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
Figue 1: Operational amplifie: circuit configured na a muitiplier
Figure 2: Winch, DO motor. and masa.
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