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 – prototype LC circuits oscillate forever LCR circuits oscillate with damping AC circuits oscillate at forced frequency AC circuits look simpler using “phasors” Currents and voltages have fixed relative phasing that differs for simple resistive, capacitive, inductive components in circuits. Series LCR Circuit driven by an AC voltage

Time dependent effects - RC and RL circuits with constant EMF Growth Phase Decay Phase Voltage across C related to Q - integral of current. Voltage will LAG current Voltage across L related to derivative of current. Voltage will LEAD current Now: LCR in same circuit + periodic EMF C R L Circuit driven by external AC frequency wD=2pf which could be the ‘resonant’ frequency. Resonant oscillations in LC circuit – ignore damping New descriptive quantities: Generalized resistances: reactances, impedance Damped oscillation in LCR circuit - not driven New effects:

Model: Resonant mechanical oscillations (SHM) Definition: oscillating system State ( t ) = state( t + T ) = ……. = state( t + NT ). T is the “period” -Time to complete one complete cycle f = 1/T is the frequency. w = 2pf is the “angular frequency” “State” can mean: position and velocity, electric & magnetic fields,… Mechanical Example: Simple harmonic oscillator (undamped) no friction Mechanical energy is conserved` Systems that oscillate obey equations like this... With solutions like this... Energy oscillates between 100% kinetic and 100% potential: Kmax = Umax What oscillates? position & velocity, acceleration, spring force,

Electrical Oscillations in an LC circuit, zero resistance Charge capacitor fully to Q0=CE then switch to “b” a L C E + b Kirchhoff loop equation: Substitute: An oscillator equation results solution: derivative: Peaks of current and charge are out of phase by 900 What oscillates? Charge, current, B & E fields, UB, UE Note: could have chosen sin rather than cos, with phase constant f adjusted

Electrical Oscillations in an LC circuit, zero resistance Total electrical potential energy is constant: The peak values UE0 and UB0 are equal:

Details: Derive Oscillator Equation from Energy Conservation The total energy: It is constant so: The definition and imply that: Either (no current ever flows) OR: oscillator equation Oscillator solution: To evaluate w0: plug the first and second derivatives of the solution into the differential equation. The resonant oscillation frequency w0 is:

Oscillations Forever? 13 – 1: What do you think will happen to the oscillations in an ideal LC circuit (versus a real circuit) over a long time? They will stop after one complete cycle. They will continue forever. They will continue for awhile, and then suddenly stop. They will continue for awhile, but eventually die away. There is not enough information to tell what will happen.

Potential Energy alternates between all electrostatic and all magnetic – two reversals per period t C is fully charged twice each cycle (opposite polarity) So is L Peaks are 90o out of phase

Use preceding solutions with f = 0 Example: A 4 µF capacitor is charged to E = 5.0 V, and then connected to a 0.3 Henry inductance in an LC circuit C L Use preceding solutions with f = 0 a) Find the oscillation period and frequency f = ω / 2π b) Find the maximum (peak) current (differentiate Q(t) ) c) When does current i(t) have its first maximum? When |sin(w0t)| = 1 Maxima of Q(t): All energy is in E field Maxima of i(t): All energy is in B field Maxima of current at T/4, 3T/4, … (2n+1)T/4 First One at: Others at:

Example Find the voltage across the capacitor in the circuit as a function of time. L = 30 mH, C = 100 mF. The capacitor has charge Q0 = 0.001 C. 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 = Q0, so chose phase angle f = 0 to match initial condition. Find the expression for the current in the circuit: c) How long until the capacitor charge is exactly reversed? That occurs every ½ period, given by:

Which Current is Greatest? 13 – 2: The expressions below could represent the charge on a capacitor in an LC circuit as a function of time. Which one has the greatest peak current magnitude? Q(t) = 2 sin(5t) Q(t) = 2 cos(4t) Q(t) = 2 cos(4t+p/2) Q(t) = 2 sin(2t) Q(t) = 4 cos(2t)

LC circuit oscillation frequencies 13 – 3: The three LC circuits below have identical inductors and capacitors. Order the circuits according to their oscillation frequency in ascending order. I, II, III. II, I, III. III, I, II. III, II, I. II, III, I. I. II. III.

