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
Similarly, output of op-amp is
Step No: 6
A KCL equation at the node , yields
Step No: 7
A KCL equation at the inverting terminal of a amplifier is
By substituting the equation (3) in equation (4), we get
Step No: 8
Substituting equation (2) in equation (5), we get
Step No: 9
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