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EGR 2201 Unit 6
Theorems: Thevenin’s, Norton’s,
Maximum Power Transfer



Read Alexander & Sadiku, Sections 4.5
to 4.11.
Homework #6 and Lab #6 due next
week.
Quiz next week.
Techniques That Can Simplify
Circuit Analysis

Chapter 4 presents several new
techniques:

Linearity
Superposition
Source transformation

Thevenin’s theorem




Norton’s theorem
Maximum-power-transfer theorem
Thevenin’s Theorem


Thevenin’s theorem says that any
linear two-terminal circuit can be
replaced by an equivalent circuit
consisting of an independent voltage
source VTh in series with a resistor
RTh.
There’s a standard procedure for
finding the values of VTh and RTh. But
first, what exactly does this mean
and why is it useful?
Load


Often an electrical circuit or system
contains an element (called the “load”)
that can be varied while the rest of the
circuit or system stays fixed.
Example: When you
unplug a lamp at
home and plug in
a hair-dryer, you’ve varied one small part of a
huge electrical circuit that extends all the way
back to the power plant. The lamp or hairdryer is the “load” part of the circuit.
What Thevenin’s Theorem Means

Thevenin’s theorem lets us replace
everything on one side of a pair of
terminals by a very simple equivalent
circuit consisting of just a voltage
source and a resistor.
Original Circuit
Equivalent Circuit
Why It’s Useful

This greatly simplifies computation
when you wish to find values of
voltage or current for several
different possible values of a load
resistance.
Original Circuit
Equivalent Circuit
Steps in Finding VTH and RTH
1.
Open the circuit at the two
terminals where you wish to
find the Thevenin-equivalent
circuit. (In the circuit shown,
this means removing the load.)
VTH is the voltage across the
two open terminals. Pay
attention to polarity!
RTH is the resistance looking
into the open terminals with all
independent sources turned off.
2.
3.
•
•
Recall that to turn off a voltage
source we replace it by a short.
To turn off a current source we
replace it by an open.
More on Step 3 (Finding RTH)
 The previous slide said
that RTH is the resistance
looking into the open
terminals with all independent
sources turned off.
 What about dependent sources? We don’t
turn those off.
 As shown in the following two examples, how
you find RTH depends on whether the circuit
contains any dependent sources.
 If no dependent sources, just combine resistors in
series and parallel to find equivalent resistance.
 If you have dependent sources, it’s trickier.
Finding the Thevenin-Equivalent Circuit
Example #1: No Dependent Sources
 Goal: Find the Theveninequivalent circuit
for the circuit in the
book’s Figure 4.27:
 Finding VTH:
 Finding RTH:
Finding the Thevenin-Equivalent Circuit Example #1:
No Dependent Sources (Conclusion)
 We found that VTH = 30 V and RTH = 4 .
 So as far as the load is concerned, the original
circuit below is equivalent to the simpler
circuit on the right.
Original Circuit
Equivalent Circuit
 Question: Suppose you need to find the load current IL
for six different values of RL. Would you rather analyze
the left-hand circuit six times or the right-hand circuit
six times?
Finding RTH For a Circuit That
Contains Dependent Sources
 To understand the steps in finding RTH for a
circuit that contains dependent sources, think
about this question:
 Suppose you have lab equipment
including a voltage source (of known
value vo ) and an ammeter.
If I give you a resistor of
unknown value, how can you find
its resistance?
 Answer: Connect the voltage source
across the resistor and measure
how much current io flows
through the resistor. Then use
Ohm’s law to compute the
𝑣
resistance: 𝑅 = 𝑜
𝑖𝑜
Finding RTH For a Circuit
Containing Dependent Sources
If a circuit contains dependent sources,
follow these steps to find RTH.
1. Turn off all independent sources as
previously described.
2. Connect an independent voltage source of
any value vo (say, 1 V) across the open
terminals.
3. Find the current io flowing into the circuit
from the voltage source that you
connected in Step 2.
4. To find RTh, divide the voltage vo by the
current io. In symbols, RTh = vo ÷ io.
Finding the Thevenin-Equivalent Circuit
Example #2: With Dependent Sources
 Goal: Find the Theveninequivalent circuit
for the circuit in the
book’s Figure 4.31:
 Finding VTH:
 Finding RTH:
Finding the Thevenin-Equivalent Circuit Example #2:
With Dependent Sources (Conclusion)
 We found that VTH = 20 V and RTH = 6 . So
we conclude that, as far as any load
connected to terminals a-b is concerned, the
original circuit on the left below is equivalent
to the much simpler circuit on the right below.
Original Circuit
Equivalent Circuit
Summary: Thevenin’s Theorem

As we’ve seen, Thevenin’s theorem
says that any linear two-terminal
circuit can be replaced by an
equivalent circuit consisting of an
independent voltage source VTh in
series with a resistor RTh.
Original Circuit
Thevenin-Equivalent Circuit
Techniques That Can Simplify
Circuit Analysis

Chapter 4 presents several new
techniques:

Linearity
Superposition
Source transformation
Thevenin’s theorem

Norton’s theorem

Maximum-power-transfer theorem



Norton’s Theorem

Norton’s theorem says that any
linear two-terminal circuit can be
replaced by an equivalent circuit
consisting of an independent current
source IN in parallel with a resistor
RN .
Original Circuit
Norton-Equivalent Circuit
Not Surprising, Is It?

