poles-and-zeros

poles-and-zeros - Complex Functions Transfer Functions A...

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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 + bm-1 sm-1 + . . . + b1 s + b0 = D(s) an sn + an-1 sn-1 + . . . + 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 pole-zero 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, Open-Loop Poles & Closed-Loop Poles January 22, 2010 31 / 34 C. S. George Lee () Poles, Zeros, Open-Loop Poles & Closed-Loop 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 s-plane 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 s-plane where the magnitude of the transfer function becomes infinite. If s = pi , then G(s)|s=pi = . H(s) The stability of a closed-loop (linear) control system is determined by the location of its closed-loop poles. The characteristic equation of the closed-loop control system is 1 + G(s)H(s) = 0. The roots of the characteristic equation are the closed-loop poles. The zeros of the characteristic equation are the closed-loop poles. The poles of the characteristic equation are the open- loop poles. C. S. George Lee () Poles, Zeros, Open-Loop Poles & Closed-Loop Poles January 22, 2010 33 / 34 C. S. George Lee () Poles, Zeros, Open-Loop Poles & Closed-Loop Poles January 22, 2010 34 / 34 ...
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