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.

__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 theslow time variationassumption, this equation reduces to

[ VdP-5 ] |

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

[ VdP-6 ] |

To studywe 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 casefrequency locking

[ 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 theassumption, this equation reduces toslow time variation

[ 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 .[1] B. van der Pol,

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 ]

This page was prepared and is maintained by R. Victor Jones,

Last updated March 16, 2000