1. Copyright R. Janow – Spring 2015 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

2. Copyright R. Janow – Spring 2015 LC and RC circuits with constant EMF - Time dependent effects R C E R L E Growth Phase Decay Phase Now LCR in same circuit, time varying EMF –> New effects C R L R L C External AC drives circuit at frequency  D =2  f which may or may not be at resonance Resonant oscillations in LC circuit Generalized resistances: reactances, impedance Damped oscillation in LCR circuit Voltage across C related to integral of current Voltage across L related to derivative of current

3. Copyright R. Janow – Spring 2015 Recall: Resonant mechanical oscillations Definition of an oscillating system: Periodic, repetitive behavior System state ( t ) = state( t + T ) = …= state( t + NT ) T = period = time to complete one complete cycle State can mean: position and velocity, electric and magnetic fields,… Can convert mechanical to LC equations by substituting: Energy oscillates between 100% kinetic and 100% potential: K max = U max With solutions like this... Systems that oscillate obey equations like this... What oscillates for a spring in SHM? position & velocity, spring force, Mechanical example: Spring oscillator (simple harmonic motion) no friction OR

4. Copyright R. Janow – Spring 2015 Electrical Oscillations in an LC circuit, zero resistance a L C E + b Charge capacitor fully to Q 0 =C E then switch to “b” Kirchoff loop equation: Substitute: Peaks of current and charge are out of phase by 90 0 An oscillator equation where solution: What oscillates? Charge, current, B & E fields, U B, U E Show: Total potential energy is constant is constant Peak Values Are Equal

5. Copyright R. Janow – Spring 2015 Details: Energy conservation used to deduce oscillations The total energy: It is constant so: The definition and imply that: Oscillator solution: To evaluate w 0 : plug the first and second derivatives of the solution into the differential equation. The resonant oscillation frequency w 0 is: Either (no current ever flows) OR: oscillator equation

6. Copyright R. Janow – Spring 2015 13 – 1: What do you think will happen to the oscillations in an ideal 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. Oscillations Forever?

7. Copyright R. Janow – Spring 2015 Potential Energy alternates between all electrostatic and all magnetic – two reversals per period  C is fully charged twice each cycle (opposite polarity)

8. Copyright R. Janow – Spring 2015 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 Use preceding solutions with  = 0 C L b) Find the maximum (peak) current (differentiate Q(t) ) a) Find the oscillation period and frequency f = ω / 2π c) When does the first current maximum occur? When |sin(  0 t)| = 1 Maxima of Q(t): All energy is in E field Maxima of i(t): All energy is in B field Current maxima at T/4, 3T/4, … (2n+1)T/4 First One at: Others at:

9. Copyright R. Janow – Spring 2015 Example a)Find the voltage across the capacitor in the circuit as a function of time. L = 30 mH, C = 100  F The capacitor is charged to Q 0 = 0.001 Coul. at time t = 0. The resonant frequency is: The voltage across the capacitor has the same time dependence as the charge: At time t = 0, Q = Q 0, so choose phase angle  = 0. b)What is the expression for the current in the circuit? The current is: c) How long until the capacitor charge is exactly reversed? That occurs every ½ period, given by:

10. Copyright R. Janow – Spring 2015 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+  /2) D. Q(t) = 2 sin(2t) E. Q(t) = 4 cos(2t) Which Current is Greatest?

11. Copyright R. Janow – Spring 2015 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. LC circuit oscillation frequencies I. II. III.

12. Copyright R. Janow – Spring 2015 LCR circuits with series resistance: Oscillations but with damping Charge capacitor fully to Q 0 =C E then switch to “b” Stored energy decays with time due to the resistance a L C E + b R Critically damped Underdamped: Q(t) Overdamped Solution is cosine with exponential decay (weak or under-damped case) Shifted resonant frequency  ’ can be real or imaginary Resistance dissipates stored energy. The power is: Oscillator equation results, but with damping (decay) term

13. Copyright R. Janow – Spring 2015 13 – 4: How does the resonant frequency   for an ideal LC circuit (no resistance) compare with  ’ 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 (  ’ >   ). B. The resonant frequency for the damped circuit is lower than for the ideal one (  ’ <   ). C. The resistance in the circuit does not affect the resonant frequency — they are the same (  ’ =   ). D. The damped circuit has an imaginary value of w’. Resonant frequency with damping

14. Copyright R. Janow – Spring 2015 Alternating Current (AC) EMF Commercial electric power (home or office) is AC, not DC. AC frequency is 60 Hz in U.S, 50 Hz in most other countries AC transmission voltage can be changed for distribution by using transformers (lower). High transmission voltage  Lower i 2 R loss than DC Sketch: a crude AC generator. Generators convert mechanical energy to electrical energy. Mechanical power can come from a water or steam turbine, windmill, diesel or turbojet engine. EMF appears in rotating coils in a magnetic field (Faraday’s Law) Slip rings and brushes take EMF from the coil without twisting the wires. Outputs are sinusoidal functions:   is the phase angle between current and EMF.  You might (wrongly) expect  = 0

