1. Van der Pol

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

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

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

5. 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. Steady State

6. 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. Frequency Locking

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