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 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 11.1 A single-input/single-output lin￾ear 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 * *