Download Lesson 9 Linearity and Superposition

Basic Electric Circuits
Linearity And
Superposition
Lesson 9
Basic Electric Circuits
Linearity and Superposition: Linearity.
Basically, a mathematical equation is said to be linear
if the following properties hold.
• homogenity
• additivity
What does this mean? We first look at the
property of homogenity.
1
Basic Electric Circuits
Linearity : Homogeneity.
Homogenity requires that if the input (excitation)
of a system (equation) is multiplied by a constant,
then the output should be obtained by multiplying
by the same constant to obtain the correct solution.
Sometimes equations that we think are linear, turn
out not be be linear because they fail the homogenity
property. We next consider such an example.
2
Basic Electric Circuits
Linearity : Homogeneity (scaling).
Illustration: Does homogenity hold for the following equation?
Given,
y = 4x
Eq 9.1
If x = 1, y = 4. If we double x to x = 2 and substitute
this value into Eq 9.1 we get y = 8.
Now for homogenity to hold, scaling should hold for y.
that is, y has a value of 4 when x = 1. If we increase
x by a factor of 2 when we should be able to multiply
y by the same factor and get the same answer and when
we substitute into the right side of the equation for
x = 2.
3
Basic Electric Circuits
Linearity : Homogeneity (scaling).
Illustration: Does homogenity hold for the following equation?
Given,
y = 4x + 2
Eq 9.2
If x = 1, then y = 6. If we double x to x=2, then y = 10.
Now, since we doubled x we should be able to double
the value that y had when x = 1 and get y = 10. In this
case we get y = (2)(6) = 12, which obviously is not 10,
so homogenity does not hold.
4
We conclude that Eq 9.2 is not a linear equation. In some
ways that goes against the gain of what we have been
taught about linear equations.
Basic Electric Circuits
Linearity : Homogeneity (scaling).
Many of us were brought-up to think that if plotting
an equation yields a straight line, then the equation
is linear. From the following illustrations we have;
homogenity
does not hold
homogenity
holds
y
y
Lnear
0
5
x
2
0
Not Linear
x
Basic Electric Circuits
Linearity : Additivity Property.
The additivity property is equivalent to the
statement that the response of a system to
a sum of inputs is the same as the responses
of the system when each input is applied
separately and the individual responses
summed (added together).
This can be explained by considering the
following illustrations.
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Basic Electric Circuits
Linearity : Additivity Property.
Illustration:
Given, y = 4x.
Let x = x1, then y1 = 4x1
Let x = x2, then y2 = 4x2
Then y = y1 + y2 = 4x1 + 4x2
Eq 9.3
Also, we note,
y = f(x1 + x2) = 4(x1 + x2) = 4x1 + 4x2
Eq 9.4
Since Equations (9.3) and (9.4) are identical,
the additivity property holds.
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Basic Electric Circuits
Linearity : Additivity Property.
Illustration:
Given, y = 4x + 2.
Let x = x1, then y1 = 4x1 + 2
Let x = x2, then y2 = 4x2 + 2
Then y = y1 + y2 = 4x1+2 + 4x2+2 = 4(x1+x2) + 4
Eq 9.5
Also, we note,
y = f(x1 + x2) = 4(x1 + x2) + 2
Eq 9.6
Since Equations (9.5) and (9.6) are not identical,
the additivity property does not hold.
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Basic Electric Circuits
Linearity :
Example 9.1: Given the circuit shown
in Figure 9.1. Use the concept of linearity
(homogeneity or scaling) to find the current I0.
11 
VS = 90 V
+
_
I0
6
12 
Figure 9.1: Circuit for Example 9.1.
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Assume I0 = 1 A. Work back to find that this gives
VS = 45 V. But since VS = 90 V this means the true
I0 = 2 A.
Basic Electric Circuits
Linearity :
Example 9.2: In the circuit shown
below it is known that I0 = 4 A when IS = 6 A. Find
I0 when IS = 18 A.
