6. CIRCUIT ANALYSIS BY LAPLACE
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
CIRCUITS by Ulaby & Maharbiz
All rights reserved. Do
not copy or distribute.
© 2013 National
Technology and
Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Second Order Circuits
A second order circuit is
characterized by a second order
differential equation

Resistors and two energy storage
elements

Determine voltage/current as a
function of time

Initial/final values of
voltage/current, and their
derivatives are needed
Initial/Final Conditions
Guidelines



vC, iL do not change
instantaneously
Get derivatives dvC/dt
and diL/dt from iC , vL
Capacitor open, Inductor
short at dc
All rights reserved. Do not copy
or distribute. © 2013
National Technology and
Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example 6-2: Determine Initial/Final Conditions
Circuit
V0  24 V
I0  4 A
R1  2 
R2  4 
R3  6 
L  0 .2 H
C  8 mF
t = 0‒
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example 6-2: Initial/Final Conditions (cont.)
t = 0+
Given:
V0  24 V
I0  4 A
R1  2 
R2 4 
R3 6 
L  0 .2 H
C  8 mF
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example 6-2: Initial/Final Conditions (cont.)
t
V0  24 V
I0  4 A
R1  2 
R2 4 
R3 6 
L  0 .2 H
C  8 mF
Series RLC Circuit:
General Response
All rights reserved. Do not copy
or distribute. © 2013 National
Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Series RLC Circuit : General Solution
Solution Outline
Transient solution
Steady State solution
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Series RLC Circuit: Natural Response
Find Natural Response Of RLC Circuit
Natural response occurs when no active sources
are present, which is the case at t > 0.
0
Series RLC Circuit: Natural Response
Find Natural Response Of RLC Circuit
Solution of Diff. Equation
0
Assume:
It follows that:
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Solution of Diff. Equation (cont.)
Invoke Initial Conditions to determine A1 and A2
0
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Circuit Response: Damping Conditions
s1 and s2 are real
s1 = s2
Damping coefficient
s1 and s2 are complex
Resonant frequency
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Overdamped Response
s1, 2  a  a 2  w0
2
a  dampingfactor
w0  resonantfrequency
a
R
2L
w0 
1
LC
Overdamped, a > w0
vt   A1es1t  A2es2t
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Underdamped Response
Damping: loss of stored energy
s1, 2  a  a 2  w0
2
a  dampingfactor
w0  resonantfrequency
R
1
a
w0 
2L
LC
Underdamped a < w0
vt   ea t D1 coswd t  D2 sin wd t 
wd  w02  a 2
Damped natural frequency
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Critically Damped Response
s1, 2  a  a 2  w0
2
a  dampingfactor
w0  resonantfrequency
a
R
2L
w0 
1
LC
Critically damped a = w0
vt   B1  B2t ea t
All rights reserved. Do
not copy or distribute.
© 2013 National
Technology and Science
Press
Example 6-3: Overdamped RLC Circuit
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Cont.
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example 6-3: Overdamped RLC Circuit
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Parallel RLC Circuit
v
dv
iC
 Is
R
dt
di
vL
dt
Is
d 2i 1 di i



