The electrical engineering handbook CH006

The electrical engineering handbook

Dorf, R.C., Wan, Z., Johnson, D.E “Laplace Transform”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

6 Laplace Transform

6.1 Definitions and Properties

Laplace Transform Integral • Region of Absolute Convergence • Properties of Laplace Transform • Time-Convolution Property • Time-Correlation Property • Inverse Laplace Transform

6.2 Applications

Differentiation Theorems • Applications to Integrodifferential Equations • Applications to Electric Circuits • The Transformed Circuit • Thévenin’s and Norton’s Theorems • Network Functions • Step and Impulse Responses • Stability

6.1 Definitions and Properties

Richard C Dorf and Zhen Wan

The Laplace transform is a useful analytical tool for converting time-domain signal descriptions into functions

of a complex variable This complex domain description of a signal provides new insight into the analysis ofsignals and systems In addition, the Laplace transform method often simplifies the calculations involved inobtaining system response signals

Laplace Transform Integral

The Laplace transform completely characterizes the exponential response of a time-invariant linear function.This transformation is formally generated through the process of multiplying the linear characteristic signal

x(t) by the signal e –st and then integrating that product over the time interval (–¥, +¥) This systematicprocedure is more generally known as taking the Laplace transform of the signal x(t)

Definition: The Laplace transform of the continuous-time signal x(t) is

The variable s that appears in this integrand exponential is generally complex valued and is therefore oftenexpressed in terms of its rectangular coordinates

s = s + j w

where s = Re(s) and w = Im(s) are referred to as the real and imaginary components of s, respectively.The signal x(t) and its associated Laplace transform X(s) are said to form a Laplace transform pair Thisreflects a form of equivalency between the two apparently different entities x(t) and X(s) We may symbolizethis interrelationship in the following suggestive manner:

X ( s ) = + [ x ( t )]

where the operator notation + means to multiply the signal x(t) being operated upon by the complex nential e –stand then to integrate that product over the time interval (–¥, +¥)

expo-Region of Absolute Convergence

In evaluating the Laplace transform integral that corresponds to a given signal, it is generally found that thisintegral will exist (that is, the integral has finite magnitude) for only a restricted set of s values

The definition of region of absolute convergence is as follows The set of complex numbers s for which themagnitude of the Laplace transform integral is finite is said to constitute the region of absolute convergencefor that integral transform This region of convergence is always expressible as

s+ < Re( s ) < s–

where s+ and s– denote real parameters that are related to the causal and anticausal components, respectively,

of the signal whose Laplace transform is being sought

Laplace Transform Pair Tables

It is convenient to display the Laplace transforms of standard signals in one table Table 6.1 displays the timesignal x(t) and its corresponding Laplace transform and region of absolute convergence and is sufficient forour needs

Example. To find the Laplace transform of the first-order causal exponential signal

x1( t ) = e–atu ( t )

where the constant a can in general be a complex number

The Laplace transform of this general exponential signal is determined upon evaluating the associated Laplacetransform integral

(6.1)

In order for X1(s) to exist, it must follow that the real part of the exponential argument be positive, that is,

If this were not the case, the evaluation of expression (6.1) at the upper limit t = +¥ would either be unbounded

if Re(s) + Re(a) < 0 or undefined when Re(s) + Re(a) = 0 On the other hand, the upper limit evaluation iszero when Re(s) + Re(a) > 0, as is already apparent The lower limit evaluation at t = 0 is equal to 1/(s + a)for all choices of the variable s

The Laplace transform of exponential signal e – at u(t) has therefore been found and is given by

Properties of Laplace Transform

Linearity

Let us obtain the Laplace transform of a signal, x(t), that is composed of a linear combination of two other

signals,

x ( t ) = a1x1( t ) + a2x2( t )

where a1 and a2 are constants

The linearity property indicates that

and the region of absolute convergence is at least as large as that given by the expression

TABLE 6.1 Laplace Transform Pairs

2. t k e –at u(–t) Re(s) > –Re(a)

3. –e –at u(–t) Re(s) < –Re(a)

4. (–t) k e –at u(–t) Re(s) < –Re(a)

Source: J.A Cadzow and H.F Van Landingham, Signals, Systems, and Transforms,

Englewood Cliffs, N.J.: Prentice-Hall, 1985, p 133 With permission.

