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JAMES W. NILSSON
&
SUSAN A. RIEDEL
ELECTRIC
CIRCUITS
EIGHTH EDITION
CHAPTER 13
THE LAPLACE
TRANSFORM IN
CIRCUIT ANALYSIS
© 2008 Pearson Education
CONTENTS
13.1 Circuit Elements in the s Domain
13.2 Circuit Analysis in the s Domain
13.3 Applications
13.4 The Transfer Function
13.5 The Transfer Function in Partial
Fraction Expansions
© 2008 Pearson Education
CONTENTS
13.6 The Transfer Function and the
Convolution Integral
13.7 The Transfer Function and the SteadyState Sinusoidal Response
13.8 The Impulse Function in Circuit
Analysis
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13.1 Circuit Elements in the s Domain
 We
can represent each of the circuit
elements as an s-domain equivalent circuit
by Laplace-transforming the voltagecurrent equation for each elements:
Resistor: V = RI
Inductor: V = s LI – LI0
Capacitor: V = (1/s C)I + V0 /s
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13.1 Circuit Elements in the s Domain
In these equations,
V = L {v}, I = L {i)
I0 = initial current through the inductor,
V0 = initial voltage across the capacitor.
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13.1 Circuit Elements in the s Domain
The resistance element.
Time domain
Frequency domain
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13.1 Circuit Elements in the s Domain
An inductor of L henrys carrying
an initial current of I0 amperes
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13.1 Circuit Elements in the s Domain
The series equivalent circuit for an inductor of L
henrys carrying an initial current of I0 amperes
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13.1 Circuit Elements in the s Domain
The parallel equivalent circuit for an inductor of
L henrys carrying an initial current of I0 amperes
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13.1 Circuit Elements in the s Domain
The s-domain circuit for an inductor
when the initial current is zero
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13.1 Circuit Elements in the s Domain
A capacitor of C farads initially
charged to V0 volts
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13.1 Circuit Elements in the s Domain
The parallel equivalent circuit for a
capacitor initially charged to V0 volts
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13.1 Circuit Elements in the s Domain
The series equivalent circuit for a
capacitor initially charged to V0 volts
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13.1 Circuit Elements in the s Domain
The s-domain circuit for a capacitor
when the initial voltage is zero
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13.1 Circuit Elements in the s Domain
We can perform circuit analysis in the sdomain by replacing each circuit element
with its s-domain equivalent circuit.
The resulting equivalent circuit is solved
by writing algebraic equations using the
circuit analysis techniques from resistive
circuits.
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13.1 Circuit Elements in the s Domain
Summary of the s-domain equivalent circuits
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13.2 Circuit Analysis in the s Domain

Circuit analysis can be performed in the s
domain by replacing each circuit element
with its s-domain equivalent circuit.
Ohm’s Law in the s-domain
V  ZI
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13.3 Applications
 Circuit
analysis in the s domain is particularly
advantageous for solving transient response
problems in linear lumped parameter circuits
when initial conditions are known.
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13.3 Applications
 It
is also useful for problems involving
multiple simultaneous mesh-current or
node-voltage equations, because it reduces
problems to algebraic rather than differential
equations.
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13.3 Applications
The Natural Response of an RC Circuit
The capacitor discharge circuit
An s-domain equivalent circuit
An s-domain equivalent circuit
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13.3 Applications
The Step Response of a Parallel Circuit
The step response of a
parallel RLC circuit
An s-domain equivalent
circuit
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13.3 Applications
The Step Response of a Multiple Mesh Circuit
The multiple-mesh RL circuit
An s-domain equivalent circuit
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13.3 Applications
The Use of Thévenin’s Equivalent
A circuit to be analyzed using
Thévenin’s equivalent in the s
domain
An s-domain model of the
circuit
A simplified version of the
circuit, using a Thévenin’s
equivalent
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13.3 Applications
The Use of Superposition
A circuit showing the use of
superposition in s-domain
analysis
The s-domain equivalent for
the above circuit
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13.3 Applications
The Use of Superposition
The circuit with Vg acting alone
The circuit with Ig acting alone
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13.3 Applications
The Use of Superposition
The circuit with energized
inductor acting alone
The circuit with energized
capacitor acting alone
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13.4 The Transfer Function
 The
transfer function is the s-domain ratio
of a circuit’s output to its input. It is
represented as
Y ( s)
H ( s) 
X ( s)
Y(s) is the Laplace transform of the output signal,
X(s) is the Laplace transform of the input signal.
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13.5 The Transfer Function in Partial
Fraction Expansions
 The
partial fraction expansion of the
product H(s)X(s) yields a term for each
pole of H(s) and X(s).
 The
H(s) terms correspond to the transient
component of the total response; the X(s)
terms correspond to the steady-state
component.
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13.5 The Transfer Function in Partial
Fraction Expansions
 If
a circuit is driven by a unit impulse, x(t) =
δ(t), then the response of the circuit equals
the inverse Laplace transform of the transfer
function,
y(t) = L
-1{H(s)}
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= h(t)
13.5 The Transfer Function in Partial
Fraction Expansions
 A time-invariant
circuit is one for which, if the
input is delayed by a seconds, the response
function is also delayed by a seconds.
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13.6 The Transfer Function and the
Convolution Integral
 The
output of a circuit, y(t), can be
computed by convolving the input, x(t), with
the impulse response of the circuit, h(t):
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13.6 The Transfer Function and the
Convolution Integral
The excitation signal of x(t)
(a)A general excitation signal
(b)Approximating x(t) with series of pulses
(c)Approximating x(t) with a series of impulses
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13.6 The Transfer Function and the
Convolution Integral
The approximation of y(t)
(a)The impulse response
(b)Summing the impulse responses
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13.6 The Transfer Function and the
Convolution Integral
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13.6 The Transfer Function and the
Convolution Integral
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13.7 The Transfer Function and the
Steady-State Sinusoidal Response
 We
can use the transfer function of a
circuit to compute its steady-state
response to a sinusoidal source.
 To
make the substitution s = jω in H(s) and
represent the resulting complex number as
a magnitude and phase angle.
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13.7 The Transfer Function and the
Steady-State Sinusoidal Response
 If
x(t) = A cos(ωt + ø),
H(jω) = |H(jω)|e jθ(ω)
then
Steady-state sinusoidal response computed
using a transfer function
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13.8 The Impulse Function
in Circuit Analysis
 Laplace
transform analysis correctly
predicts impulsive currents and voltages
arising from switching and impulsive
sources.
 The
s-domain equivalent circuits are
based on initial conditions at t = 0-, that is,
prior to the switching.
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13.8 The Impulse Function
in Circuit Analysis
A circuit showing the
creation of an
impulsive current
The s-domain
equivalent circuit
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13.8 The Impulse Function
in Circuit Analysis
The plot of i(t) versus t for
two different values of R
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13.8 The Impulse Function in Circuit
Analysis
A circuit showing the
creation of an
impulsive voltage
The s-domain
equivalent circuit
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THE END
© 2008 Pearson Education