Frequency Response

When a stable linear/invariant system has a sinusoidal input, then after some time the transients associated with the starting conditions die out, and there remains only a steady-state sinusoidal response at the same frequency as that of the input. Its ampli­tude and phase are generally different from those of the input, however, and the ex­pression of these differences is embodied in the frequency-response function.

Consider a single input/response pair, and let the input be the sinusoid a, cos cot. We find it convenient to replace this by the complex expression c = A, e,eut, of which ax cos cot is the real part. A, is known as the complex amplitude of the wave. The re­sponse sinusoid can be represented by a similar expression, x = Аге1Ш, the real part of which is the physical response. As usual, x and c are interpreted as rotating vectors whose projections on the real axis give the relevant physical variables (see Fig. 7.8a).

From Table A. 1, item 8, the transform of c is

A,

c = ——–

s — ico

Figure 7.8 (a) Complex input and response. (b) Effect of singularity close to axis.

The function G(s) is given by (7.2,9) so that

m

X 1 (i – ico)f(s)

The roots of the denominator of the r. h.s. are

A, • • • A„, іш

so that the application of the expansion theorem (A.2,10) yields the complex output

«+1 г (^ – K)N(s) I

*(0 =AX 7———– ■ ,v4

r=l L (S – l0>)f(S) J,= A,

Since we have stipulated that the system is stable, all the roots A, ••• A„ of the charac­teristic equation have negative real parts. Therefore eK, t —» 0 as t —> °° for r = 1 ••• n, and the steady-state periodic solution is

N(iw)

x(t) = At——ela“, t-* oo f(iw)

or

x(t) = AtG(iw)eia“
= А2еш

Thus

A2 = AtG(ito) (7.5,3)

is the complex amplitude of the output, or

Giito) = — (7.5,4)

At

the frequency response function, is the ratio of the complex amplitudes. In general, G{ito) is a complex number, varying with the circular frequency w. Let it be given in polar form by

G{iw) = KMeiv (7.5,6)

where К is the static gain (7.4,5). Then

An.

— = KMe‘v (7.5,7)

Ai

From (7.5,7) we see that the amplitude ratio of the steady-state output to the input is! A2M| = KM: that is, that the output amplitude is a2 = КМаъ and that the phase re-

Locus of Me* (semicircle)

lation is as shown on Fig. 7.8a. The output leads the input by the angle <p. The quan­tity M, which is the modulus of G(ioj) divided by K, we call the magnification factor, or dynamic gain, and the product KM we call the total gain. It is important to note that M and q> are frequency-dependent.

Graphical representations of the frequency response commonly take the form of either vector plots of Me,<p (Nyquist diagram) or plots of M and cp as functions of fre­quency (Bode diagram). Examples of these are shown in Figs. 7.9 to 7.13.