Signals and Systems - Electrical Engineering

Problems 157

to the theory of differential equations and shown some features that will come back when we apply
transforms.

Our next step is to do the analysis of systems with continuous-time signals by means of transforms. In
Chapter 3 we discuss the Laplace transform that allows transient as well as steady-state analysis and
that will convert the solution of differential equations into an algebraic problem. More important,
it will provide the concept of transfer function that connects with the impulse response and the
convolution integral covered in this chapter. The Laplace transform is very significant in the area of
classic control.

Problems............................................................................................

2.1. Temperature measuring system—MATLAB

The op-amp circuit shown in Figure 2.19 is used to measure the changes of temperature in a system. The

output voltage is given by

vo(t)=−R(t)vi(t)

Suppose that the temperature in the system changes cyclically aftert= 0 , so that

R(t)=[ 1 +0.5 cos( 20 πt)]u(t)

Let the input bevi(t)=1 volt.

FIGURE 2.19

Problem 2.1.

−

+

−

+ +

−

1 Ω

Thermistor R(t)

vi(t)

t 0

vo(t)

(a) Assuming that the switch closes att 0 =0 sec, use MATLAB to plot the output voltagevo(t)for 0 ≤

t≤0.2 secin time intervals of 0.01 sec.

(b)If the switch closes att 0 =50 msec, plot the output voltagev 0 (t)for 0 ≤t≤0.2 secin time intervals

of 0.01 sec.

(c)Use the above results to determine if this system is time invariant. Explain.

2.2. Zener diode—MATLAB

A zener diode circuit is such that the output corresponding to an inputvs(t)=cos(πt)is a “clipped”

sinusoid

x(t)=

{

0.5 |vs(t)|>0.5

vs(t) otherwise

as shown in Figure 2.20 for a few periods. Use MATLAB to generate the input and the output signals and

plot them in the same plot for 0 ≤t≤ 4 at time intervals of 0.001.

(a) Is this system linear? Compare the output obtained fromvs(t)with that obtained from0.3vs(t).

(b)Is the system time invariant? Explain.