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EE 422G Notes: Chapter 5 Instructor: Zhang
Chapter 5 The Laplace Transform
5-1 Introduction
(1) System analysis
static system: y(t) = ax(t) => easy (simple processing)
dynamic system:
Can we determine y(t) for given u(t) easily?
Easier solution method
(1) we have systematic way to obtain H(f) based on the differential equation (2) we can obtain X(f)
Fourier transform: an easier way
(2) Problem Fourier transform of the input signal:
if x(t) does not go to zero when and
Page 5-1
DynamicSystem
x(t) input y(t) output
A processor which processes the input signal to produce the output
DynamicSystem
X(f) Y(f )
H(f ) Algebraic equation, not differential equation
EE 422G Notes: Chapter 5 Instructor: Zhang
X(f) typically does not exist! (the existence of X(f) if not guaranteed)
A very strong condition, can not be satisfied by many signals!
(3) Solution:
Why should we care about t<0 for system analysis? We do not care! x(t) does not go to zero (when ), but may! (Will be much easier. )
use “single-sided” Fourier transform of , instead of “double-sided” Fourier transform of x(t).
very useful, let’s use a new name for it: Laplace transform.
(4) Laplace Transform
Definition: Laplace transform of x(t)
What is a Laplace transform of x(t)? A time function? No, t has been eliminated by the integral with respect to t!
A function of s ( s is complex variable)
(5) System analysis using Laplace transform
(6) How is the inverse transform defined?
Page 5-2
DynamicSystem
X(s) Y(s )
G(s )
Inverse Laplace transform
EE 422G Notes: Chapter 5 Instructor: Zhang
(7) Will we often use the definition of the inverse transform to find time function? No! What will we do?
Express X(s) as sum of terms for which we know the inverse transforms!
5-2 Examples of Evaluating Laplace Transforms using the definition
(1) x(t)=1 and step function x(t)=u(t)
(2)
Define a new complex variable
we know
(3)
Page 5-3
EE 422G Notes: Chapter 5 Instructor: Zhang
No constraint on s.
5-2B Discussion: Convergence of the Laplace Transform
(1) To assure converge, must be psotive
enough such that goes to zero when t goes to positive infinite
(2) Region of absolute convergence and pole
(3) How to obtain Fourier transform form Laplace transform:
Important: why introduce Laplace transform; definition of Laplace transform as a modification of Fourier transform; find the Laplace transforms of the three basic functions based on the (mathematical) definition of Laplace transform.
Page 5-4
EE 422G Notes: Chapter 5 Instructor: Zhang
Chapter structure
Part one: Definition (5.1)Part two: Direct evaluation of Laplace transforms of simple time-
domain function.Easy? Not at all!
complex (not simple) functions: even harder tools for application of Laplace transform in system analysis
Part three: Rest of the chapter What tools: tools for easier Laplace transform evaluation
tools for easy inverse transform
5-3 Some Laplace Transform theorems (Tools for evaluating Laplace transform based on the Laplace
transforms of the basic functions)
5.3.1 LinearityAssume ( and are time independent)
then
HW#2-1: Assume , and .
(a) Is equal to ? (Answer: Yes or No)
(b) If , is it true ?
(Answer: Yes or No)
Example 5.1(1) Find
Key to solution : express as linear combination of , , and/or :
Page 5-5
EE 422G Notes: Chapter 5 Instructor: Zhang
let
let
Can we use and to express ?
(2) Find
5-3-2 Transforms of Derivatives
Assume
Then
Proof:
(1) Definition
(2) Integration by parts: General equation:
Page 5-6
EE 422G Notes: Chapter 5 Instructor: Zhang
(3) Use the above equation Why? If we assume
must go to zero. Otherwise,
does not exist !
use as lower limit =>
HW#2-2: Assume . Prove
HW#2-3: Express using
Example 5-2
Page 5-7
EE 422G Notes: Chapter 5 Instructor: Zhang
Find i(t) using Laplace transform method for t>0
Solution:(1) Before switched from 1 to 2 at t=0
(2) System equation (t>0)
KVL:
(3) Solve system equation using Laplace transform
5-3-3 Laplace Transform of an integral
Assume
Then
where
Proof :
(1)
Page 5-8
EE 422G Notes: Chapter 5 Instructor: Zhang
(2)
(3)
Proved!
Example 5.3:
Find I(s) = L(i(t))
Solution:(1) Differential equationKVL :
(2) Laplace transform
Page 5-9
0
1
EE 422G Notes: Chapter 5 Instructor: Zhang
5-3-4. Complex Frequency shift (s-shift) Theorem
Assume
Then
=>
Example 5-4 Find
Solution:
Page 5-10
EE 422G Notes: Chapter 5 Instructor: Zhang
5-3-4 Delay Theorem question: How to express delayed function?
