10. LAPLACE TRANSFORM TECHNIQUE

CIRCUITS by Ulaby & Maharbiz

Overview

Analysis Techniques

Circuit Excitation Method of SolutionChapters

1. 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 [−∞,∞]

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

Singularity Functions (cont.)

Unit Impulse Function

For any function f(t):

Laplace Transform Definition

Laplace Transform of Singularity Functions

For A = 1 and T = 0:

Laplace Transform of Delta Function

For A = 1 and T = 0:

Properties of Laplace Transform

1. Time Scaling

2. Time Shift

Example

Properties of Laplace Transform (cont.)

3. Frequency Shift

4. Time Differentiation

Example

Properties of Laplace Transform (cont.)

5. Time Integration

6. Initial and Final-Value Theorems

Example 10-5: Initial and Final Values

Properties of Laplace Transform (cont.)

7. Frequency Differentiation 8. Frequency Integration

Circuit Analysis

Example

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

1.Partial Fractions Distinct Real Poles

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

2. Partial Fractions Repeated Real Poles

2. Partial Fractions Repeated Real Poles

Example

Cont.

2. Partial Fractions Repeated Real Poles

Example cont.

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

3. Distinct Complex Poles (Cont.)

Next, we combine the last two terms:

4. Repeated Complex Poles: Same procedure as for repeated real poles

Property #3a in Table 10-2:

Hence:

s-Domain Circuit Models

Under zero initial conditions:

Example 10-11: Interrupted Voltage Source

Initial conditions:

Voltage Source

(s-domain)Cont.

Example 10-11: Interrupted Voltage Source (cont.)

Cont.

Example 10-11: Interrupted Voltage Source (cont.)

Cont.

Example 10-11: Interrupted Voltage Source (cont.)

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−.

Transfer Function (cont.)

Multiplication in s-domain

Convolution in time domain

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 t

Assumes x(t) = 0 for t < 0

Definition of convolution

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

Useful Recipe

Cont.

Same result as Method 1

Integral can be computed graphically at successive values of t.

0.86 @1s

0.11@ 2s

Example 10-16: Graphical Convolution

Summary