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) 4
Mass on a Spring (1) (2) What is Motion? F = kx = ma = m d 2 x (3) (4) m d 2 x dt + kx = 0 Simple Harmonic Motion dt 2 x(t) = x 0 cos(ω 0 t + φ) x 0 : Amplitude of Motion φ: Phase (time offset) ω 0 = 0 k = Angular frequency m 5
Mass on a Spring: Energy (1) Spring (2) Mass (3) Spring (4) Mass x(t) ( ) = x cos(ω( t + φ) 0 0 dx = v (t) = ω x sin(ω t + x dt φ) 0 0 0 Energy has 2 parts: (Mass) Kinetic and (Spring) Potential 2 = 1 dx K m 2 dt 2 kx sin (ω t + φ) 0 0 2 2 2 U s = 1 2 kx2 = 1 2 kx 0 = 1 kx 2 sin 2 (ω t + φ) Energy sloshes back 2 cos 2 (ω t + ) and forth 0 6
Simple Harmonic Motion Amplitude (x 0 ) 1 Period = T = 1 frequency f 2π Period = T = 2π angular frequency ω ω φ x(t) = x cos( (ω φ) 0 0 t + ) Phase Shift( φ) = π 2 7
Electronic Analog: LC Circuits 8
Analog: LC Circuit Mass doesn t like to accelerate Kinetic energy associated with motion F = ma = m dv = m d 2 x ; E = 1 dt 2 mv2 dt 2 Inductor doesn t like to have current change Energy associated with current 2 ε = L di dt = L d q ; E = 1 dt 2 2 LI 2 9
Analog: LC Circuit Spring doesn t like to be compressed/extended Potential energy associated with compression F = kx; E = 1 2 kx2 Capacitor doesn t like to be charged (+ or -) Energy associated with stored charge 1 11 ε = q; E = q 2 C 2 C F ε; x q; v I; m L; k C 1 10
LC Circuit 1. Set up the circuit above with capacitor, inductor, resistor, and battery. 2. Let the capacitor become fully charged. 3. Throw the switch from a to b 4. What happens? 11
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) 12
Concept Question Questions: LC Circuit it 13
Concept Question: LC Circuit Consider the LC circuit at right. At the time shown the current has its maximum value. At this time 1. The charge on the capacitor has its maximum value 2. The magnetic field is zero 3. The electric field has its maximum value 4. The charge on the capacitor is zero 5. Don t have a clue 14
Concept Question: LC Circuit In the LC circuit at right the current ti is in the direction shown and the charges on the capacitor have the signs shown. At this time, 1. I is increasing and Q is increasing 2. I is increasing and Q is decreasing 3. I is decreasing and Q is increasing 4. I is decreasing and Q is decreasing 5. Don t have a clue 15
LC Circuit Q C L di dt = 0 ; I = dq dt d 2 Q + 1 2 Q = 0 dt LC Simple Harmonic Motion Q(t) = cos( t + = 1 ) Q ω φ ) ω 0 0 0 Q 0 : Amplitude of Charge Oscillation φ: Phase (time offset) LC 16
LC Oscillations: Energy Notice relative phases Q 2 U E = Q2 2 2C = Q 0 2C cos2 ω 0 t U B = 1 2 LI 2 = 1 2 2 LI 2 Q sin 2 ω 0 0 t = 0 2C sin2 ω 0 t U = U E + U B = Q2 + 1 LI 2 = Q 2 0 2C 2 2C Total energy is conserved!! 17
Summary: The Ideal LC Circuit
Adding Damping: RLC Circuits 19
The Real RLC Circuit: Energy Considerations i Include finite resistance: Multiply by I and after a little work: Q C + IR + L di dt = 0 d Q 2 1 2 dt 2C + 2 L I =-I I 2 R d dt TotalEnergy =-I 2 R
Damped LC Oscillations Resistor dissipates energy and system rings down over time 2 Also, frequency decreases: ω ' = ω 0 R 2L 2 21
E Experiment i t8: 8 Part 2 Undriven RLC Circuits 22
Concept Question: Expt. 8 In today s lab the battery turns on and off. Which circuit diagram is most representative of our circuit? 1. 2. 1. 1 2. 2 3. 3 4. 4 3. 4. Load lab while waiting
Concept Question Questions: Undriven Circuits it 24
Concept Question: LC Circuit The plot shows the charge on a capacitor (black curve) 0.5Q 0 and the current through it 0.0Q (red curve) after you turn 0 off the power supply. If you put a core into the inductor what will happen to the time T Lag? Capacitor Charge on 1. It will increase 2. It will decrease 3. It will stay the same 4. I don t know 1.0Q 0 1.0I 0 T Current lag Charge 0.5I 0 0.0I 0-0.5Q 0-0.5I 0-1.0Q 0-1.0I 0 0 40 80 120 Time (ms) Current throu ugh Capacito or
Concept Question: LC Circuit If you increase the 0.5Q resistance in the circuit 0 what will happen to rate 0.0Q 0 of decay of the pictured amplitudes? Capacitor Charge on 1.0Q 0 1.0I 0 T Current lag Charge 0.5I 0 0.0I 0-0.5Q 0-0.5I 0-1.0Q 0-1.0I 0 0 40 80 120 Time (ms) 1. It will increase (decay more rapidly) 2. It will decrease (decay less rapidly) 3. It will stay the same 4. I don t know Current throu ugh Capacito or 26
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