IEEE MULTIDISCIPLINARY ENGINEERING EDUCATION MAGAZINE, VOL. 2, NO. 1, MARCH 2007
1
Analyzing Sensitivity in Electric Circuits
Emmanuel A. Gonzalez, Graduate Student Member, IEEE, Martin Christian G. Leonor,
Leonard U. Ambata, and Cornelio S. Francisco
Abstract—This lecture note presents a well-known sensitivity
analysis technique using Taylor Series expansion that can be
used by students, engineers, and scientists in analyzing electric
circuits. The technique presented, which is also available in [1], is
discussed in detail, having a voltage-divider circuit as an
example.
Index Terms—Sensitivity, Taylor series expansion.
S
I. INTRODUCTION
ensitivity analysis is very important especially in the fields
of electrical and electronics engineering due to the fact
that circuits practically have tolerance values. Some
examples are high-performance synchronous circuit [9],
power electronic circuits [10], and even transmission lines
[11]. In fact, there are a number of texts dealing with circuit
sensitivity analysis tools [2]-[6]. In reality, there is no such
thing as a “perfect” component, and each element in an
electric circuit will always have tolerance values, specifically
minimum and maximum values. For example, if we consider a
simple 100-ohm resistor, it will always have a tolerance value
of, for example, five percent (5%). Thus, the minimum and
maximum values of this passive electric component are 95 and
105 ohms, respectively, without also considering temperature
effects. Even the most precise electric components available in
the market have their own tolerances. Thus, there is a need to
understand the basics of sensitivity and its application to
electric circuit analysis.
In this lecture note, the authors present a well-known
sensitivity analysis tool that is used by engineers and scientists
in analyzing the sensitivity of an electric circuit [1]. In this
note, we are restricted to memoryless circuits, i.e. circuits that
do not store any information at all, e.g. voltage and current.
The basic analysis technique used in this note is rather
straightforward and does not require too much calculus, unlike
other techniques available in some science and engineering
literature. A brief application of the Taylor series expansion is
presented and is used as the key component in sensitivity
analysis.
II. WHAT IS SENSITIVITY ANALYSIS
Sensitivity, generally, is the ability of an entity to be
susceptible to stimulation. If we are talking about electric
circuits, the sensitivity of that circuit is its ability to react with
changes in certain parameters. It is a measure of how the
system reacts to any stimulation, either internal or external.
For example, sensitivity can be the measure of how a voltagedivider circuit composed of a constant dc voltage source and
two resistors in series reacts in terms of the output voltage,
with respect to perturbations in one resistor. When we say
perturbation, we mean that the resistor changes from within its
minimum and maximum levels. Sensitivity Analysis, on the
other hand, is the study of the sensitivity in a system. If we are
applying mathematical and scientific concepts in analyzing the
variations in the output voltage of a voltage-divider circuit
with perturbations in one resistor, then we can say that we are
performing a sensitivity analysis on the circuit.
Generally, sensitivity can be described as the limiting ratio
of the fractional change in a measured value to the fractional
change in value of a certain parameter. Thus, if the change in
a measured value is too small even if the change in a certain
parameter is large, then we can say that the system is not
sensitive or has a low sensitivity. On the other hand, if the
change in a measured value is too large even if the change in a
certain parameter is small, then we say that the system is too
sensitive or has a high sensitivity. Sometimes, we want to
have a system that has a low sensitivity, such as electronic
amplifiers and oscillators in the electronic systems
perspective, and mechanical shock absorbers in mechanical
systems perspective. Furthermore, a high sensitive system is
needed especially if we are dealing with measurements and
data acquisition. Examples are sensors that require higher
sensitivities such as chemical sensors and vibration sensors.
III. TAYLOR SERIES EXPANSION
Before we proceed to the discussion on sensitivity analysis,
a review on Taylor series expansion must be discussed first.
Taylor series expansion is an expansion technique used in
order to approximate a certain function. The Taylor series
expansion of f ( x ) about x = x 0 is defined as
The authors are with the Department of Electronics and Communications
Engineering, College of Engineering, De La Salle University - Manila
({gonzaleze, leonorm, ambatal, franciscoc} @dlsu.edu.ph).
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IEEE MULTIDISCIPLINARY ENGINEERING EDUCATION MAGAZINE, VOL. 2, NO. 1, MARCH 2007
IV. SENSITIVITY ANALYSIS OF A SIMPLE VOLTAGE
DIVIDER CIRCUIT
R1 = 50 ± 5% Ω
+
vi = 5 V
+
R2 = 100 ± 2% Ω
−
vo
−
Fig. 1. An example of a voltage-divider circuit.
f (x ) = f (x 0 ) + f ′ (x 0 ) (x − x 0 )
+
+
f ′′ ( x 0 ) ( x − x 0 )
2
(1)
2!
f ′′′ ( x 0 ) ( x − x 0 )
3
3!
+ ".
We now present a simple sensitivity analysis technique for a
memoryless electric circuit, which is also available in [1].
Consider a voltage divider circuit shown in Fig. 1. The circuit
is composed of a constant dc voltage source and two resistors
in series. The output is the voltage across the 100-ohm
resistor, while the input is a 5-volt dc voltage source. The
value of the series resistance is 50 ohms with a tolerance of
5%, while the 100-ohm resistor has a tolerance of 2%. Thus, if
the 50-ohm resistor has a 5% tolerance, then its value can vary
from 47.5 to 52.5 ohms. On other hand, if the 100-ohm
resistor has a 2% tolerance, then its value can vary from 98 to
102 ohms.
