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1. (Multiplier) Consider the operational amplifier circuit in Figure 1. Draw an equivalent circuit using the model for an operational amplifier including input resistance Ri, output resistance Ro and open loop gain A (given in Figure 2.5 of the lecture notes). Analyze this circuit to obtain a relationship between the input voltage signal xand output voltage signal y. By taking limits as Ri-> 00, A - 00 and Ro 0 find an expression relating x and y assuming that the operational amplifier is ideal. Obtain the same expression directly using the rules for analysing idea] operational amplifiers. Is the system that describes this circuit stable? Is it regular? 2. (DC motor and winch) Suppose that a DC motor and winch is used to tow a body of mass M as depicted in Figure 2. The mass (or moment of inertia) of the winchis negligible compared with M. The relationship between the force f at the circumference of the winch and the turning force (torque) I is T rf where T is the radius of the winch. The winch is connected to the mass by an elastic cable modelled as a spring with constant K. As the motor turns the length of the cable l changes. The angle of the motor e is related the length of the cable by - 2nre. The force due to the spring is modelled as K(l - p). A force due to friction fropposses the motion of the mass so that fr - B,Dp where B, is the coefficient of friction. Supposing that the inductance within the motor L - 0, find a differential equation that relates the input voltage V with the position of the mass p. Find the transfer function of this system. Is this system stable? Now assume that the constants satisfy 7 - B, - R - Kt - Kb - M - K - 1. Under these assumptions: find the poles and zeros, draw a pole zero plot, and find and plot the impulse response. ail 5615 y!t) 12kn o Figue 1: Operational amplifie: circuit configured na a muitiplier m i R 9 K u is motor Winch M2 7 F /: Figure 2: Winch, DO motor. and masa.

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