I0
Rx
IS
4
2
Figure 9.2: Circuit for Example 9.2
Since IS NEW = 3xIS OLD we conclude I0 NEW = 3xI0 OLD.
Thus, I0 NEW = 3x4 = 12 A.
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Basic Electric Circuits
Linearity : Question.
For Example 9.2, one might ask, “how do we know the
circuit is linear?” That is a good question. To answer,
we assume a circuit of the same form and determine if we
get a linear equation between the output current and the
input current. What must be shown for the circuit below?
RX
IS
I0
R1
R2
Figure 9.3: Circuit for investigating linearity.
Basic Electric Circuits
Linearity : Question, continued.
We use the current splitting rule (current division)
to write the following equation.
( I S )( R1 )
I0 
 ( K )( I s )
( R1  R2 )
Eq 9.7
The equation is of the same form of y = mx, which
we saw was linear. Therefore, if R1 and R2 are constants
then the circuit is linear.
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Basic Electric Circuits
Superposition :
One might read (hear) the following regarding superposition.
(1) A system is linear if superposition holds.
(2) Superposition holds if a system is linear.
This sounds a little like the saying of which comes
first, “the chicken or the egg.”
Of the two statements, I believe one should remember
that if a system is linear then superposition applies (holds).
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Basic Electric Circuits
Superposition: Characterization of Superposition.
Let inputs f1 and f2 be applied to a system y such that,
y = k 1 f1 + k 2f2
Where k1 and k2 are constants of the systems.
Let f1 act alone so that, y = y1 = k1f1
Let f2 act alone so that, y = y2 = k2f2
The property of superposition states that if f1 and f2
Are applied together, the output y will be,
y = y1 + y2 = k1f1 + k2f2
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Basic Electric Circuits
Superposition: Illustration using a circuit.
Consider the circuit below that contains two voltage sources.
R1
V2
_+
I
V1
R2
+
_
R3
Figure 9.4: Circuit to illustrate superposition
We assume that V1and V2 acting together
produce current I.
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Basic Electric Circuits
Superposition: Illustration using a circuit.
R1
R1
V2
_ +
V1
+
_
I1
R2
R3
V1 produces current I1
I2
R2
R3
V2 produces current I2
Superposition states that the current, I, produced by both
sources acting together (Fig 9.4) is the same as the sum of
the currents, I1 + I2, where I1 is produced by V1 and I2 is
produced by V2.
16
Basic Electric Circuits
Superposition: Example 9.3. Given the circuit below.
Demonstration by solution that superposition holds.
2
VB
_
3
+
IT
VA
+
_
_
VC +
VA = 10 V, VB = 5 V, VC = 15 V
Figure 9.5: Circuit for Example 9.3
With all sources acting: IL = 6 A
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Basic Electric Circuits
Superposition: Example 9.3. Given the circuit below.
Demonstration by solution that superposition holds.
2
IT
VA
_
3
+
VB
+
VC
_
VA = 10 V, VB = 5 V, VC = 15 V
With VA + VB acting,,VC = 0:
IA+B = 3 A
With VC acting, VA + VB = 0:
IC = 3 A
We see that superposition holds.
18
_
+
Basic Electric Circuits
Superposition: Example 9.4. Given the circuit below.
Find the current I by using superposition.
12 
I
IS = 3 A
6
+
_
VS = 54 V
Figure 9.5: Circuit for Example 9.4.
First, deactivate the source IS and find I in the 6  resistor.
Second, deactivate the source VS and find I in the 6  resistor.
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Sum the two currents for the total current.
Basic Electric Circuits
Superposition: Example 9.4. Given the circuit below.
Find the current I by using superposition.
12 
IVs
6
IVs = 3 A
20
+
_
VS = 54 V
Basic Electric Circuits
Superposition: Example 9.4. Given the circuit below.
Find the current I by using superposition.
Is
IS = 3 A
12 
6
3x12
IS 
2 A
(3  12)
Total current I: I = IS + Ivs = 5 A
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CIRCUITS
End of Lesson 9
Linearity and Superposition