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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 secondorder systems are given in Fig. 27.23. Bode diagrams for higherorder
systems are obtained by adding these first and secondorder 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 higherfrequency 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
a°
^ 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 STATEVARIABLE METHODS
Statevariable methods use the vector state and output equations introduced in Section 27.4 for
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 Spring '10
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 Mechanical Engineering, The Land

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