1. INTRODUCTION

By the term frequency response, we mean the steady-state response of a system to a sinusoidal input. In frequency-response methods, we vary the frequency of the input signal over a certain range and study the resulting response. In this and the next chapter we present frequency-response approaches to the analysis and design of control systems. The information we get from such analysis is different from what we get from root-locus analysis. In fact, the frequency response and root- locus approaches complement each other. One advantage of the frequency-response approach is that we can use the data obtained from measurements on the physical system without deriving its mathematical model. In many practical designs of control systems both approaches are employed. Control engineers must be familiar with both.

Frequency-response methods were developed in 1930s and 1940s by Nyquist, Bode, Nichols, and many others. The frequency-response methods are most powerful in conventional control theory. They are also indispensable to robust control theory.

The Nyquist stability criterion enables us to investigate both the absolute and relative stabilities of linear closed-loop systems from a knowledge of their open-loop frequency- response characteristics. An advantage of the frequency-response approach is that frequency-response tests are, in general, sirnple and can be made accurately by use of readily available sinusoidal signal generators and precise measurement equipment. Often the transfer functions of complicated components can be determined experimentally by frequency-response tests. In addition, the frequency-response approach has the advantages that a system may be designed so that the effects of undesirable noise are negligible and that such analysis and design can be extended to certain nonlinear control systems.

Although the frequency response of a control system presents a qualitative picture of the transient response, the correlation between frequency and transient responses is indirect, except for the case of second-order systems. In designing a closed-loop system, we adjust the frequency-response characteristic

of the open-loop transfer function by using several design criteria in order to obtain acceptable transient-response characteristics for the system.

Presenting Frequency-Response Characteristics in Graphical Forms. The sinusoidal transfer function, a complex function of the frequency w, is characterized by its magnitude and phase angle, with frequency as the parameter. There are three commonly used representations of sinusoidal transfer functions:

1. Bode diagram or logarithmic plot

2. Nyquist plot or polar plot

3. Log-magnitude-versus-phase plot (Nichols plots)

BODE DIAGRAMS

Bode Diagrams or Logarithmic Plots. A Bode diagram consists of two graphs: One is a plot of the logarithm of the magnitude of a sinusoidal transfer function; the other is a plot of the phase angle; both are plotted against the frequency on a logarithmic scale.

The standard representation of the logarithmic magnitude of G(jw) is 20 logl~(jw)

where the base of the logarithm is 10. The unit used in this representation of the magnitude is the decibel, usually abbreviated dB. In the logarithmic representation, the curves are drawn on semilog paper, using the log scale for frequency and the linear stale for either magnitude (but in decibels) or phase angle (in degrees). (The frequency range of interest determines the number of logarithmic cycles required on the abscissa.)

The main advantage of using the Bode diagram is that multiplication of magnitudes can be converted into addition. Furthermore, a simple method for sketching an approximate log-magnitude curve is available. It is based on asymptotic approximations.

Such approximation by straight-line asymptotes is sufficient if only rough information on the frequency-response characteristics is needed. Should the exact curve be desired, corrections can be made easily to these basic asymptotic plots. Expanding the low-frequency range by use of a logarithmic scale for the frequency is highly advantageous since characteristics at low frequencies are most important in practical systems.

Although it is not possible to plot the curves right down to zero frequency because of the logarithmic frequency (log0 = -m), this does not create a serious problem. Note that the experimental determination of a transfer function can be made simple if frequency-response data are presented in the form of a Bode diagram.

Basic Factors of G( jco)H(jco). As stated earlier, the main advantage in using the logarithmic plot is the relative ease of plotting frequency-response curves. The basic factors that very frequently occur in an arbitrary transfer function G(jw)H(jw) are

1. Gain K

2. Integral and derivative factors (j~)~'

3. First-order factors (1 + jw~)"

4. Quadratic factors [I + 2[(jw/w,) + (jw/~,)~]"

Once we become familiar with the logarithmic plots of these basic factors, it is possible to utilize them in constructing a composite logarithmic plot for any general form of G(jw)H(jw) by sketching the curves for each factor and adding individual curves graphically, because adding the logarithms of the gains corresponds to multiplying them together.

