30079_27b

30079_27b - Frequency Response Plots The frequency response...

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Frequency Response Plots The frequency response of a fixed linear system is typically represented graphically, using one of three types of frequency response plots. A polar plot is simply a plot of the vector H(jcS) in the complex plane, where Re(o>) is the abscissa and Im(cu) is the ordinate. A logarithmic plot or Bode diagram consists of two displays: (1) the magnitude ratio in decibels Mdb(o>) [where Mdb(w) = 20 log M(o))] versus log w, and (2) the phase angle in degrees <£(a/) versus log a). Bode diagrams for normalized first- and second-order systems are given in Fig. 27.23. Bode diagrams for higher-order systems are obtained by adding these first- and second-order terms, appropriately scaled. A Nichols diagram can be obtained by cross plotting the Bode magnitude and phase diagrams, eliminating log a). Polar plots and Bode and Nichols diagrams for common transfer functions are given in Table 27.8. Frequency Response Performance Measures Frequency response plots show that dynamic systems tend to behave like filters, "passing" or even amplifying certain ranges of input frequencies, while blocking or attenuating other frequency ranges. The range of frequencies for which the amplitude ratio is no less than 3 db of its maximum value is called the bandwidth of the system. The bandwidth is defined by upper and lower cutoff frequencies o)c, or by o> = 0 and an upper cutoff frequency if M(0) is the maximum amplitude ratio. Although the choice of "down 3 db" used to define the cutoff frequencies is somewhat arbitrary, the bandwidth is usually taken to be a measure of the range of frequencies for which a significant portion of the input is felt in the system output. The bandwidth is also taken to be a measure of the system speed of response, since attenuation of inputs in the higher-frequency ranges generally results from the inability of the system to "follow" rapid changes in amplitude. Thus, a narrow bandwidth generally indicates a sluggish system response. Response to General Periodic Inputs The Fourier series provides a means for representing a general periodic input as the sum of a constant and terms containing sine and cosine. For this reason the Fourier series, together with the super- position principle for linear systems, extends the results of frequency response analysis to the general case of arbitrary periodic inputs. The Fourier series representation of a periodic function f(t) with period 2T on the interval t* + 2T > t > t* is jv N ^ i n/Trt i • n7rt\ /(O = -T + Zr I an cos — + bn sin — I 2, n=l \ i i I where 1 r+2^ nirt j an = ~ J^ /(O cos — dt bn = J'L f(f} sin T^dt If f(t) is defined outside the specified interval by a periodic extension of period 27, and if f(t) and its first derivative are piecewise continuous, then the series converges to /(O if f is a point of con- tinuity, or to l/2 [f(t+) + /(*-)] if t is a point of discontinuity. Note that while the Fourier series in general is infinite, the notion of bandwidth can be used to reduce the number of terms required for a reasonable approximation. 27.6 STATE-VARIABLE METHODS State-variable methods use the vector state and output equations introduced in Section 27.4 for
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30079_27b - Frequency Response Plots The frequency response...

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