Neudorfer, P. “Frequency Response” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 11 Frequency Response 11.1 Introduction 11.2 Linear Frequency Response Plotting 11.3 Bode Diagrams 11.4 A Comparison of Methods 11.1 Introduction The IEEE Standard Dictionary of Electrical and Electronics Terms defines frequency response in stable, linear systems to be “the frequency-dependent relation in both gain and phase difference between steady-state sinu- soidal inputs and the resultant steady-state sinusoidal outputs” [IEEE, 1988]. In certain specialized applications, the term frequency response may be used with more restrictive meanings. However, all such uses can be related back to the fundamental definition. The frequency response characteristics of a system can be found directly from its transfer function. A single-input/single-output linear time-invariant system is shown in Fig. 11.1. For dynamic linear systems with no time delay, the transfer function H(s) is in the form of a ratio of polynomials in the complex frequency s, where K is a frequency-independent constant. For a system in the sinusoidal steady state, s is replaced by the sinusoidal frequency jw (j = ) and the system function becomes H(jw) is a complex quantity. Its magnitude, *H(jw)*, and its argument or phase angle, argH(jw), relate, respectively, the amplitudes and phase angles of sinusoidal steady-state input and output signals. Using the terminology of Fig. 11.1, if the input and output signals are x(t) = X cos (wt + Q x ) y(t) = Y cos (wt + Q y ) then the output’s amplitude Y and phase angle Q y are related to those of the input by the two equations Y = *H(jw)*X Q y = argH(jw) + Q x Hs K Ns Ds () () () = -1 Hj K Nj Dj Hje jHj () () () () () w w w w w ==** arg Paul Neudorfer Seattle University ? 2000 by CRC Press LLC The phrase frequency response characteristics usually implies a complete description of a system’s sinusoidal steady-state behavior as a function of frequency. Because H(jw) is complex and, therefore, two dimensional in nature, frequency response characteristics cannot be graphically dis- played as a single curve plotted with respect to frequency. Instead, the magnitude and argument of H(jw) can be sep- arately plotted as functions of frequency. Often, only the magnitude curve is presented as a concise way of character- izing the system’s behavior, but this must be viewed as an incomplete description. The most common form for such plots is the Bode diagram (developed by H.W. Bode of Bell Laboratories), which uses a logarithmic scale for frequency. Other forms of frequency response plots have also been developed. In the Nyquist plot (Harry Nyquist, also of Bell Labs), H(jw) is displayed on the complex plane, Re[H(jw)] on the horizontal axis, and Im[H(jw)] on the vertical. Frequency is a parameter of such curves. It is sometimes numerically identified at selected points of the curve and sometimes omitted. The Nichols chart (N.B. Nichols) graphs magnitude versus phase for the system function. Frequency again is a parameter of the resultant curve, sometimes shown and sometimes not. Frequency response techniques are used in many areas of engineering. They are most obviously applicable to such topics as communications and filters, where the frequency response behaviors of systems are central to an understanding of their operations. It is, however, in the area of control systems where frequency response techniques are most fully developed as analytical and design tools. The Nichols chart, for instance, is used exclusively in the analysis and design of feedback control systems. The remaining sections of this chapter describe several frequency response plotting methods. Applications of the methods can be found in other chapters throughout the Handbook. 11.2 Linear Frequency Response Plotting Linear frequency response plots are prepared most directly by computing the magnitude and phase of H(jw) and graphing each as a function of frequency (either f or w), the frequency axis being scaled linearly. As an example, consider the transfer function Formally, the complex frequency variable s is replaced by the sinusoidal frequency jw and the magnitude and phase found. The plots of magnitude and phase are shown in Fig. 11.2. FIGURE 11.1A single-input/single-output lin- ear system. Hs ss () , = ++ 160,000 2 220 160000 Hj jj Hj Hj () , () () , () , (, )() arg ( ) tan , w ww w ww w w w = ++ = -+ =- - - 160000 220 160000 160000 160000 220 220 160000 2 22 2 1 2 * * ? 