The Gain K. A number greater than unity has a positive value in decibels, while a number smaller than unity has a negative value.The log-magnitude curve for a constant gain K is a horizontal straight line at the magnitude of 20 logK decibels.The phase angle of the gain K is zero. The effect of varying the gain K in the transfer function is that it raises or lowers the log-magnitude curve of the

transfer function by the corresponding constant amount, but it has no effect on the phase curve.

A number-decibel conversion line is given in Figure 8-4. The decibel value of any number can be obtained from this line. As a number increases by a factor of 10, the corresponding decibel value increases by a factor of 20. This may be seen from the following:

Note that, when expressed in decibels, the reciprocal of a number differs from its value only in sign; that is, for the number K,

1

20 log K = -20 log -

K

Integral and Derivative Factors (j~)~'. The logarithmic magnitude of l/jw in decibels is The phase angle of l/jw is constant and equal to -90".

In Bode diagrams, frequency ratios are expressed in terms of octaves or decades. An octave is a frequency band from w1 to 2w1, where w1 is any frequency value. A decade is a frequency band from w, to low,, where again w, is any frequency. (On the logarithmic scale of semilog paper, any given frequency ratio can be represented by the same horizontal distance. For example, the horizontal distance from w = 1 to w = 10 is equal to that from w = 3 to w = 30.)

If the log magnitude -20 log w dB is plotted against w on a logarithmic scale, it is a straight line.To draw this straight line, we need to locate one point (0 dB, w = 1) on it. Since (-20 log low) dB = (-20 log w - 20) dB the slope of the line is -20 dB/decade (or -6 dB/octave).

Similarly, the log magnitude of jw in decibels is 20 log 1 jol = 20 log w Db

The phase angle of jw is constant and equal to 90°.The log-magnitude curve is a straight line with a slope of 20 dB/decade. Figures 8-5(a) and (b) show

frequency-response curves for l/jw and jw, respectively. We can clearly see that the differences in the frequency responses of the factors l/jw and jo lie in the signs of the slopes of the log- magnitude curves and in the signs of the phase angles. Both log magnitudes become equal to 0 dB at w = 1.

If the transfer function contains the factor (lljw)" or (jw)", the log magnitude becomes, respectively,

20 log --

l (jb1.I =

-n X 20 log 1 jwl = -20n log w dB

20 log 1 (jw)"l = n X 20 log 1 jwl = 20n log w dB

The slopes of the log-magnitude curves for the factors (l/jw)" and (jw)" are thus -20n dB/decade and 20n dB/decade, respectively. The phase angle of (lljw)" is equal to -90" X n over the entire frequency range, while that of (jw)" is equal to 90' X n over the entire frequency range. The magnitude curves will pass through the point

(0 dB, w = 1).

First-Order Factors (1 + juT)". The log magnitude of the first-order factor

1/(1 + joT) is

1

2010gil + jwTl

= -20 log v'ixw dB

For low frequencies, such that w < 1/T, the log magnitude may be approximated by

Thus, the log-magnitude curve at low frequencies is the constant 0-dB line. For high frequencies, such that w B 1 /T,

-20 log vm = -20 log wT dB

This is an approximate expression for the high-frequency range. At w = 1/T, the log magnitude equals 0 dB; at w = 10/T, the log magnitude is -20 dB. Thus, the value of -20 log wT dB decreases by 20 dB for every decade of o. For w S 1/T, the log-magnitude curve is thus a straight line with a slope of -20 dB/decade (or -6 dB/octave).

Our analysis shows that the logarithmic representation of the frequency-response curve of the factor 1/(1 + jwT) can be approximated by two straight-line asymptotes, one a straight line at 0 dB for the frequency range 0 < o < 1/T and the other a straight line with slope -20 dBIdecade (or -6 dB/octave) for the frequency range 1/T < w < co.

The exact log-magnitude curve, the asymptotes, and the exact phase-angle curve are shown in Figure .

The frequency at which the two asymptotes meet is called the corner frequency or break frequency. For the factor 1/(1 + jwT), the frequency w = 1/T is the corner frequency since at w = 1/T the two asymptotes have the same value.

