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.


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 ždetuning termÓ 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 ]
[1] 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