Signals and Systems - Electrical Engineering

410 C H A P T E R 6: Application to Control and Communications

(c) By mistake we input the received signal into the demodulator, and the resulting signal into the

cascade of the band-pass and the low-pass filters. If you use the band-pass filter obtained above,

determine the recovered signal (i.e., the output of the low-pass filter). Would you get the same result

regardless of whatm(t)is? Explain.

6.3. Op-amps as feedback systems

An ideal operational amplifier circuit can be shown to be equivalent to a negative-feedback system. Con-

sider the amplifier circuit in Figure 6.28 and its two-port network equivalent circuit to obtain a feedback

system with inputVi(s)and outputV 0 (s). What is the effect ofA→∞on the above circuit?

−Av−

Ro 0

+

+ +

+

+

+

R 1

R 2

R 1

R 2

v− v−

v+

v+

Ri

vi(t)

vo(t) vi(t)

vo(t)

→

→

−

−

−

−

−

−

∞

FIGURE 6.28

6.4. RC circuit as feedback system

Consider a series RC circuit with input a voltage sourcevi(t)and output the voltage across the capacitor

vo(t).

(a) Draw a negative-feedback system for the circuit using an integrator, a constant multiplier, and an

adder.

(b) Let the input be a battery (i.e.,vi(t)=Au(t)). Find the steady-state errore(t)=vi(t)−vo(t).

6.5. RLC circuit as feedback system

A resistorR, a capacitorC, and an inductorLare connected in series with a sourcevi(t). Consider the

output of the voltage across the capacitorvo(t). LetR= 1 ,C= 1 F andL= 1 H.

(a) Use integrators and adders to implement the differential equation that relates the inputvi(t)and the

outputvo(t)of the circuit.

(b) Obtain a negative-feedback system block diagram with inputVi(s)and outputV 0 (s). Determine the

feedforward transfer functionG(s)and the feedback transfer functionH(s)of the feedback system.

(c) Find an equation for the errorE(s)=Vi(s)−V 0 (s)H(s)and determine its steady-state response when

the input is a unit-step signal (i.e.,Vi(s)= 1 /s).

6.6. Ideal and lossy integrators

An ideal integrator has a transfer function 1 /s, while a lossy integrator has a transfer function 1 /(s+K).

(a) Determine the feedforward transfer functionG(s)and the feedback transfer function H(s)of a

negative-feedback system that implements the overall transfer function

Y(s)

X(s)

=

K

K+s

whereX(s)andY(s)are the Laplace transforms of the inputx(t)and the outputy(t)of the feedback

system. Sketch the magnitude response of this system and determine the type of filter it is.