In this problem, we explore the use of positive feedback for generating oscillating sign...

In this problem, we explore the use of positive feedback for generating oscillating signals.

(a) Consider the system illustrated in Figure P11.54(a). Show that if

Suppose that we connect terminals 1 and 2 in Figure P11.54(a) and make . Then the output of the system should remain unchanged if we

satisfy eq. (P11.54-1). The system now produces an output without any input. Therefore, the system shown in Figure P11.54(b) is an oscillator, provided that eq. (P11.54-1) is satisfied.

(b) A commonly used oscillator in practice is the sinusoidal oscillator. For such an oscillator, we may rewrite the condition of eq. (P11.54-1) as

What is the value of the closed-loop gain for the system shown in Figure P11.54(b) at when eq. (P11.54-2) is satisfied?

(c) A sinusoidal oscillator may be constructed on the basis of the principle out-lined above by using the circuit shown in Figure P11.54(c). The input to the

amplifier is the difference between the voltages . In this circuit, the amplifier has a gain of A and an output resistance of , and are impedances. (That is, each is the system function of an LTI system whose input is the current flowing through the impedance element and whose output is the voltage across the element.) It can be shown that, for this circuit,