This preview shows page 1. Sign up to view the full content.
Unformatted text preview: remain unchanged whether the zero(s) are in t he
L HP or the RHP. However, the phase of t he R HP zero a t s = a is L(jw - a) =L - (a -jw) = + tan-l (-aw ) =
IT IT - tan-l(~) 490 7 Frequency Response a nd Analog Filters whereas t he p hase of t he L HP pole a t s = - a is - tan-1(w/a).
T he c omplex conjugate zeros in R HP give rise t o a t erm 8 2 - 2(W n 8 +w~, which
is identical t o t he t erm 8 2 + 2 (W n 8 + w~ w ith a sign change in ( . Hence, from Eqs.
(7.23) a nd (7.24) i t follows t hat t he amplitudes are identical, b ut t he phases are of
opposite signs for the two terms. 7.2-1 The Transfer Function From t he Frequency Response In t he p revious section we were given t he t ransfer function o f a system. From
a knowledge o f t he t ransfer function, we developed techniques for determining t he
s ystem response to sinusoidal inputs. We can also reverse t he p rocedure t o determine t he t ransfer function of a system if t he s ystem response to a sinusoidal i nput is
known. This problem has significant practical utility. I f we a re given a system in a
black box w ith only t he i nput a nd o utput t erminals available, t he t ransfer function
has t o b e d etermined by experimental measurements a t t he i nput a nd o utput t erminals. T he f requency response to sinusoidal inputs is one of t he possibilities t hat
is very a ttractive because of t he simple n ature of t he m easurements involved. O ne
o nly needs t o a pply a sinusoidal signal a t t he i nput a nd observe t he o utput. We find
t he a mplitude gain IH(jw)1 a nd t he o utput p hase shift L.H(jw) ( with respect t o t he
i nput s inusoid) for various values of w over t he e ntire range from 0 t o 0 0. T his information yields t he frequency response plots (Bode plots) when plotted against log
w. From t hese p lots we d etermine t he a ppropriate asymptotes by taking advantage
o f t he fact t hat t he slopes of all asymptotes must be multiples of ± 20 d B/decade if
t he t ransfer function is a r ational function (function which is n ot necessarily a ratio
of two polynomials in 8). From t he a symptotes, t he corner frequencies are obtained.
C orner frequencies determine t he poles a nd zeros of t he t ransfer function. 7 .3 Control System Design Using Frequency Response
F igure 7 .10a shows a basic closed-loop system, whose open-loop transfer function (transfer function when t he loop is opened) is K G(8)H(8). T he closed-loop
transfer function is [see Eq. (6.69) 7.3 C ontrol System Design Using Frequency Response Sec. 6.7 a nd t he frequency response method should be considered as c omplementary
r ather t han as alternatives or as rivals.
Frequency response information can be presented in a various forms of which
Bode plot is one. T he same information can be presented by t he N yquist p lot"
also known as t he p olar p lot or by t he N ichols p lot also known as log-magnitude
versus angle plot. Here, we shall discuss t he techniques using Bode a nd N yquist
plots only. Figure 7.lOb shows Bode plots for t he open-loop transfer function
View Full Document