Transcript Unit 5 PowerPoint Slides

EGR 2201 Unit 5
Linearity, Superposition, & Source
Transformation
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Read Alexander & Sadiku, Sections 4.1
to 4.4.
Homework #5 and Lab #5 due next
week.
Quiz next week.
Techniques That Can Simplify
Circuit Analysis
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Nodal analysis and mesh analysis are
powerful tools for analyzing circuits.
But you should also know other
techniques that may simplify the
equations that you get when you
perform nodal or mesh analysis, or
may save you from having to
perform nodal or mesh analysis at
all.
Example of a Simplifying
Technique
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One such technique is our method of
finding the equivalent resistance of
several series-parallel resistors.
Example: In this
circuit, if you want
to find i, using this
technique will get
you the answer
much faster than
doing a full nodal
or mesh analysis of the circuit.
Other Useful Techniques
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The current-divider rule and voltagedivider rule are two other useful tools
that can save you work when
analyzing circuits.
Chapter 4 presents several new
techniques:
 Linearity
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Superposition
Source transformation
Thevenin’s theorem
Norton’s theorem
Maximum-power-transfer theorem
Linear Circuits
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A linear circuit is a circuit whose output is
linearly related (or directly proportional) to
its input.
In other words, in a linear circuit, if the
input voltage or current increases by a
factor of k, then the output voltage or
current must increase by that same factor.
Typically the input
is a voltage source
or current source
and the output is a
resistor’s voltage or
current.
Why Linearity is Useful
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Linearity is a useful property because
it lets us easily answer certain “whatif” questions about linear circuits.
Example: Suppose we know that in the
linear circuit shown, i = 2 A when
vs = 10 V.
Then we know
that doubling
vs to 20 V will
also double
i to 4 A.
How to Tell if a Circuit is Linear
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Any circuit that contains only linear
elements (which include resistors,
capacitors, and inductors),
independent sources, and linear
dependent sources is a linear circuit.
All of the circuits that we will study in
this course are linear circuits.
In other courses you will study nonlinear circuits.
Caution: Power Is Not Related
Linearly to Voltage and Current
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Recall that a resistor’s power is nonlinearly related to its current and
voltage by the equations
p=i
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2
R
p=v
Therefore, for example,
doubling vs in
the circuit
shown will
not double
R’s power.
2
R
Techniques That Can Simplify
Circuit Analysis
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Chapter 4 presents several new
techniques:
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Linearity
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Superposition
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Source transformation
Thevenin’s theorem
Norton’s theorem
Maximum-power-transfer theorem
Superposition Principle (1 of 2)
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The superposition principle says that
to find the total effect of two or more
independent sources in a linear
circuit, you can find the effect of
each source acting alone, and then
combine those effects.
Superposition Principle (2 of 2)
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To find the effect of an independent
source acting alone, you must turn
off all of the other independent
sources.
 To turn off a voltage source,
replace it by a short circuit.
 To turn off a current source,
replace it by an open circuit.
Steps in Using the Superposition
Principle
1. Select one independent source and
turn off the others, as explained
above. Find the desired voltage or
current due to this source using
techniques from earlier chapters.
2. Repeat step 1 for each of the other
independent sources.
3. Add the values obtained from the steps
above, paying careful attention to the
signs (+ or ) of each value.
Example: Using Superposition (1 of 2)
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Suppose we wish
to find v in the
circuit shown.
With the voltage
source acting alone:
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With the current
source acting alone:
Example: Using Superposition (2 of 2)
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In this circuit, use
voltage-divider rule
to find v1 = 2 V.
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In this circuit, use
current-divider rule
to find i3 = 2 A, Ohm’s
law to find v2 = 8 V.
Therefore, in the
original circuit,
v = 2 V + 8 V = 10 V.
Wouldn’t Nodal Analysis Be
Easier?
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For this circuit, you
might argue that
nodal (or mesh)
analysis is easier
than superposition.
And you might be right.
But looking ahead to the second half
of this course, if one source were a
DC source and the other were an AC
source, we would have no choice but
to use superposition.
Techniques That Can Simplify
Circuit Analysis
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Chapter 4 presents several new
techniques:

Linearity
Superposition
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Source transformation
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Thevenin’s theorem
Norton’s theorem
Maximum-power-transfer theorem
Combining Series or Parallel
Sources
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Before looking at source
transformation, note the following:
Just as we can combine series or
parallel resistors into a single
equivalent resistor, we combine series
or parallel sources into a single
equivalent source.
In particular:
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We can combine series voltage sources
into an equivalent voltage source.
We can combine parallel current
sources into an equivalent current source.
Series-Aiding Voltage Sources
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If two series-connected voltage sources
produce current in the same direction, they
are said to be series-aiding.
The net effect on the
circuit is the same
as that of a single
source whose
voltage equals the
sum of the two
voltage sources.
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Example: Most flashlights
have two 1.5-V batteries
in series to give 3 V.
Series-Opposing Voltage Sources
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If two series-connected sources
produce current in opposite
directions, they’re said to be seriesopposing.
The effect is the same
as that of a single
source equal in
magnitude to the
difference between
the source voltages
and having the same
polarity as the larger of the two.
Parallel Voltage Sources?
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Although there are exceptions, in
general we don’t connect voltage
sources in parallel with each other.
To see why not, think about this:
what is the voltage between points a
and b in the circuit below?
Parallel Current Sources
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Current sources can be connected in
parallel. The parallel-connected
sources can be replaced by a single
equivalent current source that
produces a current equal to the
algebraic sum of the sources.
Series Current Sources?
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In general we don’t connect current
sources in series with each other.
To see why not, think about this: how
much current flows around the loop
in the circuit below?
Source Transformation
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Sometimes when
you're analyzing
a circuit, it can
be useful to
substitute a
voltage source (and a resistor in series with
it) by a current source (and a resistor in
parallel with it), or vice versa.
Either substitution is called a source
transformation.
Two questions:
1. Why would this be useful?
2. Are we allowed to do this?
Example of Why Source
Transformation Can Be Useful
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Suppose we wish to
find vo in the circuit
to the right.
This is not difficult,
but it requires a bit
of work.
We may get the
answer more easily if
we can first replace
the current-source-andparallel-resistor with
a voltage-source-and-series-resistor.
Are We Allowed to Do This?
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Yes, we’re allowed
to make this
substitution,
as long as we
use the right
values for the elements.
The resistor R in series with the voltage
source must be the same size as the one in
parallel with the current source.
The current source’s and voltage source’s
values must satisfy the following:
𝑣𝑠
𝑣𝑠 = 𝑖𝑠 𝑅
or
𝑖𝑠 =
𝑅
Source Transformation: A More
Complicated Example (1 of 2)
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In some cases we can use source
transformation repeatedly to convert a
complicated problem into a much simpler one.
The book’s Example 4.6 shows such a case:
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Original circuit:
After transforming
both sources:
Continued on next slide…
Source Transformation: A More
Complicated Example (2 of 2)
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…From previous slide:
After combining
4- with 2- and
transforming 12-V
source:
After combining
6- with 3-
and combining
2-A with 4-A: