Module 25: Outline Resonance & Resonance Driven & LRC Circuits Circuits 2

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1 Module 25: Driven RLC Circuits 1

2 Module 25: Outline Resonance & Driven LRC Circuits 2

3 Driven Oscillations: Resonance 3

4 Mass on a Spring: Simple Harmonic Motion A Second Look 4

5 Mass on a Spring (1) (2) We solved this: (3) (4) F = kx = ma = m d 2 x m d 2 x + kx = 0 dt 2 dt 2 Simple Harmonic Motion ω + φ x(t) = x 0 cos( 0 t ) Moves at natural frequency What if we now move the wall? Push on the mass? 5

6 Demonstration: Driven Mass on a Spring Off Resonance 6

7 Driven Mass on a Spring F(t) F = F t Now we get: () kx = ma = m d 2 x () m d 2 x + kx = F t dt 2 dt 2 Assume harmonic force: F(t) = F 0 cos(ωt) Simple Harmonic Motion ω φ x(t) = x cos( t + ) max Moves at driven frequency 7

8 Resonance x(t) ( ) = x cos(ωt + φ) max Now the amplitude, x max, depends on how close the drive frequency is to the natural frequency x max Let s See ω 0 ω 8

9 Demonstration: Driven Mass on a Spring 9

10 Resonance x(t) = x cos(ωt + φ) max x max depends on drive frequency x max Many systems behave like this: Swings Some cars Kendall T Station ω 0 ω 10

11 Famous Resonance Examples 11

12 Electronic Analog: RLC Circuits 12

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 13

14 Demonstration: RLC with Light Bulb 14

15 Concept Q.: RLC Circuit w/ Light bulb As I slide the core into the inductor the light bulb changes brightness. Why? I am driving the circuit through resonance by 1. continuously increasing the frequency of current oscillations in the circuit 2. continuously decreasing the frequency of current oscillations in the circuit 3. continuously increasing the natural frequency of oscillations in the circuit 4. continuously decreasing the natural frequency of oscillations in the circuit 5. I don t know 15

16 Start t at Beginning: i AC Circuits 16

17 Problem: Discovery Mathlet t Driven RLC Circuits 17

current oscillates sinusoidal voltage source V (t) = V

18 Alternating-Current Circuit direct current (dc) current flows one way (battery) alternating current (ac) current oscillates sinusoidal voltage source V (t) = V 0 sin ωt ω = 2π f : angular frequency V : voltage amplitude 0 18

19 AC Circuit: Single Element V l t = V element = V sin ωt 0 I(t) = I 0 sin(ωt φ) Questions: 1. What is I 0? 2. What is φ? 19

20 AC Circuit: Resistors V R = I R R I = V R = V 0 sin ωt I = V 0 R 0 R R R = I 0 sin ωt 0 ϕ = 0 ( ) 0 I R and V R are in phase 20

21 AC Circuit: Capacitors V = C Q I C (t) = dq dt I 0 = ωcv 0 = ωcv 0 cos ωt φ = π /2 C = I 0 sin(ωt π /2) I C leads V C by π/2 Q(t) ) = CV = CV sinωt C 0 21

22 di L V L = L dt AC Circuit: Inductors V I L (t) = V 0 L = V 0 sin ωt dt V 0 I = 0 cos ωt 0 ω L ω L / 2 = I 0 sin(ωt π /2) φ = π / 2 I L lags V L by π/2 di V V L = L dt L = 0 sin ωt L 22

23 AC Circuits: Summary Current Element I 0 vs. Voltage Resistance Reactance Resistor V 0 R R In Phase R = R Capacitor Leads X = 1 ωcv 0C C ωcc V 0 L Inductor Lags X = ω L L ω L Although h derived d from single element circuits, it these relationships hold generally! 23

24 Concept Question: Leading or The plot shows the driving voltage V (black curve) and the current I (red curve) in a driven RLC circuit. In this circuit, Lagging? 1. The current leads the voltage 2. The current lags the voltage 3. Don t have a clue. 24

25 Concept Question: Leading or The graph shows current versus voltage in a driven RLC circuit at a given driving frequency. In this plot Lagging? 1. The current leads the voltage by about 45º 2. The current lags the voltage by about 45º 3. The current and the voltage are in phase 4. Don t have a clue 25

26 Phasor Diagram: Resistor V 0 = I 0 R φ = 0 I R and V R are in phase 26

27 Phasor Diagram: Capacitor V 0 = I 0 X C = I 0 1 ωc φ = π /2 I C leads V C by π/2 27

28 Phasor Diagram: Inductor V = I X 0 0 L = I 0 ω L φ = π /2 I L lags V L by π/2 28

29 Put it all together: th Driven RLC Circuits 29

30 Question of Phase We had fixed phase of voltage: V element = V 0 sin ωt I(t) = I 0 sin(ωt φ) It s the same to write: V element = V 0 sin(ωt + φ) I(t) = I 0 sin ωt (Just shifting zero of time) 30

31 Driven RLC Series Circuit I(t) = I 0 sin(ωt) V R = V R0 sin ( ωt) V L = V L0 sin ( ωt + π /2) V S = V 0S sin ( ωt + ) φ V C = V C0 sin ( ωt π /2) What is I (and V = I R, V = I X V = I X )? 0, R0 0 L0 0 L, C 0 0 C What is φ? Does the current lead or lag V s? Must Solve: V S = V R + V L + V C 31

