Physics 121 Fall 2007

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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 et /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 106 ) 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 (3102H)(104F) 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.

103C V (t)  cos(577t)  10cos(577t) volts C 104F

b) What is the expression for the current in the circuit? The current is: dQ i   Q w sin(w t)  (103 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

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’.

2 2 1/2 ω'   ω0  (R / 2 L) 

• AC is easier to transmit than DC • AC transmission voltage can be changed by using a transformer. • Commercial electric power (home or office) is AC, not DC. • The U.S., the AC frequency is 60 Hz. Most other countries use 50 Hz.

• Sketch: a crude AC generator. • EMF appears in a rotating a coil of wire in a magnetic field (Faraday’s Law) • Slip rings and brushes allow the EMF to be taken off the coil without twisting the wires. • Generators convert mechanical energy to electrical energy. Power to rotate the coil can come from a water or steam turbine, windmill, or turbojet engine.

Represent outputs as ind(t)  m sinwdt sinusoidal functions: i  i0 sin(wdt  )

External, sinusoidal, instantaneous EMF applied to load: I(t) E(t )  Emax sin(ωDt ) Emax  amplitude E(t) The potential across the load V ( t )  E(t) load load

ωD  the driving frequency

ωD  resonant frequency w0, in general

Current in load flows with same frequency wD ...but may be retarded or advanced (relative to E) by “phase angle” F (due to inertia L and stiffness 1/C) Current has the same amplitude and phase everywhere in a branch

I( t )  I max sin ( ωDt  Φ ) Example: Series LCR Circuit R Everything oscillates at driving frequency wD

At “resonance”: ωD  ω0  1/LC E C F is zero at resonance – circuit acts purely resistively.

Otherwise F is + or – (current leads or lags applied EMF) Copyright R. Janow – Fall 2013 L Instantaneous, peak, average, and RMS quantities for AC circuits i( t )  i sin ( ω t  Φ ) E(t )  Emax sin(ωDt ) max D Instantaneous voltages and currents: • depend on time through argument: wt • periodic, repetitive, oscillatory • possibly advanced or retarded relative to each other by phase angles • represented by rotating “phasors” – see below Peak voltage and current amplitudes are just the coefficients out front i Emax max Simple time averages of periodic quantities are zero (and useless). • Example: Integrate over a whole number of periods – one  is enough (wt=2p) 1  1   Eav   E(t )dt  Emax  sin(ωDt )dt  0 sin(wt  F) dt  0  0  0 0 1  1  Integrand is odd i  i(t )dt  i sin(ω t  F )dt  0 av  0 max  0 D “RMS” averages are used the way instantaneous quantities were in DC circuits • “RMS” means “root, mean, squared”. 1/ 2 1/ 2  2   1  2  Emax Erms  E  E (t )dt   av   0  1  2    2  sin (wt  F) dt  1/ 2 1/ 2 1/ 2  0  2   1  2  imax Integrand is even irms  i (t )   i (t )dt    av   Copyright 0 R. Janow – Fall2 2013 Phasor Picture: Show current and potentials as y vectors rotating at frequency wD

Imax • The measured instantaneous values of i( t ) and E( t ) F are projections of the phasors on the y-axis. Emax • The lengths of the vectors are the peak amplitudes. Φ  “phase angle” measures when peaks pass wDt- F Φ and w t are independent x D • Current is the same (phase included) everywhere E(t )  E sin(ω t ) max D in a single branch of any circuit. I( t )  i max sin ( ωDt  Φ ) • EMF E(t) applied to the circuit can lead or lag the current by a phase angle F in the range [-p/2, +p/2]. Series LCR circuit: Relate internal voltage drops to phase of the current

• Voltage across R is in phase with the current. • Voltage across C lags the current by 900. 0 • Voltage across L leads the current by 90Copyright. R. Janow – Fall 2013 AC circuit, resistance only  current and voltage in phase

Kirchoff loop rule: ε - VR(t )  0 i VR ( t ) V (t )  sin(ω t ) ε R ε(t)  εM D

Current:iR (t )  iRmax sin(ωDt - Φ)

Voltage drop across R: VR (t)  iR (t) R (definition of resistance)

Em  Φ  0 i  R V R VR (t ) R

I I (t ) R Peak current and peak voltage are in phase in a purely ω t resistive part of a circuit, rotating at the driving frequency wD D

