< Secrets of the Phase-shift Oscillator – revealed!

## Secrets of the Phase-shift Oscillator - revealed!

-- by: L. Coyle

It's pretty easy to make an oscillator - sometimes it's hard not to make one (like when you're trying to build an amplifier).

Here's how it's done: Take an amplifier and feed some of its output back into the input, forming a loop. If the total gain around the loop is one or more, and if the phase-shift around the loop at some frequency is zero or 360° (or some multiple of 360°) ... then, congratulations! You've made an oscillator.

### The Barkhausen Criterion

What you have actually done is fulfilled the "Barkhausen criterion" , which states that the conditions for oscillation in a linear electronic circuit containing a feedback loop are:

|Aβ| = 1

where, A = forward gain of the amplifying element, and β = fraction of the output fed back to the input.

Note that both A and β can (and usually do) vary with frequency.

This is the Barkhausen criterion in block diagram form:

### Circuit Elements of the Oscillator

In a conventional phase-shift oscillator, a separate phase shifting network is included to put the phase shift under the control of the designer, and the gain around the loop is controlled by the amplifier.

The phase-shift network accounts for 180° of phase at the design frequency and the inverting amplifier adds the remaining 180° for a 360° total. The job of the amplifier is to add enough gain to compensate for any amplitude loss in the phase-shift network and bring the total gain around the loop to unity.

### The Phase Shift Network

The phase-shift network is (usually) a three-stage resistor-capacitor (RC) low-pass network, in which each stage contributes about 60° of phase shift. Since it is impossible to achieve more than 90° of phase shift in a single RC section, this represents a reasonable trade-off between phase shift and signal loss.

### The Complete Oscillator Circuit

In its final form, the phase-shift oscillator looks like this:

The nominal frequency of oscillation of this circuit is given by:

$\large&space;frequency&space;(Hz)=&space;\frac{\sqrt{6}}{2\pi&space;Rp&space;Cp}$

and, yes, it will oscillate if the gain of inverting amplifier Ua (set by the ratio of Ra to Rb) is made high enough to overcome the signal loss through the phase-shift network - about 30 db.

The above equation is most nearly exact for frequencies well below the closed-loop bandwidth of the amplifier. As we get close to that limit, the amplifier itself contributes some phase shift. Remember, the total phase shift around the loop must be 360° for the Barkhausen criterion to apply, so if there is additional phase-shift around the loop, the oscillator will find a frequency somewhat different than that given by the equation.