An integrator has as its frequency response where the impulse at ω = o is a result o...

An integrator has as its frequency response

where the impulse at ω = o is a result of the fact that the integration of a constant input from t = -∞ results in an infinite output. Thus, if we avoid inputs that are

In other words, the Bode plot for an integrator, as illustrated in Figure P6.31, consists of two straight-line plots. These plots reflect the principal characteristics of an integrator: a phase shift of —90° at all positive values of frequency and the amplification of low frequencies

(a) A useful, simple model of an electric motor is an LTI system with input equal to the applied voltage and output given by the motor shaft angle. This system can be visualized as the cascade of a stable LTI system (with the voltage as input and shaft angular velocity as output) and an integrator (representing the integration of the angular velocity). Often, a model of first-order system is used for the first part of the cascade. Assuming, for example, that this first-order system has a time constant of 0.1 second, we obtain an overall motor frequency response of