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COMPLEX VARIABLES AND THE LAPLACE TRANSFORM FOR ENGINEERS
By Wilbur L. LePage Dover Publications, Inc.
Copyright © 1961 Wilbur R. LePage
All rights reserved.
ISBN: 978-0-486-13644-8
CHAPTER 1
CONCEPTUAL STRUCTURE OF SYSTEM ANALYSIS
1-1. Introduction. It is worthwhile for a serious student of the analytical approach to engineering to recognize that one important facet of his education consists in a transition from preoccupation with techniques of problem solving, with which he is usually initially concerned, to the more sophisticated levels of understanding which make it possible for him to approach a subject more creatively than at the purely manipulative level. Lack of adequate motivation to carry out this transition can be a serious deterrent to learning. This chapter is directed at dealing with this matter. Although it is assumed that you are familiar with the Laplace transform techniques of solving a problem, at least to the extent covered in a typical undergraduate curriculum, it cannot be assumed that you are fully aware of the importance of functions of a complex variable or of the wide applicability of the Laplace transform theory.
Since motivation is the primary purpose of this chapter, for the most part we shall make little effort to attain a precision of logic. Our aim is to form a bridge between your present knowledge, which is assumed to be at the level described above, and the more sophisticated level of the relatively carefully constructed logical developments of the succeeding chapters. In this first chapter we briefly use several concepts which are reintroduced in succeeding chapters. For example, we make free use of complex numbers in Chap. 1, although they are not defined until Chap. 2. Presumably a student with no background in electric-circuit theory or other applications of the algebra of complex numbers could study from this book; but he would probably be well advised to start with Chap. 2.
Most of Chap. 1 is devoted to a review of the roles played by complex numbers, the Fourier series and integral, and the Laplace transform in the analysis of linear systems. However, the theory ultimately to be developed in this book has applicability beyond the purely linear system, particularly through the various convolution theorems of Chap. 11 and the stability considerations in Chaps. 6, 7, and 13.
1-2. Classical Steady-state Response of a Linear System. A brief summary of the essence of the sinusoidal steady-state analysis of the response of a linear system requires a prediction of the relationship between the magnitudes A and B and initial angles α and β for two functions such as
va = A cos (ωt + α) vb = B cos (ωt + β) (1-1)
where va, for example, is a driving function and vb is a response function. From a steady-state analysis we learn that it is convenient to define two complex quantities
Va = Aejα Vb = Bejβa (1-2)
which are related to each other through a system function H(jw) by the equation
Vb = H(jω)Va (1-3)
H(jw), a complex function of the real variable w, provides all the information required to determine the magnitude relationship and the phase difference between input and output sinusoidal functions. Presently we shall point out that H(jw) also completely determines the nonperiodic response of the system to a sudden disturbance.
In the example of Fig. 1-1, the H(jw) function is
H(jω) = jωRC/1 - ω2LC + jωRC (1-3a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-3b)
Equation (1-3a) emphasizes the fact that H(jw) is a rational function (ratio of polynomials) of the variable jW, and Eq. (1-3b) places in evidence the factors of H(jw) which are responsible for changing the magnitude and angle of Va, to give Vb. Evaluation of the steady-state properties of a system is usually in terms of magnitude and angle functions given in Eq. (1-3b), but the rational form is more convenient for analysis.
This brief summary leaves out the details of the procedure for finding H(jw) from the differential equations of a system. It should be recognized that H(jw) is a rational function only for systems which are described by ordinary linear differential equations with constant coefficients.
