Question

What is the formula of frequency of oscillation for rc phase shift oscillators?

Answer 1

op-amp oscillator

My aim is to find the oscillation frequency of a Phase Shift Oscillator.



I start by finding transfer function of the cascaded RC network.

Vo(s)=1C3sR3+1C3sV2(s)=11+R3C3sV2(s).Vo(s)=1C3sR3+1C3sV2(s)=11+R3C3sV2(s).

Similarly,

V2(s)=11+R2C2sV1(s)andV1(s)=11+R1C1sVi(s).V2(s)=11+R2C2sV1(s)andV1(s)=11+R1C1sVi(s).

Then the transfer function is:

H(s)=Vo(s)Vi(s)==1(1+R1C1s)(1+R2C2s)(1+R3C3s)1R1R2R3C1C2C3s3+…⋯(R1R2C1C2+R2R3C2C3+R1R3C1C3)s2+…⋯(R1C1+R2C2+R3C3)s+1H(s)=Vo(s)Vi(s)=1(1+R1C1s)(1+R2C2s)(1+R3C3s)=1R1R2R3C1C2C3s3+…⋯(R1R2C1C2+R2R3C2C3+R1R3C1C3)s2+…⋯(R1C1+R2C2+R3C3)s+1

So the frequency response is:

H(jω)=1jω[(R1C1+R2C2+R3C3)−R1R2R3C1C2C3ω2]+…⋯[1−(R1R2C1C2+R2R3C2C3+R1R3C1C3)ω2]H(jω)=1jω[(R1C1+R2C2+R3C3)−R1R2R3C1C2C3ω2]+…⋯[1−(R1R2C1C2+R2R3C2C3+R1R3C1C3)ω2]

Now, we are looking for a special ωω value, ω0ω0, for which the argument of H(wj)H(wj) will be ±180o±180o. Clearly, it happens when

R1C1+R2C2+R3C3=R1R2R3C1C2C3ω2∣∣ω=ω0.R1C1+R2C2+R3C3=R1R2R3C1C2C3ω2|ω=ω0.

Hence we find the oscillation frequency as

ω0f0==R1C1+R2C2+R3C3R1R2R3C1C2C3−−−−−−−−−−−−−−−−−−√12πR1C1+R2C2+R3C3R1R2R3C1C2C3−−−−−−−−−−−−−−−−−−√.ω0=R1C1+R2C2+R3C3R1R2R3C1C2C3f0=12πR1C1+R2C2+R3C3R1R2R3C1C2C3.

When R1=R2=R3=RR1=R2=R3=R and C1=C2=C3=CC1=C2=C3=C:

f0=3–√2πRCf0=32πRC

However, according to all online articles including Wikipedia, the formula for the oscillation frequency is

f0=12πRC6–√.