Parametric Oscillation in a Damped Resonant System

Parametric oscillation can be induced in a simple passive resonant system with losses for certain ranges of values of the pump frequency
\omega_p
, provided a threshold value of the reactance variation has been exceeded. Assuming the reactance variation to be of the form
C(t) = C_{0} + C_{1}\cos \omega_{p} t
. and the depth of modulation
C_1/C_0 \not> 0.2
, Mathieu's equation is taken to apply and iso-
\mu
curves have been computed and plotted in the first unstable region on an
a - q
chart (after McLachlan) for values of
q
from 0 to 0.3. Using a simple semigraphical method, it was found that to just permit oscillation the threshold variation
C_1/C_0
varied from 0.024 at
Q = 100
to 0.2 at
Q = 20
, while the pump-frequency range over which oscillation could be induced varied from 0 to
0.2 \omega_0
for modulation depths from 0.024 to 0.2 at
Q = 100
and from 0 to
0.17 \omega_0
for modulation depths from 0.99 to 0.2 at
Q = 20
.