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ECE 3301
General Electrical Engineering
Section 18
Thevenin Equivalent Circuit Theorem
1
Thevenin Equivalent Circuit
• Any linear, two-terminal network with any
number of independent or dependent
sources, may be replaced with an equivalent
circuit consisting of a voltage source in
series with a resistance.
a
a

Any
Network
RL
RTH
VTH
RL

b
b
2
Thevenin Equivalent Circuit
• Proof: Consider a Network with n + 1
Nodes described by the set of n Node
Equations:
G11
 G21

.
.
 .
 Gn1
G12 ... G1n

G22 ... G2n 
..
..
.
. 
Gn2 ... Gnn 
V1
V2
..
.
Vn
I1
 I2 
= 
..
 .
 In 
3
Thevenin Equivalent Circuit
• The principle system matrix describes the
self-conductance and mutual-conductances
connected to each Node. The source vector
shows a current source connected between
Nodes.
G11
 G21

.
.
 .
 Gn1
G12 ... G1n

G22 ... G2n 
..
..
.
. 
Gn2 ... Gnn 
V1
V2
..
.
Vn
I1
 I2 
= 
..
 .
 In 
4
Thevenin Equivalent Circuit
• Consider extracting the kth current source
from the network and observing the
network properties at the terminals of that
current source.
Any
Network
a

Vk

b
Ik
5
Thevenin Equivalent Circuit
• Vk is the voltage at the kth node with respect
to the reference node. This voltage can be
calculated by the formula:
G11
 G21
 .
 ..
 Gn1
G12 ... G1n

G22 ... G2n 
.
..
..
. 
Gn2 ... Gnn 
V1

V2 
.. =
. 
Vn 
I1

I2 
..
. 
In 
k
Vk =

6
Thevenin Equivalent Circuit
•  is the determinant of the system matrix and k
is the determinant of the system matrix with the
current source vector substituted for the kth
column. The determinant k may be expanded
down the kth column to reveal :
k = I11k + I22k + … + Ikkk + … + Innk
7
Thevenin Equivalent Circuit
• The node voltage Vk is thus:
I11k + I2 2k + … + Ikkk + … + In nk
Vk =

• These terms may be arranged to the form:
Ikkk
Vk =


+

I11k I22k
Ik – 1 k – 1 k – 1
+
+… +



Ik + 1 k + 1 k + 1
Innk
+
+… +





8
Thevenin Equivalent Circuit
• The quotient ii/ has the units of resistance.
Consequently the sum of terms in brackets
represents an effective voltage (current times
resistance).
Ikkk
Vk =


+

I11k I22k
Ik – 1 k – 1 k – 1
+
+… +



Ik + 1 k + 1 k + 1
Innk
+
+… +





9
Thevenin Equivalent Circuit
• The quantity in the brackets is defined as the
Thevenin equivalent voltage:
VTH

=

I11k I22k
Ik – 1 k – 1 k – 1
+
+… +



Ik + 1 k + 1 k + 1
Innk
+
+… +





10
Thevenin Equivalent Circuit
• The quotient kk/ has the units of resistance. The
Thevenin equivalent resistanceis defined:
kk
RTH =

• The node voltage Vk is:
Vk = IkRTH + VTH
11
Thevenin Equivalent Circuit
• The Network may be replaced by this equivalent
circuit:
RTH

VTH


 a 
Vk
b 
Ik
Vk = IkRTH + VTH
12
Thevenin Equivalent Circuit
• If the terminals a and b are open-circuited, that is,
if Ik = 0, the voltage measured at the terminals is
called the open-circuit voltage. This voltage is
VOC = VTH
• This is illustrated in the following Figure.
RTH

a 

VOC
b 
VTH
13
Thevenin Equivalent Circuit
• If a short-circuit link is installed between
terminals a and b, the short-circuit current can be
calculated. This is illustrated in the following
Figure.
RTH

a
VTH
ISC

b
14
Thevenin Equivalent Circuit
• The short-circuit current at terminals a and
b is:
VTH
ISC =
RTH
RTH

a
VTH
ISC

b
15
Thevenin Equivalent Circuit
• The Thevenin Equivalent Resistance is:
VTH
RTH =
ISC
RTH

a
VTH

b
16
Thevenin Equivalent Circuit
Any
Network
a 
Any
Network
VOC
b 
ISC
b
RTH
VTH = VOC

