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Unformatted text preview: Complex Functions Transfer Functions
A rational transfer function is a complex function that can be described either as a ratio of 2 polynomials in s G(s) = N(s) bm sm + bm1 sm1 + . . . + b1 s + b0 = D(s) an sn + an1 sn1 + . . . + a1 s + a0 A complex function F (s), which is a function of s, has a real part and an imaginary part as F (s) = real part + imaginary part = Fx + jFy where Fx and Fy are real numbers. The magnitude and the phase angle of F (s) are, respectively, 1 Fy 2 + F2 ; F (s) = Fx F (s) = = tan y Fx or in a factored polezero form as: K m (s  zi ) N(s) K (s  z1 )(s  z2 ) (s  zm ) G(s) = = n i=1 = D(s) (s  p1 )(s  p2 ) (s  pn ) j=1 (s  pj ) where K is the gain of the transfer function. C. S. George Lee () Poles, Zeros, OpenLoop Poles & ClosedLoop Poles January 22, 2010 31 / 34 C. S. George Lee () Poles, Zeros, OpenLoop Poles & ClosedLoop Poles January 22, 2010 32 / 34 Poles and Zeros Poles and Characteristic Equation Things That May Be Confusing, But Important!
R(s) + C(s) _ G(s) The roots of the numerator z1 , z2 , . . . , zm are called the finite zeros of the system. The zeros are locations in the splane where the transfer function is zero. If s = zi , then G(s)s=zi = 0. The roots of the denominator p1 , p2 , . . . , pn are called the poles of the system. The poles are locations in the splane where the magnitude of the transfer function becomes infinite. If s = pi , then G(s)s=pi = . H(s) The stability of a closedloop (linear) control system is determined by the location of its closedloop poles. The characteristic equation of the closedloop control system is 1 + G(s)H(s) = 0. The roots of the characteristic equation are the closedloop poles. The zeros of the characteristic equation are the closedloop poles. The poles of the characteristic equation are the open loop poles. C. S. George Lee () Poles, Zeros, OpenLoop Poles & ClosedLoop Poles January 22, 2010 33 / 34 C. S. George Lee () Poles, Zeros, OpenLoop Poles & ClosedLoop Poles January 22, 2010 34 / 34 ...
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 Spring '08
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