Introduction to Circuit Theory
Circuit Theorems
2012-09-12
Jieh-Tsorng Wu
National Chiao-Tung University
Department of Electronics Engineering
Outline
1.
2.
3.
4.
5.
6.
7.
Linearity Property
Superposition
Source Transformation
Thevenin’s Theorem
Norton’s Theorem
Maximum Power Transfer
Source Modeling
4. Circuit Theorems
2
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1
Linearity Property
 Linearity = Homogeneity (Scaling) + Additivity
 A linear circuit is one whose output is linearly related to its input.
 A linear circuit consists of only linear elements, linear dependent sources, and
independent sources.
For f() with homoneneity property:
If y  f ( x), then a y  f (a  x) where x is a constant.
For f() with additivity property:
If y1  f ( x1 ) and y 2  f ( x2 ), then y1  y2  f ( x1  x2 ).
Thus, for a linear function f(),
If y1  f ( x1 ) and y 2  f ( x2 ), then a1 y1  a2 y2  f (a1 x1  a2 x2 )
where a1 and a 2 are constants.
4. Circuit Theorems
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Linear Circuit Example
4. Circuit Theorems
vs  12 V

vs  24 V

4
12
A
76
24
Io 
A
76
Io 
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Superposition
 In a linear circuit, the voltage across (current through) an element is the algebraic
sum of the voltage across (current through) that element due to each independent
source acting alone.
 A turned-off voltage source = a short circuit
 A turned-off current source = an open circuit
v(V1 ,..., VN ; I1 ,..., I M )
 v(V1 ,...,0;0,...,0)  ...  v(0,..., VN ;0,...,0)
 v(0,...,0; I1 ,...,0)  ...  v(0,...,0;0,..., I M )
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Superposition Example 1
=
+
v  v1  v2
4
1
3
12
1 4 1 8
 2  8  10 (V)
6
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Superposition Example 2
i0  i0'  i0"
+
=
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Superposition Example 3
i  i1  i2  i3
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Source Transformation
 An equivalent circuit is one whose v-i characteristics are identical with the original
circuit.
 The source transformation is not possible when R=0 for voltage source and R=∞
for current source.
Independent Source
vs  is  R
or
is 
vs
R
Dependent Source
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Source Transformation
1
i1
i2
+
+
vab
2
vab
_
_
For any vab , i1  i2
1 1 
vs
 is  vab   
R1
 R1 R2 
vs
1 1
 is  0 and

0
R1
R1 R2
vs  vab
v
 is  ab
R1
R2


 R1  R2  R vs  is  R
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Source Transformation Example 1
4 3
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Source Transformation Example 2
 3  5i  v x  18  0
(1)
 3  1i  v x  0
(2)
 i  4.5 A
 v x  7. 5 V
4 0.25v x
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Thevenin’s Theorem
 A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a
voltage source Vth in series with a resistor Rth.
 Vth is the open-circuit voltage at the terminals, and Rth is the input resistance at the
terminals when the independent sources are turned off.
4. Circuit Theorems
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Circuit with a Load
IL 
VTh
RTh  RL
VL  VTh 
+
VL
−
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RL
RTh  RL
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Find VTh and RTh
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Thevenin Circuit Example 1
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Thevenin Circuit Example 2
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Norton’s Theorem
 A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a
current source IN in parallel with a resistor RN.
 IN is the short-circuit current through the terminals, and IN is the input resistance at
the terminals when the independent sources are turned off.
I N  VTh RTh
Rin  RN  RTh  voc / isc
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Norton Circuit Example 1
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Norton Circuit Example 2
2vx
i
+ 
+
3vx

6
ix
2
+
vx

2
+
vx

1V
+

2vx
+ 
6
4. Circuit Theorems
10 A
Isc
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Maximum Power Transfer
2
 VTh 
p  i  RL  
  RL
 RTh  RL 
R  RL
dp
 VTh2  Th
 0  RL  RTh
dRL
( RTh  RL )3
2
4. Circuit Theorems
pmax
VTh2

4 RTh
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Source Modeling
Voltage source
Rs  0
vL 
RL
vs
Rs  RL
vL 
Current source
vL 
vL 
RL
vs
Rs  RL
R
RL
vs
Rs  RL
p
 
iL 
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RL
vs
Rs  RL
Rp
R p  RL
is
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