3. 4.1 Introduction
Circuit Theorems 3
A large
complex circuits
Simplify
circuit analysis
Circuit Theorems
‧Thevenin’s theorem ‧ Norton theorem
‧Circuit linearity ‧ Superposition
‧source transformation ‧ max. power transfer
4. 4.2 Linearity Property
Circuit Theorems 4
Homogeneity property (Scaling)
iR
v
i
kiR
kv
ki
Additivity property
R
i
v
i 2
2
2
R
i
v
i 1
1
1
2
1
2
1
2
1
2
1 )
( v
v
R
i
R
i
R
i
i
i
i
5. A linear circuit is one whose output is linearly
related (or directly proportional) to its input
Fig. 4.1
Circuit Theorems 5
V0
I0
i
6. Linear circuit consist of
● linear elements
● linear dependent sources
● independent sources
Circuit Theorems 6
mA
1
mV
5
A
2
.
0
V
1
A
2
V
10
i
v
i
v
i
v
s
s
s
nonlinear
R
v
R
i
p :
2
2
7. Example 4.1
For the circuit in fig 4.2 find I0 when vs=12V and
vs=24V.
Circuit Theorems 7
Eastern Meiterranean University
8. Example 4.1
KVL
Eqs(4.1.1) and (4.1.3) we get
Circuit Theorems 8
0
4
12 2
1
s
v
i
i
0
3
16
4 2
1
s
x v
v
i
i
1
2i
vx
becomes
)
2
.
1
.
4
(
0
16
10 2
1
s
v
i
i
(4.1.1)
(4.1.2)
(4.1.3)
2
1
2
1 6
0
12
2 i
i
i
i
9. Example 4.1
Eq(4.1.1), we get
When
When
Showing that when the source value is doubled, I0
doubles.
Circuit Theorems 9
76
0
76 2
2
s
s
v
i
v
i
A
76
12
2
0
i
I
V
12
s
v
A
76
24
2
0
i
I
V
24
s
v
10. Example 4.2
Assume I0 = 1 A and use linearity to find the
actual value of I0 in the circuit in fig 4.4.
Circuit Theorems 10
11. Example 4.2
Circuit Theorems 11
A,
2
4
/
V
8
)
5
3
(
then
A,
1
If
1
1
0
1
0
v
I
I
v
I
A
3
0
1
2
I
I
I
A
2
7
,
V
14
6
8
2 2
3
2
1
2
V
I
I
V
V
A
5
2
3
4 I
I
I A
5
S
I
A
5
1
0
S
I
A
I
A
15
A
3
0
S
I
I
12. 4.3 Superposition
The superposition principle states that the voltage
across (or current through) an element in a linear
circuit is the algebraic sum of the voltages across
(or currents through) that element due to each
independent source acting alone.
Turn off, killed, inactive source:
● independent voltage source: 0 V (short circuit)
● independent current source: 0 A (open circuit)
Dependent sources are left intact.
Circuit Theorems 12
13. Steps to apply superposition principle:
1. Turn off all independent sources except one source.
Find the output (voltage or current) due to that active
source using nodal or mesh analysis.
2. Repeat step 1 for each of the other independent
sources.
3. Find the total contribution by adding algebraically all
the contributions due to the independent sources.
Circuit Theorems 13
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14. How to turn off independent sources
Turn off voltages sources = short voltage sources;
make it equal to zero voltage
Turn off current sources = open current sources;
make it equal to zero current
Circuit Theorems 14
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15. Superposition involves more work but simpler
circuits.
Superposition is not applicable to the effect on
power.
Circuit Theorems 15
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16. Example 4.3
Use the superposition theorem to find in the
circuit in Fig.4.6.
Circuit Theorems 16
17. Example 4.3
Since there are two sources,
let
Voltage division to get
Current division, to get
Hence
And we find
Circuit Theorems 17
2
1 V
V
V
V
2
)
6
(
8
4
4
1
V
A
2
)
3
(
8
4
8
3
i
V
8
4 3
2
i
v
V
10
8
2
2
1
v
v
v
18. Example 4.4
Find I0 in the circuit in Fig.4.9 using superposition.
Circuit Theorems 18
21. 4.5 Source Transformation
A source transformation is the process of replacing
a voltage source vs in series with a resistor R by a
current source is in parallel with a resistor R, or
vice versa
Circuit Theorems 21
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22. Fig. 4.15 & 4.16
R
v
i
R
i
v s
s
s
s
or
Circuit Theorems 22
Eastern Mediterranean University
24. Arrow of the current source
positive terminal of voltage source
Impossible source Transformation
● ideal voltage source (R = 0)
● ideal current source (R=)
Circuit Theorems 24
25. Example 4.6
Use source transformation to find vo in the circuit
in Fig 4.17.
