Signals and Systems - Electrical Engineering

Problems 55

0.4. Backward difference—MATLAB

Another definition for the finite difference is the backward difference:

1 [x(nTs)]=x(nTs)−x((n− 1 )Ts)

( 1 [x(nTs)]/Tsapproximates the derivative ofx(t).)

(a) Indicate how this new definition connects with the finite difference defined earlier in this chapter.

(b) Solve Problem 0.3 with MATLAB using this new finite difference and compare your results with the

ones obtained there.

(c) For the value ofTs=0.1, use the average of the two finite differences to approximate the derivative of

the analog signalx(t). Compare this result with the previous ones. Provide an expression for calculating

this new finite difference directly.

0.5. Differential and difference equations—MATLAB

Find the differential equation relating a current sourceis(t)=cos( 0 t)with the currentiL(t)in an inductor,

with inductanceL=1 H, connected in parallel with a resistor ofR= 1 (see Figure 0.26). Assume a zero

initial current in the inductor.

(a) Obtain a discrete equation from the differential equation using the trapezoidal approximation of an

integral.

(b) Create a MATLAB script to solve the difference equation forTs=0.01and three frequencies for

is(t), 0 =0.005π, 0.05π, and0.5π. Plot the input current sourceis(t)and the approximate solution

iL(nTs)in the same figure. Use the MATLAB functionplot. Use the MATLAB functionfilterto solve the

difference equation (usehelpto learn aboutfilter).

(c) Solve the differential equation using symbolic MATLAB when the input frequency is 0 =0.5π.

(d) Use phasors to find the amplitude ofiL(t)when the input isis(t)with the given three frequencies.

FIGURE 0.26
Problem 0.5. RL circuit: inputis(t)and output
iL(t).

is(t)

iL(t)

1 Ω 1H

0.6. Sums and Gauss—MATLAB

Three rules in the computation of sums are

n Distributive law:

∑

k

cak=c

∑

k

ak

n Associative law:

∑

k

(ak+bk)=

∑

k

ak+

∑

k

bk