VOC
RTH =
ISC
a
a
VTH

b
17
Norton Equivalent Circuit
• Perform a source transformation on a
Thevenin Equivalent Circuit to get a Norton
Equivalent Circuit.
RTH

a
a
IN
VTH

b
Thevenin Equivalent Circuit
RN
b
Norton Equivalent Circuit
18
Norton Equivalent Circuit
Any
Network
a 
Any
Network
a
VOC
b 
ISC
b
a
IN = ISC
IN
VOC
RN =
ISC
RN
b
19
Thevenin/Norton Equivalent Circuits
RTH

a
a
IN
VTH

RN
b
b
Thevenin Equivalent Circuit
Norton Equivalent Circuit
VOC = VTH = IN RN
ISC
VTH
=
= IN
RTH
RTH
VOC
= RN =
ISC
20
How to find the Thevenin
Equivalent Circuit
• Find the open-circuit voltage at the
terminals:
VTH = VOC
• Find the short-circuit current at the
terminals:
IN = ISC
21
How to find the Thevenin
Equivalent Circuit
• The Norton/Norton resistance may be found
in one of three ways:
• If VOC and ISC have been determined:
VOC
RTH = RN =
ISC
22
How to find the Thevenin
Equivalent Circuit
• If the network has only independent voltage and
current sources, turn off all of the independent
sources (replace voltage sources with shortcircuits; replace current sources with open
circuits), and determine the input resistance of the
network at the terminals.
Any Network
(All independent
sources disabled)
a
RTH
b
23
How to find the Thevenin
Equivalent Circuit
• If the network contains independent and
dependent sources, turn off the independent
sources, leave the dependent sources intact, apply
a test voltage source to the terminals and
determine the resulting current into the terminals.
The Thevenin/Norton resistance is
RTH = VTest/ITest.
ITEST
Any Network
(All independent sources
disabled. Dependent
sources intact)
a

VTEST
b

24
How to find the Thevenin
Equivalent Circuit
• If the network contains independent and
dependent sources, turn off the independent
sources, leave the dependent sources intact, apply
a test current source to the terminals and
determine the resulting voltage at the terminals.
The Thevenin/Norton resistance is
RTH = VTest/ITest.
Any Network
(All independent sources
disabled. Dependent
sources intact)
a

VTEST
 b
ITEST
25
How to find the Thevenin
Equivalent Circuit
• If the network contains only independent
voltage sources and independent current
sources, the Thevenin/Norton equivalent
circuit may also be found by performing
source transformations until a Thevenin or
Norton circuit is revealed.
26
Example
• Find the Thevenin Equivalent circuit at terminals a
and b.
R1
V1
R3

VS
R2

IS
a

VOC

b
27
Example
• Proceed by finding the open-circuit voltage at the
terminals.
• We note that the open-circuit voltage is the same
as the voltage at Node 1.
R1
V1
R3

VS
R2

IS
a

VOC

b
28
R1
V1
R3

VS
R2
IS

a

VOC

b
VOC = V1
• This voltage may be determined by writing
a node-voltage equation:
VS – V1
V1
+ IS =
R1
R2
29
R1
V1
R3

VS
R2
IS

a

VOC

b
VS
1 
 1
+ IS = V1 + 
R1
 R1 R2 
VS
R1 + R2
+ IS = V1
R1
R1R2
VS
 R1R2
V1 =  + IS
= VTH
R
R
+
R
 1
 1
2
30
R1
V1
R3
a

VS
R2
ISC
IS

b
• The short-circuit current is determined
through the short-circuit link.
VS – V1
V1 V1
+ IS = +
R1
R2 R3
31
R1
V1
R3
a

VS
R2
ISC
IS

b
VS
1 1 
 1
+ IS = V1 + + 
R1
 R1 R2 R3 
VS
R2R3 + R1R3 + R1R2
+ IS = V1
R1
R1R2R3
R1R2R3
 VS

V1 =  + IS 
 R1
 R2R3 + R1R3 + R1R2
32
R1
V1
R3
a

VS
R2
ISC
IS

b
• The short-circuit current is found by:
V1  VS
R1R2

ISC = =  + IS
R3  R1
 R2R3 + R1R3 + R1R2
33
R1
V1
R3

VS
R2
IS

a

VOC

b
• The Thevenin Resistance is:
VOC
RTH =
ISC
VS
 R1R2
R + IS  R + R
 1
 1
2
RTH =
R1R2
 VS

