Module 25: Driven RLC Circuits 1
Module 25: Outline Resonance & Driven LRC Circuits 2
Driven Oscillations: Resonance 3
Mass on a Spring: Simple Harmonic Motion A Second Look 4
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
Demonstration: Driven Mass on a Spring Off Resonance 6
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
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
Demonstration: Driven Mass on a Spring 9
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
Famous Resonance Examples 11
Electronic Analog: RLC Circuits 12
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
Demonstration: RLC with Light Bulb 14
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
Start t at Beginning: i AC Circuits 16
Problem: Discovery Mathlet t Driven RLC Circuits 17
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
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
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
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
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
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
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
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
Phasor Diagram: Resistor V 0 = I 0 R φ = 0 I R and V R are in phase 26
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
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
Put it all together: th Driven RLC Circuits 29
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
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
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
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
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
Plot I, V s vs. Time V S V C V L V R I 0 -π/2 0 1 2 +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
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
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
RLC Circuits: it Resonances 38
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
Resonance V V I 0 = 0 0 1 = ; 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
Demonstration: RLC with Light Bulb 41
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
Problem: RLC Circuit Current (ma) Current (ma A) 5.0 2.5 0.0-2.5-5.0 Time 1.0 0.5 0.0-0.5 Cap. Volta age (V) 1.0 0.5 0.0-0.5-1.0-1.0 100 1.0 1.0 50 0-50 0.5 0.0-0.5 Cap. Voltage (V) 0.5 0.0-0.5-100 -1.0-1.0 Time Cu urrent (ma) 2 04 0.4 10 1.0 1 0-1 -2 Time 0.2 0.0-0.2-0.4 Voltage (mv) Cap. 0.5 0.0-0.5-1.0 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
Experiment 9: Driven RLC Circuit 44
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
Concept Question Questions: Resonance 46
RLC Circuits: it leading/lagging 47
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
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
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
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