Colpitts oscillator theory

Question

I am working on oscillators at the moment and need clarification. So far, I understand that the barkhausan criteria determines if the oscillator will oscillate or not. The rules are: 1)make the loop gain, gain of feedback network times gain of amplifier, 1. 2)Make the phase angle around the loop 0 degrees. To me, these two rules are used to determine the formula for oscillation frequency, which is necessary. However, this says nothing about the mechanism used to stabilize the gain.

Upon further investigation, I found that the poles of the loop gain need to move to the right half plan to start oscillation and a non-linear amplitude stabilization mechanism pulls the poles back to the imaginary axis. I don't understand how this works. Do the right half-plane poles mean that the gain is greater than 1 (positive feedback), left half plane poles mean the gain is less than 1 (negative feedback) and the imaginary axis indicates unity gain?

Answer 1

Quote: "I found that the poles of the loop gain need to move to the right half plan to start oscillation and a non-linear amplitude stabilization mechanism pulls the poles back to the imaginary axis. I don't understand how this works. Do the right half-plane poles mean that the gain is greater than 1 (positive feedback), left half plane poles mean the gain is less than 1 (negative feedback) and the imaginary axis indicates unity gain?"

• Perhaps a typing error? The poles of the closed-loop gain need to be in the right half of the s-plane to safely start oscillations. And yes - this corresponds to a loop gain which is (slightly) greater then unity at he desired frequency of oscillation.

• In the time domain, this situation corresponds to a positive exponent of the e-function (solution of the diff. equation) which describes the rising amplitude.

• When the loop contains a non-linear and amplitude-sensible mechanism (additional part, property of the amplifier or hard-limiting due to finite supply voltage) the loop gain decreases with rising amplitudes and the closed-loop pole moves to the imaginary axis.

• However, due to some inherent delay (inertia of the system) the poles do not stop at the imag. axis but move slighly into the left half of the s-plane (slighly decreasing amplitudes) - until the non-linearity (again rising loop gain) brings the pole again back (through the imag. axis) to the right half.

• As a consequence, during oscillations, the pole "swings" around the imag. axis - a kind of amplitude modulation, which however can be kept very small (nearly unvisible) for a good design.