Electric Circuits I “Dr. Ahmed El-Shenawy”
Electrical Circuits I
Lecture 8
<Dr Ahmed El-Shenawy>
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin and Norton theorems
•
•
Thevenin theorem tells us that we can replace the entire network,
exclusive of the load resistor, by an equivalent circuit that contains
only an independent voltage source in series with a resistor in such a
way that the current-voltage (i-v) relationship at the load resistor is
unchanged.
Norton theorem tells us that we can replace the entire network,
exclusive of the load resistor, by an equivalent circuit that contains
only an independent current source in parallel with a resistor in such
a way that the current-voltage (i-v) relationship at the load resistor is
unchanged.
io
+
vo
-
io
io
+
+
vo
vo
-
(a)
A complex network
including a load resistor RL.
(b)
A Thevenin equivalent
network
(c)
A Norton equivalent
network
2
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
•How to determine vTH(t) and RTH for a particular
circuit.
•It is helpful to note that if we connect no load and
therefore io(t) = 0, then we can determine vTH(t)
from
io
vTH ( t )  v opencircui t ( t )  v oc ( t )
+
vo
-
where voc(t) is called the open circuit voltage
•If we short circuit the two terminals to force vo(t)
= 0, then we get
v TH ( t )
 i shortcircu it ( t )  i sc ( t )
RTH
•If vTH(t) ≠ 0, then ishortcircuit(t) ≠ 0 and we find
R TH
v oc ( t )

i sc ( t )
3
Electric Circuits I “Dr. Ahmed El-Shenawy”
Case 1 example 1
Find the Thevenin equivalent circuit at
terminal pair a and b for the circuit
shown.
+
This specific problem can be solved by
using different approaches. We solve the
problem by using source transformation
technique.
-
Thus we have
RTH = 12
and
vTH = -8V
4
Thevenin Theorem
Network
1
Electric Circuits I “Dr. Ahmed El-Shenawy”
A
B
• We now deactivate all sources of Network 1.
• To deactivate a voltage source, we remove
the source and replace it with a short circuit.
• To deactivate a current source, we remove
the source.
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
Find VX by first finding VTH and RTH to the left of A-B.
4
12 

_
30 V +
6
A
+
VX
2
_

B
First remove everything to the right of A-B.
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
4
12 

_
30 V +
A
6

B
(30)(6)
VAB 
 10V
6  12
Notice that there is no current flowing in the 4  resistor
(A-B) is open. Thus there can be no voltage across the
resistor.
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
We now deactivate the sources to the left of A-B and find
the resistance seen looking in these terminals.
4
12 

A
RTH
6

We see,
RTH = 12||6 + 4 = 8 
B
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
After having found the Thevenin circuit, we connect this
to the load in order to find VX.
RTH
A

8
VTH
+
_
10 V
+
2
VX
_
B

(10)( 2)
VX 
 2V
28
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
For the circuit below, find VAB by first finding the Thevenin
circuit to the left of terminals A-B.
1.5 A
5
 A
10 
20 V _+
20 
17 

B
We first find VTH with the 17  resistor removed.
Next we find RTH by looking into terminals A-B
with the sources deactivated.
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
1.5 A
5
 A
10 
20 V _+
20 

B
20(20)
VOS  VAB  VTH  (1.5)(10) 
(20  5)
 VTH  31V
Thevenin Theorem
Electric Circuits I “Dr. Ahmed El-Shenawy”
5
 A
10 
20 

RTH
B
5(20)
 10 
 14 
(5  20)
Electric Circuits I “Dr. Ahmed El-Shenawy”
Thevenin Theorem
RTH
A

14 
VTH
+
_
31 V
+
17 
VAB
_
B

We can easily find that,
VAB  17 V
Electric Circuits I “Dr. Ahmed El-Shenawy”
Norton theorem
io
•Based on source transformation we have learned, we
can determine iN(t) and RN
+
vo
(c)
R
N
 R TH
v oc ( t )

i sc ( t )
vTH
i N (t ) 
 ishortcircu it (t )  isc (t )
RTH
Electric Circuits I “Dr. Ahmed El-Shenawy”
Norton theorem
Find the Norton equivalent circuit to the left of terminals A-B
for the network shown below. Connect the Norton equivalent
circuit to the load and find the current in the 50  resistor.
10 A
20 
40 
A

+
_
50 V
60 
50 

B
Electric Circuits I “Dr. Ahmed El-Shenawy”
Norton theorem
10 A
20 
+
_
50 V
40 
60 
ISS
It can be shown by standard circuit analysis that
I SS 10.7 A
Electric Circuits I “Dr. Ahmed El-Shenawy”
Norton theorem
It can also be shown that by deactivating the sources,
We find the resistance looking into terminals A-B is
RN  55 
RN and RTH will always be the same value for a given circuit.
The Norton equivalent circuit tied to the load is shown below.
10.7 A
55 
50 
Electric Circuits I “Dr. Ahmed El-Shenawy”
Norton theorem
Electric Circuits I “Dr. Ahmed El-Shenawy”
Norton theorem