ENGI 252
20 March 2004
Laplace Transforms
ENGI 252
Euler’s Formula
e
jθ
= cos θ + j sin θ < a >
e
-jθ
= cos θ - j sin θ
<b>
Euler’s Formula
Now add < a > and < b > and solve for cos θ
1 jθ
( e + e-jθ )
2
2 cos θ = ( e jθ + e-jθ )
cos θ =
Subtract < b > from < a > and solve for sin θ
sin θ =
20 March 2004
Laplace Transforms
1 jθ
( e - e-jθ )
2j
ENGI 252 - Laplace Transforms
2
1
ENGI 252
20 March 2004
Laplace Transforms
z
The Laplace Transform is a technique that transforms
circuit analysis from the time domain into the frequency
domain
„
z
z
z
The mathematical solution in the frequency domain is easier
For most circuits or systems whose behavior is described
by a 2nd Order or higher Differential Equation, solving for
the output may be either impossible or very difficult
It is easier to solve algebraic equations rather than
Differential Equations
The Laplace Transform is:
„
„
„
„
a powerful mathematical tool
It will allow us to transform differential equations into algebraic equations
We can then use all the circuit analysis techniques that we developed for
passive circuits
It allows us to find the complete time response for any circuit with any
input
20 March 2004
ENGI 252 - Laplace Transforms
3
Methods for Solving Differential Equations
x(t)
y(t)
Time
Domain
Time Domain
Circuit
Circuit
Laplace
L
Transform
L
−1
s-Domain
X(s)
Circuit
Inverse
Laplace
Transform
Y(s)
s = α + jω = Complex Frequency
2 Types of s-Domain Circuits
With and Without Initial Conditions
20 March 2004
Laplace Transforms
ENGI 252 - Laplace Transforms
4
2
ENGI 252
20 March 2004
Laplace Transform Basics
z
The Laplace transform analysis provides insight into
transient phenomena in the circuit
„
Laplace transform methodology allows us to analyze much more
complex circuits than is possible with SS methods
L{f(t)} =
z
0
f(t) e-st dt
The power of e must be dimensionless.
„
„
z
∫
∞
Therefore s must have the dimensions of 1/t, or frequency
Thus the s-domain is a frequency domain
We refer to the Laplace transform of a function f(t) as
F(s), or:
L{f(t)} = F(s)
20 March 2004
ENGI 252 - Laplace Transforms
5
Laplace Transform Basics
z
The Laplace Transform of f(t) = F(s)
L [ f(t)] = F(s) =
z
z
z
z
∞
0
f(t) e-st dt
We will use tables to find F(s) from f(t)
Since the power of e must be dimensionless, then s must
have the dimensions of 1/t, or frequency. Thus the sdomain is a frequency domain.
Using Laplace transforms can greatly ease the solution of
many time-domain electrical circuit problems.
Integral and differential equations in the time domain are
reduced to algebraic equations in the s-domain, making
their solutions much easier.
20 March 2004
Laplace Transforms
∫
ENGI 252 - Laplace Transforms
6
3
ENGI 252
20 March 2004
Laplace Transform Basics
z
z
Once a problem is solved in the s-domain, we can
use the inverse Laplace transform to determine the
time-domain solution (similar to phasor analysis)
The Inverse Laplace Transform of F(s) = f(t)
L 1 [ F(s) ] = f(t) =
z
1
j2π
∫
α + jω
α - jω
F(s) est ds
We will use either tables of Inverse Laplace
Transform or Partial Fraction Expansion to find
f(t) from F(s)
20 March 2004
ENGI 252 - Laplace Transforms
7
Laplace Transform Basics
z
z
z
z
In this course, we will use the Laplace Transform as a
unilateral transform - it will be valid only for t > 0
Not all functions have a Laplace Transform
We normally use 0 as the lower limit of the integral
The upper limit of the integral is ∞, which means that
some transforms diverge
„
z
„
„
z
most problems in electrical engineering will have a convergent solution
There are two types of Laplace Transforms:
Functional transforms – Laplace Transforms of a specific
mathematical function (sine or cosine, exponentials, algebraic
expressions, etc)
Operational transforms – Laplace Transforms of mathematical
operations (derivative, integral, product, etc.)
Now we need an review of some mathematical functions
that are extremely important
20 March 2004
Laplace Transforms
ENGI 252 - Laplace Transforms
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4
ENGI 252
20 March 2004
Representation of the Resistor
A resistor represented:
(a) in the time domain
(b) in the frequency domain using the Laplace transform
20 March 2004
ENGI 252 - Laplace Transforms
9
Representation of the Capacitor
A capacitor represented:
(a) in the time domain
(b) in the frequency domain using the Laplace Transform
(c) in an alternate frequency-domain representation
20 March 2004
Laplace Transforms
ENGI 252 - Laplace Transforms
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5
ENGI 252
20 March 2004
Representation of the Inductor
An inductor represented:
(a) in the time domain
(b) in the frequency domain using the Laplace Transform
(c) in an alternate frequency-domain representation
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ENGI 252 - Laplace Transforms
11
Poles and Zeroes in the s-Domain
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Laplace Transforms
ENGI 252 - Laplace Transforms
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6
ENGI 252
20 March 2004
Plot of Poles and Zeroes on the s Plane
Example of a Typical Pole – Zero S-Plane Plot
20 March 2004
Laplace Transforms
ENGI 252 - Laplace Transforms
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7