15. Copyright R. Janow – Spring 2015 External AC EMF driving a circuit External, instantaneous EMF acts on load: Current in load has same frequency  D... but may be retarded or advanced (relative to E ) by “phase angle”  for the whole circuit  ( due to inertia L and stiffness 1/C ). load the driving frequency of EMF resonant frequency  0, in general USE KIRCHHOFF LOOP & JUNCTION RULES, INSTANTANEOUS QUANTITIES R L C In a Series LCR Circuit: Everything oscillates at driving frequency  D  is zero at resonance – circuit acts purely resistively. At “resonance ”: Otherwise  is positive (current lags applied EMF) or negative (current leads applied EMF) Current is the same everywhere (including phase) Current has the same amplitude and phase everywhere in a branch Voltage drops have the same amplitude and phase across parallel branches At junctions, instantaneous currents in & out are conserved

16. Copyright R. Janow – Spring 2015 Phasor Picture: Show current and potentials as vectors all rotating at frequency  D The lengths of the vectors are the peak amplitudes. The measured instantaneous values of i( t ) and E ( t ) are projections of the phasors on the x-axis. “phase angle” measured when peaks pass and are independent Current is the same (phase included) everywhere in a single (series) branch of any circuit Use rotating current vector as reference. EMF E (t) applied to the circuit can lead or lag the current by a phase angle  in the range [-  /2, +  /2] DtDt  x y In SERIES LCR circuit: Relate voltage peaks to phase of the current VRVR VCVC VLVL Voltage across R is in phase with the current. Voltage across C lags the current by 90 0. Voltage across L leads the current by 90 0. X L >X C X C >X L

17. Copyright R. Janow – Spring 2015 The meaningful quantities in AC circuits may be instantaneous, peak, or average Instantaneous voltages and currents: oscillatory, depend on time through argument “  t” possibly advanced or retarded relative to each other by phase angles represented by rotating “phasors” – see slides below Peak voltage and current amplitudes are just the coefficients out front Simple time averages of periodic quantities are zero (and useless). Example: Integrate over a whole number of periods – one  is enough ( w  =2  ) Integrand is odd on a full cycle So how should “average” be defined?

18. Copyright R. Janow – Spring 2015 Averaging Definitions for AC circuits “RMS” averages are used the way instantaneous quantities were in DC circuits: “RMS” means “root, mean, squared”. Integrand is positive on a whole cycle after squaring So-called “rectified average values” are seldom used in AC circuits: Integrate |cos(  t)| over one full cycle or cos(wt) over a positive half cycle Prescription: RMS Value = Peak value / Sqrt(2) NOTE:

19. Copyright R. Janow – Spring 2015 For a Resistor only: AC current i(t) and voltage v R (t) are in phase v R ( t ) i(t) Peak current and peak voltage are in phase in a purely resistive part of a circuit, rotating at the driving frequency  D Voltage drop across R: The ratio of the AMPLITUDES (peaks) V R to I is the resistance: The phases of v R (t) and i(t) coincide Kirchoff loop rule: Current has time dependence: Applied EMF:

20. Copyright R. Janow – Spring 2015 For an Inductor only: AC current i(t) lags the voltage v L (t) by 90 0 Voltage drop across L is due to Faraday Law: v L (t) L i(t) Note: The RATIO of the peaks (AMPLITUDES) V L to I is the inductive reactance X L : Definition: inductive reactance so...phase angle  for inductor Voltage phasor leads current by +  /2 in inductive part of a circuit (F positive) Inductive Reactance limiting cases   0: Zero reactance. Inductor acts like a wire.   infinity: Infinite reactance. Inductor acts like a broken wire. Kirchoff loop rule: Current has time dependence: Applied EMF: Sine from derivative

21. Copyright R. Janow – Spring 2015 For a Capacitor only: AC current i(t) leads the voltage v C (t) by 90 0 Note: The RATIO of the AMPLITUDES V C to I is the capacitive reactance X C : Definition: capacitive reactance so...phase angle  for capacitor Voltage phasor lags the current by  /2 in a pure capacitive circuit branch (F negative) i v C ( t ) C Capacitive Reactance limiting cases   0: Infinite reactance. DC blocked. C acts like broken wire.   infinity: Reactance is zero. Capacitor acts like a simple wire Kirchoff loop rule: Current has time dependence: Applied EMF: Voltage drop across C is proportional to q(t): Sine from integral

22. Copyright R. Janow – Spring 2015 Current & voltage phases in pure R, C, and L circuits Apply sinusoidal current i (t) = I m cos(  D t) For pure R, L, or C loads, phase angles for voltage drops are 0,  /2, -  /2 “Reactance” means ratio of peak voltage to peak current (generalized resistances). V R & I m in phase Resistance V L leads I m by  /2 Inductive Reactance V C lags I m by  /2 Capacitive Reactance Phases of voltages in series components are referenced to the current phasor currrent Same Phase

23. Copyright R. Janow – Spring 2015 Same frequency dependance as E (t) Same phase for the current in E, R, L, & C, but...... Current leads or lags E (t) by a constant phase angle  Phasor Model for a Series LCR circuit, AC voltage R L C E vRvR vCvC vLvL Apply EMF: ImIm EmEm  DtDt VLVL VCVC VRVR Show component voltage phasors rotating at  D with phases relative to the current phasor I m and magnitudes below : V R has same phase as I m V C lags I m by  /2 V L leads I m by  /2 along I m perpendicular to I m ImIm EmEm   D t+  DtDt Phasors all rotate CCW at frequency  D Lengths of phasors are the peak values (amplitudes) The “x” components are instantaneous values measured Kirchoff Loop rule for series LRC: The current i(t) is the same everywhere in the single branch in the circuit: