Laplace Transform Solutions of Transient Circuits

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Laplace Transform
Solutions of Transient
Circuits

Dr. Holbert
March 5, 2008

Lect13             EEE 202        1
Introduction
• In a circuit with energy storage elements,
voltages and currents are the solutions to
linear, constant coefficient differential
equations
• Real engineers almost never solve the
differential equations directly
• It is important to have a qualitative
understanding of the solutions

Lect13              EEE 202                    2
Laplace Circuit Solutions
• In this chapter we will use previously
established techniques (e.g., KCL, KVL,
nodal and loop analyses, superposition,
source transformation, Thevenin) in the
Laplace domain to analyze circuits
• The primary use of Laplace transforms
here is the transient analysis of circuits

Lect13               EEE 202                   3
Laplace Circuit Element Models
• Here we develop s-domain models of circuit
elements
• DC voltage and current sources basically remain
unchanged except that we need to remember
that a dc source is really a constant, which is
transformed to a 1/s function in the Laplace
domain

Lect13                EEE 202                   4
Resistor
• We start with a simple (and trivial) case, that of the
resistor, R
• Begin with the time domain relation for the element
v(t) = R i(t)
• Now Laplace transform the above expression
V(s) = R I(s)
• Hence a resistor, R, in the time domain is simply
that same resistor, R, in the s-domain

Lect13                    EEE 202                          5
Capacitor
• Begin with the time domain relation for the element
d v(t)
i(t)  C
dt
• Now Laplace transform the above expression
I(s) = s C V(s) – C v(0)
• Interpretation: a charged capacitor (a capacitor with
non-zero initial conditions at t=0) is equivalent to an
uncharged capacitor at t=0 in parallel with an
impulsive current source with strength C·v(0)

Lect13                    EEE 202                           6
Capacitor (cont’d.)
• Rearranging the above expression for the capacitor
I(s) v(0)
V(s)      
sC     s
• Interpretation: a charged capacitor can be replaced
by an uncharged capacitor in series with a step-
function voltage source whose height is v(0)
• Circuit representations of the Laplace transformation
of the capacitor appear on the next page

Lect13                   EEE 202                       7
Capacitor (cont’d.)
iC(t)
Time                         +
Domain                     vC(t)                 C
–

IC(s)
IC(s)
+
1/sC                   +              Cv(0)
1/sC
VC(s)                               VC(s)
+   v(0)
–    s                 –
–
Laplace (Frequency) Domain Equivalents
Lect13                            EEE 202                                8
Inductor
• Begin with the time domain relation for the
element                  d i(t)
v(t)  L
dt
• Now Laplace transform the above expression
V(s) = s L I(s) – L i(0)
• Interpretation: an energized inductor (an
inductor with non-zero initial conditions) is
equivalent to an unenergized inductor at t=0 in
series with an impulsive voltage source with
strength L·i(0)
Lect13                 EEE 202                      9
Inductor (cont’d.)
• Rearranging the above expression for the
inductor           V(s) i(0)
I(s)         
sL        s
• Interpretation: an energized inductor at t=0 is
equivalent to an unenergized inductor at t=0 in
parallel with a step-function current source with
height i(0)
• Circuit representations of the Laplace
transformation of the inductor appear on the
next page
Lect13                     EEE 202                    10
Inductor (cont’d.)

Time                        +
Domain                     vL(t)              L
iL(0)
–

IL(s)                            IL(s)
+                                   +
sL
VL(s)                                          sL   i(0)
VL(s)            s
–
Li(0)              –
–            +

Laplace (Frequency) Domain Equivalents
Lect13                              EEE 202                         11
Analysis Techniques
• In this section we apply our tried and
tested analysis tools and techniques to
perform transient circuit analyses
– KVL, KCL, Ohm’s Law
– Voltage and Current division
– Loop/mesh and Nodal analyses
– Superposition
– Source Transformation
– Thevenin’s and Norton’s Theorems
Lect13                 EEE 202              12
Transient Analysis
• Sometimes we not only must Laplace transform
the circuit, but we must also find the initial
conditions

Capacitor   I = 0; open circuit
Inductor    V = 0; short circuit

Lect13                 EEE 202                     13
Class Examples
• Drill Problems P6-4, P6-5

Lect13              EEE 202   14

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