Class 26: Outline

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

2. Last Time:Undriven RLC Circuits

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

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

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

7. Demonstration:Driven Mass on a SpringOff Resonance

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

9. Resonance • Now the amplitude, xmax, depends on how close the drive frequency is to the natural frequency xmax Let’s See… w0 w

10. Demonstration:Driven Mass on a Spring

11. Resonance • xmax depends on drive frequency xmax Many systems behave like this: Swings Some cars Musical Instruments … w0 w

12. Electronic Analog:RLC Circuits

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. Demonstration:RLC with Light Bulb

15. Start at Beginning:AC Circuits

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

17. AC Circuit: Single Element • Questions: • What is I0? • What is f ?

18. AC Circuit: Resistors IR and VR are in phase

19. AC Circuit: Capacitors ICleadsVC by p/2

20. AC Circuit: Inductors ILlagsVL by p/2

21. AC Circuits: Summary Although derived from single element circuits, these relationships hold generally!

22. PRS Question:Leading or Lagging?

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. Demonstration:Phasors

25. Phasor Diagram: Resistor IR and VR are in phase

26. Phasor Diagram: Capacitor ICleadsVC by p/2

27. Phasor Diagram: Inductor ILlagsVL by p/2

28. PRS Questions:Phase

29. Put it all together:Driven RLC Circuits

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

31. Driven RLC Series Circuit

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

33. Driven RLC Series Circuit Impedance

34. Plot I, V’s vs. Time

35. PRS Question:Who Dominates?

36. RLC Circuits:Resonances

37. Resonance At very low frequencies, C dominates (XC>>XL): it fills up and keeps the current low At very high frequencies, L dominates (XL>>XC): the current tries to change but it won’t let it At intermediate frequencies we have resonance I0 reaches maximum when

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

39. Demonstration:RLC with Light Bulb

40. PRS Questions:Resonance

41. Experiment 11: Driven RLC Circuit

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 I0 vs. w • Part III • Use exp11c.ds • Plot I(t) vs. V(t) for several frequencies