Editing Wien bridge oscillator (section)

==Automatic gain control dynamics==
[[File:Root Locus.png|thumb|400px|Root locus plot of Wien bridge oscillator pole positions for R<sub>1</sub> = R<sub>2</sub> = 1 and C<sub>1</sub> = C<sub>2</sub> =1 versus K = (R<sub>b</sub> + R<sub>f</sub>)/R<sub>b</sub>.  The numerical values of K are shown in a purple font.  The trajectory of the poles for K=3 is perpendicular to the imaginary (β) axis.  For K >> 5, one pole approaches the origin and the other approaches K.<ref>{{Harvnb|Strauss|1970|p=667}}</ref>]]

Small perturbations in the value of R<sub>b</sub> cause the dominant poles to move back and forth across the jω (imaginary) axis.  If the poles move into the left half plane, the oscillation dies out exponentially to zero.   If the poles move into the right half plane, the oscillation grows exponentially until something limits it.  If the perturbation is very small, the magnitude of the equivalent Q is very large so that the amplitude changes slowly.  If the perturbations are small and reverse after a short time, the envelope follows a ramp.  The envelope is approximately the integral of the perturbation.  The perturbation to envelope transfer function rolls off at 6&nbsp;dB/octave and causes −90° of phase shift.

The light bulb has thermal inertia so that its power to resistance transfer function exhibits a single pole low pass filter.  The envelope transfer function and the bulb transfer function are effectively in cascade, so that the control loop has effectively a low pass pole and a pole at zero and a net phase shift of almost −180°.  This would cause poor [[transient response]] in the control loop due to low [[phase margin]].  The output might exhibit [[squegging]].  Bernard M. Oliver<ref>{{harvnb|Oliver|1960}}</ref> showed that slight compression of the gain by the amplifier mitigates the envelope transfer function so that most oscillators show good transient response, except in the rare case where non-linearity in the [[vacuum tube]]s canceled each other producing an unusually linear amplifier.