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Physics 121 Fall 2007

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Physics 121 Fall 2007 Electricity and Magnetism Lecture 13 - Physics 121 Electromagnetic Oscillations in LC & LCR Circuits, Y&F Chapter 30, Sec. 5 - 6 Alternating Current Circuits, Y&F Chapter 31, Sec. 1 - 2 • Summary: RC and LC circuits • Mechanical Harmonic Oscillator • LC Circuit Oscillations • Damped Oscillations in an LCR Circuit • AC Circuits, Phasors, Forced Oscillations • Phase Relations for Current and Voltage in Simple Resistive, Capacitive, Inductive Circuits. • Summary Copyright R. Janow – Fall 2013 Recap: LC and RC circuits with constant EMF - – Time dependent effects R R E Q(t) E di Vc(t) EL(t) L C C L dt C RC capacitive time constant L L/R inductive time constant Growth Phase E t /RC t / L Q(t) Q 1 e Q CE i(t) iinf 1 e iinf inf inf R Decay Phase t / E Q(t) Q et /RC Q CE i(t) i e L i 0 0 0 0 R Now LCR in same circuit, time varying EMF –> New effects R • Generalized Resistances: Reactances, Impedance R C 1/(w C) L wL [ Ohms ] 1/ 2 L 2 2 Z R L C C • New behavior: Resonant Oscillation in LC Circuit 1/ 2 wres (LC) R • New behavior: Damped Oscillation in LCR Circuit L C • External AC can drive circuit, frequency wD=2pf Copyright R. Janow – Fall 2013 Recall: Resonant mechanical oscillations Definition of • Periodic, repetitive behavior an oscillating • System state ( t ) = state( t + T ) = …= state( t + NT ) • T = period = time to complete one complete cycle system: • State can mean: position and velocity, electric and magnetic fields,… Mechanical example: Spring oscillator (simple harmonic motion) Hooke's Law restoring force : F kx ma OR 1 2 1 2 Emech constant Kblock Usp mv kx 2 2 2 dEmech d x dx dx dx no friction 0 m kx v 0 dt dt 2 dt dt dt Systems that oscillate d2x w2x w k/m obey equations like this... dt 2 0 0 With solutions like this... x(t) x0 cos(w0t ) What oscillates for a spring in SHM? position & velocity, spring force, Energy oscillates between 100% kinetic and 100% potential: Kmax = Umax Substitutions can convert mechanical to LC equations: 2 2 x Q v i m L k 1/C w0 k /m w0 1/LC Q2 U 1 kx2 U 1 K 1 mv2 U 1 Li2 el 2 C 2 C block 2 CopyrightL 2 R. Janow – Fall 2013 Electrical Oscillations in an LC circuit, zero resistance a Charge capacitor fully to Q =C then switch to “b” 0 E + b Kirchoff loop equation: VC - EL 0 2 L Substitute: Q di d Q E C VC and EL L L C dt dt 2 2 d Q(t) 1 An oscillator Q(t) 1/LC w0 dt 2 LC equation where solution: Q(t) Q0 cos(w0t ) Q0 CE dQ(t) i(t) i sin(w t ) i w Q dt 0 0 0 0 0 1 What oscillates? Charge, current, B & E fields, U , U w2 B E 0 Peaks of current and charge are out of phase by 900 c L where c RC, L L /R Show: Total potential energy is constant Utot UE(t) UB(t) 2 2 2 Li2 Q (t) Q0 2 Li (t) 0 2 U cos (w t ) U sin (w t ) E 2C 2C 0 B 2 2 0 Li2 L 2Q2 LQ2 Q2 Peak Values 0 w0 0 0 1 0 UB UE Are Equal 0 2 2 2 LC 2C 0 UE UB U 0 0 tot is constant Copyright R. Janow – Fall 2013 Details: use energy conservation to deduce oscillations 1 Q(t)2 • The total energy: U U U Li(t)2 B E 2 2C dU di Q dQ • It is constant so: 0 Li dt dt C dt dQ di d2Q dQ d2Q Q • The definitions i and imply that: (L ) 0 dt dt dt 2 dt dt 2 C 2 dQ d Q Q oscillator • Either 0 (no current ever flows) or: 0 dt dt 2 LC equation • Oscillator solution: Q Q0 cos(w0t ) • To evaluate w0: plug the first and second derivatives of the solution into the differential equation. dQ 2 Q0w0 sin(w0t ) d Q Q 1 dt Q cos(w t ) w2 0 0 0 0 2 dt 2 LC LC d Q 2 Q0w0 cos(w0t ) dt 2 1 • The resonant oscillation frequency w0 is: w0 LCCopyright R. Janow – Fall 2013 Oscillations Forever? 