1. ELECTRIC CIRCUITS EIGHTH EDITION
JAMES W. NILSSON
&
SUSAN A. RIEDEL
ELECTRIC CIRCUITS EIGHTH EDITION

2. INTRODUCTION TO THE LAPLACE TRANSFORM
CHAPTER 12
INTRODUCTION TO THE LAPLACE TRANSFORM
© 2008 Pearson Education

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4. CONTENTS 12.1 Definition of the Laplace Transform
12.2 The Step Function
12.3 The Impulse Function
12.4 Functional Transforms
12.5 Operational Transforms
12.6 Applying the Laplace Transform
12.7 Inverse Transforms
12.8 Poles and Zeros of F(s)
12.9 Initial- and Final-Value Theorems
© 2008 Pearson Education

5. 12.1 Definition of the Laplace Transform
The Laplace transform is a tool for converting time-domain equations into frequency-domain equations, according to the following general definition:
f(t) = the time-domain expression
F(s) = the frequency-domain expression
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6. 12.1 Definition of the Laplace Transform
A continuous and discontinuous function at the origin.
f(t) is continuous at the origin
f(t) is discontinuous at the origin.
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7. 12.2 The Step Function
The step function Ku(t) describes a function that experiences a discontinuity from one constant level to another at some point in time.
K is the magnitude of the jump; if K=1, Ku(t) is the unit step function.
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12.2 The Step Function

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12.2 The Step Function

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12.2 The Step Function

12. sol.:
at t = 0 ~ 1, +2t
at t = 1 ~ 3, -2t + 4
at t = 3 ~ 4, +2t – 8

13. 12.3 The Impulse Function The impulse function Kδ(t) is defined
Where K is the strength of the impulse; if K=1, Kδ(t) is the unit impulse function.
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14. 12.3 The Impulse Function
A magnified view of the discontinuity, assuming a linear transition between –ε and +ε
The derivative of the function
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15. The property of impulse fn.

16. A variable-parameter function used to generate an impulse function
12.3 The other impulse Function
A variable-parameter function used to generate an impulse function
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12.3 The Impulse Function

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20. The first derivative of the impulse function.
The impulse-generating function used to define the first derivative of the impulse
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21. The first derivative of the impulse function.
The first derivative of the impulse-generating function that approaches δ’(t) as ε→0
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22. The derivative of the impulse generating fn

23. The impulse function as the derivative of the step function.
f(t) → u(t) as ε→0
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24. The impulse function as the derivative of the step function.
f’(t) → δ(t) as ε→0
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25. The Laplace transform of unit step fn

26. 12.4 Functional Transforms
A functional transform is the Laplace transform of a specific function.
A decaying exponential function
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27. 12.4 Functional Transforms
A sinusoidal function for t > 0
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28. 12.4 Functional Transforms
Important functional transform pairs are summarized in the table below.
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29. 12.5 Operational Transforms
Operational transforms define the
general mathematical properties of the
Laplace transform.
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31. Derivation:
Then,
Therefore,

32. Let’s
Laplace transform of g(t) is

33. In general,

37. Let’s

39. 12.5 Operational Transforms
An abbreviated list of operational transforms
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40. 12.6 Applying the Laplace Transform
A parallel RLC circuit
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41. 12.6 Applying the Laplace Transform
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42. 12.6 Applying the Laplace Transform
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44. 12.7 Inverse Transforms
If F(s) is a proper rational function, the inverse transform is found by a partial fraction expansion.
Example of a proper rational function:
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51. Check!

52. See:

54. We 1st multiply both sides by
Next we differentiate both sides once with respect to s
& then evaluate at s = - 5:

55. We 1st multiply both sides by
Next we differentiate both sides twice with respect to s
& then evaluate at s = - 5:

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59. See:
that is,
Conjugate pair is
Inverse transform is

60. 12.7 Inverse Transforms Four useful transform pairs
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61. If F(s) is an improper rational function, it can be inverse transformed by first expanding it into a sum of a polynomial and a proper rational function.

64. 12.8 Poles and Zeros of F(s) Plotting poles and zeros on the s plane
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65. 12.8 Poles and Zeros of F(s)
F(s) can be expressed as the ratio of two factored polynomials.
The roots of the denominator are called poles and are plotted as Xs on the complex s plane.
The roots of the numerator are called zeros and are plotted as 0s on the complex s plane.
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66. 12.9 Initial- and Final-Value Theorems
Initial value theorem
Final value theorem
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67. Proof: Initial value theorem
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Proof: Initial value theorem

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69. Proof: final-value theorem
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Proof: final-value theorem

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73. 12.9 Initial- and Final-Value Theorems
The theorem is valid only if the poles of F(s), except for a first-order pole at the origin, lie in the left half of the s plane.
The initial- and final-value theorems allow us to predict the initial and final values of f(t) from a s-domain expression.
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77. 제출기일을 지키지않는 레포트는 사정에서 제외함
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Home work
Prob
제출기한:
다음 요일 수업시간 까지
제출기일을 지키지않는 레포트는 사정에서 제외함
제출기한:
다음 요일 수업시간 까지
제출기일을 지키지않는 레포트는 사정에서 제외함
Prob
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78. THE END
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