 R + IS R R + R R + R R
 1
 2 3
1 3
1 2
34
R1
V1
R3

VS
R2
IS

a

VOC

b
• The Thevenin Resistance is:
R2R3 + R1R3 + R1R2
RTH =
R1 + R2
35
• The Thevenin Equivalent Circuit is:
R2R3 + R1R3 + R1R2
R1 + R2
VS + IS R1R2
R1  R1 + R2


a
b
36
• The Norton Equivalent Circuit is:
R2R3 + R1R3 + R1R2
R1 + R2
R1R2
 VS + IS
 R1
 R2R3 + R1R3 + R1R2
a
b
37
• The Thevenin Resistance may be found by
turning off all of the independent sources
and finding the equivalent resistance as
illustrated in the following figure.
R1
V1
R3
a
RTH
R2
b
38
R1
V1
R3
a
RTH
R2
b
R1R2
RTH = R3 +
R1 + R2
R3R1 + R3R2 + R1 R2
RTH =
R1 + R2
39
• The Thevenin Equivalent Circuit may also
be found by performing source
transformations on the original circuit:
R1
R3
a

VS
R2
IS

b
40
• The Thevenin Equivalent Circuit may also
be found by performing source
transformations on the original circuit:
R3
VS
R1
R1
R2
a
IS
b
41
• The Thevenin Equivalent Circuit may also
be found by performing source
transformations on the original circuit:
R3
VS
+I
R1 S
a
R1R2
R1 + R2
b
42
• The Thevenin Equivalent Circuit may also
be found by performing source
transformations on the original circuit:
R1R2
R1 + R2
R3
a

VS + IS R1R2
R1  R1 + R2 
b
43
• The Thevenin Equivalent Circuit may also
be found by performing source
transformations on the original circuit:
R3R1 + R3R2 + R1R2
R1 + R2
a

VS + IS R1R2
R1  R1 + R2 
b
44
• The Norton Equivalent Circuit may also be
found by performing source transformations
on the original circuit:
a
R1R2
VS + IS
R1  R3R1 + R3R2 + R1R2
R3R1 + R3R2 + R1R2
R1 + R2
b
45
Example
• Find the Thevenin Equivalent circuit at terminals a
and b.
 V1

IS
V1


R2
a

R1
VOC

b
46
 V1


IS
V1
R2
a

R1

VOC

b
• The open-circuit voltage is:
VOC = V1 + V1 = V1(1 + )
• In the open-circuit condition, IS flows through R1,
so V1 = ISR1, and
VOC = ISR1(1 + )
47
 V1

R2
a

IS
V1
R1
ISC

b
• The short-circuit current is found from this circuit.
• Writing a node-voltage equation for voltage V1
gives:
V1 V1 + V1
IS = R +
R2
1
R1 + R2 + R1
IS = V1
R1R2
48
 V1

R2
a

IS
V1
R1
ISC

b
• Solving for V1.
ISR1R2
V1 =
R1(1 + ) + R2
• The short-circuit current is:
V1 + V1
1+
ISC =
= V1 R
R2
2
49
 V1

R2
a

IS
V1
R1
ISC

b
• The short-circuit current is:
ISR1(1 + 
ISC =
R1(1 + ) + R2
50
 V1


IS
V1
R2
a

R1

VOC

b
• The Thevenin Resistance is:
VOC
RTH = I
SC
RTH =
ISR1(1 + )
ISR1(1 + 
R1(1 + ) + R2
51
 V1


IS
V1
R2
a

R1

VOC

b
• The Thevenin Resistance is:
RTH = R1(1 + ) + R2
52
• The Thevenin Equivalent Circuit is:
R1(1 + ) + R2

ISR1(1 + )

a
b
53
• The Norton Equivalent Circuit is:
a
ISR1(1 + 
R1(1 + ) + R2
R1(1 + ) + R2
b
54
• The Thevenin resistance may be found by
turning off the independent sources and
applying a test voltage.
 V1

R2
a ITest

V1

VTest
R1


b
55
 V1

R2
a ITest

V1

VTest
R1


b
• A single mesh-current equation may be
written:
VTest = ITestR2 + V1 + V1
VTest = ITestR2 +  ITestR1 + ITestR1
VTest = ITest(R2 + R1 + R1)
56
 V1

R2
a ITest

V1

VTest
R1


b
• Solve for the Thevenin Resistance:
VTest = ITest(R2 + R1 + R1)
VTest
RTH = I
= R1(1 + ) + R2
Test
57