32 Recall: Phasor Diagram Nice way of tracking magnitude & phase: V (t) = V 0 sin(ωt) ω V 0 ωt Notes: (1) As the phasor (red vector) rotates, the projection (pink vector) oscillates (2) Do both for the current and the voltage 32

33 Driven RLC Series Circuit I(t) V S V 0L V 0C V 0S I 0 V0R Now Solve: V = V + V + V S R L C Now we just need to read the phasor diagram! 33

34 Driven RLC Series Circuit V 0L V 0S I(t) = I sin(ωt φ) 0 V S = V 0S sin(ωt) V 0C I0 φ V0R V 0S = V 2 R0 + (V L0 V C0 ) 2 = I 0 R 2 + ( X L X C ) 2 I 0 Z V 0S I 0 = Z Z = R 2 + ( X 2 L X C ) Impedance φ = tan 1 X L X C R 34

35 Plot I, V s vs. Time V S V C V L V R I 0 -π/ φ I(t) = I 0 sin(ω t) V R 0 (t) = I 0 R sin(ωt) 0 (t) = I 0 X L sin(ωt + π /2) +π/2 0 (t) = I 0 X C sin(ωt π /2) 0 Time (Periods) V L V C V S (t) = V S 0 sin(ωt + φ) 3 35

36 Concept Q.: Who Dominates? The graph shows current & voltage vs. time in a driven RLC circuit at a particular driving frequency. At this frequency, the circuit is dominated by its 1. Inductance 2. Capacitance 3. I don t know 36

37 Concept Q.: What Frequency? The graph shows current & voltage vs. time in a driven RLC circuit at a particular driving frequency. Is this frequency above or below the resonance frequency of the circuit? 1. Above the resonance frequency 2. Below the resonance frequency 3. I don t know 37

38 RLC Circuits: it Resonances 38

39 Resonance = V 0 V I = 0 ; X = = 1 0 ω L, X Z R 2 + ( X L X C ) 2 L C ωc 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 I 0 reaches maximum when X L = X C 1 ω = 0 LC 39

40 Resonance V V I 0 = = ; X L = ω L, X C = Z R 2 + ( X X ) 2 ωc C-like: φ <0 L-like: I leads φ > 0 I lags L C φ = tan 1 X L X C R ω 0 = 1 LC 40

41 Demonstration: RLC with Light Bulb 41

42 Concept Question: RLC Circuit With Light Bulb Imagine another light bulb connected in parallel to this LRC circuit. With the core pulled out that light bulb would be flashing: 1. before the LRC light bulb (leading) 2. after the LRC light bulb (lagging) 3. in time with the LRC light bulb 4. not at all 5. I don t know 42

43 Problem: RLC Circuit Current (ma) Current (ma A) Time Cap. Volta age (V) Cap. Voltage (V) Time Cu urrent (ma) Time Voltage (mv) Cap P.S. Volta age (V) (V) P.S. Voltage P. S. Voltage (V) Consider plots of V C, V S and I made at 3 frequencies: a very low angular frequency (100 s -1 ), a very high one (10 5 s -1 ) and the resonance frequency, which is somewhere in between Each plot allows you to find one of R,L,C. In that order, which plot do you use, which frequency is it and what are the values of R, L & C? 43

44 Experiment 9: Driven RLC Circuit 44

45 What to Learn from Lab 1) Properties of resonance? How can you tell when you are on resonance? 2) From plot I & V vs. t OR I vs. V: Which is leading (I or V)? L-like or C-like? Above or below resonance? 45

46 Concept Question Questions: Resonance 46

47 RLC Circuits: it leading/lagging 47

48 Concept Question: Leading or The graph shows current versus voltage in a driven RLC circuit at a given driving frequency. In this plot Lagging? 1. Current lags voltage by ~90º 2. Current leads voltage by ~90º 3. Current and voltage are almost in phase 4. Not enough info (but they aren t in phase!) 5. I don t know. 48

49 Concept Question: Leading or Lagging The graph shows the current versus the voltage in a driven RLC circuit at a given driving frequency. In this plot 1. Current lags voltage by ~90º 2. Current leads voltage by ~90º 3. Current and voltage are almost in phase 4. We don t have enough information (but they aren t in phase!) 5. I don t know 49

50 Concept Question: What d You Do? The graph shows current & voltage vs. time in a driven RLC circuit. We had been in resonance a second ago but then either put in or took out the core from the inductor. Which was it? 1. Put in the core 2. Took out the core 3. I don t know 50

51 MIT OpenCourseWare SC Physics II: Electricity and Magnetism Fall 2010 For information about citing these materials or our Terms of Use, visit:

Module 24: Outline. Expt. 8: Part 2:Undriven RLC Circuits

Module 24: Outline. Expt. 8: Part 2:Undriven RLC Circuits Module 24: Undriven RLC Circuits 1 Module 24: Outline Undriven RLC Circuits Expt. 8: Part 2:Undriven RLC Circuits 2 Circuits that Oscillate (LRC) 3 Mass on a Spring: Simple Harmonic Motion (Demonstration)