Van der Pol Negative Resistance Oscillator (pdf)

Van der Pol's analysis[1]of "negative resistance" (]e.g., tunnel diode) oscillators provides a valuable framework for treating with relative simplicity important features of oscillatory systems.

Consider the following negative resistance oscillatory circuit:

By simple circuit analysis, it is a straightforword proposition to find the following simple circuit equation which is the fundamental van der Pol oscillator equation:

[ VdP-1 ]

where .

If a is small, it is reasonable to take

[ VdP-2 ]

Then Equation [ VdP-1 ] becomes without approximation

[ VdP-3 ]

If we ignore harmonic generation, Equation [ VdP-3 ] may be approximated as

[ VdP-4 ]

If we make the slow time variation assumption, this equation reduces to

[ VdP-5 ]

where . This essential Equation [ VI-25b ] in the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory. We saw there that the general steady state solution is given by

[ VdP-6 ]

To study frequency locking we suppose that a driving source (to be precise a current source in parallel with the negative resistance) and then the van der Pol equation becomeshe driven case

[ VdP-7 ]

In this case, it is reasonable to take

[ VdP-8 ]

If we again ignore harmonic geneation, Equation [ VdP-7 ] becomes

[ VdP-9 ]

Again under the slow time variation assumption, this equation reduces to

[ VdP-10 ]

If we take , this equation separate into the following pair of equations:

[ VdP-11a ]

[ VdP-11b ]

where (the "detuning term") and  (the "locking coefficient"). For small  we can decouple the equations and take  from Equation [ VdP-6 ] so that .
 If  the relative phase angle changes linearly in time at the rate  .   As  decreases toward unity, the "locking term" subtracts from the ìdetuning termî in one half of a cycle and adds in the other half.  At  there are two values of the phase angle that yield the "mode locking" condition  -- viz.
 

[ VdP-12 ]

We can test the stability of these solutions by taking  and therefore Equation [VdP-11b] becomes
 

[ VdP-13 ]

and the solutions are stable if

	[ VdP-14a ]
	[ VdP-14b ]