What will happen to the output voltage if the resistances in the
circuit change? Or in other words, how accurate will the
output voltage be with changes in resistor values? To answer
these questions, we need to determine first the change of
values in the resistors. Let the difference between the actual
resistor value and its ideal value be the change in resistance
value, i.e. ∆R Ractual − Rideal . Thus, we can have
∆R1 R1,actual − R1,ideal and ∆R2 R2,actual − R2,ideal . If we
The ideas involved in Taylor series for functions of one
variable can be generalized. For example, the Taylor series of
a two-variable function f ( x , y ) about x = x 0 and y = y 0
assume that ∆R1 and ∆R2 are small enough compared to
their ideal values, then the Taylor series expansion
v 0 ( R1 + ∆R1, R2 + ∆R2 ) =
can be expressed as
f (x, y )
= f (x 0, y0 ) +
+
∂f ( x 0 , y 0 )
∂f ( x 0 , y 0 )
∂x

+2
∂x
2
2
∂ f (a, b )
∂x ∂y
(x
∂ f (x 0, y0 )
∂yx
+
(x
− x0 )
+
2
(2)
− x 0 ) (y − y 0 )
of
2 ∂R1
2
1 ∂ v0
∂R2
∆R2
2
∆R1 +
2
∂R1∂R2
∆R1∆R2
(4)
(y − y )  + ".

2
∂f ( x 0 , y 0 )
∂f ( x 0 , y 0 )
∂x
(y − y ) .
(x
2 ∂R2
2
∆R2 + "
2
v 0 ( R1 + ∆R1, R2 + ∆R2 )
can be approximated by
neglecting second- and higher-order terms, thus having the
expression
0
2
f (x, y ) ≈ f (x 0 , y0 ) +
∂y
∂ v0
2
v 0 ( R1 + ∆R1 , R2 + ∆R2 ) ≈
However, if one wants to make a linear approximation of (2),
then the second- and higher-ordered terms can be eliminated,
having the expression
+
1 ∂ v0
∂R1
∂v 0
∆R1 +
2
2
+
− x0 )
(y − y )
1  ∂ f (x 0 , y0 )
2
∂v 0
v 0 ( R1 , R2 ) +
0
∂y
2
+
(x
2
v 0 ( R1 , R2 ) +
∂v 0
∂R1
∆R1 +
∂v 0
∂R2
∆R2 .
(5)
Consequently, the deviation of the output voltage from its
nominal value is then
− x0 )
(3)
0
A good text containing information in Taylor series expansion
can be found in [6] and [7].
v 0 ( R1 + ∆R1 , R2 + ∆R2 ) − v 0 ( R1 , R2 ) ≈
∂v 0
∂R1
∆R1 +
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∂v 0
∂R2
∆R2 .
(6)
IEEE MULTIDISCIPLINARY ENGINEERING EDUCATION MAGAZINE, VOL. 2, NO. 1, MARCH 2007
Thus, if we consider the circuit given in Figure 1, then
substituting the values will result to
v 0 ( R1 , R2 )
=
=
∂v 0
∂R1
R2
R1 + R2
10
3
= −
100
vi =
50 + 100
( 5)
1
analysis by eliminating second- and higher-order terms, thus,
having a linear approximation of the sensitivity. Furthermore,
a voltage-divider circuit is used as an example to appreciate
the said analysis technique.
ACKNOWLEDGMENT
V;
(R
3
R2
+ R2 )
2
vi = −
100
( 50 + 100 ) 2
(5)
The authors would like give thanks to the anonymous
reviewers for their helpful and insightful comments and
suggestions about the contents of this lecture note.
≈ −0.0222 V / Ω ; and
∂v 0
∂R2
(R
1
R1
+ R2 )
2
vi =
50
( 50 + 100 )
2
(5)
≈ 0.0111 V / Ω
Furthermore, we can now obtain a linear approximation of the
output voltage v 0 ( R1 , R2 ) in the worst case scenario, having
the largest values of ∆R1 and ∆R2 with equal signs. Hence,
∆v 0
max
can be obtained by using Eq. (5), having
∆v 0
max
=
∂v 0
∂R1
∆R1,max +
∂v 0
∂R2
∆R2,max
which is approximately equal to 0.0777 volts. With this value,
it can be assumed that if the values of both resistors are at
their maxima, the output voltage, i.e. voltage across the 100ohm resistor, will increase from 3.3333 V to approximately
3.4110 V.
V. CONCLUSION
In this lecture note, a simple sensitivity analysis technique
used in the science and engineering fields is presented for its
application to memoryless electric circuits. It was shown that a
simple Taylor series expansion can be used in sensitivity
REFERENCES
[1]
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[3] J. W. Bandler and R. E. Seviora, “Computation of sensitivities for
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[7] M. R. Spiegel, Schaum’s Outline of Theories and Problems of Advanced
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[8] E. Kreysig, Advanced Engineering Mathematics, 8th Ed., Singapore:
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[9] X. Liu, M. C. Papaefthymiou, and E. G. Friedman, “Minimizing
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[11] J.-F. Mao and E. S. Kuh, “Fast simulation and sensitivity analysis of
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