Van der Pol

Convergence  The damped driven oscillator has both transient and steady-state behavior. Transient dies outTransient dies out Converges to steady stateConverges to steady state

Equivalent Circuit  Oscillators can be simulated by RLC circuits. Inductance as mass Resistance as damping Capacitance as inverse spring constant v in v C L R

Negative Resistance  Devices can exhibit negative resistance. Negative slope current vs. voltageNegative slope current vs. voltage Examples: tunnel diode, vacuum tubeExamples: tunnel diode, vacuum tube  These were described by Van der Pol. R. V. Jones, Harvard University

Steady State  Assume an oscillating solution. Time varying amplitude V Slow time variation  The equation for V follows from substitution and approximation.  The steady state is based on the relative damping terms.

Frequency Locking  The amplitude term can be separated. Two coupled equations Detuning term d Locking coefficient l  The detuning is roughly the frequency difference.  For small driving force the locking coefficient depends on the relative damping.

Relaxation Oscillator  The Van der Pol oscillator shows slow charge build up followed by a sudden discharge.  The oscillations are self sustaining, even without a driving force. Wolfram Mathworld

Limit Cycle  The phase portraits show convergence to a steady state.  This is called a limit cycle. next