Barkhausen criteria and RC oscillator using op-amp

Barkhausen criteria: An amplifier with feedback produces sustained sinusoidal oscillation at the frequency for which the loop gain Aβ satisfies both:

1. Magnitude condition: |Aβ| = 1 (loop gain is exactly unity)
2. Phase condition: the total phase shift around the loop is 0° or 360° (positive feedback), that is, ∠Aβ = 0°, 360°, ...

Here A is the amplifier gain and β is the feedback factor.

Is the principle applicable to RC oscillators using an op-amp?

Yes. An RC oscillator is still a feedback system: an op-amp amplifier (gain A) and an RC feedback network (factor β). The oscillation condition is the same: loop gain |Aβ| = 1 with total phase shift 0° (360°) at one frequency. The RC network provides the frequency selectivity (it has the right phase only at one frequency), instead of an LC tank. The op-amp supplies the gain and needed phase (inverting or non-inverting).

Frequency and gain

Two common op-amp RC oscillators:

(a) RC phase-shift oscillator (inverting amplifier + three RC sections)

The three-section network gives 180° phase shift and an attenuation of 29 at one frequency. With the inverting amplifier of 180°, the total is 360°.

f = 1/(2π √6 RC) ≈ 0.065/(RC)

Required gain: |A| = Rf/R ≥ 29

(b) Wien-bridge oscillator (non-inverting amplifier + Wien network)

The network gives β = 1/3 and 0° phase at ω = 1/RC.

f = 1/(2π RC)

Required gain: A = 1 + Rf/R1 = 3 (Rf = 2 R1; slightly more than 3 to start the oscillation).

Example: R = 10 kΩ, C = 10 nF: phase-shift f = 650 Hz with gain 29; Wien f = 1.59 kHz with gain 3.