1-3. Characterization of the System Function as a Function of a Complex Variable. The material of the preceding section provides our first point of motivation for a study of functions of a complex variable. In the first place, purely for convenience of writing, it is simpler to write
H(s) = RCs/1 + RCs + LCs2 (1-4)
which reduces to Eq. (1-3a) if we make the substitution s = jw. However, wherever we write an expression like this, with s indicated as the variable, we understand that s is a complex variable, not necessarily jw. In fact, throughout the text we shall use the notation s = σ + jw. Another advantage of Eq. (1-4) is recognized when it appears in the factored form
H(s) = (R/L)s/(s -s1)(s -s2) (1-5)
Carrying these ideas a bit further, we observe that the general steady-state-system response function can be characterized as a rational function
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-6)
Various systems differ with respect to the degree of numerator and denominator of Eq. (1-6), in the factor K, and in the locations of the critical values s1 s2, s3, etc. The quantities s1, s3, etc., in the numerator are called zeros of the function, and the corresponding s2, s4, etc., in the denominator are poles of the function. In general, these critical values of s, where H(s) becomes either zero or infinite, are complex numbers, emphasizing the need to deal with complex numbers in the analysis of a linear system.
Equation (1-6) provides an example of the importance of becoming accustomed to thinking in terms of a function of a complex variable, since, with s = jw and w variable, this function represents the variation of system response as a function of frequency. In particular, the variation of response magnitude with frequency is often important, as in filter design; and Eq. (1-6) provides a convenient vehicle for obtaining this functional variation. Geometrically, each factor in the numerator or denominator of Eq. (1-6) has a magnitude represented graphically by line AB in Fig. 1-2a, shown for the particular case where sk is a negative real number. Except for the real multiplying factor K, for any complex value of s the complex number H(s) has a magnitude which can be calculated as a product and quotient of line lengths like AB in the figure and an angle which is made up of sums and differences of angles like αk, Thus, a plot in the complex s plane provides a pictorial aid in understanding the properties of the function H(s). In particular, steady-state response for variable frequency is characterized by allowing point s to move along the vertical axis.
This formulation is also helpful when we are concerned with variation of the magnitude 1H(jw)l as a function of w. The function H(s)H(—s) plays a central role in this question. H(-s) is made up of products and quotients of factors like—8—Sk, one of which is portrayed by magnitude|—s—skl and angle α'k in Fig. 1-2b. Thus, if each sk is real, H(s)H(—s) is formed from the product and quotient of factors like
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The geometry of Fig. 1-2a makes it evident that when s = jw (placing s on the vertical axis) the sum of angles αk + α'k is zero and therefore H(jw)H(-jw) is real. Furthermore, AB = AC when s = jw, and therefore H(jw)H(—jw) is the square of the magnitude of H(jw). It can be shown, from physical considerations, that, if sk is complex, the factor s—sk is accompanied by a companion factor s - [??]k, where [??]k is the complex conjugate of Sk, as illustrated in Fig. 1-2c. In that case, both factors are considered together, with the conclusion that the product of four factors (s - sk)(s - [??]k)-s -sk)(- s -sk) is real when s = jw. Thus, since K is real, we find generally that the function H(s)(—sH) is a function of a complex variable which has the peculiar property of being real when s = j ω and furthermore of being the square of the magnitude of H(jω). We summarize by writing
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-7)
Equation (1-4) can be used for an illustration, where
H(-s) = -RCs/1 -RCs + LCs2 (1-8)
giving
H(s)H(-s) = -R2C2s2/(1 + LCs2)2 - R2C2s2 (1-9)
When s = jω, since (jω)2 = -ω2, we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-10)
which is the square of the magnitude factor in Eq. (1-3b).
The function H(s)H(—s) is particularly important in the design of filter and corrective networks because of the property just demonstrated. Again we can say that analytical work is easier if we deal with the complex function H(s)H(—s) than if we deal only with the real function 1H(jω)l.
1-4. Fourier Series. The sinusoidal function described in Sec. 1-3 plays a vital role beyond the sinusoidal case for which it is defined. The reason is provided by the Fourier series, whereby a periodic function va(t), of angular frequency ωa, can be described as a sum of sinusoidal components. One way to write the Fourier series for the driving function is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-11)
each term of which is like Eqs. (1-1). Assuming that the principle of superposition is applicable, the response is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-12)
Upon comparing with Eqs. (1-2) and (1-3), it is evident that An and Bn and αn and βn are related by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-13)
The important concept here is the emergence of the idea of a signal spectrum (a line spectrum in the case of a periodic function) represented by the sequence of complex numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], etc., and of the modification of this signal spectrum by the system function H(jnωa) to give the spectrum of the output signal. Emphasis is on the importance of H(jω) as a function of ω, where in this case the function is used only at discrete values of ω.