2000 by CRC Press LLC 11.3 Bode Diagrams A Bode diagram consists of plots of the gain and phase of a transfer function, each with respect to logarithmically scaled frequency axes. In addition, the gain of the transfer function is scaled in decibels according to the definition This definition relates to transfer functions which are ratios of voltages and/or currents. The decibel gain between two powers has a multiplying factor of 10 rather than 20. This method of plotting frequency response information was popularized by H.W. Bode in the 1930s. There are two main advantages of the Bode approach. The first is that, with it, the gain and phase curves can be easily and accurately drawn. Second, once drawn, features of the curves can be identified both qualitatively and quantitatively with relative ease, even when those features occur over a wide dynamic range. Digital computers have rendered the first advantage obsolete. Ease of interpretation, however, remains a powerful advantage, and the Bode diagram is today the most common method chosen for the display of frequency response data. A Bode diagram is drawn by applying a set of simple rules or procedures to a transfer function. The rules relate directly to the set of poles and zeros and/or time constants of the function. Before constructing a Bode diagram, the transfer function is normalized so that each pole or zero term (except those at s = 0) has a dc gain of one. For instance: Figure 11.2 Linear frequency response curves of H(jw). ** * *HH Hj dB dB ==20 10 log ( )w Hs K s ss Ks ss K s ss z p z p z p z p () () / (/ ) ( ) = + + = + + = ￠ + + w w w w w w t t 1 1 1 1 ? 2000 by CRC Press LLC In the last form of the expression, t z =1/w z and t p =1/w p . t p is a time constant of the system and s = –w p is the corresponding natural frequency. Because it is understood that Bode diagrams are limited to sinusoidal steady- state frequency response analysis, one can work directly from the transfer function H(s) rather than resorting to the formalism of making the substitution s = jw. Bode frequency response curves (gain and phase) for H(s) are generated from the individual contributions of the four terms K￠, st z + 1, 1/s, and 1/(st p + 1). As described in the following paragraph, the frequency response effects of these individual terms are easily drawn. To obtain the overall frequency response curves for the transfer function, the curves for the individual terms are added together. The terms used as the basis for drawing Bode diagrams are found from factoring N(s) and D(s), the numerator and denominator polynomials of the transfer function. The factorization results in four standard forms. These are (1) a constant K; (2) a simple s term corresponding to either a zero (if in the numerator) or a pole (if in the denominator) at the origin; (3) a term such as (st + 1) corresponding to a real valued (nonzero) pole or zero; and (4) a quadratic term with a possible standard form of [(s/w n ) 2 + (2z/w n )s + 1] corresponding to a pair of complex conjugate poles or zeros. The Bode magnitude and phase curves for these possibilities are displayed in Figs. 11.3–11.5. Note that both decibel magnitude and phase are plotted semilogarithmically. The frequency axis is logarithmically scaled so that every tenfold, or decade, change in frequency occurs over an equal distance. The magnitude axis is given in decibels. Customarily, this axis is marked in 20-dB increments. Positive decibel magnitudes correspond to amplifications between input and output that are greater than one (output amplitude larger than input). Negative decibel gains correspond to attenuation between input and output. Figure 11.3 shows three separate magnitude functions. Curve 1 is trivial; the Bode magnitude of a constant K is simply the decibel-scaled constant 20 log 10 K, shown for an arbitrary value of K = 5 (20 log 10 5 = 13.98). Phase is not shown. However, a constant of K > 0 has a phase contribution of 0° for