E.P. 155.3: Electric and Magnetic Circuits I
Lecture 19
March 17th, 2005
Maximum Power Transfer Theorem
Reading:
Boylestad, 3rd Canadian Edition
Chapter 9.5, Pages 256 - 264
Assignment:
Assignment #9
Due: March 24th, 2005
E.P. 155.3: Electric and Magnetic Circuits I
Maximum Power Transfer Theorem
A load will receive maximum power from a linear
bilateral dc network when its total resistive value
is exactly equal to the Thévenin resistance of the
network as “seen” by the load.
Consider the following circuit:
RTH
ETH
RL
This is nothing more than a Thévenin equivalent
circuit with a variable load resistance, RL, attached to the
two terminals.
For the circuit shown
IL =
ETH
(RTH + RL ) ,
and therefore
ETH 2
2
−2
PL = I L RL =
2 RL = ETH ( RTH + RL ) .
(RTH + RL )
2
March 17th, 2005
Maximum Power Transfer Theorem
2
E.P. 155.3: Electric and Magnetic Circuits I
Substituting
x = RL ,
and using the following relationship (Product Rule)
[ f ( x) g ( x) ]' =
f ' ( x ) g ( x ) + f ( x ) g ' (x ) ,
take the derivative of PL (x) and set it equal to
0
dPL ( x)
= 1ETH 2 (RTH + x )− 2 − 2 ETH 2 x(RTH + x )− 3 = 0
dx
[
]
= ETH 2 ( RTH + x )−2 − 2 x( RTH + x )−3 = 0 .
Simplifying
1−
2x
=0
RTH + x
RTH + x = 2 x
RTH = x .
Therefore (after back substituting RL = x )
RL = RTH .
The load resistance is equal to the
Thévenin resistance for maximum power
transfer.
March 17th, 2005
Maximum Power Transfer Theorem
3
E.P. 155.3: Electric and Magnetic Circuits I
Example #1:
What is the value of RL for the maximum power
transfer?
2Ω
10V
a
2Ω
RL
b
March 17th, 2005
Maximum Power Transfer Theorem
4
E.P. 155.3: Electric and Magnetic Circuits I
Example #2:
What the value for the maximum power that can be
transferred to the load?
RTH
ETH
RL
From previous discussion we know that the value of
RL required for maximum power transfer is when it is
equal to RTH.
Therefore it is to our benefit to calculate a general
expression for the value of the PMAX.
March 17th, 2005
Maximum Power Transfer Theorem
5
E.P. 155.3: Electric and Magnetic Circuits I
Example #3:
What is the value of R2 for maximum power in R4? Be
careful here. This is not a test of whether you know the
Maximum Power Transfer Theorem as such. It is intended
to see if you understand Power Transfer in general.
100V
March 17th, 2005
R1
R3
75Ω
25Ω
R2
R4
Maximum Power Transfer Theorem
6
E.P. 155.3: Electric and Magnetic Circuits I
Example #4:
Find the maximum power that can be transferred to
the resistor R.
3Ω
12V
March 17th, 2005
2Ω
6Ω
2A
Maximum Power Transfer Theorem
R
7
E.P. 155.3: Electric and Magnetic Circuits I
Real World Experiment:
1.
Using Microsoft Excel you can calculate the
power delivered to the load and plot the results.
The location of the power peak should
correspond to a RL value equal to RTH.
The following Thévenin equivalent circuit was
used:
2Ω
10V
RL
The following is a graph from Excel showing a
plot of the power delivered to the load as the
resistance is varied. The peak of 12.5 watts
occurs at RL = 2Ω.
Maximum Power Transfer
Load Power (watts)
14.000
12.000
10.000
8.000
6.000
4.000
2.000
0.000
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
Load Resistance (ohms)
March 17th, 2005
Maximum Power Transfer Theorem
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E.P. 155.3: Electric and Magnetic Circuits I
Efficiency
The efficiency, ç, in percent (%) of power transfer
from a circuit to a load is defined as
η=
PL
∗ 100%
PS
where
PL is the power delivered to the load.
PS is the power supplied by the circuit source(s).
Substituting
I 2 RL
RL
η= 2
∗ 100% =
∗ 100%
(
)
I RTotal
RTH + RL
Note:
RL >> RTH
ç à 100%
RL << RTH
çà
RL = RTH
ç à 50%
March 17th, 2005
0%
Maximum Power Transfer Theorem
9
E.P. 155.3: Electric and Magnetic Circuits I
The following is a graph from Excel showing a plot of
the efficiency of the power delivered to the load as the
load resistance, RL, is varied for the circuit considered
previously.
Note that at the maximum power transfer point, RL =
2Ω, only 50% of the power sourced is delivered to the
load.
Efficiency
70.0
Efficiency (%)
60.0
50.0
40.0
30.0
20.0
10.0
0.0
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
Load Resistance (ohms)
March 17th, 2005
Maximum Power Transfer Theorem
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