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Additional resources for Digital Control of Electrical Drives (Power Electronics and Power Systems)
The resulting traces are shown in Fig. 6. The traces are obtained assuming that the friction B is negligible compared with KP . 5 at a constant value. The proportional gain KP is varied, affecting the damping factor ξ (Eq. 38) and changing the closed-loop zero z1 = −KI /KP . Fig. 6. Step response of the closed-loop system in Fig. 2. The natural frequency ωn and the gain KI are kept constant (Eq. 38), while the gain KP assumes four different values, affecting the damping factor ξ and changing the value of the transfer function zero z1 = –KI /KP .
The complex image of the given load disturbance TL(t) can be obtained by the Laplace transform. It is assumed that TL(t) = 0 for t < 0. The complex image TL(s) can be expressed in terms of the numerator NL(s) and denominator DL(s): 38 2 Basic Structure of the Speed Controller TL (s ) = L [T (t )] = ND ((ss)) . L L L With TL(t > 0) = TL(∞) = 1, one obtains TL(s) = 1/s, with NL(s) = 1 and DL(s) = s + 0. In case of a ramp signal, TL(t) = t, NL(s) = 1 and DL(s) = s2 + 0·s + 0. The speed controller WSC (s) has to remove the impact of TL(t) on the steady-state value of the output ω (∞).
27, it is possible to observe the impact of the feedback gains on ξ and ωn. 27 can be rewritten by multiplying both the numerator and denominator by KI /J. The denominator assumes the form shown in Eq. 38. Within the same expression, it is evident that the integral gain KI determines the natural frequency ωn of the closed-loop poles, while the proportional gain KP can be used for tuning the damping factor ξ. 38) The effects of the damping factor ξ on the step response are investigated in Figs. 5.