-- 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.
|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:
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 nominal frequency of oscillation of this circuit is given by:
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.