The circuit in Figure P15.28 is an alternative configuration of a phase- shift oscillator, (a) Assume that R1 = R2 = R3 = RA1 = RA2 = RA3 ≡ R and C1 = C2 = C3 = C. Show that the frequency of oscillation is(b) Assume equal magnitudes of gain in each amplifier stage. What is the minimum magnitude of gain required in each stage to sustain oscillation?

Step No: 1

The given phase-shift oscillator circuit is

Step No: 2

Redraw the given circuit by labeling the voltages as

Step No: 3

a) Given that,

Assume that and and let .

A KCL equation at the node yields,

Step No: 4

Amplifier gain of first op-amp is

Step No: 5

Hence, the output voltages of op-amp are

…… (1)

Similarly, output of op-amp is

…… (2)

Step No: 6

A KCL equation at the node , yields

…… (3)

Step No: 7

A KCL equation at the inverting terminal of a amplifier is

…… (5)

By substituting the equation (3) in equation (4), we get

Step No: 8

Substituting equation (2) in equation (5), we get

Step No: 9

…… (6)

Substituting equation (1) in equation (6), we get

To find the frequency of oscillation, set and the imaginary term must be zero.

Step No: 10

Consider the imaginary values then

Therefore, the required frequency of oscillation is

Step No: 11

b) The condition for oscillation is

from the above expression.

Hence, the required minimum magnitude of gain in each stage to oscillation is