LCR Series Circuit adds series resistance: Shifted Oscillations but with Damping Charge capacitor fully to Q0=CE then switch to “b” a L C E + b R The resistor dissipates stored energy. The power is: Find transient solution: oscillations with exponential decay (under- damped case) Modified oscillator equation has damping (decay) term: Shifted resonant frequency w’ can be real or imaginary “damped frequency” Critically damped Underdamped: Q(t) Overdamped

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 LCR circuit whose resistance is not negligible? The resonant frequency for the damped circuit is higher than for the ideal one (w’ > w0). The resonant frequency for the damped circuit is lower than for the ideal one (w’ < w0). The resistance in the circuit does not affect the resonant frequency—they are the same (w’ = w0). The damped circuit has an imaginary value of w’.

Apply Sinusoidal AC Source to circuit load the driving frequency resonant frequency STEADY STATE RESPONSE (after transients die away): . Current in load is sinusoidal, has same frequency as source ... but . Current may be retarded or advanced relative to E by “phase angle” F for the whole circuit (due to inertia L and stiffness 1/C). KIRCHHOFF RULES APPLY: . In a series branch, current has the same amplitude and phase everywhere. . Across parallel branches, voltage drops have the same amplitudes and phases . At nodes (junctions), instantaneous net currents in & out are conserved peak applied voltage peak current DEFINE: IMPEDANCE IS THE RATIO OF PEAK (or RMS) EMF TO PEAK (or RMS) CURRENT. Impedance for a circuit is a function of resistances, reactances, and how they are interconnected

represent currents and potentials as vectors rotating at frequency wD in complex plane Phasors Imaginary Y wDt F Real X Peak amplitudes  lengths of the phasors. Measurable, real, instantaneous values of i( t ) and E( t ) are projections of the phasors onto the x-axis. “phase angle”: the angle from current phasor to EMF phasor in the driven circuit. and are independent Current is the same (phase included) everywhere in a single essential branch of any circuit. Current vector is reference for series LCR circuit. Applied EMF Emax(t) leads or lags the current by a phase angle F in the range [-p/2, +p/2] Relative phases for individual series circuit elements (to be shown): VR VC VL Voltage across R is in phase with the current. Voltage across C lags the current by 900. Voltage across L leads the current by 900. XL>XC XC>XL

So how should “average” be defined? The meaningful types of quantities in AC circuits can be instantaneous, peak, or average (RMS) Instantaneous voltages and currents refer to a specific time: oscillatory, depend on time through argument “wt” possibly advanced or retarded relative to each other by phase angle represented by x-component of rotating “phasors” – see slides below Peak voltage and current amplitudes are just the coefficients out front represented by lengths of rotating phasors Simplistic time averaging of periodic quantities gives zero (and useless) results). example: integrate over a whole number of periods – one t is enough (wt=2p) Integrand is odd over a full cycle So how should “average” be defined?

Definitions of Average AC Quantities “RMS averages” are used the way instantaneous quantities were in DC circuits: “RMS” means “root, mean, squared”. Integrand on a whole cycle is positive after squaring Prescription: RMS Value = Peak value / Sqrt(2) NOTE: “Rectified average values” are useful for DC but seldom used in AC circuits: Integrate |cos(wt)| over one full cycle or cos(wt) over a positive half cycle

Current/Voltage Phases in Resistors, Capacitors, and Inductors Sinusoidal current i (t) = Imcos(wDt). Peak is Im vR(t) ~ i(t) vL(t) ~ derivative of i(t) Vc(t) ~ integral of i(t) Peak voltages across R, L, or C lead/lag current by 0, p/2, -p/2 radians Ratios of peak voltages to peak currents are called “reactances” in C’s and L’s Phases of voltages in a series branch are relative to the current phasor. VR& Im in phase Resistance currrent Same Phase VL leads Im by p/2 Inductive Reactance VC lags Im by p/2 Capacitive Reactance