(The low-frequency asymptotic expression at w = 1/T is 20 log 1 dB = 0 dB, and the high-frequency asymptotic expression at w = 1/T is also 20 log 1 dB = 0 dB.) The corner frequency divides the frequency-response curve into two regions: a curve for the low-frequency region and a curve for the high-frequency region. The corner frequency is very important in sketching logarithmic frequency-response curves.

The exact phase angle 6 of the factor 1/(1 -k jwT) is

4 = -tan-' wT

At zero frequency, the phase angle is 0". At the corner frequency, the phase angle is

T

4 = -tan-' - = -tan-' 1 = -45"

T

At infinity, the phase angle becomes -90". Since the phase angle is given by an inverse- tangent function, the phase angle is skew symmetric about the inflection point at

6 = -45".

Quadratic Factors [l + 25(jw/w,) + (j~lw,)~]~'. Control systems often possess quadratic factors of the form If J > 1, this quadratic factor can be expressed as a product of two first-order factors with real poles. If 0 < J < 1, this quadratic factor is the product of two complex- conjugate factors. Asymptotic approximations to the frequency-response curves are not accurate for a factor with low values of 6. This is because the magnitude and phase of the quadratic factor depend on both the corner frequency and the damping ratio 6.

The asymptotic frequency-response curve may be obtained as follows: Since for low frequencies such that w < w,,, the log magnitude becomes

-20 log1 = 0dB

The low-frequency asymptote is thus a horizontal line at 0 dB. For high frequencies such

that w 9 w,, the log magnitude becomes

w2 w

-20 log - = -40 log - dB

0: w n

The equation for the high-frequency asymptote is a straight line having the slop(

-40 dB/decade since

Fig. Log-magnitude curves, together with the asymptotes, and phase-angle curves of the quadratic transfer function.

phase-angle curves for the quadratic factor given by Equation (8-7) with several values

of J. If corrections are desired in the asymptotic curves, the necessary amounts of cor-

rection at a sufficient number of frequency points may be obtained from Figure 8-9.

The phase angle of the quadratic factor [l f 2J(jw/w,) + (j~/o,)~]-l is

4 = = -tan-' (8-8)

The phase angle is a function of both o and J. At w = 0, the phase angle equals 0". At the corner frequency o = w,, the phase angle is -90" regardless of J, since

At o = m, the phase angle becomes -180". The phase-angle curve is skew symmetric about the inflection point-the point where 4 = -90°.There are no simple ways to sketch such phase curves. We need to refer to the phase-angle curves shown in Figure 8-9.

The frequency-response curves for the factor can be obtained by merely reversing the sign of the log magnitude and that of the phase angle of the factor

To obtain the frequency-response curves of a given quadratic transfer function, we must first determine the value of the corner frequency w, and that of the damping ratio 5.

Then, by using the family of curves given in Figure 8-9, the frequency-response curves

can be plotted.

The Resonant Frequency w, and the Resonant Peak Value Mr. The magnitude of

POLAR PLOTS

The polar plot of a sinusoidal transfer function G(jw) is a plot of the magnitude of G(jw) versus the phase angle of G(jw) on polar coordinates as w is varied from zero to infinity.Thus, the polar plot is the locus of vectors l~(jw)I /G(jw) as w is varied from zero to infinity. Note that in polar plots a positive (negative) phase angle is measured counter- clockwise (clockwise) from the positive real axis.The polar plot is often called the Nyquist plot.An example of such a plot is shown in Figure 8-26. Each point on the polar plot of G(jw) represents the terminal point of a vector at a particular value of w. In the polar plot, it is important to show the frequency graduation of the locus. The projections of G(jw) on the real and imaginary axes are its real and imaginary components.

An advantage in using a polar plot is that it depicts the frequency-response characteristics of a system over the entire frequency range in a single plot. One disadvantage is that the plot does not clearly indicate the contributions of each individual factor of the open-loop transfer function.

Integral and Derivative Factors (j~)~'. The polar plot of G(jo) = l/jw is the negative imaginary axis since The polar plot of G(jw) = jw is the positive imaginary axis.

