ee 529 circuit and systems analysis lecture 6
DESCRIPTION
Mathematical Models of Electrical Components A. Nullator a bTRANSCRIPT
EASTERN MEDITERRANEAN UNIVERSITY
EE 529 Circuit and Systems
AnalysisLecture 6
Mustafa Kemal Uyguroğlu
Mathematical Models of Electrical Components
A
B
O
( ) 0( ) 0v ti t
A. Nullator
a
b
( )( )v ti t
Mathematical Models of Electrical Components
A
B
0 0 ( )0 0 ( )
v ti t
0
B. Noratora
b
( )( )v ti t
Nullator and Norator are conceptual elements. They are used to represent some electrical elements in different ways.
Mathematical Models of Electrical Components
E C
B
1 2
1 2
1
2
1 1
1 1
kv t kv tES R CS
kv t kv tF ES CS
i t I e I e
i t I e I e
C. Three Terminal and two-port Circuit Elements
C.1 TRANSISTORe
b
c
1 2
Ebers-Moll Equations
Mathematical Models of Electrical Components
1 : n
C
i1 i2A B+
v 1
-
+
v2
-
C. Three Terminal and two-port Circuit Elements
C.2 IDEAL TRANSFORMER
a
c
b
1 2 1 1
2 2
10
1 0
v ini v
n
1 1
2 2
00
i vnv in
Mathematical Models of Electrical Components
1 : ni1 i2
A
B
C
D
+
v1
-
+
v2
-
C. Three Terminal and two-port Circuit Elements
C.2 IDEAL TRANSFORMER
1 1
2 2
10
1 0
v ini v
n
1 1
2 2
00
i vnv in
c
d
2
a
b
1
Mathematical Models of Electrical Components
C
i1 i2A B+
v 1
-
+
v2
-
k
C. Three Terminal and two-port Circuit Elements
C.3 IDEAL GYRATOR
a
c
b
1 2
1 1
2 2
00
v ikv ik
1 1
2 2
10
1 0
i vki v
k
Mathematical Models of Electrical Components
i1 i2A
k
B
C
D
+
v1
-
+
v2
-
C. Three Terminal and two-port Circuit Elements
C.3 IDEAL GYRATOR
1 1
2 2
00
v ikv ik
1 1
2 2
10
1 0
i vki v
k
c
d
2
a
b
1
Mathematical Models of Electrical Components
1 : ni1 i2
A
B
C
D
+
v1
-
+
v2
-
1 1
2 2
00
i vnv in
Representation of Ideal Transformer with Dependent Sourcesc
d
2
a
b
1
1 1 1
2 2 2
1 1 1
2 2 2
1 1 1 1
2 2 2 2
0 0 00 0 0
or
0 0 00 0 0
i v vnv i in
i i iv v v
i v i vnv i v in
Mathematical Models of Electrical Components
1 1 1
2 2 2
1 1 1
2 2 2
1 1 1 1
2 2 2 2
0 0 00 0 0
or
0 0 00 0 0
i v vnv i in
i i iv v vi v i vnv i v in
Representation of Ideal Transformer with Dependent Sources
A
B
C
D
i2
+
v'2
_
i'1
+
v1
_
nv1
A
B
C
D
i2
+
v''2
_
+
v1
_
i''1
ni2
i1
+
v1
-
Mathematical Models of Electrical Components
i1 i2A
B
C
D
+
v1
-
+
v2
-
nv1-ni2
1 1
2 2
00
i vnv in
Representation of Ideal Transformer with Dependent Sources
1 : ni1 i2
A
B
C
D
+
v1
-
+
v2
-
Mathematical Models of Electrical Components
i1 i2A
B
C
D
+
v1
-
+
v2
-
-ki1ki2
Representation of Ideal Gyrator with Dependent Sources
i1 i2A
k
B
C
D
+
v1
-
+
v2
-
1 1
2 2
00
v ikv ik
Mathematical Models of Electrical Components
i1 i2A
B
C
D
+
v1
-
+
v2
--1/k v2 1/k v1
Representation of Ideal Gyrator with Dependent Sources
i1 i2A
k
B
C
D
+
v1
-
+
v2
-
1 1
2 2
10
1 0
i vki v
k
Mathematical Models of Electrical ComponentsOperational Amplifier
A1A2
A3
A0
1 1
2 2
3 3
0 0 00 0 0
0
i vi vv A A i
a1
1 2
a2 a3
a0
3
Mathematical Models of Electrical ComponentsOperational Amplifier
A1A2
A3
A0
1 1
2 2
3 3
0 0 00 0 0
0
i vi vv A A i
a1
1 2
a2 a3
a0
3
A: open loop gain, very big!
2 1 very small!v t v t
Mathematical Models of Electrical ComponentsOperational Amplifier
A1 A3
A0
1 1
3 3
0 00
i vv iA
a1
1
a3
a0
3
A: open loop gain, very big!
1 0v
Mathematical Models of Electrical Components
A1 A3
A0
O
A1 A3A2
A0A2
A1 A3
A0
O
A1 A3
A0
• Representation of OP-AMP with Nullator and Norator.
