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IntroductIon
We solved first-order and second-order circuits using the differential equation model in earlier
chapters. Differential equation model can be used to solve multi-mesh, multi-node circuits containing
R, L, C, M
, linear dependent sources and independent sources too.
A linear time-invariant circuit containing
R
,
L
,
C
,
M
, linear dependent sources and a single input
source is described by a linear ordinary differential equation with constant coefficients. The differential
equation for such a circuit can be expressed in a standard format as below.
d y
dt
a
d
y
dt
a
dy
dt
a y
b
d x
dt
b
d
x
dt
n
n
n
n
n
m
m
m
m
m
m
+
+ +
+
=
+
−
−
−
−
−
−
1
1
1
1
0
1
1
1
++ +
+
b
dx
dt
b x
1
0
Chapter
13
13.2
Analysis of Dynamic Circuits by Laplace Transforms
The variable
y
is any circuit voltage or current variable chosen as the
describing variable
for the
circuit and
x
is the input source function. Standard mesh analysis or nodal analysis technique along
with variable elimination will help us to arrive at this equation. However, the variable elimination
involved can be considerably tricky in the case of large circuits containing many energy storage
elements. This is a serious shortcoming of time-domain analysis by differential equation model.
The
order
of a circuit is equal to the order of the differential equation that describes it. Order of
the circuit will be equal to the total number of independent inductors and capacitors – (number of all-
inductor nodes
+
number of all-capacitor loops in the circuit).
The order of a circuit depends also on the kind and location of input. The same circuit will have
different order if voltage source input is replaced by current source input.
The coefficients
a
n
-
1
…
a
0
and
b
m
…
b
0
are decided by the circuit parameters. They are real-valued.
a
n
-
1
…
a
0
are positive real numbers if the circuit is passive
i.e.,
if it contains only
R
,
L
,
C
and
M
. They
can be zero or negative real numbers if the circuit contains dependent sources.
b
m
…
b
0
can be positive
or negative or zero in all circuits.
The format of left-side of the differential equation that describes a circuit is independent of the
particular circuit variable chosen as the describing variable in general. That is,
a
n
-
1
…
a
0
will remain
the same even if some other circuit variable is used as the variable
y
. However,
b
m
…
b
0
will depend on
the variable chosen.
This
n
th
-order differential equation requires
n
initial values for solving it if the input function is
known only for a range of values of
t
rather than over entire
t
-axis. The required initial values are
y
dy
dt
d y
dt
d
y
dt
n
n
( ),
,
,
(
)
(
)
(
)
0
0
2
2
0
1
1
0
+
−
−
+
+
+
…
The differential equation can be solved using
x
(
t
) for
t
≥
0
+
and these initial values.
The initial values available in a circuit are the initial current values for all inductors and initial
voltage values for all capacitors. It requires considerable effort to translate these values into the
required initial values in the case of a circuit containing many energy storage elements. This is another
serious shortcoming of the differential equation approach.
Laplace transformation technique converts a linear differential equation with constant coefficients
into an algebraic equation. Thus, the task of solving a collection of simultaneous differential equations
will be reduced to a much simpler task of solving a set of simultaneous algebraic equations involving
Laplace transforms of input signals and Laplace transform of desired output signal. This set of
equations may be solved for the Laplace transform of output and the time-function may be obtained
by inverting the transformation. Moreover, we will see later that the initial conditions specified for
inductor currents and capacitor voltages can be used directly in Laplace transform method of solving
a circuit. This makes it sound as if the technique of Laplace transforms is just a mathematical artifice
for solving linear differential equations. Just as Logarithm is not a mathematical artifice for converting
multiplication into addition, Laplace transform is not just a mathematical artifice to make solution of
differential equations easier.
There is a very compelling reason why we take Laplace transform of a function. The reason is that
(i) complex exponential signals are eigen functions of linear time-invariant circuits, (ii) linear time-
invariant circuits obey superposition principle and (iii) Laplace transform expresses a given arbitrary
input function as a sum of complex exponential signals.
Therefore, we commence our study of Laplace transform method of solving a circuit by examining
the circuit response to a complex exponential input.
Circuit Response to Complex Exponential Input
13.3
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