all frequencies. For K < 0, the contribution would be ±180° (Recall that –cos q = cos (q ± 180°). Curve 2 shows the magnitude frequency response curve for a pole at the origin (1/s). It is a straight line with a slope of –20 dB/decade. The line passes through 0 dB at w = 0 rad/s. The phase contribution of a simple pole at the origin is a constant –90°, independent of frequency. The effect of a zero at the origin (s) is shown in Curve 3. It is again a straight line that passes through 0 dB at w = 0 rad/s; however, the slope is +20 dB/decade. The phase contribution of a simple zero at s = 0 is +90°, independent of frequency. Figure 11.3Bode magnitude functions for (1) K = 5, (2) 1/s, and (3) s. ? 2000 by CRC Press LLC Note from Fig. 11.3 and the foregoing discussion that in Bode diagrams the effect of a pole term at a given location is simply the negative of that of a zero term at the same location. This is true for both magnitude and phase curves. Figure 11.4 shows the magnitude and phase curves for a zero term of the form (s/w z + 1) and a pole term of the form 1/(s/w p + 1). Exact plots of the magnitude and phase curves are shown as dashed lines. Straight line approximations to these curves are shown as solid lines. Note that the straight line approximations are so good that they obscure the exact curves at most frequencies. For this reason, some of the curves in this and later figures have been displaced slightly to enhance clarity. The greatest error between the exact and approximate magnitude curves is ±3 dB. The approximation for phase is always within 7° of the exact curve and usually much closer. The approximations for magnitude consist of two straight lines. The points of intersection between these two lines (w = w z for the zero term and w = w p for the pole) are breakpoints of the curves. Breakpoints of Bode gain curves always correspond to locations of poles or zeros in the transfer function. In Bode analysis complex conjugate poles or zeros are always treated as pairs in the corresponding quadratic form [(s/w n ) 2 + (2z/w n )s + 1]. 1 For quadratic terms in stable, minimum phase systems, the damping ratio z (Greek letter zeta) is within the range 0 < z < 1. Quadratic terms cannot always be adequately represented by straight line approximations. This is especially true for lightly damped systems (small z). The traditional approach was to draw a preliminary representation of the contribution. This consists of a straight line of 0 dB from dc up to the breakpoint at w n followed by a straight line of slope ±40 dB/decade beyond the breakpoint, depending on whether the plot refers to a pair of poles or a pair of zeros. Then, referring to a family of curves as shown in Fig. 11.5, the preliminary representation was improved based on the value of z. The phase contribution of the quadratic term was similarly constructed. Note that Fig. 11.5 presents frequency response contributions for a quadratic pair of poles. For zeros in the corresponding locations, both the magnitude and phase curves would be negated. Digital computer applications programs render this procedure unnecessary for purposes of constructing frequency response curves. Knowledge of the technique is still valuable, however, in the qualitative and quantitative interpretation of frequency response curves. Localized peaking in the gain curve is a reflection of the existence of resonance in a system. The height of such a peak (and the corresponding value of z) is a direct indication of the degree of resonance. Bode diagrams are easily constructed because, with the exception of lightly damped quadratic terms, each contribution can be reasonably approximated with straight lines. Also, the overall frequency response curve is found by adding the individual contributions. Two examples follow. 1 Several such standard forms are used. This is the one most commonly encountered in controls applications. Figure 11.4Bode curves for (1) a simple pole at s = –w p and (2) a simple zero at s = –w z . ? 