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Unformatted text preview: s ystem frequency response. So t he u ltimate p roblem reduces t o r elating t he
f requency response of t he closed-loop system t o t hat o f t he the o pen-loop system.
To do t his, we shall consider t he case of unity feedback system, where t he feedback
t ransfer f unction is H (s) = 1. t T he closed-loop transfer function in this case is -4.0 -3.0 M =1.6/ -2.0 -1.0 ...(l.5 " ', o 1.0 (a) 1m \ \00 = 10 Re K G(s)
T (s) = 1 + K G(s) (b) a nd T ' _ K G(jw)
(Jw) - 1 + K G(jw) 00=1 L et T (jw) = M eja(w), a nd K G(jw)=x(w)+jy(w) t The r esults for t he u nity feedback c an b e e xtended t o n onunity feedback systems. o F ig. 7 .12 Relationship between the open-loop and the closed-loop frequency response. 496 7 Frequency Response and Analog Filters 7.4 Filter Design by Placement of Poles and Zeros
M eja(w) = 497 t + jy
1 + x + jy 1m x S traightforward manipulation of this equation yields ( x+ M~~ 1) 2 + y2 = (M~~ 1)2 T his is a n e quation of a circle centered a t [ - M/.f~l 0] a nd of radius Mf_l in
t he K G(jw) p lane. Figure 7.12a shows circles for various values of M . Because
M is t he closed-loop system amplitude response, these circles are t he contours o f
c onstant a mplitude response of t he closed-loop system. For example, t he p oint
A = - 2 - j1.85 lies on t he circle M = 1.3. This means, a t a frequency where t he
open-loop t ransfer function is G (jw) = - 2 - j1.85, t he corresponding closed-loop
transfer function amplitude response is 1.3. t
To o btain t he closed-loop frequency response, we superimpose on these contours
t he Nyquist p lot o f t he open-loop transfer function K G(jw). For each point o f
K G(jw), we c an d etermine t he corresponding value of M , t he closed-loop amplitude
response. F rom similar contours for constant a ( the closed-loop phase response),
we can determine t he closed loop phase response. Thus, t he complete closed-loop
frequency response can b e o btained from this plot. We a re primarily interested
in finding M p , t he p eak value of M a nd wp , t he frequency where i t occurs. Figure
7.12b indicates how these values may be determined. T he circle t o which t he Nyquist
plot is t angent corresponds t o M p , a nd t he frequency a t which t he Nyquist plot is
tangential t o t his circle is wp. For t he system, whose Nyquist plot appears in Fig.
7.12b, Mp = 1 .6 and wp = 2. From these values, we can estimate ( a nd wn , a nd
determine t he t ransient p arameters P O, t r a nd t •.
I n d esigning systems, we first determine Mp a nd wp required t o m eet t he given
transient specifications from Eqs. (7.27). T he Nyquist plot in conjunction with M
circles suggests how these values of Mp a nd wp may b e realized. I n many cases, a
mere change in gain K of t he open-loop transfer function will suffice. Increasing K
e xpands t he N yquist plot and changes t he values Mp a nd wp correspondingly. I f t his
is n ot enough, we should consider some form of compensation such as lag a nd/or
lead networks. Using a computer...
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