2
dt RC dt LC LC
Same form of diff. equation
as series RLC
s1, 2  a  a 2  w0
a
1
2 RC
w0 
2
1
LC
Overdamped (a > w0)
it   i  A1es1t  A2es2t
Critically Damped (a = w0)
it   i  B1  B2t ea t
Underdamped (a < w0)
it   i  ea t D1 coswdt  D2 sin wdt 
Oscillators
If R=0 in a series or parallel
RLC circuit, the circuit becomes
an oscillator
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example 6-5 (cont.)
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Analysis Techniques
Circuit Excitation
1. dc (w/ switches)
2. ac
Method of Solution
Chapters
Transient analysis
5&6
Phasor-domain analysis 7 -9
( steady state only)
3. any waveform
(single-sided)
4. Any waveform
(double-sided)
Single-sided: defined over [0,∞]
Laplace Transform This Chapter
(transient + steady state)
Fourier Transform
12
(transient + steady state)
Double-sided: defined over [−∞,∞]
Singularity Functions
A singularity function is a function
that either itself is not finite
everywhere or one (or more) of
its derivatives is (are) not finite
everywhere.
Unit Step Function
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Singularity Functions (cont.)
Unit Impulse Function
For any function f(t):
Sampling Property
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Review of Complex Numbers
We will find it is useful to represent
sinusoids as complex numbers
j  1
z  x  jy
Rectangular coordinates
z  z   z e j
Polar coordinates
Rez   x
Im(z )  y
Relations based
on Euler’s Identity
e j  cos  j sin 
Relations for Complex Numbers
Learn how to
perform these
with your
calculator/computer
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Laplace Transform Technique
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Laplace Transform Definition
All rights reserved.
Do not copy or
distribute.
© 2013 National
Technology and
Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Laplace Transform of Singularity Functions
For A = 1 and T = 0:
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Laplace Transform of Delta Function
For A = 1 and T = 0:
All rights reserved. Do not
copy or distribute.
© 2013 National
Technology and Science
Press
Properties of Laplace Transform
Time Scaling
Example
Time Shift
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Properties of Laplace Transform (cont.)
Frequency Shift
Example
Time Differentiation
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Properties of Laplace Transform (cont.)
Time Integration
Frequency Differentiation
Frequency Integration
All rights reserved. Do not copy
or distribute.
© 2013 National Technology
and Science Press
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
Circuit Analysis
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not copy or distribute.
Example
© 2013 National Technology and Science Press
Partial Fraction Expansion
Partial fraction expansion facilitates inversion
of the final s-domain expression for the
variable of interest back to the time domain.
The goal is to cast the expression as the sum
of terms, each of which has an analog in
Table 10-2.
Example
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
1.Partial Fractions
Distinct Real Poles
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
1. Partial Fractions
Distinct Real Poles
Example
The poles of F(s) are s = 0, s = −1, and s = −3.
All three poles are real and distinct.
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
2. Partial Fractions
Repeated Real Poles
All rights reserved. Do
not copy or distribute.
© 2013 National
Technology and Science
Press
2. Partial Fractions
Repeated Real Poles
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
Example
Cont.
2. Partial Fractions
Repeated Real Poles
Example cont.
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
3. Distinct Complex Poles



Procedure similar to “Distinct Real
Poles,” but with complex values for s
Complex poles always appear in
conjugate pairs
Expansion coefficients of conjugate
poles are conjugate pairs themselves
Example
Note that B2 is the complex conjugate of B1.
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
3. Distinct Complex Poles (Cont.)
Next, we combine the last two terms:
4. Repeated Complex Poles:
Same procedure as for repeated real poles
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Property #3a in Table 10-2:
Hence:
s-Domain Circuit Models
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
Under zero initial conditions:
All rights reserved. Do not copy
or distribute. © 2013 National
Technology and Science Press
All rights reserved.
Do not copy or
distribute. © 2013
National Technology
and Science Press
All rights reserved. Do not
copy or distribute.
© 2013 National
Technology and Science
Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
All rights reserved. Do not
copy or distribute.
© 2013 National Technology
and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example : Interrupted Voltage Source
Initial conditions:
Voltage Source
(s-domain)
Cont.
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example : Interrupted Voltage Source (cont.)
Cont.
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example : Interrupted Voltage Source (cont.)
Cont.
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Example : Interrupted Voltage Source (cont.)
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
All rights reserved. Do not copy or
distribute. © 2013 National
Technology and Science Press
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press
Multisim Example of RLC Circuit
All rights reserved. Do not copy or distribute.
RFID Circuit
© 2013 National Technology and Science Press
Tech Brief 10:
Micromechanical
Sensors and
Actuators
All rights reserved. Do
not copy or distribute.
© 2013 National
Technology and Science
Press
Tech Brief 11: Touchscreens and Active Digitizers
All rights
reserved. Do not
copy or
distribute. ©
2013 National
Technology and
Science Press
Summary
All rights reserved. Do not copy or distribute.
© 2013 National Technology and Science Press