2000 by CRC Press LLC Example 1 In Fig. 11.6, the individual contributions of the four factored terms of A(s) are shown as long dashed lines. The straight line approximations for gain and phase are shown with solid lines. The exact curves are presented with short dashed lines. Example 2 Note that the damping factor for the quadratic term in the denominator is z = 0.35. If drawing the response curves by hand, the resonance peak near the breakpoint at w = 100 would be estimated from Fig. 11.5. Figure 11.7 shows the exact gain and phase frequency response curves for G(s). Figure 11.5Bode diagram of 1/[(s/w n ) 2 + (2z/w n )s + 1]. 10 20 0 -10 -20 -30 -40 0.1 0.2 0.3 0.4 0.50.6 0.8 1.0 2 3 4 5 6 78910 z = 0.05 0.10 0.15 0.20 0.25 0.3 0.4 0.5 0.6 0.8 1.0 u = w/w n = Frequency Ratio 20 log |G| Phase Angle, De grees w/w n = Frequency Ratio (a) (b) -20 0 -40 -60 -80 -100 -120 -140 -160 -180 0.1 0.2 0.3 0.4 0.50.6 0.8 1.0 2 3 4 5 6 7 8910 z = 0.05 0.10 0.15 0.20 0.25 0.3 0.4 0.5 0.6 0.8 1.0 As s ss s ss s ss ()= 10 4 25 4 1 1100 10 10 100 1000 10 100 1 1000 1 ++ = ++ = ++ - ()()(/)(/) Gs s ss s ss () () , (/ ) (/ ) (.)(/ ) = + ++ = + ++ 1000 500 70 10000 50 500 1 100 2035 100 1 22 ? 2000 by CRC Press LLC 11.4 A Comparison of Methods This chapter concludes with the frequency response of a simple system function plotted in three different ways. Example 3 Figure 11.8 shows the direct, linear frequency response curves for T(s). Corresponding Bode and Nyquist diagrams are shown, respectively, in Figs. 11.9 and 11.10. Figure 11.6Bode diagram of A(s). Figure 11.7Bode diagram of G(s). Ts sss () ()()() = +++ 10 100 200 300 7 ? 2000 by CRC Press LLC Figure 11.8 Linear frequency response plot of T(s). Figure 11.9 Bode diagram of T(s). Figure 11.10 Nyquist plot of T(s). ? 2000 by CRC Press LLC Defining Terms Bode diagram: A frequency response plot of 20 log gain and phase angle on a log-frequency base. Breakpoint: A point of abrupt change in slope in the straight line approximation of a Bode magnitude curve. Damping ratio: The ratio between a system’s damping factor (measure of rate of decay of response) and the damping factor when the system is critically damped. Decade: Synonymous with power of ten. In context, a tenfold change in frequency. Decibel: A measure of relative size. The decibel gain between voltages V 1 and V 2 is 20 log 10 (V 1 /V 2 ). The decibel ratio of two powers is 10 log 10 (P 1 /P 2 ). Frequency response: The frequency-dependent relation in both gain and phase difference between steady- state sinusoidal inputs and the resultant steady-state sinusoidal outputs. Nichols chart: Control systems — a plot showing magnitude contours and phase contours of the return transfer function referred to as ordinates of logarithmic loop gain and abscissae of loop phase angle. Nyquist plot: A parametric frequency response plot with the real part of the transfer function on the abscissa and the imaginary part of the transfer function on the ordinate. Resonance: The enhancement of the response of a physical system to a steady-state sinusoidal input when the excitation frequency is near a natural frequency of the system. Related Topics 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals ? 100.3 Frequency Response Methods: Bode Diagram Approach References R.C. Dorf, Modern Control Systems, 4th ed., Reading, Mass.: Addison-Wesley, 1986. IEEE Standard Dictionary of Electrical and Electronics Terms, 4th ed., The Institute of Electrical and Electronics Engineers, 1988. D.E. Johnson, J.R. Johnson, and J.L. Hilburn, Electric Circuit Analysis, 2nd ed., Englewood Cliffs, N.J.: Prentice- Hall, 1992. B.C. Kuo, Automatic Control Systems, 4th ed., Englewood Cliffs, N.J.: Prentice-Hall, 1982. K. Ogata, System Dynamics, Englewood Cliffs, N.J.: Prentice-Hall, 1992. W.D. Stanley, Network Analysis with Applications, Reston, Va.: Reston, 1985. M.E. Van Valkenburg, Network Analysis, 3rd ed., Englewood Cliffs, N.J.: Prentice-Hall, 1974. Further Information Good coverage of frequency response techniques can be found in many undergraduate-level electrical engi- neering textbooks. Refer especially to classical automatic controls or circuit analysis books. Useful information can also be found in books on active network design. Examples of the application of frequency response methods abound in journal articles ranging over such diverse topics as controls, acoustics, electronics, and communications. ? 2000 by CRC Press LLC