 k
+ v -
4 k 6 mA
Chord Currents Method
Draw the circuit graph.
v = v3
6 k 9 V
18 mA 4 k
12 k
+ v -
4 k 6 mA
a
b c
d
a
b c
d1
2
3
4 5
6
7
Chord Currents Method
Select a proper tree6 k 9 V
18 mA 4 k
12 k
+ v -
4 k 6 mA
a
b c
d
a
b c
d1
2
3
4 5
6
7
Chord Currents Method
Write the fundamental circuit equations for the chords which are not current sources.
a
b c
d1
2
3
4 5
6
7
2 3 6v v v
4 3 5v v v
Chord Currents Method
Write all resistor voltages in terms of terminal currents using terminal equations.
a
b c
d1
2
3
4 5
6
7
3 3
5
4 4
2 2
6 6
12
9 V
6
4
4
v ki
v
v ki
v ki
v ki
Chord Currents Method
Substitute the terminal equations to fundamental circuit equations
a
b c
d1
2
3
4 5
6
7
2 3 6
2 3 6
2 3 6
4 12 4
3
v v v
ki ki ki
or
i i i
4 3 5
4 3
4 3
6 12 9
2 4 3
and
v v v
ki ki
or
i i m
Chord Currents Method
Write fundamental cut-set equations for i3 and i6.
3 1 2 4 2 418i i i i m i i
6 1 2 7 2 218 6 24i i i i m i m i m
a
b c
d1
2
3
4 5
6
7
Chord Currents Method
Substitute the fundamental cut-set equations to fundamental circuit equations
a
b c
d1
2
3
4 5
6
7
2 3 6
2 2 4 2
2 4
3
3 18 24
5 3 78 ...........(1)
i i i
i m i i i m
i i m
4 3
4 2 4
2 4
2 4 3
2 4 18 3
4 6 69 .............(2)
and
i i m
i m i i m
i i m
Chord Currents Method
Substitute the fundamental cut-set equations to fundamental circuit equations
a
b c
d1
2
3
4 5
6
7
2 3 6
2 2 4 2
2 4
3
3 18 24
5 3 78 ...........(1)
i i i
i m i i i m
i i m
4 3
4 2 4
2 4
2 4 3
2 4 18 3
4 6 69 .............(2)
and
i i m
i m i i m
i i m
Chord Currents Method
Using Eqns.(1) and (2), i2 and i4 can be calculated as
2
4
3 3
14.5
11
6
1112 12 18 14.5 20V
6
i mA
i mA
v v ki k m m m
Chord Currents Method
Consider the following circuit. Find v0.
+
v1
_2 i1 V
5
20 10 10 V 5 A
+
v0
_
i1 0.4 v1 A
ab c
d e