Norton’s theorem follows from Thevenin’s
theorem, since we can substitute a voltagesource-plus-series-resistor by a currentsource-plus-parallel-resistor, or vice versa.
(Remember source transformation?)
Thevenin-Equivalent Circuit
Original Circuit
Norton-Equivalent Circuit
Finding IN and RN
 The book describes how to find IN and RN.
The procedure is similar to how we find VTh
and RTh.
 But you don’t need to learn this new
procedure. Instead, just find VTh and RTh, and
then apply a source transformation:
𝑅N = 𝑅Th
Thevenin-Equivalent Circuit
𝑉Th
and 𝐼N =
𝑅Th
Norton-Equivalent Circuit
Techniques That Can Simplify
Circuit Analysis

Chapter 4 presents several new
techniques:

Linearity
Superposition
Source transformation
Thevenin’s theorem
Norton’s theorem

Maximum-power-transfer theorem




Source and Load



In many cases, we can think of an
electrical system as being composed of
a source of power and a load
connected to that source.
Examples of sources: amplifier,
generator, power supply.
Examples of loads: loudspeaker, electric
motor, antenna.
Maximizing the Load Power

Replacing the source with its Theveninequivalent circuit, we have the following
situation:
Thevenin-equivalent
of source

Variable load
resistance
In many applications, we wish to
maximize the power transferred from a
fixed source to a variable load.
The Load’s Power Depends on the
Load Resistance
For this circuit, the
load resistor’s power
is given by:
2
𝑉𝑇ℎ
2
𝑝 = 𝑖 𝑅𝐿 =
𝑅𝐿
𝑅𝑇ℎ + 𝑅𝐿


Question: For fixed values of VTh and
RTh, what value of RL will result in
maximum load power?

The answer is not obvious, since RL appears
in both the numerator and the denominator.
Variation in the Load’s Power as
RL Varies

A graph of the equation
2
 VTh 
p
 RL
 RTh  RL 
looks like this:


Note that power approaches 0 as RL
approaches 0. Also, power approaches
0 as RL approaches .
What is the value of RL where the graph
peaks?....
Maximum Power Transfer
Theorem


The maximum power transfer
theorem says that
maximum power is
transferred to a
load when the load
resistance equals
the source’s Thevenin
resistance (RL = RTh).
To see this, find
and solve for RL.
𝑑𝑝
,
𝑑𝑅𝐿
set it equal to 0,
Matching Source and Load


When source and load have the same
resistance, they are said to be
matched.
In many practical applications, a
major part of a circuit designer’s
effort is to ensure that components
are matched.
An Important Application of Thevenin’s
Theorem: Source Modeling

We’ve been using ideal voltage
sources. These are theoretical
models that maintain the same
output voltage no matter what
you attach to their terminals.


Symbols for
ideal voltage
sources:
Such sources do not exist in real life.
What about real voltage sources,
such as batteries or power supplies?

To a first approximation, we treat them
as ideal voltage sources.
Source Modeling


For a better
approximation, we
treat a real voltage
source (also called a
“practical” voltage
source) as an ideal voltage source in
series with a resistor.
This resistor is called the source’s
internal resistance: the lower it is,
the better.
Using Source Modeling to Analyze a
Circuit



Suppose we connect a load resistor across
a real voltage source rated at 10 V.
If we treat the real source
as an ideal source, we’ll
conclude that the voltage
across the load resistor is 10 V.
But if we take into account
the source’s internal
resistance, we’ll find that
the voltage across the load
resistor is less than 10 V.
A Practical Implication of This

1.
2.
3.
We’ve seen that a real voltage source’s
voltage will decrease when you connect it
to resistors. Therefore, which is the better
way to build a circuit on the breadboard?
Method A
Adjust the voltage
source to the correct
value.
Build the circuit.
Connect the voltage
source to the circuit.
1.
2.
3.
Method B
Build the circuit.
Connect the voltage
source to the circuit.
Adjust the voltage
source to the correct
value.
It’s Still Just an Approximation

This model does not
perfectly predict all
aspects of a real
voltage source’s
behavior, but it gives
more accurate results
than treating the real
source as an ideal
source.
Making Graphs in Word 2013 (1 of 4)
1.
Select Insert > Chart
on Word’s menu bar.
2.
Select X Y (Scatter).
3.
Select Scatter with
Smooth Lines and
Markers.
4.
Click OK.
Making Graphs in Word 2013 (2 of 4)
5.
Type your data values
in this window.
6.
You can create a new
plot on the same chart
by typing a new column
of data.
Making Graphs in Word 2013 (3 of 4)
7.
Close the data-editing
window by clicking X.
8.
If you need to re-open that window to edit your
data, select Edit Data on Word’s menu bar.
Making Graphs in Word 2013 (4 of 4)
9.
Add axis titles and a chart title by clicking the + and
checking the boxes.
10.
Edit your axis titles and chart title by clicking them and
typing.