The responses of a system to any of a number of input disturbances are useful for studying the dynamic behavior of the system. Typical choices might be a pulse, a step, or a ramp. Flowever, the frequency response, which in­volves the response of a selected system output due to a sinusoidal input, is particularly useful. As will be shown below, the output is a sinusoid with the same frequency as the input, but shifted by some phase angle ф. The ratio of the amplitude of the output to the amplitude of the input and the phase

angle completely specify the frequency response. A typical pair of waveforms appears in Fig. 2.4.

(a) Relation between Frequency Response and Transfer Function

We may now determine the relation between the frequency response and the system transfer function. (Many authors use the terms interchangeably. Here the term transfer function refers to a mathematical quantity, a ratio of Laplace transforms. The term frequency response refers to a physically observable quantity.) For an input 5I(t) = b sin cot, the Laplace transform (see Table 2.1) is bco/(s2 + со2). The Laplace transform of the output is obtained using Eq. (2.3.1).

<50(s) = G(s) 8I(s) = G(s)bco/(s2 + со2) (2.4.1)

We can use the method of residues to determine the output:

where s, is the pole of G(s) and j = ч/— 1. (This development assumes simple poles, but the result is the same for systems with multiple poles.) For a stable system, all the st have negative real parts. Therefore, after a sufficient time, all the terms containing e**’ will have vanished, and only the first two terms in Eq. (2.4.2) will remain:

*>(.) = j (2,4.3)

The terms G(jco) and G( — jco) are complex quantities. A complex quantity (a + jfS) always may be written as a magnitude and a phase, G(j(o)ei*. This is demonstrated in Fig. 2.5. This shows that G(— jco) = |G(ycu)|е~ІФ.

Fig. 2.5. Complex plane representation of G(jto).

Thus Eq. (2.4.3) may be written

gjtaW + tfO _ e~j(o>t + tli)

2/

We use Euler’s formulas (eJX = cosx + jsinx, e~jx = cos x — ;’sinx) to give

50(t) = bG(jco) sin(cut + ф) (2.4.5)

This verifies the earlier assertion that the output for a sinusoidal input is a sine wave with the same frequency as the input, but shifted by a phase angle ф. The amplitude of the output is |G(yat)| times the amplitude of the input.

This development has shown that the theoretical frequency response is obtained simply by substituting jco for s in the transfer function and carry­ing out the complex arithmetic. One of the main reasons that frequency response analysis and testing is commonly used is that such a simple link between the theoretical transfer function and the experimental frequency response exists.

(b) Frequency Response Plots

The result of a frequency response calculation is a complex number, which may be represented as a magnitude and a phase:

G(jco) = a (jco) + jP(jco) = |G(yw)|eJlM“) (2.4.6)

where

I G(jco) = {[a(ycu)]2 + mjco)]2}1/2 ‘ (2.4.7)

and

{//(jco) = tan” ‘[P(joj)/a(joj)] (2.4.8)

The most common way to plot the frequency response is to show separate plots of amplitude and phase as a function of frequency. This is called a Bode plot. A log-log plot is used for the amplitude curve and a semilog plot is used for the phase curve. These are demonstrated in Fig. 2.6 for the transfer function l/(s + 1). It is also common to define the magnitude in terms of the decibel (dB), given by

|G(yco)| (in decibels) = 20log10|G(yco)| (in absolute units) (2.4.9)

A few values are shown in tabular form to illustrate the relation between absolute gain and gain in decibels.

Absolute gain	Gain in dB
0.1	20 log 0.1 = -20
1	20 log 1 = 0
2	20 log 2 = 6.02
10	20 log 10 = 20

Another common plotting procedure presents the frequency response on a single polar plot. In the complex plane, the frequency response at some frequency is given by a vector as shown in Fig. 2.7. At some other frequency, the vector will have a different orientation and a different length. A curve traced out by the tip of the vector as the frequency changes is a complete description of the frequency response. A polar plot for the transfer function l/(s + 1) is shown in Fig. 2.8.

A third method of graphical presentation of frequency response data is by a plot of gain versus phase. Such a plot is called a Nichols plot. The gain may be expressed in absolute units or in decibels. A Nichols plot for the transfer function l/(s + 1) is shown in Fig. 2.9.

(b) Frequency (radians/sec.)

Fig. 2.6. (a) Amplitude for transfer function 1 /(s + 1); (b) phase shift for transfer function!/(*+ !)•

Fig. 2.7. Complex plane plot of G(jco) at a single frequency со.

Fig. 2.8. Polar plot for transfer function l/(s + 1).

Of course the Bode plot, the polar plot, and the Nichols plot are only different methods for presenting the same data. The purpose for which the data are used determines the most appropriate plotting method.

The Bode plot is convenient for furnishing certain information about system dynamics. The Bode plot for the system transfer function is inspected for “break frequencies” and resonance peaks.

Let us first consider the break frequency. Take the transfer function G(s) = l/(s + a). The real and imaginary parts of the frequency response are given by

Re[G(jco)] = a/(a2 + a>2), Im[G(ya>)] = — to/(a2 + a>2)

The amplitude and phase are then

|G(;w)| = [1 /(a2 + ш2)]1/2, іИМ = tan-1(-aVa)

It is informative to examine the asymptotic value of |G(yw)| and i/d/cu) for very small frequencies and for very large frequencies. For very small fre­quencies,

|G(;cu)| s (l/a2)1/2 = 1/a, іj> s tan" ‘(0) = 0

For very large frequencies,

|G(;cu)| s (l/w2)1/2 = 1 /со, ф s tan-‘(—со) = -90°

This shows that the amplitude has a constant value of 1/a at low frequencies and varies as 1/ш at high frequencies. The phase has a constant value of 0

at low frequency and —90° at high frequency. These low-frequency and high-frequency approximations are shown in Fig. 2.10 along with the exact curves. Note that the two amplitude curves intersect at со = a. This frequency is called the break frequency.