1 ε  10V, f  60 Hz applied to a 20 Ω Resistor. Find the current i( t ) at t  s Example M 240

εM wD = 2pf = 120p  = 1/60 s. t = /4 ε Em i(t)  iRmax sin(ωDt )  sin(120p . 1/240) 1 R t  240 10 π 1 i(t )  sin  A t 20 2 2 1 1 τ   Copyright R. Janow – Fall 2013 f 60 Capacitance-only AC circuit: current leads voltage by p/2

Kirchoff loop rule: ε(t)  VC (t)  0 i Vc(t )  VCmax sin(ωDt ) ; VCmax  Em ε VC ( t ) C Charge: Q(t )  C VC(t )  C VCmax sin(ωDt ); dQ(t ) Current: i (t )   ω C V cos(ω t )  i sin(w t  F) C dt D Cmax D Cmax D π Note: cos( ω t )  sin(ω t  ) so...phase angle F   p/2 for capacitor D D 2

VCmax  iCmax C ( recall VR  iR R ) Definition: 1 capacitive   (Ohms ) Reactances are ratios of peak values C reactance ωDC

Current leads the voltage by p/2 in a pure VCmax π  i C(t )  sin(ωDt  ) capacitive part of a circuit (F negative) C 2 Phasor Picture Capacitive Reactance limiting cases V (t ) C • w  0: Infinite reactance. DC blocked. C acts like VCmax iC(t ) broken wire. i Cmax • w  infinity: Reactance is ωD t zero. Capacitor acts like a   900 simple wire Copyright R. Janow – Fall 2013 Inductance-only AC circuit: current lags voltage by p/2

Kirchoff loop rule: E(t)  VL(t )  0 i V (t )   E sin(ω t ) ; E  V L V L m D m Lmax ε L d i d i V (t) Faraday Law: V (t )  L L  L   L L dt dt L V V L max L max Current: i (t )   di   sin(ω t) dt   cos(ω t)  i max sin(w t  F) L L L D ω L D L D π D Note: cos(ωDt )   sin(ωDt  ) so...phase angle F   p/2 for inductor 2 Definition: VLmax  iLmax L inductive L  ωDL (Ohms) Reactances are ratios of peak values reactance

V Current lags the voltage by p/2 in a pure L max π iLmax(t )  sin(ωDt  ) inductive part of a circuit (F positive) L 2

Phasor Picture Inductive Reactance

V limiting cases VL (t ) Lmax • w  0: Zero reactance. ω t Inductance acts like a wire.   900 • w  infinity: Infinite reactance. Inductance acts like a broken iL(t ) i Lmax wire. Copyright R. Janow – Fall 2013 Current & voltage phases in pure R, C, and L circuits Current is the same everywhere in a single branch (including phase) Phases of voltages in elements are referenced to the current phasor

• Apply sinusoidal voltage E (t) = Emsin(wDt) • For pure R, L, or C loads, phase angles are 0, p/2, -p/2 • Reactance” means ratio of peak voltage to peak current (generalized resistances).

VR& iR in phase VC lags iC by p/2 VL leads iL by p/2 Resistance Capacitive Reactance Inductive Reactance 1 V /i  R Vmax /iC  C  Vmax /iL  L  wDL max R wDC

Copyright R. Janow – Fall 2013 Series LCR circuit driven by an external AC voltage Apply EMF: VR E(t)  Emax sin(wDt) wD is the driving frequency E R L V The current i(t) is the same everywhere in the circuit L i(t)  i sin(w t  F) C max D • Same frequency dependance as E (t) but... • Current leads or lags E (t) by a constant phase angle F VC • Same phase for the current in E, R, L, & C

Phasors all rotate CCW at frequency wD F • Lengths of phasors are the peak values (amplitudes) E im m • The “y” components are the measured values. t wD wDt-F Plot voltages in components with phases

relative to current phasor im :

• VR has same phase as im VR  imR • V lags i by p/2 C m VC  imXC V L • VL leads im by p/2 Em VL  imXL im F Kirchoff Loop rule for potentials (measured along y)     E(t)  V (t)  V (t)  V (t)  0 wDt-F R L C VR       0 Em  VR  (VL  VC) VC lags VL by 180 V C along im perpendicularCopyright R. Janow to i –m Fall 2013 Summary: Lecture 13/14 Chapter 31 - LCR & AC Circuits, Oscillations