1 (s a+ )

k

s a k

! ( + ) +1

1

s

d t dt

k k

, – ,

2

s

w w

0 2 0

s +

s

s2 0

+ w w w (s a+ ) 2 +

where the pairs (s1

+; s+2) < Re(s) < min(s– ; s– ) identify the regions of convergence for the Laplace transforms

X1(s) and X2(s), respectively.

Time-Domain Differentiation

The operation of time-domain differentiation has then been found to correspond to a multiplication by s in the Laplace variable s domain.

The Laplace transform of differentiated signal dx(t)/dt is

Furthermore, it is clear that the region of absolute convergence of dx(t)/dt is at least as large as that of x(t).

This property may be envisioned as shown in Fig 6.1

Time Shift

The signal x(t – t0) is said to be a version of the signal x(t) right shifted (or delayed) by t0 seconds Right shifting

(delaying) a signal by a t0 second duration in the time domain is seen to correspond to a multiplication by e –s t 0

in the Laplace transform domain The desired Laplace transform relationship is

where X(s) denotes the Laplace transform of the unshifted signal x(t) As a general rule, any time a term of the form e –s t 0 appears in X(s), this implies some form of time shift in the time domain This most important

property is depicted in Fig 6.2 It should be further noted that the regions of absolute convergence for the

signals x(t) and x(t – t0) are identical

FIGURE 6.1 Equivalent operations in the (a) time-domain operation and (b) Laplace transform-domain operation.

(Source: J.A Cadzow and H.F Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall,

ë

ù û

ú =

+ [ ( )] ( ) x t t e - = -st X s

Time-Convolution Property

The convolution integral signal y(t) can be expressed as

where x(t) denotes the input signal, the h(t) characteristic signal identifying the operation process.

The Laplace transform of the response signal is simply given by

where H(s) = + [h(t)] and X(s) = + [x(t)] Thus, the convolution of two time-domain signals is seen to correspond to the multiplication of their respective Laplace transforms in the s-domain This property may be

envisioned as shown in Fig 6.3

Time-Correlation Property

The operation of correlating two signals x(t) and y(t) is formally defined by the integral relationship

The Laplace transform property of the correlation function fxy(t)is

in which the region of absolute convergence is given by

FIGURE 6.3 Representation of a time-invariant linear operator in (a) the time domain and (b) the s-domain (Source:

J A Cadzow and H F Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p 144.

-Autocorrelation Function

The autocorrelation function of the signal x(t) is formally defined by

The Laplace transform of the autocorrelation function is

and the corresponding region of absolute convergence is

Other Properties

A number of properties that characterize the Laplace transform are listed in Table 6.2 Application of theseproperties often enables one to efficiently determine the Laplace transform of seemingly complex time functions

TABLE 6.2 Laplace Transform Properties

Signal x(t) Laplace Transform Region of Convergence of X(s)

Linearity a 1x1(t) + a2x2(t) a1X1(s) + a2X2(s) At least the intersection of the region of

convergence of X1(s) and X2(s)

convergence of H(s) and X(s)

X(–s)X(s) max(– sx –, sx+ ) < Re(s) < min(–sx +, sx –)

Source: J A Cadzow and H.F Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs,

N.J.: Prentice-Hall, 1985 With permission

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k k

ò

Inverse Laplace Transform

Given a transform function X(s) and its region of convergence, the procedure for finding the signal x(t) that generated that transform is called finding the inverse Laplace transform and is symbolically denoted as

The signal x(t) can be recovered by means of the relationship

In this integral, the real number c is to be selected so that the complex number c + jw lies entirely within the region of convergence of X(s) for all values of the imaginary component w For the important class of rational

Laplace transform functions, there exists an effective alternate procedure that does not necessitate directly

evaluating this integral This procedure is generally known as the partial-fraction expansion method.

Partial Fraction Expansion Method

As just indicated, the partial fraction expansion method provides a convenient technique for reacquiring thesignal that generates a given rational Laplace transform Recall that a transform function is said to be rational

if it is expressible as a ratio of polynomial in s, that is,

The partial fraction expansion method is based on the appealing notion of equivalently expressing this rational

transform as a sum of n elementary transforms whose corresponding inverse Laplace transforms (i.e., generating

signals) are readily found in standard Laplace transform pair tables This method entails the simple five-stepprocess as outlined in Table 6.3 A description of each of these steps and their implementation is now given

I Proper Form for Rational Transform. This division process yields an expression in the proper form asgiven by

TABLE 6.3 Partial Fraction Expansion Method for Determining the Inverse Laplace Transform

I Put rational transform into proper form whereby the degree of the numerator polynomial is less than or equal to that of the denominator polynomial.

II Factor the denominator polynomial.

III Perform a partial fraction expansion.

IV Separate partial fraction expansion terms into causal and anticausal components using the associated region of absolute convergence for this purpose.

V Using a Laplace transform pair table, obtain the inverse Laplace transform.

Source: J A Cadzow and H.F Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985,

n n n

( )

+ + × × × + +-

- 1 1

1 1

in which Q(s) and R(s) are the quotient and remainder polynomials, respectively, with the division made so that the degree of R(s) is less than or equal to that of A(s).

II Factorization of Denominator Polynomial. The next step of the partial fraction expansion method entails

the factorizing of the nth-order denominator polynomial A(s) into a product of n first-order factors This factorization is always possible and results in the equivalent representation of A(s) as given by

The terms p1, p2, , pn constituting this factorization are called the roots of polynomial A(s), or the poles of X(s).

III Partial Fraction Expansion. With this factorization of the denominator polynomial accomplished, the

rational Laplace transform X(s) can be expressed as

(6.2)

We shall now equivalently represent this transform function as a linear combination of elementary transform

functions

Case 1: A(s) Has Distinct Roots.

where the ak are constants that identify the expansion and must be properly chosen for a valid representation

and

The expression for parameter a0 is obtained by letting s become unbounded (i.e., s = +¥) in expansion (6.2).

Case 2: A(s) Has Multiple Roots.

The appropriate partial fraction expansion of this rational function is then given by

1other elementary terms due to the

The coefficient a0 may be expediently evaluated by letting s approach infinity, whereby each term on the

right side goes to zero except a0 Thus,

The aq coefficient is given by the convenient expression

(6.3)

The remaining coefficientsa1, a2, … , aq–1 associated with the multiple root p1 may be evaluated by solving

Eq (6.3) by setting s to a specific value.

IV Causal and Anticausal Components. In a partial fraction expansion of a rational Laplace transform X(s)

whose region of absolute convergence is given by

it is possible to decompose the expansion’s elementary transform functions into causal and anticausal functions

(and possibly impulse-generated terms) Any elementary function is interpreted as being (1) causal if the real

component of its pole is less than or equal to s+ and (2) anticausal if the real component of its pole is greater

than or equal to s–

The poles of the rational transform that lie to the left (right) of the associated region of absolute convergencecorrespond to the causal (anticausal) component of that transform Figure 6.4 shows the location of causal andanticausal poles of rational transform

V Table Look-Up of Inverse Laplace Transform. To complete the inverse Laplace transform procedure, oneneed simply refer to a standard Laplace transform function table to determine the time signals that generateeach of the elementary transform functions The required time signal is then equal to the same linear combi-nation of the inverse Laplace transforms of these elementary transform functions

FIGURE 6.4 Location of causal and anticausal poles of a rational transform (Source: J.A Cadzow and H.F Van ham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p 161 With permission.)

1 1

1 1

1

s+ < Re( ) s < s

-Defining Terms

Laplace transform: A transformation of a function f(t) from the time domain into the complex frequency

domain yielding F(s).

where s = s + jw.

Region of absolute convergence: The set of complex numbers s for which the magnitude of the Laplace

transform integral is finite The region can be expressed as

where s+ and s– denote real parameters that are related to the causal and anticausal components,respectively, of the signal whose Laplace transform is being sought

E Kamen, Introduction to Signals and Systems, 2nd Ed., Englewood Cliffs, N.J.: Prentice-Hall, 1990.

B.P Lathi, Signals and Systems, Carmichael, Calif.: Berkeley-Cambridge Press, 1987.

6.2 Applications1

David E Johnson

In applications such as electric circuits, we start counting time at t = 0, so that a typical function f(t) has the property f(t) = 0, t < 0 Its transform is given therefore by

which is sometimes called the one-sided Laplace transform Since f(t) is like x(t)u(t) we may still use Table 6.1

of the previous section to look up the transforms, but for simplicity we will omit the factor u(t), which is

1Based on D.E Johnson, J.R Johnson, and J.L Hilburn, Electric Circuit Analysis, 2nd ed., Englewood Cliffs, N.J.:

Prentice-Hall, 1992, chapters 19 and 20 With permission.