A similar analysis may be made for transfer functions of the form G(s) = s + a. The Bode plot for this transfer function appears in Fig. 2.11. This break frequency is again at со = a, but in this case the amplitude rises after the break. Also, the phase goes to +90° at high frequencies.

In general, the amplitude will break downward at frequencies correspond­ing to real poles and upward at frequencies corresponding to real zeros.

Lumped systems in which all poles and zeros have negative real parts are called minimum-phase systems. An exact relation between the gain curve and the associated phase curve for minimum-phase systems is provided by Bode’s first theorem (2). This theorem gives the phase at all frequencies if the gain at all frequencies is known. This is useful for obtaining detailed
phase information from gain results or vice versa, but our main interest is in methods for making rough approximations.

It is sometimes useful to determine the asymptotic gain and the associated asymptotic phase for a system. The asymptotic gain and phase are the gain and phase that would occur if the poles and zeros were widely separated. For a minimum-phase system with real poles and zeros, the results are very simple:

1. The asymptotic slope of the gain for a transfer function with negative real poles and zeros is given by

= (Na — Da) (2.4.10)

where Sa is the logarithmic slope of the gain curve at frequency со (in decades of change in gain per decade increase in со), Na the number of zeros with numerical values less than со, and Dm the number of poles with numerical values less than со.

2. The asymptotic phase shift is given by

Фю = S<o x 90° (2.4.11)

These properties may be used to construct approximate Bode plots for specified transfer functions or to construct approximate transfer functions for specified Bode plots. This is best illustrated by an example.

Fig. 2.12. Components of G(s) — (s + l)/[(s + 0.1)(s + 10)].

Example 2.4.1. Consider the transfer function

w (s + 0.1)(s + 10)

Each term contributes to the amplitude as shown in Fig. 2.12. The resulting amplitude is shown in Fig. 2.13a. Because of the relation between phase angles and slopes on amplitude plots, the approximate phase is as shown in Fig. 2.13b. Of course the phase curve will not display sharp changes, but will gradually change as shown in Fig. 2.13b.

(a)

Fig. 2.13. (a) Asymptotic amplitude approximation for (s + l)/[(s + 0.1 )(s + 10)] and exact results; (b) asymptotic phase approximation for s + l/[(s + 0.1)(s + 10)] and exact results.

The situation is somewhat different for the common case of complex poles or zeros. Figure 2.14 shows the gain and phase for the following transfer function:

G(s) = l/(s2 + 2£s + 1)

This transfer function has complex poles for £ > 0. We observe that Eqs.

(2.4.10)

and (2.4.11) are not valid in the region of the peak in the gain. In

general, systems with complex poles can have resonance peaks and Eqs.

(2.4.10) and (2.4.11) are not valid near the peaks. However, these relations are valid at frequencies well away from the peak as is demonstrated in Fig. 2.14.

(c) Stability Analysis

The Nyquist stability criterion uses a frequency response obtained from the open-loop transfer function. The criterion for negative feedback systems whose closed-loop transfer function is given by Eq. (2.3.4) is:

The closed loop system whose open loop transfer function is GlH(s) is stable if and only if

R = P

where R is the number of clockwise encirclements of the (— 1, JO) point by the locus GiHijco) as a> varies from — 00 to +00, and P the number of poles of G1H with positive real parts.

For most nuclear reactor applications, the transfer functions Gj and H have no poles with positive real parts. For this case, the Nyquist criterion may be stated:

The closed loop system whose open loop transfer function has no poles with positive real parts is stable if and only if

R = 0

Typical Nyquist plots for a stable system and an unstable system appear in Fig. 2.15. This figure also illustrates the procedure for connecting the 0~ point (the point approached as a> approaches zero from the negative side) and the 0+ point (the point approached as a> approaches zero from the positive side). The general procedure is to close the locus with an infinite — radius clockwise trajectory from the 0“ point to the 0+ point if the 0“ and 0+ points do not coincide. Clearly, the proximity of the locus to (—1,_/0) obtained for a stable system is a measure of the stability margin. Two measures are used to assess the stability margin:

Phase margin—the angle between the negative real axis and the line pass­ing from the origin through the point where the Nyquist locus has a magni­tude of unity.

Gain margin—the factor by which GlH would have to be multiplied to cause the intercept of the negative real axis to pass through ( — 1, j’0).

Fig. 2.15. Nyquist plots for a stable system and an unstable system. Arrows indicate increasing frequency.

These concepts are demonstrated in Fig. 2.16. Clearly, these measures are meaningless for loci that do not cross the negative real axis or that have very complicated shapes near the origin.

The phase margin is the most common of the two stability measures for Nyquist plots. A rule of thumb is that a phase margin of at least 20° is desirable. The phase margin may be obtained by calculation or by experi­ment.

Fig. 2.16. Portion of a Nyquist plot near the origin, where *F is the phase margin and 1 fa the gain margin.