EEE 302
Electrical Networks II
Dr. Keith E. Holbert
Summer 2001
Lecture 16
1
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
Lecture 16
2
LC Behavior
• Recall some facts on the behavior of LC elements
• Inductors (L):
– The current in an inductor cannot change abruptly in zero
time; an inductor makes itself felt in a circuit only when
there is a changing current
– An inductor looks like a short circuit to d.c.
• Capacitors (C):
– The voltage across a capacitor cannot change
discontinuously; a capacitor makes itself felt only when
there exists a changing potential (voltage) difference
– A capacitor looks like an open circuit to d.c.
Lecture 16
3
Laplace Circuit Element Models
• Here we develop s-domain models of circuit
elements
• 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
• Note on subsequent slides how without initial
conditions, we could have used the substitution s=j
Lecture 16
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 (this is very similar
to how we derived an impedance relation for R also)
Lecture 16
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)
Lecture 16
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 stepfunction voltage source whose height is v(0)
• A circuit representation of the Laplace transformation
of the capacitor appears on the next page
Lecture 16
7
Capacitor (cont’d.)
+
Time
Domain
vC(t)
C
–
IC(s)
+
VC(s)
–
IC(s)
+
1/sC
+
–
1/sC
Cv(0)
VC(s)
v(0)
s
–
Frequency Domain Equivalents
Lecture 16
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)
Lecture 16
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)
• A circuit representation of the Laplace transformation
of the inductor appears on the next page
Lecture 16
10
Inductor (cont’d.)
+
Time
Domain
vL(t)
iL(0)
L
–
IL(s)
+
VL(s)
–
IL(s)
+
sL
–
+
VL(s)
–
Li(0)
sL
i(0)
s
Frequency Domain Equivalents
Lecture 16
11
Class Examples
• Extension Exercise E14.1
Lecture 16
12