Ali A.M.
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
3.3 Norton's Theorem :
Norton's theorem states that : "any electrical circuit can be replaced by an equivalent circuit
consisting of a current source ( ) in parallel with a resistor (
current through the terminals and (
), where ( ) is the short-circuit
) is the input or equivalent resistance at the terminals when the
independent sources are turned off ".
To find Norton's equivalent circuit, the following steps must be followed :
1) Remove the portion of the circuit across which the Norton's equivalent circuit to be found.
2) Mark the terminals of the remaining circuit.
3) Calculate (
) by setting all sources to zero (voltage sources are replaced by short circuits
and current sources are replaced by open circuits), and then find the resultant resistance
between the marked terminals.
4) Calculate ( ) by first returning all sources to their original positions, and then finding the
short circuit current between the two marked terminals.
5) Draw the equivalent circuit shown above and return the already removed part of the circuit to
the equivalent one.
Ex11: Find the Norton's equivalent circuit of the circuit shown below, to the left of the terminals (ab). Then find the current through (RL = 6 Ω, 16 Ω, and 36Ω) .
9
Ali A.M.
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
Solution :

Steps 1 & 2 :

Step 3 :
RN

Step 4 :
By using Ohm's law :

Step 5 :
10
Ali A.M.
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
Ex12: Find the voltage through the (3Ω) resistor in the circuit below using Norton's theorem.
Ex13: Determine the current through the (5Ω) resistor in the circuit below using Norton's theorem.
3.4 Relation between Thevenin's equivalent circuit and Norton's equivalent
circuit :
The Thevenin and Norton equivalent circuits can be found from each other by using the source
conversion method previously discussed, as shown below :
11
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
3.4 Maximum Power Transfer Theorem :
The maximum power transfer theorem states the following : " A load will receive maximum
power from a DC network when its total resistive value is exactly equal to the Thevenin's resistance
of the network as seen by the load. "
Consider the circuit shown. The power delivered to the
load is :
To prove maximum power transfer, we differentiate the
equation above with respect to (
) and set the result
equal to zero. We obtain :
[
]
]
[
[
By varying the load resistance (
]
), the power delivered to the load varies also as sketched in the
figure below. We notice that power is small for small or large values of (
when (
). So, the maximum power
becomes :
12
), but maximum only
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
When dealing with Norton's equivalent circuit, maximum power transfer occurs when:
Ex14: Find the value of (
) for maximum power transfer in the circuit below. Then find the
maximum power.
Solution:

Calculating (
):

Calculating (
):
Using mesh analysis:
Loop1:
Applying KVL around the outer
loop:
13
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
So, for maximum power transfer :
and the maximum power is :
Ex15: Find the value of (
) for maximum power transfer in the circuit below. Then find the
maximum power.
Solution:

Calculating (

Calculating (
):
):
By applying KVL:
So, for maximum power:
and
14
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
Homework:
1) Find the maximum power that can be delivered to the resistor (R) in the circuits below:
[(1.6Ω , 0.625W),(7.2Ω , 1.25W)]
2) Compute the value of (R) that results in maximum power transfer to the (10 Ω) resistor in
the circuit below:
[20Ω]
3) (a) For the circuit below, obtain the Thevenin equivalent at terminals a-b.
(b) Calculate the current when
.
(c) Find ( ) for maximum power deliverable.
(d) Determine that maximum power.
4) For the circuit below, determine the value of (R) such that the maximum power delivered to
the load is (3 mW).
[1KΩ]
1
15
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
3.5 Millman's Theorem :
This Theorem is used for finding the common voltage across any network which contains a
number of parallel voltage sources as shown below. Then common voltage VAB which appears
across the output terminals A and B is affected by the voltage sources E1, E2 and E3. The value of
the voltage is given by:
( ⁄ )
( ⁄ )
This voltage represents Thevenin voltage
( ⁄
)
( ⁄ )
( ⁄ )
( ⁄ )
. The Thevenin resistance can be found as usual by
replacing each voltage source by a short circuit.
Ex16: Use Millman’s theorem, to find the voltage across terminals A and B and the load current in
the circuit below.
16
Basics of Electrical Engineering Lectures
University of Missan / College of Engineering
Solution:
( )
( )
(
( )
( )
( )
(
)
)
Ex17: Use Millman’s theorem, to find the voltage across and current through the load resistor (
) in
the circuit below.
Solution:
First thing to do is to convert the current source into a voltage source. Then, the circuit becomes as
follows :
(
)
(
|
)
(
)
|
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