Circuit Theorems 25
27. Example 4.6
we use current division in Fig.4.18(c) to get
and
Circuit Theorems 27
A
4
.
0
)
2
(
8
2
2
i
V
2
.
3
)
4
.
0
(
8
8
i
vo
28. Example 4.7
Find vx in Fig.4.20 using source transformation
Circuit Theorems 28
29. Example 4.7
Applying KVL around the loop in Fig 4.21(b) gives
(4.7.1)
Appling KVL to the loop containing only the 3V
voltage source, the resistor, and vx yields
(4.7.2)
Circuit Theorems 29
0
18
5
3
x
v
i
1
i
v
v
i x
x
3
0
1
3
30. Example 4.7
Substituting this into Eq.(4.7.1), we obtain
Alternatively
thus
Circuit Theorems 30
A
5
.
4
0
3
5
15
i
i
A
5
.
4
0
18
4
i
v
i
v x
x
V
5
.
7
3
i
vx
31. 4.5 Thevenin’s Theorem
Thevenin’s theorem states that a linear two-
terminal circuit can be replaced by an equivalent
circuit consisting of a voltage source VTh in series
with a resistor RTh where VTh is the open circuit
voltage at the terminals and RTh is the input or
equivalent resistance at the terminals when the
independent source are turn off.
Circuit Theorems 31
32. Property of Linear Circuits
Circuit Theorems 32
i
v
v
i
Any two-terminal
Linear Circuits
+
-
Vth
Isc
Slope=1/Rth
34. How to Find Thevenin’s Voltage
Equivalent circuit: same voltage-current relation
at the terminals.
Circuit Theorems 34
:
Th oc
v
V b
a
at
ltage
circuit vo
open
35. How to Find Thevenin’s Resistance
Circuit Theorems 35
:
in
Th R
R
b.
a
at
circuit
dead
the
of
resistance
input
circuited
open
b
a
sources
t
independen
all
off
Turn
36. CASE 1
If the network has no dependent sources:
● Turn off all independent source.
● RTH: can be obtained via simplification of either parallel
or series connection seen from a-b
Circuit Theorems 36
37. Fig. 4.25
CASE 2
If the network has dependent
sources
● Turn off all independent sources.
● Apply a voltage source vo at a-b
● Alternatively, apply a current
source io at a-b
Circuit Theorems 37
o
o
i
v
R
Th
o
o
Th
i
v
R
38. The Thevenin’s resistance may be negative,
indicating that the circuit has ability providing
power
Circuit Theorems 38
40. Example 4.8
Find the Thevenin’s equivalent circuit of the
circuit shown in Fig 4.27, to the left of the
terminals a-b. Then find the current through RL =
6,16,and 36 .
Circuit Theorems 40
41. Find Rth
Circuit Theorems 41
short
source
voltage
V
32
:
Th
R
open
source
current
A
2
4
1
16
12
4
1
12
||
4
Th
R
42. Find Vth
Circuit Theorems 42
analysis
Mesh
)
1
(
:
Th
V
A
2
,
0
)
(
12
4
32 2
2
1
1
i
i
i
i
A
5
.
0
1
i
V
30
)
0
.
2
5
.
0
(
12
)
(
12 2
1
Th
i
i
V
Analysis
Nodal
ely,
Alternativ
)
2
(
12
/
2
4
/
)
32
( Th
Th V
V
V
30
Th
V
43. Example 4.8
Circuit Theorems 43
Fig. 4.29
transform
source
ely,
Alternativ
)
3
(
V
30
24
3
96
12
2
4
32
TH
TH
TH
TH
TH
V
V
V
V
V
44. Example 4.8
Circuit Theorems 44
:
get
To L
i
L
L
L
R
R
R
V
i
4
30
Th
Th
6
L
R A
3
10
/
30
L
I
16
L
R A
5
.
1
20
/
30
L
I
A
75
.
0
40
/
30
L
I
36
L
R
45. Example 4.9
Find the Thevenin’s equivalent of the circuit in Fig.
4.31 at terminals a-b.
Circuit Theorems 45
46. Example 4.9
(independent + dependent source case)
Circuit Theorems 46
Fig(a)
:
find
To Th
R
0
source
t
independen
intact
source
dependent
,
V
1
o
v
o
o
o
i
i
v
R
1
Th
47. Example 4.9
For loop 1,
Circuit Theorems 47
2
1
2
1 or
0
)
(
2
2 i
i
v
i
i
v x
x
2
1
4
But i
i
v
i x
2
1 3i
i
48. Example 4.9
Circuit Theorems 48
:
3
and
2
Loop
0
)
(
6
)
(
2
4 3
2
1
2
2
i
i
i
i
i
0
1
2
)
(
6 3
2
3
i
i
i
gives
equations
these
Solving
.
