Neudorfer, P. "Frequency Resp The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Neudorfer, P. “Frequency Response” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
Frequency response 11.1 Introduction 11.2 Linear Frequency Response Plotting Paul Neudorfer 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, ne 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 jo(=v-1)and the system function becomes H(o)=k No)= h(o) jo D() ment or phase angle, arg HGjo), relat respectively, the amplitudes and phase angles of sinusoidal steady-state input and output signals. Using the terminology of Fig. 11. 1, if the and output x(t)=X cos(ot +O) y(t)=Y cos(ot +e) then the output's amplitude Y and phase angle e, are related to those of the input by the two equations H(jo)X O,=argHGjO)+ O c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 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 sinusoidal 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 + Qx) y(t) = Y cos (wt + Qy) then the output’s amplitude Y and phase angle Qy are related to those of the input by the two equations Y = *H(jw)*X Qy = argH(jw) + Qx H s K N s D s ( ) ( ) ( ) = -1 H j K N j D j H j ej H j ( ) ( ) ( ) ( ) ( ) w w w w w = = * * arg Paul Neudorfer Seattle University
The phrase fre implies a complete description of a systems sinusoidal steady-state behavior as a function of frequency. Because Hgjo) is complex and, therefore, two dimensional in nature,x H(s) frequency response characteristics cannot be graphically dis- played as a single curve plotted with respect to frequency. Instead, the magnitude and argument of H(jo) can be sep- FIGURE 11.1 A single-inpu arately plotted as functions of frequency. Often, only the magnitude curve is presented as a concise way of character ing the systems 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 ale for frequency. Other forms of frequency response plots have also been developed. In the Nyquist plot (Harry Nyquist, also of Bell Labs), H(o)is displayed on the complex plane, Re[Hgjo)] on the horizontal axis,and Im( HGio)] on the vertical Frequency is a parameter of uch curves. It is sometimes numerically identified at selected points of the curve and sometimes omitted. The Nichols chart(N B. Nichols) graph 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 HGjo) and graphing each as a function of frequency (either f or o), the frequency axis being scaled linearly. As an example, consider the transfer function 160.00 H(s) +220s+160,000 Formally, the complex frequency variable s is replaced by the sinusoidal frequency j@ and the magnitude and 160,000 (jo)2+220(j)+160,000 H(jo)I 160000 (160,00092+2002 The plots of magnitude and phase are shown in Fig. 11.2. e 2000 by CRC Press LLC
© 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 displayed as a single curve plotted with respect to frequency. Instead, the magnitude and argument of H(jw) can be separately plotted as functions of frequency. Often, only the magnitude curve is presented as a concise way of characterizing 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 11.1 A single-input/single-output linear system. H s s s ( ) , = + + 160,000 2 220 160 000 H j j j H j H j ( ) , ( ) ( ) , ( ) , ( , ) ( ) arg ( ) tan , w w w w w w w w w = + + = - + = - - - 160 000 220 160 000 160 000 160 000 220 220 160 000 2 2 2 2 1 2 * *
1.5 200 600 800 1000 Radian Frequency o 180 1000 Figure 11.2 Linear frequency response curves of HGjo 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 hlds haB 20 logo(jo) 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 fund on. 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 H(s)=Ks+o1-=02so2+1 sτ.+1 s(s+op)0ps(s/02+1) s(sτn+1) e 2000 by CRC Press LLC
© 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). * * H H *H j * dB = dB = 20 10 log ( w) H s K s s s K s s s K s s s z p z p z p z p ( ) ( ) / ( / ) ( ) = + + = + + = ¢ + + w w w w w w t t 1 1 1 1
40dB r 9+20dB -20dB 40dB 1og【] Figure 11.3 Bode magnitude functions for(1)K=5, (2)1/s, and (3)s In the last form of the expression, t,=1/@, and tp=1/@ tp is a time constant of the system and s=-0, 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= jo Bode frequency response curves(gain and phase) for H(s) are generated from the individual contributions of the four terms K, st,+ 1, 1/s, and 1/(st, 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 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/o)2+(25/@ )s 1] corresponding to a pair of complex conjugate poles or zeros. The Bode magnitude and phase curves for these possibilities are isplayed 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 utput Figure 11.3 shows three separate magnitude functions. Curve I is trivial; the Bode magnitude of a constant is simply the decibel-scaled constant 20 logo K, shown for an arbitrary value of K= 5(20 logo 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 t180(Recall that-cos 0=cos(0+ 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 hrough 0 dB at @=0 rad/s The phase contribution of a simple pole at the origin is a constant-90o, 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 @=0 rad/s; however, the slope is +20 dB/decade. The phase contribution of a simple zero at ent of frequen e 2000 by CRC Press LLC
© 2000 by CRC Press LLC In the last form of the expression, tz =1/wz and tp =1/wp. tp is a time constant of the system and s = –wp is the corresponding natural frequency. Because it is understood that Bode diagrams are limited to sinusoidal steadystate 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¢, stz + 1, 1/s, and 1/(stp + 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/wn)2 + (2z/wn)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 log10 K, shown for an arbitrary value of K = 5 (20 log10 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 11.3 Bode magnitude functions for (1) K = 5, (2) 1/s, and (3) s