1. 10. Laplace TransforM Technique
CIRCUITS by Ulaby & Maharbiz

2. Overview

3. Analysis Techniques Circuit Excitation Method of Solution Chapters
1. dc (w/ switches) Transient analysis 5 & 6
2. ac Phasor-domain analysis
( steady state only)
3. any waveform Laplace Transform This Chapter
(single-sided) (transient + steady state)
4. Any waveform Fourier Transform
(double-sided) (transient + steady state)
Single-sided: defined over [0,∞] Double-sided: defined over [−∞,∞]

4. 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

5. Singularity Functions (cont.)
Unit Impulse Function
For any function f(t):

6. Laplace Transform Definition

7. Laplace Transform of Singularity Functions
For A = 1 and T = 0:

8. Laplace Transform of Delta Function
For A = 1 and T = 0:

10. Properties of Laplace Transform
1. Time Scaling
Example
2. Time Shift

11. Properties of Laplace Transform (cont.)
3. Frequency Shift
Example
4. Time Differentiation

12. Properties of Laplace Transform (cont.)
5. Time Integration
6. Initial and Final-Value Theorems

13. Example 10-5: Initial and Final Values

14. Properties of Laplace Transform (cont.)
7. Frequency Differentiation
8. Frequency Integration

18. Circuit Analysis

19. Example

20. 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

21. 1.Partial Fractions Distinct Real Poles

22. 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.

23. 2. Partial Fractions Repeated Real Poles

24. 2. Partial Fractions Repeated Real Poles
Example
Cont.

25. 2. Partial Fractions Repeated Real Poles
Example cont.

26. 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.

27. 3. Distinct Complex Poles (Cont.)
Next, we combine the last two terms:

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

29. Property #3a in Table 10-2:
Hence:

30. s-Domain Circuit Models
Under zero initial conditions:

32. Example 10-11: Interrupted Voltage Source
Initial conditions:
Voltage Source
(s-domain)
Cont.

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

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

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

36. 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−.

37. Transfer Function (cont.)
Convolution in time domain
Multiplication in s-domain

38. Convolution Integral Impulse Response h(t): output of linear system when input is a delta function
y(t) cannot depend on excitations occurring after time t
Assumes x(t) = 0 for t < 0
Definition of convolution

39. 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

40. Useful Recipe

42. Cont.

43. Same result as Method 1

44. Integral can be computed graphically at successive values of t.

45. 2s

46. Example 10-16: Graphical Convolution

47. Summary