Van der Pol

1
Van der Pol
2
Convergence

• The damped driven oscillator has both transient
  and steady-state behavior.
• Transient dies out
• Converges to steady state

3
Equivalent Circuit

• Oscillators can be simulated by RLC circuits.
• Inductance as mass
• Resistance as damping
• Capacitance as inverse spring constant

L
vin
v
C
R
4
Negative Resistance

• Devices can exhibit negative resistance.
• Negative slope current vs. voltage
• Examples tunnel diode, vacuum tube
• These were described by Van der Pol.

R. V. Jones, Harvard University
5
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.

6
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.

7
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
8
Limit Cycle

• The phase portraits show convergence to a steady
  state.
• This is called a limit cycle.

next