P Class 26: Outline Hour 1: Driven Harmonic Motion (RLC) Hour 2: Experiment 11: Driven RLC Circuit

P Last Time: Undriven RLC Circuits

P LC Circuit It undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass)

P Damped LC Oscillations Resistor dissipates energy and system rings down over time

P Mass on a Spring: Simple Harmonic Motion` A Second Look

P Mass on a Spring (1) (2) (3) (4) We solved this: Simple Harmonic Motion What if we now move the wall? Push on the mass? Moves at natural frequency

P Demonstration: Driven Mass on a Spring Off Resonance

P Driven Mass on a Spring Now we get: Simple Harmonic Motion F(t)F(t) Assume harmonic force: Moves at driven frequency

P Resonance Now the amplitude, x max, depends on how close the drive frequency is to the natural frequency   x max Let’s See…

P Demonstration: Driven Mass on a Spring

P Resonance x max depends on drive frequency   x max Many systems behave like this: Swings Some cars Musical Instruments …

P Electronic Analog: RLC Circuits

P Analog: RLC Circuit Recall: Inductors are like masses (have inertia) Capacitors are like springs (store/release energy) Batteries supply external force (EMF) Charge on capacitor is like position, Current is like velocity – watch them resonate Now we move to “frequency dependent batteries:” AC Power Supplies/AC Function Generators

P Demonstration: RLC with Light Bulb

P Start at Beginning: AC Circuits

P Alternating-Current Circuit sinusoidal voltage source direct current (dc) – current flows one way (battery) alternating current (ac) – current oscillates

P AC Circuit: Single Element Questions: 1.What is I 0 ? 2.What is  ?

P AC Circuit: Resistors I R and V R are in phase

P AC Circuit: Capacitors I C leads V C by  /2

P AC Circuit: Inductors I L lags V L by  /2

P ElementI0I0 Current vs. Voltage Resistance Reactance Impedance ResistorIn Phase CapacitorLeads InductorLags AC Circuits: Summary Although derived from single element circuits, these relationships hold generally!

P PRS Question: Leading or Lagging?

P Phasor Diagram Nice way of tracking magnitude & phase: Notes: (1) As the phasor (red vector) rotates, the projection (pink vector) oscillates (2) Do both for the current and the voltage

P Demonstration: Phasors

P Phasor Diagram: Resistor I R and V R are in phase

P Phasor Diagram: Capacitor I C leads V C by  /2

P Phasor Diagram: Inductor I L lags V L by  /2

P PRS Questions: Phase

P Put it all together: Driven RLC Circuits

P Question of Phase We had fixed phase of voltage: It’s the same to write: (Just shifting zero of time)

P Driven RLC Series Circuit

P Driven RLC Series Circuit I(t) Now we just need to read the phasor diagram! VSVS

P Driven RLC Series Circuit Impedance

P Plot I, V’s vs. Time

P PRS Question: Who Dominates?

P RLC Circuits: Resonances

P Resonance I 0 reaches maximum when At very low frequencies, C dominates (X C >>X L ): it fills up and keeps the current low At very high frequencies, L dominates (X L >>X C ): the current tries to change but it won’t let it At intermediate frequencies we have resonance

P Resonance C-like:  < 0 I leads L-like:  > 0 I lags

P Demonstration: RLC with Light Bulb

P PRS Questions: Resonance

P Experiment 11: Driven RLC Circuit

P Experiment 11: How To Part I Use exp11a.ds Change frequency, look at I & V. Try to find resonance – place where I is maximum Part II Use exp11b.ds Run the program at each of the listed frequencies to make a plot of I 0 vs.  Part III Use exp11c.ds Plot I(t) vs. V(t) for several frequencies