Saturday, February 27, 2016

Van der Pol Oscillator

The original neon-bulb oscillator circuit studied by Van der Pol and Van der Mark in their 1927 paper consists of a neon-bulb connected to a capacitor, resistor, a bias voltage source and a sinuosoidal signal voltage source.  The operation of this circuit without the sinusoidal signal can be described as follows: the neon bulb is not conducting and the voltage source charges up the the capacitor until a certain point at which the neon bulb starts conducting (and glowing) and discharges the capacitor until the voltage is low enough to turn off the neon bulb and the cycle starts anew.  Looking at the $v$-$i$ characteristic of the neon bulb we see that it is a function with two local extrema just like the cubic polynomial.  The role of the bias voltage source is to add power to the system, since the neon bulb by itself does not supply any power.  In particular, by looking at the equivalent $v$-$i$ characteristic of the neon bulb in parallel with the bias voltage, it effectively shift the characteristic so it looks more like the cubic and thus can oscillate.

Except for the external driving source, this circuit is similar to the circuits considered in a previous post.  To write state equations, we model the neon-bulb as a nonlinear resistor in parallel with an small linear inductor and a small linear capacitor.

When the sinusoidal source is turned off, the oscillator oscillates freely with a frequency $\omega_1$.  The addition of the sinusoidal voltage source add another frequency $\omega_2$ to the system.  One can think of the Poincaré map (generated by sampling the flow at frequency $\omega_2$) as a map on the circle, the circle being the limit cycle.  In fact, for certain parameters, we see behavior similar to those obtained from the circle map.

The complete state equations of the circuit is given by:
\[ \begin{array}{lcl} \frac{dv}{dt} & = &
-\frac{1}{RC}v - \frac{1}{C}i + \frac{E}{RC} \\
\frac{di}{dt} & = & \frac{1}{L}v - \frac{f(i)}{L}
- \frac{E_0 \sin(\omega t)}{L}

By adjusting the capacitor $C$, we change the free running frequency of the oscillator, i.e. changing the ratio between the two frequency. We observe frequency locking where the oscillator oscillates at a submultiple of the driving frequency for a range of capacitor values.  If we vary the driving frequency, we observe the Devil's staircase, similar to the one observed in the circle map.

Van der Pol, B. and Van der Mark, J., “Frequency demultiplication”, Nature, 120, 363-364, 1927.
Kennedy, M. P. & Chua, L. O., “Van der Pol and chaos,” IEEE Trans. Circuits Syst. CAS-33, 974–980, 1986.
Kennedy, M. P., Krieg, K. R. & Chua, L. O., “The Devil’s staircase: The electrical engineer’s fractal,” IEEE Trans. Circuits Syst. CAS-36, 1133–1139, 1989.

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