First-Order Factors (1 + juT)". For the sinusoidal transfer function the values of G(jw) at w = 0 and w = 1/T are, respectively,

G(j0) = 1 IO" and G

If w approaches infinity, the magnitude of G(jw) approaches zero and the phase angle approaches -90". The polar plot of this transfer function is a semicircle as the frequency w is varied from zero to infinity, as shown in Figure 8-27(a). The center is located at 0.5 on the real axis, and the radius is equal to 0.5.

To prove that the polar plot of the first-order factor G(jw) = 1/(1 + jwT) is a semicircle, define

G(jw) = X + jY

where

X =

I

= real part of Gfjw)

1 + w2T2

-wT

Y = = imaginary part of G(jw)

1 + w2T2

Then we obtain

Quadratic Factors [I + 2<(jco/con) + (j~/co,)~]'~. The low- and high-frequency portions of the polar plot of the following sinusoidal transfer function

are given, respectively, by limG(jw) = 1/0" and lim G(jw) = 0/-180"

w+O w+m

The polar plot of this sinusoidal transfer function starts at 1 /0" and ends at 0/-180" as w increases from zero to infinity. Thus, the high-frequency portion of G(jw) is tangent to the negative real axis.

General Shapes of Polar Plots. The polar plots of a transfer function of the form where n > m or the degree of the denominator polynomial is greater than that of the numerator, will have the following general shapes:

1. For A = 0 or type 0 systems: The starting point of the polar plot (which corresponds to w = 0) is finite and is on the positive real axis. The tangent to the polar plot at w = 0 is perpendicular to the real axis. The terminal point, which corresponds to w = co, is at the origin, and the curve is tangent to one of the axes.

2. For A = 1 or type 1 systems: the jw term in the denominator contributes -90" to the total phase angle of G(jw) for 0 % w 5 co.At w = 0, the magnitude of G(jw) is infinity, and the phase angle becomes -90". At low frequencies, the polar plot is asymptotic to a line parallel to the negative imaginary axis. At w = co, the

magnitude becomes zero, and the curve converges to the origin and is tangent to one of the axes.

3. For A = 2 or type 2 systems: The (jw)' term in the denominator contributes -180" to the total phase angle of G(jw) for 0 5 w 5 co. At w = 0, the magnitude of G(jw) is infinity, and the phase angle is equal to -180'. At low frequencies, the polar plot is asymptotic to a line parallel to the negative real axis. At w = oo, the magnitude becomes zero, and the curve is tangent to one of the axes.

The general shapes of the low-frequency portions of the polar plots of type 0, type

Polar Plots of Simple Transfer Functions

LOG-MAGNITUDE-VERSUS-PHASE PLOTS

Another approach to graphically portraying the frequency-response characteristics is to use the log-magnitude-versus-phase plot, which is a plot of the logarithmic magnitude in decibels versus the phase angle or phase margin for a frequency range of interest. [The phase margin is the difference between the actual phase angle r$ and -180"; that is, 4 - (-180") = 180" + 4.1 The curve is graduated in terms of the frequency w. Such log-magnitude-versus-phase plots are commonly called Nichols plots.

In the Bode diagram, the frequency-response characteristics of G(jw) are shown on semilog paper by two separate curves, the log-magnitude curve and the phase-angle curve, while in the log-magnitude-versus-phase plot, the two curves in the Bode diagram are combined into one. In the manual approach the log-magnitude-versus-phase plot can easily be constructed by reading values of the log magnitude and phase angle from the Bode diagram. Notice that in the log-magnitude-versus-phase plot a change in the gain constant of G(jw) merely shifts the curve up (for increasing gain) or down (for decreasing gain), but the shape of the curve remains the same.

Advantages of the log-magnitude-versus-phase plot are that the relative stability of the closed-loop system can be determined quickly and that compensation can be worked out easily.

The log-magnitude-versus-phase plot for the sinusoidal transfer function G(jw) and that for l/G(jw) are skew symmetrical about the origin since

Log-Magnitude-versus-Phase Plots of Simple Transfer Functions

REFERENCES