Van der PolConvergenceEquivalent CircuitNegative ResistanceSteady StateFrequency LockingRelaxation OscillatorLimit CycleVan der PolVan der PolConvergenceConvergenceThe damped driven oscillator The damped driven oscillator has both transient and has both transient and steady-state behavior.steady-state behavior.•Transient dies outTransient dies out•Converges to steady stateConverges to steady stateqqtfqqqcos220 22222022022042arctancoscostftaeqtEquivalent CircuitEquivalent CircuitOscillators can be simulated Oscillators can be simulated by RLC circuits.by RLC circuits.•Inductance as massInductance as mass•Resistance as dampingResistance as damping•Capacitance as inverse Capacitance as inverse spring constantspring constantvinvCLRCqvvCout22dtqdLdtdiLvLdtdqRRivR0RCLinvvvvtVvLCdtdvLRdtvdsin10222LR2LC120Negative ResistanceNegative ResistanceDevices can exhibit negative Devices can exhibit negative resistance.resistance.•Negative slope current vs. Negative slope current vs. voltagevoltage•Examples: tunnel diode, Examples: tunnel diode, vacuum tubevacuum tubeThese were described by Van These were described by Van der Pol.der Pol.R. V. Jones, Harvard University tVvvvdtddtvdsin0220322Steady StateSteady StateAssume an oscillating Assume an oscillating solution.solution.•Time varying amplitude Time varying amplitude VV•Slow time variationSlow time variationThe equation for V follows The equation for V follows from substitution and from substitution and approximation.approximation.The steady state is based on The steady state is based on the relative damping terms.the relative damping terms...)(21ccetVvti 02432121022220ViVViViV342VFrequency LockingFrequency LockingThe amplitude term can be The amplitude term can be separated.separated.•Two coupled equationsTwo coupled equations•Detuning term Detuning term dd•Locking coefficient Locking coefficient llThe detuning is roughly the The detuning is roughly the frequency difference.frequency difference.For small driving force the For small driving force the locking coefficient depends locking coefficient depends on the relative damping.on the relative damping.)()(tieVtV sin22cos24321022002VVVVVV sinld83200VVVl Relaxation OscillatorRelaxation OscillatorThe Van der Pol oscillator The Van der Pol oscillator shows slow charge build up shows slow charge build up followed by a sudden followed by a sudden discharge.discharge.The oscillations are self The oscillations are self sustaining, even without a sustaining, even without a driving force.driving force.Wolfram Mathworld 012 yyyyLimit CycleLimit CycleThe phase portraits show The phase portraits show convergence to a steady convergence to a steady state.state.This is called a This is called a limit cyclelimit
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