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

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

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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

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 © 2008 Pearson Education

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. © 2008 Pearson Education

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. © 2008 Pearson Education

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EE141 12.2 The Step Function

EE141 12.2 The Step Function

EE141 12.2 The Step Function

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

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. © 2008 Pearson Education

12.3 The Impulse Function A magnified view of the discontinuity, assuming a linear transition between –ε and +ε The derivative of the function © 2008 Pearson Education

The property of impulse fn.

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 © 2008 Pearson Education

EE141 12.3 The Impulse Function

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The first derivative of the impulse function. The impulse-generating function used to define the first derivative of the impulse © 2008 Pearson Education

The first derivative of the impulse function. The first derivative of the impulse-generating function that approaches δ’(t) as ε→0 © 2008 Pearson Education

The derivative of the impulse generating fn

The impulse function as the derivative of the step function. f(t) → u(t) as ε→0 © 2008 Pearson Education

The impulse function as the derivative of the step function. f’(t) → δ(t) as ε→0 © 2008 Pearson Education

The Laplace transform of unit step fn

12.4 Functional Transforms A functional transform is the Laplace transform of a specific function. A decaying exponential function © 2008 Pearson Education

12.4 Functional Transforms A sinusoidal function for t > 0 © 2008 Pearson Education

12.4 Functional Transforms Important functional transform pairs are summarized in the table below. © 2008 Pearson Education

12.5 Operational Transforms Operational transforms define the general mathematical properties of the Laplace transform. © 2008 Pearson Education

Derivation: Then, Therefore,

Let’s Laplace transform of g(t) is

In general,

Let’s

12.5 Operational Transforms An abbreviated list of operational transforms © 2008 Pearson Education

12.6 Applying the Laplace Transform A parallel RLC circuit © 2008 Pearson Education

12.6 Applying the Laplace Transform © 2008 Pearson Education

12.6 Applying the Laplace Transform © 2008 Pearson Education

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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: © 2008 Pearson Education

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Check!

See:

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

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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See: that is, Conjugate pair is Inverse transform is

12.7 Inverse Transforms Four useful transform pairs © 2008 Pearson Education

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.

12.8 Poles and Zeros of F(s) Plotting poles and zeros on the s plane © 2008 Pearson Education

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. © 2008 Pearson Education

12.9 Initial- and Final-Value Theorems Initial value theorem Final value theorem © 2008 Pearson Education

Proof: Initial value theorem EE141 Proof: Initial value theorem

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Proof: final-value theorem EE141 Proof: final-value theorem

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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. © 2008 Pearson Education

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제출기일을 지키지않는 레포트는 사정에서 제외함 EE141 Home work Prob. 12.6 12.13 12.18 12.19 12.21 12.23 12.27 12.28 12.31 12.34 12.37 12.39 12.40 12.41 12.42 12.43 제출기한: 다음 요일 수업시간 까지 제출기일을 지키지않는 레포트는 사정에서 제외함 제출기한: 다음 요일 수업시간 까지 제출기일을 지키지않는 레포트는 사정에서 제외함 Prob. 9.1 9.3 9.5 9.8 9.9 9.11 9.22 9.23 9.24 9.27 9.29 9.33 9.45 9.47 9.51 9.54 9.57 9.61 9.62 9.64 9.72 9.74 9.79 EE141

THE END © 2008 Pearson Education