INTRODUCTION TO LAPLACE TRANSFORM Advanced Circuit Analysis Technique.
10. Laplace TransforM Technique
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Transcript of 10. Laplace TransforM Technique
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10. LAPLACE TRANSFORM TECHNIQUE
CIRCUITS by Ulaby & Maharbiz
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Overview
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Analysis Techniques Circuit Excitation Method of Solution
Chapters1. dc (w/ switches) Transient analysis 5 & 62. ac Phasor-domain analysis 7 -9 ( steady state only)3. any waveform Laplace Transform This
Chapter (single-sided) (transient + steady state)4. Any waveform Fourier Transform 11 (double-sided) (transient + steady state)
Single-sided: defined over [0,∞] Double-sided: defined over [−∞,∞]
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Singularity FunctionsA 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
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Singularity Functions (cont.) Unit Impulse Function
For any function f(t):
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Laplace Transform Definition
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Laplace Transform of Singularity Functions
For A = 1 and T = 0:
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Laplace Transform of Delta Function
For A = 1 and T = 0:
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Properties of Laplace Transform
1. Time Scaling
2. Time Shift
Example
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Properties of Laplace Transform (cont.)
3. Frequency Shift
4. Time Differentiation
Example
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Properties of Laplace Transform (cont.)
5. Time Integration
6. Initial and Final-Value Theorems
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Example 10-5: Initial and Final Values
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Properties of Laplace Transform (cont.)
7. Frequency Differentiation 8. Frequency Integration
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Circuit Analysis
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Example
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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
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1.Partial Fractions Distinct Real Poles
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1. Partial Fractions Distinct Real Poles
The poles of F(s) are s = 0, s = −1, and s = −3. All three poles are real and distinct.
Example
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2. Partial Fractions Repeated Real Poles
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2. Partial Fractions Repeated Real PolesExample
Cont.
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2. Partial Fractions Repeated Real Poles
Example cont.
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3. Distinct Complex Poles
Example
Note that B2 is the complex conjugate of B1.
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
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3. Distinct Complex Poles (Cont.)
Next, we combine the last two terms:
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4. Repeated Complex Poles: Same procedure as for repeated real poles
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Property #3a in Table 10-2:
Hence:
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s-Domain Circuit Models
Under zero initial conditions:
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Example 10-11: Interrupted Voltage Source
Initial conditions:
Voltage Source
(s-domain)Cont.
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Example 10-11: Interrupted Voltage Source (cont.)
Cont.
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Example 10-11: Interrupted Voltage Source (cont.)
Cont.
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Example 10-11: Interrupted Voltage Source (cont.)
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Transfer Function
In the s-domain, the circuit is characterized by a transfer function H(s), defined as the ratio of the output Y(s) to the input X(s), assuming that all initial conditions relating to currents and voltages in the circuit are zero at t = 0−.
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Transfer Function (cont.)
Multiplication in s-domain
Convolution in time domain
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Convolution IntegralImpulse Response h(t): output of linear system when input is a delta function
Input Output
y(t) cannot depend on excitations occurring after time tAssumes x(t) = 0 for t < 0
Definition of convolution
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Convolution Integral Can be used to determine output
response entirely in the time domain Can be useful when input is a sequence of
experimental data or not a function with a definable Laplace transform
Convolution can be performed by shifting h(t) or x(t):
h(t) shifted x(t) shifted
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Useful Recipe
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Cont.
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Same result as Method 1
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Integral can be computed graphically at successive values of t.
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0.86 @1s
0.11@ 2s
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Example 10-16: Graphical Convolution
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Summary