# Van der Pol Negative Resistance Oscillator (pdf)

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

The characteristic curve of a "negative resistance" device 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 ]
 B. van der Pol, Radio Rev. 1, 704-754, 1920 and B. van der Pol, Phil. Mag. 3, 65, 1927

This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated March 16, 2000