## Transcribed Text

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

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