Phasors applied to a Series LCR circuit Applied EMF: Current: R L C E vR vC vL Same current including phase everywhere in the single branch Refer voltage phasors to current phasor Everything oscillates at the driving frequency wD Same phase for the current in all the component, but...... Current leads or lags E (t) by a constant phase angle F Im Em F wDt+F wDt Phasors for VR, VL, & VC all rotate at wD : VR has same phase as Im VC lags Im by p/2 VL leads Im by p/2 Instantaneous voltages add normally (Loop Rule) Im Em F wDt VL VC VR along Im perpendicular to Im Phasor magnitudes add like vectors

Applies to a single series branch with L, C, R Impedance functions for a Series LCR Circuit Voltage peak addition rule in series circuit: Reactances: Same peak current flows in each component: Z Im Em F wDt VL-VC VR XL-XC R Divide each voltage in Em by (same) peak current: Magnitude of Z: Applies to a single series branch with L, C, R Phase angle F: See phasor diagram R ~ 0  Im normal to Em  F ~ +/- p/2  tiny losses, no power absorbed RESONANCE: XL=XC  Im parallel to Em  F = 0  Z = R  maximum current F measures the power absorbed by the circuit: F is zero  circuit acts purely resistively. Otherwise: F >0 (current lags EMF) or F<0 (current leads EMF)

Summary: AC Series LCR Circuit Z Im Em F wDt VL-VC VR XL-XC R sketch shows XL > XC L C E vR vC vL VL = ImXL +90º (p/2) Lags VL by 90º XL=wdL L Inductor VC = ImXC -90º (-p/2) Leads VC by 90º XC=1/wdC C Capacitor VR = ImR 0º (0 rad) In phase with VR R Resistor Amplitude Relation Phase Angle Phase of Current Resistance or Reactance Symbol Circuit Element

SUPPLEMENTARY MATERIAL

Alternating External Voltage (AC) Source Real, instantaneous current flows Apply external voltage “Steady State” sinusoidal signal has frequency wD: “Phase angle” F is between current and EMF in the driven circuit. You might (wrongly) expect F = 0 F is positive when Emax leads Imax and negative otherwise (convention) Commercial electric power (home or office) is AC, with frequency = 60 Hz in U.S, 50 Hz in most other countries. High transmission voltage  Lower i2R loss than DC, and is reduced for distribution to customers by using transformers. Sinusoidal EMF appears in rotating coils in a magnetic field (Faraday’s Law) Slip rings and brushes take EMF from the coil and produce sinusoidal AC voltage and current. External circuits are driven by sinusoidal EMF Current in external circuit may lead or lag EMF according to character of the load connected.

From voltage drop across R: AC current i(t) and voltage vR(t) in a resistor are in phase Extra Discussion From Kirchhoff loop rule: Time dependent current: Applied EMF: vR( t ) i(t) From voltage drop across R: The phases of vR(t) and i(t) coincide Peak current and peak voltage are in phase across a resistance, rotating at the driving frequency wD The ratio of the AMPLITUDES (peaks) VR to Im is the resistance:

AC current i(t) lags the voltage vL(t) by 900 in an inductor From Kirchhoff loop rule: Time dependent current: Applied EMF: vL(t) L i(t) Sine from derivative From voltage drop across L (Faraday Law): Note: so...phase angle F = + p/2 for inductor Voltage phasor across an inductor leads current by +p/2 (F positive) The RATIO of the peaks (AMPLITUDES) VL to Im is the inductive reactance XL: Definition: inductive reactance Inductive Reactance limiting cases w  0: Zero reactance. Inductor acts like a wire. w  infinity: Infinite reactance. Inductor acts like a broken wire. Extra Discussion

AC current i(t) leads the voltage vC(t) by 900 in a capacitor From Kirchhoff loop rule: Time dependent current: Applied EMF: i vC ( t ) C From voltage drop across C (proportional to Q(t)): Sine from integral Note: so...phase angle F = - p/2 for capacitor Voltage phasor across capacitor lags the current by p/2 (F negative) Extra Discussion The RATIO of the AMPLITUDES VC to Im is the capacitive reactance XC: Capacitive Reactance limiting cases w  0: Infinite reactance. DC blocked. C acts like broken wire. w  infinity: Reactance is zero. Capacitor acts like a simple wire Definition: capacitive reactance