1. P26 - 1 Class 26: Outline Hour 1: Driven Harmonic Motion (RLC) Hour 2: Experiment 11: Driven RLC Circuit

2. P26 - 2 Last Time: Undriven RLC Circuits

3. P26 - 3 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)

4. P26 - 4 Damped LC Oscillations Resistor dissipates energy and system rings down over time

5. P26 - 5 Mass on a Spring: Simple Harmonic Motion` A Second Look

6. P26 - 6 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

7. P26 - 7 Demonstration: Driven Mass on a Spring Off Resonance

8. P26 - 8 Driven Mass on a Spring Now we get: Simple Harmonic Motion F(t)F(t) Assume harmonic force: Moves at driven frequency

9. P26 - 9 Resonance Now the amplitude, x max, depends on how close the drive frequency is to the natural frequency   x max Let’s See…

10. P26 - 10 Demonstration: Driven Mass on a Spring

11. P26 - 11 Resonance x max depends on drive frequency   x max Many systems behave like this: Swings Some cars Musical Instruments …

12. P26 - 12 Electronic Analog: RLC Circuits

13. P26 - 13 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

14. P26 - 14 Demonstration: RLC with Light Bulb

15. P26 - 15 Start at Beginning: AC Circuits

16. P26 - 16 Alternating-Current Circuit sinusoidal voltage source direct current (dc) – current flows one way (battery) alternating current (ac) – current oscillates

17. P26 - 17 AC Circuit: Single Element Questions: 1.What is I 0 ? 2.What is  ?

18. P26 - 18 AC Circuit: Resistors I R and V R are in phase

19. P26 - 19 AC Circuit: Capacitors I C leads V C by  /2

20. P26 - 20 AC Circuit: Inductors I L lags V L by  /2

21. P26 - 21 ElementI0I0 Current vs. Voltage Resistance Reactance Impedance ResistorIn Phase CapacitorLeads InductorLags AC Circuits: Summary Although derived from single element circuits, these relationships hold generally!

22. P26 - 22 PRS Question: Leading or Lagging?

23. P26 - 23 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

24. P26 - 24 Demonstration: Phasors

25. P26 - 25 Phasor Diagram: Resistor I R and V R are in phase

26. P26 - 26 Phasor Diagram: Capacitor I C leads V C by  /2

27. P26 - 27 Phasor Diagram: Inductor I L lags V L by  /2

28. P26 - 28 PRS Questions: Phase

29. P26 - 29 Put it all together: Driven RLC Circuits

30. P26 - 30 Question of Phase We had fixed phase of voltage: It’s the same to write: (Just shifting zero of time)

31. P26 - 31 Driven RLC Series Circuit

32. P26 - 32 Driven RLC Series Circuit I(t) Now we just need to read the phasor diagram! VSVS

33. P26 - 33 Driven RLC Series Circuit Impedance

34. P26 - 34 Plot I, V’s vs. Time

35. P26 - 35 PRS Question: Who Dominates?

36. P26 - 36 RLC Circuits: Resonances

37. P26 - 37 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

38. P26 - 38 Resonance C-like:  < 0 I leads L-like:  > 0 I lags

39. P26 - 39 Demonstration: RLC with Light Bulb

40. P26 - 40 PRS Questions: Resonance

41. P26 - 41 Experiment 11: Driven RLC Circuit

42. P26 - 42 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