A
6
/
1
3
i
A
6
1
But 3
i
io
6
1
Th
o
i
V
R
49. Example 4.9
Circuit Theorems 49
0
)
(
2
2 2
3 i
i
vx
5
1
i
Fig(b)
:
get
To Th
V
2
3 i
i
vx
analysis
Mesh
0
6
)
(
2
)
(
4 2
1
2
1
2 i
i
i
i
i 0
2
4
12 3
1
2
i
i
i
.
3
/
10
2
i
V
20
6 2
Th
i
v
V oc
x
v
i
i
)
(
4
But 2
1
50. Example 4.10
Determine the Thevenin’s
equivalent circuit in
Fig.4.35(a).
Solution
Circuit Theorems 50
)
case
only
source
dependent
(
o
o
i
v
R
Th
0
Th
V
:
anaysis
Nodal
4
/
2 o
x
x
o v
i
i
i
51. Example 4.10
Circuit Theorems 51
2
2
0 o
o
x
v
v
i
But
4
4
2
4
o
o
o
o
x
o
v
v
v
v
i
i
o
o i
v 4
or
:
4
Thus Th
o
o
i
v
R power
Supplying
54. 4.6 Norton’s Theorem
Norton’s theorem states that a linear two-terminal
circuit can be replaced by equivalent circuit
consisting of a current source IN in parallel with a
resistor RN where IN is the short-circuit current
through the terminals and RN is the input or
equivalent resistance at the terminals when the
independent source are turn off.
Circuit Theorems 54
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56. How to Find Norton Current
Thevenin and Norton
resistances are equal:
Short circuit current
from a to b :
Circuit Theorems 56
Th
R
RN
Th
Th
R
V
i
I sc
N
57. Thevenin or Norton equivalent circuit :
The open circuit voltage voc across terminals a and
b
The short circuit current isc at terminals a and b
The equivalent or input resistance Rin at terminals
a and b when all independent source are turn off.
Circuit Theorems 57
oc
Th v
V
N
I
Th
Th N
Th
V
R R
R
sc
i
58. Example 4.11
Find the Norton equivalent circuit of the circuit in
Fig 4.39.
Circuit Theorems 58
59. Example 4.11
Circuit Theorems 59
:
)
(
40
.
4
Fig a
4
25
5
20
20
||
5
)
8
4
8
(
|
|
5
N
R
N
R
find
To
60. Example 4.11
Circuit Theorems 60
N
i
find
To
.
and
terminals
circuit
short b
a
))
(
40
.
4
.
Fig
( b
:
Mesh 0
4
20
,
A
2 2
1
2
1
i
i
i
i
N
sc I
i
i
A
1
2
61. Example 4.11
Circuit Theorems 61
N
I
for
method
e
Alternativ
Th
Th
N
R
V
I
voltage
circuit
open
:
Th
V b
a and
:
))
(
40
.
4
( c
Fig
:
analysis
Mesh
0
12
4
25
,
2 3
4
3
i
i
A
i
A
8
.
0
4
i
terminals
across
V
4
5 4
i
V
v Th
oc
63. Example 4.12
Using Norton’s theorem, find RN and IN of the
circuit in Fig 4.43 at terminals a-b.
Circuit Theorems 63
64. Example 4.12
Circuit Theorems 64
N
R
find
To )
(
44
.
4
. a
Fig
shorted
resistor
4
Parallel
:
2
||
||
5 x
o i
v
Hence, 2
.
0
5
/
1
5
/
o
x v
i
5
2
.
0
1
o
o
N
i
v
R
65. Example 4.12
Circuit Theorems 65
N
I
find
To )
(
44
.
4
. b
Fig
x
i
v 2
||
5
||
10
||
4
Parallel
:
.5A,
2
4
0
10
x
i
A
7
2(2.5)
5
10
2
x
x
sc i
i
i
7A
N
I
66. 4.8 Maximum Power Trandfer
Circuit Theorems 66
L
L R
R
R
V
R
i
p
2
L
TH
TH
2
Fig 4.48
67. Fig. 4.49
Maximum power is transferred to the load when
the load resistance equals the Thevenin resistance
as seen the load (RL = RTH).
Circuit Theorems 67
71. Example 4.13
Circuit Theorems 71
W
R
V
p
R
R
V
V
V
i
i
A
i
i
i
L
TH
TH
L
TH
TH
i
44
.
13
9
4
22
4
9
22
0
)
0
(
2
3
1
6
12
2
,
12
18
12
2
2
max
2
2
2
1