Assume Then
(If , it will not be a delay!) Proof :
Question: will be true if
Page 5-11
EE 422G Notes: Chapter 5 Instructor: Zhang
No! (it will not be a delay)
Example 5-5: Square wave beginning at t = 0
5-3-5 Convolution Signal 1: Signal 2 :
if
if
Therefore, if
Page 5-12
EE 422G Notes: Chapter 5 Instructor: Zhang
Look at
Then
5-3-7 Product5-3-8 Initial Value Theorem
Example: A demonstration where x(0) is obvious
It is evident: Using Laplace transform
Page 5-13
t
EE 422G Notes: Chapter 5 Instructor: Zhang
5-3-9 Final Value Theorem: if and are Laplace transformable, then
(condition: has no poles on or in the right-half s-plan or exists)
5-3-10 Scalinga>0: x(at) a times fast (if a>1)
or slow (if a<1) as x(t)
What do we expect on ?
5-4 Inversion of Rational Functions(1) Ways to find from
(1) (Contour Integral)
(2) Transform pair
Therefore
Page 5-14
EE 422G Notes: Chapter 5 Instructor: Zhang
(2) All kinds of Laplace Transform ? No!
We almost only see
rational function X(s) Delay
Consider Rational Functions only!
(3) Non proper Rational Function proper Rational Function
Non proper m>=n proper m<n
Non proper => proper + Polynomial (using long division)
How to find inverse Laplace transform for polynomials?
consider proper rational functions only!
(4) Proper Rational Functions: Partial Fraction Expansion
Page 5-15
EE 422G Notes: Chapter 5 Instructor: Zhang
sum of
Let’s look at examples, and then summarize! Techniques:
Common Denominator Factorize first! Specific value of s Expand second! Heaviside’s Expansion Find coefficients third! Matlab
Example 5-9: Simple Factors
Solution:
(1) Factorize and expand
(2) Common Denominator Methods
specific values of s
Page 5-16
EE 422G Notes: Chapter 5 Instructor: Zhang
Can you solve for A and B?
Heaviside Expansion
and
Example 5-10 Imaginary Roots
Solution: what do we have:
A1+A2 must be real number(-A1+A2)j must be real number
Heaviside Expansion => A3=1 and A4= - 2.
s=1 =>
Page 5-17
Same as real roots!
EE 422G Notes: Chapter 5 Instructor: Zhang
s=2 =>
Can we solve for c1 and c2?c1=1 c2=1
=>Too complex: use MATLAB
Example 5-11 Repeated linear Factors
Example 5-12
Example 5-13 Complex - Conjugate Factors
Example 5-14 Repeated Quadratic Factors
* Summary of Partial–Fraction Expansion
Page 5-18
EE 422G Notes: Chapter 5 Instructor: Zhang
(1) Expansion Structure:Simple Roots (including complex conjugate)
=> could be complex.
Repeated Roots: m multiplicity
=>
real number or complex number
(2) Avoid complex numberFor complex conjugates:
(3) Inverse Laplace transform
Matlab use for Partial – Fraction Expansion.
Have to be memorized: (1) Table 5-2 all except for No. 7(2) Table 5-3 No. 1-No. 7
Page 5-19
EE 422G Notes: Chapter 5 Instructor: Zhang
Appendix: Partial-Faction Expansion with MatLab
1. Command (MatLab Help)2. An Introduction 3. An Example: Complex Conjugate
1. Command (MatLab Help)» help residue
RESIDUE Partial-fraction expansion (residues). [R,P,K] = RESIDUE(B,A) finds the residues, poles and direct term of a partial fraction expansion of the ratio of two polynomials B(s)/A(s). If there are no multiple roots, B(s) R(1) R(2) R(n) ---- = -------- + -------- + ... + -------- + K(s) A(s) s - P(1) s - P(2) s - P(n) Vectors B and A specify the coefficients of the numerator and denominator polynomials in descending powers of s. The residues are returned in the column vector R, the pole locations in column vector P, and the direct terms in row vector K. The number of poles is n = length(A)-1 = length(R) = length(P). The direct term coefficient vector is empty if length(B) < length(A), otherwise length(K) = length(B)-length(A)+1. If P(j) = ... = P(j+m-1) is a pole of multplicity m, then the expansion includes terms of the form R(j) R(j+1) R(j+m-1) -------- + ------------ + ... + ------------ s - P(j) (s - P(j))^2 (s - P(j))^m [B,A] = RESIDUE(R,P,K), with 3 input arguments and 2 output arguments, converts the partial fraction expansion back to the polynomials with coefficients in B and A. Warning: Numerically, the partial fraction expansion of a ratio of polynomials represents an ill-posed problem. If the denominator polynomial, A(s), is near a polynomial with multiple roots, then small changes in the data, including roundoff errors, can make arbitrarily large changes in the resulting poles and residues. Problem formulations making use of state-space or zero-pole representations are preferable. See also POLY, ROOTS, DECONV.
»
2. An Introduction
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EE 422G Notes: Chapter 5 Instructor: Zhang
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EE 422G Notes: Chapter 5 Instructor: Zhang
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EE 422G Notes: Chapter 5 Instructor: Zhang
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EE 422G Notes: Chapter 5 Instructor: Zhang
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EE 422G Notes: Chapter 5 Instructor: Zhang
3. An Example: Complex Conjugate
Page 5-25
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