Analysis of Circuits Containing Multi-terminal Components The terminal equations of resistors are
The terminal equations of multi-terminal components are similar to two-terminal components but the coefficient matrices are full.
,or
,
where , , , and are diagonal matrices
t t t t
t t t t
b b b c c c
b b b c c c
b c b c
v R i v R i
i G v i G v
R R G G
Analysis of Circuits Containing Multi-terminal Components
or
t tt t
t tt t
b bb bc b
c cb cc c
b bb bc b
c cb cc c
v R R iv R R i
i G G vi G G v
where
vb : branch voltages
vc : Chord voltages
ib : branch currents
ic : Chord currents
Analysis of Circuits Containing Multi-terminal Components
(A) Branch Voltages Method
1 2
2
or
where : current sources
tt
t
tt
t
cb
s
b1 s
c
s
ii A A
i
iU A A i
i
i
2 ......................(1)t
tt
bb bc b1 s
cb cc c
G G vU A A i
G G v
By using the terminal equations of the multiterminal components, the above equation can be written as
Analysis of Circuits Containing Multi-terminal Components On the other, the chord voltages can be
written in terms of branch voltages by using the fundamental circuit equations.
If Eq.(2) is substituted into (1) and the known quantities are collected on the right hand side then the following equation is obtained:
or
.................(2)
tt
t
t tt t
sc 1 2
b
b s
c b1 2
vv B B
v
v v0 Uv vB B
Analysis of Circuits Containing Multi-terminal Components
21
1 1
where
t t t
bb bc bc T1 b 1 1 s sT
cb cc cc
T
T1 1
G G GUU A v U A A v A i
G G GB
B A
A B
Analysis of Circuits Containing Multi-terminal Components
Example : In the following figure, the circuit contains a 3-terminal component. The terminal equation of the 3-terminal component is:
Using the branch voltages method, obtain the circuit equations
1 11 12 1
2 21 22 2
i g g vi g g v
3-terminal
A B
C D
Ra
Rb
Is
Vs 1 2
a b
c
1 2
a b
c
IS
VS
(Ra)(Rb)
Analysis of Circuits Containing Multi-terminal Components
Example : In the following figure, the circuit contains a 3-terminal component. The terminal equation of the 3-terminal component is:
Using the branch voltages method, obtain the circuit equations
1 11 12 1
2 21 22 2
.........(1)i g g vi g g v
3-terminal
A B
C D
Ra
Rb
Is
Vs 1 2
a b
c
1 2
a b
c
IS
VS
(Ra)(Rb)
Analysis of Circuits Containing Multi-terminal Components
The fundamental cut-set equations for tree branches 1 and 2:
1 2
a b
c
IS
VS
(Ra)(Rb)
1
2
........(2)a s
s b
i IiI ii
The terminal equations of the resistors:
............(3)a a a
b b b
i G vi G v
Subst. of Eqs.(3) and (1) into (2) yields:
Analysis of Circuits Containing Multi-terminal Components
va and vb can be expressed in terms of branch voltages using fundamental circuit equations.
Subst. of Eq.(5) into (4) gives:
11 12 1
21 22 2
1.........(4)
1a a
sb b
G vg g vI
G vg g v
1 1
2 2
01 0.....(5)
0 1a
b s s
v v vv v V Vv
11 12 1 1
21 22 2 2
0 0 10 1a
s sb b
Gg g v vV I
G Gg g v v
or
11 12 1
21 22 2
0 11
a s
b b s
g G g Vvg g G G Iv
Analysis of Circuits Containing Multi-terminal Components
Example : In the following figure, the circuit contains a 2-port gyrator and a 3-terminal voltage controlled current source. The terminal equations of these components are:
Using the branch voltages method, obtain the circuit equations
1 1 3 3
2 2 4 4
0 0 0, ...........(1)
0 0i v i vi v i vk
B
Ra
Rb
i4(t)
A
Vs
CD
a
d
a
d
bb
c
1 2
3
4
Analysis of Circuits Containing Multi-terminal Components
B
Ra
Rb
i4(t)
A
Vs
CD
B
Ra
Rb
A
Vs
CD
2 - port
3-terminal
1 2
3
4
Vs
(Ra)
(Rb)
Analysis of Circuits Containing Multi-terminal Components
31
3 42
........(2)a b
b
i i iii i ii
1 2
3
4
Vs
(Ra)
(Rb)
The terminal equations of the resistors:
..........(3)a a a
b b b
i G vi G v
Subst. of Eqs.(3) and (1) into (2) yields:
Analysis of Circuits Containing Multi-terminal Components
13
2
0 0.............(4)
00a b a
b b
G G vvv
G vv k
va , vb and v3 can be expressed in terms of branch voltages using
fundamental circuit equations.
11
1 22
3 1 2
1 0 01 1 .....(5)1 1
a
b s s
s s
v vv
v v V v Vv
v v V v V
•Subst. of Eq.(5) into (4) gives:
1 1 1
2 2 2
1 1
2 2
00 1 0 01 1
00 1 1
00
a bs
b s
a b b bs
b b b
G Gv v vV
G Vv v vk
G G G Gv vV
G k G k G kv v