1. Copyright R. Janow – Fall 2015 Electricity and Magnetism Lecture 13 - Physics 121 (for ECE and non-ECE) 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 – Prototypical Oscillator LC Circuit - Natural Oscillations LCR Circuit - Damped Oscillations AC Circuits, Phasors, Forced Oscillations Phase Relations for Current and Voltage in Simple Resistive, Capacitive, Inductive Circuits.

2. Copyright R. Janow – Fall 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, periodic 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 – Fall 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 circuit 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 – Fall 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 derivative:

5. Copyright R. Janow – Fall 2015 Details: Oscillator Equation follows from Energy Conservation 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 – Fall 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 – Fall 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 – Fall 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 – Fall 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 – Fall 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 – Fall 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 – Fall 2015 LCR Circuits with Series Resistance: “Natural” Oscillations but with Damping Charge capacitor fully to Q 0 =C E then switch to “b” a L C E + b R Critically damped Underdamped: Q(t) Overdamped Find transient solution Modified oscillations, i.e., cosine with exponential decay (weak or under damped case) Shifted resonant frequency  ’ can be real or imaginary Modified oscillator equation has damping (decay) term: Resistance dissipates stored energy. The power is: “damped radian frequency”

13. Copyright R. Janow – Fall 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 – Fall 2015 Alternating Voltage (AC) Source 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 shows 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. “Steady State” Outputs are sinusoidal functions:   is the “phase angle” between EMF and current in the driven circuit.  You might (wrongly) expect  = 0   is positive when E max leads I max Real, instantaneous

15. Copyright R. Janow – Fall 2015 Sinusoidal AC Source driving a circuit STEADY STATE RESPONSE (after transients die away):. Current in load is sinusoidal, has same frequency  D as source... but. Current 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 an essential branch. Voltage drops across parallel branches have the same amplitudes and phases. At nodes (junctions), instantaneous currents in & out are conserved

16. Copyright R. Janow – Fall 2015 Phasors: Represent currents and potentials as vectors rotating at frequency  D in complex plane 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  Real X Imaginary Y In SERIES LCR circuit: Relate timing of voltage to current peaks 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 – Fall 2015 The meaningful quantities in AC circuits are instantaneous, peak, & average (RMS) 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 – Fall 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 – Fall 2015 AC current i(t) and voltage v R (t) in a resistor 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 – Fall 2015 AC current i(t) lags the voltage v L (t) by 90 0 in an inductor 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 – Fall 2015 AC current i(t) leads the voltage v C (t) by 90 0 in a capacitor 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 – Fall 2015 Relative Current/Voltage Phases in pure R, C, and L elements 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 currrent Same Phase Phases of voltages in series components are referenced to the current phasor

23. Copyright R. Janow – Fall 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  Simple 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 voltage phasors all 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:

24. Copyright R. Janow – Fall 2015 R ~ 0  tiny losses, no power absorbed  I m normal to E m   ~ +/-  /2 X L =X C  I m parallel to E m    Z=R  maximum current (resonance)  measures the power absorbed by the circuit: Magnitude of E m in series circuit: Reactances: Same current amplitude in each component: Z ImIm EmEm  DtDt V L -V C VRVR X L -X C R For series LRC circuit, divide each voltage in | E m | by (same) peak current peak applied voltage peak current Magnitude of Z: Applies to a single series branch with L, C, R Phase angle  : See phasor diagram Impedance magnitude is the ratio of peak EMF to peak current: Voltage addition rule for series LRC circuit