1-5. Fourier Integral. It is a short intuitive step from the Fourier series to the Fourier integral. In this case, υa(t) is not periodic, but we can define a periodic function, of period π/Δω,
ωt = va(t) |t| < π/Δω (1-14)
and for all t,
ω(t) = ω(t [+ or -] 2πΔω) (1-15)
as illustrated in Fig. 1-3. The period of w(t) can be made as large as desired by making Δω arbitrarily small. As another indication of the usefulness of complex quantities, it is also known that we can write
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-16)
thereby putting into evidence the fact that ω(t) can be represented as a complex Fourier series involving a summation over negative as well as positive n, as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-17)
where each An is complex, rather than real and positive as in Eq. (1-11). The function w(t) is identical with υa(t) over the arbitrarily large interval —π/Δω < t< π/Δω and will have a response r(t) given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-18)
where
Bn = H(jn ΔΩ)An (1-19)
The last two equations are similar to Eqs. (1-12) and (1-13), with the exception that An and Bn are complex in Eq. (1-19). As defined, r(t) is the response to a fictitious periodic function which approximates υa(t) over the interval -π/Δω < t< π/Δω. If υa(t) reduces essentially to zero for t outside this interval, it is intuitively reasonable to expect r(t) to be an approximation for υb(t), valid over the same interval. Assuming this to be the case, one might expect to be able to take the limit as Δω goes to zero, thereby obtaining exact expressions which we formally write for the two functions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-20a)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-20b)
Of course, according to the well-known theory of Fourier series, the coefficients An are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-21)
We now allow Δω to approach zero, with no concern for the question of whether the limits exist. Equation (1-21), which is a function of the discrete variable n Δω, becomes a function of a continuous variable jω, and we formally write
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-22)
The summations in Eqs. (1-20) become integrals, as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-23a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-23b)
where
υb(jω) = H(jωυa(jω) (1-24)
takes the place of Eq. (1-19). The system response function now "acts on" the continuous-spectrum function υa(jω) of the excitation, to produce the continuous spectrum of the response vb(jω).
The above brief outline gives the essential ideas of the Fourier integral treatment of a linear system, showing how the response to a general nonperiodic excitation is determined by the system function H(s), again emphasizing the importance of this function.
1-6. The Laplace Integral. The Fourier integral approach, although powerful, is not satisfying for solving certain practical problems and does not provide as general a basis for theoretical analysis as we should like. Its shortcomings are two in number:
1. An integral like Eq. (1-22) does not exist for most υa(t) functions of practical interest. For example, it will not handle such a simple case as the unit-step function.
2. The formulation does not conveniently take into consideration the transient effects when energy stored in system components is suddenly released. That is, arbitrary initial conditions cannot be handled.
Both these problems are dealt with by making two simple modifications. The excitation function is replaced by a function which is defined to be zero for negative t; and υa(t) and υb(t) are multiplied by a "converging factor" e-σt, For positive values of the real number σ, this function approaches zero fast enough to allow many integrals like
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
to converge when they do not converge in the absence of the e-σt factor. For example, if υa(t) is the unit step,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and each integral on the right exists if σ > .0. This permits definition of a function of σ + jω, which bears the same relationship to υa(t)e-αt as υ a(jω) bears to υa(t), with the additional stipulation that υa(t) is now zero for t< 0. Thus, we define
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-25)
and a formula corresponding to Eq. (1-23a) can be derived, giving
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
which is more conveniently written
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-26)
Similar expressions apply for υb(t), for which we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-27)
as the Fourier integral of υb(t)e-σt, where υb(t) = 0 when t< 0; and also, in similarity with Eq. (1-26),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-28)
(Continues...)
Excerpted from COMPLEX VARIABLES AND THE LAPLACE TRANSFORM FOR ENGINEERS by Wilbur L. LePage. Copyright © 1961 Wilbur R. LePage. Excerpted by permission of Dover Publications, Inc..
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