13 – 1: What do you think will happen to the oscillations in a true LC circuit (versus a real circuit) over a long time? A.They will stop after one complete cycle. B.They will continue forever. C.They will continue for awhile, and then suddenly stop. D.They will continue for awhile, but eventually die away. E.There is not enough information to tell what will happen. Copyright R. Janow – Fall 2013 Potential Energy alternates between all electrostatic and all magnetic – two reversals per period C is fully charged twice each cycle (opposite polarity) Copyright R. Janow – Fall 2013 Example: A 4 µF capacitor is charged to E = 5.0 V, and then discharged through a 0.3 Henry inductance in an LC circuit C L Use preceding solutions with = 0 a) Find the oscillation period and frequency f = ω / 2π 1 1 ω0 LC (0.3)( 4 x 106 ) f ω0 2π 145 Hz 1 2π 913 rad/s T Period f 6.9 ms w0 b) Find the maximum (peak) current (differentiate Q(t) ) dQ d Q(t) Q0 cos(w0t) i(t) Q cos(w t) - Q w sin (w t) dt 0 dt 0 0 0 0 0 Q0 CE 6 imax Q0w0 CEw0 4x10 x 5 x 913 18 mA c) When does the first current maximum occur? When |sin(w0t)| = 1 First One: Maxima of Q(t): All energy is in E field 1 T Maxima of i(t): All energy is in B field T f 6.9 ms 4 1.7 ms Current maxima at T/4, 3T/4, … (2n+1)T/4 Others at: 3.4 ms increments Copyright R. Janow – Fall 2013 Example a) Find the voltage across the capacitor in the circuit as a function of time. L = 30 mH, C = 100 mF The capacitor is charged to Q0 = 0.001 Coul. at time t = 0. The resonant frequency is: 1 1 w0 577.4 rad/s LC (3102H)(104F) The voltage across the capacitor has the same time dependence as the charge: Q(t) Q cos(w t ) V (t) 0 0 C C C At time t = 0, Q = Q0, so choose phase angle = 0. 103C V (t) cos(577t) 10cos(577t) volts C 104F b) What is the expression for the current in the circuit? The current is: dQ i Q w sin(w t) (103 C)(577 rad/s)sin(577t) 0.577sin(577t) amps dt 0 0 0 c) How long until the capacitor charge is reversed? That happens every ½ period, given by: T p 5.44 ms Copyright R. Janow – Fall 2013 2 w0 Which Current is Greatest? 13 – 2: The expressions below could be used to represent the charge on a capacitor in an LC circuit. Which one has the greatest maximum current magnitude? A. Q(t) = 2 sin(5t) B. Q(t) = 2 cos(4t) C. Q(t) = 2 cos(4t+p/2) D. Q(t) = 2 sin(2t) E. Q(t) = 4 cos(2t) dQ(t) Q(t) Q0 cos(w0t ) i dt Copyright R. Janow – Fall 2013 Time needed to discharge the capacitor in LC circuit 13 – 3: The three LC circuits below have identical inductors and capacitors. Order the circuits according to their oscillation frequency in ascending order. A. I, II, III. B. II, I, III. C. III, I, II. D. III, II, I. E. II, III, I. I. II. III. 2p 1 1 w0 1/LC Cpara Ci T Cser Ci Copyright R. Janow – Fall 2013 LCR circuits: Add series resistance Circuits still oscillate but oscillation is damped Charge capacitor fully to Q =CE then switch to “b” 0 a R Stored energy decays with time due to resistance 2 2 + Q (t) Li (t) b U U (t) U (t) tot E B 2C 2 E L C Resistance dissipates stored dU tot i2(t)R energy. The power is: dt 2 Oscillator equation results, but d Q(t) R dQ 2 w0Q(t) 0 w0 1/LC with damping (decay) term dt 2 L dt Solution is cosine with Rt/ 2L exponential decay (weak Q(t) Q0e cos(w't ) Q0 CE or under-damped case) Shifted resonant frequency w’ ω' ω2 (R / 2L)2 1/2 can be real or imaginary 0 Underdamped: Critically damped Overdamped 2 R 2 ' 2 R 2 ' 2 R 2 ω0 ( ) ω 0 ω0 ( ) ω is imaginary ω0 ( ) 2L 2L 2L Q(t) complicated - ω t e 0 exponential ω' t t 2π Copyright R. Janow – Fall 2013 Resonant frequency with damping 13 – 4: How does the resonant frequency w0 for an ideal LC circuit (no resistance) compare with w’ for an under-damped circuit whose resistance cannot be ignored? A. The resonant frequency for the non-ideal, damped circuit is higher than for the ideal one (w’ > w0). B. The resonant frequency for the damped circuit is lower than for the ideal one (w’ < w0). C. The resistance in the circuit does not affect the resonant frequency— they are the same (w’ = w0). D. The damped circuit has an imaginary value of w’.
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