Physics 121 - Electricity and Magnetism Lecture 14 - AC Circuits, Resonance Y&F Chapter 31, Sec. 1 - 8 AC Circuits, Phasors, Forced Oscillations Phase Relations for Current and Voltage in Simple Resistive, Capacitive, Inductive Circuits Phasor Diagrams for Voltage and Current The Series RLC Circuit. Amplitude and Phase Relations Impedance and Phasors for Impedance Resonance Power in AC Circuits, Power Factor Transformers

Sinusoidal AC Source driving a circuit USE KIRCHHOFF LOOP & JUNCTION RULES, INSTANTANEOUS QUANTITIES load the driving frequency of EMF resonant frequency w0, in general STEADY STATE RESPONSE (after transients die away): . Current in load is sinusoidal, has same frequency wD 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). Application of KIRCHHOFF RULES: . 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 R L C In a Series LCR Circuit: Everything oscillates at driving frequency wD F is zero at resonance  circuit acts purely resistively. At “resonance”: Otherwise F can be positive (current lags applied EMF) or negative (current leads applied EMF) Current is the same everywhere (including phase)

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 “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 “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 Prescription: RMS Value = Peak value / Sqrt(2) NOTE:

Current/Voltage Phasing in pure R, C, and L circuit elements Sinusoidal current i (t) = Imcos(wDt). Peak is Im vR(t) ~ i(t) vL(t) ~ di(t)/dt Peak voltage drops across R, L, or C lead/lag current by 0, p/2, -p/2 radians Reactances (generalized resistances) are ratios of peak voltages to peak currents Phases of voltages in a series branch are referenced to the current phasor VC lags Im by p/2 Capacitive Reactance VL leads Im by p/2 Inductive Reactance VR& Im in phase Resistance currrent Same Phase

From voltage drop across R: AC current i(t) and voltage vR(t) in a resistor are in phase 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 leads current by +p/2 in inductive part of a circuit (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.

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 lags the current by p/2 in a pure capacitive circuit branch (F negative) The RATIO of the AMPLITUDES VC to Im is the capacitive reactance XC: Definition: capacitive reactance 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

Phasors applied to a Series LCR circuit Applied EMF: R L C E vR vC vL Current: Recall: Phasors rotate CCW at frequency wD Lengths of phasors are the peak values (amplitudes) Instantaneous values are the “x” components Refer voltage phasors to current phasor 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 F Same current everywhere in the single branch Im Em F wDt+F wDt Apply Loop Rule to instantaneous voltages: Im Em F wDt VL VC VR Voltage phasors for VR, VL, & VC all rotate at wD : VC lags Im by p/2 VR has same phase as Im VL leads Im by p/2 along Im perpendicular to Im Voltage phasor magnitudes add like vectors

Applies to a single series branch with L, C, R Voltage addition rule for series LRC circuit Magnitude of Em in series circuit: Z Im Em F wDt VL-VC VR XL-XC R Reactances: Same current amplitude in each component: peak applied voltage peak current Define: Impedance is the ratio of peak EMF to peak current. 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  tiny losses, no power absorbed  Im normal to Em  F ~ +/- p/2 XL=XC  Im parallel to Em  F = 0  Z=R  maximum current (resonance) F measures peak power absorbed by the circuit:

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

Example 1: Analyzing a series RLC circuit A series RLC circuit has R = 425 Ω, L = 1.25 H, C = 3.50 μF. It is connected to an AC source with f = 60.0 Hz and εm= 150 V. Determine the impedance of the circuit. Find the amplitude of the current (peak value). Find the phase angle between the current and voltage. Find the instantaneous current across the RLC circuit. Find the peak and instantaneous voltages across each circuit element.

Example 1: Analyzing a Series RLC circuit A series RLC circuit has R = 425 Ω, L = 1.25 H, C = 3.50 μF. It is connected to an AC source with f = 60.0 Hz and εm=150 V. Determine the impedance of the circuit. Angular frequency: Resistance: Inductive reactance: Capacitive reactance: (B) Find the peak current amplitude: Current phasor Im leads the Voltage Em Phase angle will be negative XC > XL (Capacitive) Find the phase angle between the current and voltage.

Example 1: Analyzing a series RLC circuit - continued A series RLC circuit has R = 425 Ω, L = 1.25 H, C = 3.50 μF. It is connected to an AC source with f = 60.0 Hz and εm=150 V. Find the instantaneous current in the RLC circuit. (E) Find the peak and instantaneous voltages in each circuit element. VR in phase with Im VR leads Em by |F| = 0 VL leads VR by p/2 VC lags VR by p/2 Add voltages above: What’s wrong? Voltages add with proper phases:

Why should fD make a difference? Example 2: Resonance in a series LCR Circuit: R = 3000 W L = 0.33 H C = 0.10 mF Em = 100 V. Find |Z| and F for fD = 200 Hertz, fD = 876 Hz, & fD = 2000 Hz R L C E vR vC vL Why should fD make a difference? 200 Hz Capacitive Em lags Im - 68.3º 8118 W 415 W 7957 W 3000 W Frequency f Circuit Behavior Phase Angle F Impedance |Z| Reactance XL Reactance XC Resistance R 876 Hz Resistive Max current 0º Resonance 1817 W 2000 Hz Inductive Em leads Im +48.0º 4498 W 4147 W 796 W Im Em F < 0 F=0 F > 0

Resonance in a series LCR circuit Vary wD: At resonance maximum current, minimum impedance inductance dominates current lags voltage capacitance dominates current leads voltage width of resonance (selectivity, “Q”) depends on R. Large R  less selectivity, smaller current at peak damped spring oscillator near resonance

Power in AC Circuits Resistors always dissipate power, but the instantaneous rate varies as i2(t)R No power is lost in pure capacitors and pure inductors in an AC circuit Capacitor stores energy during two 1/4 cycle segments. During two other segments energy is returned to the circuit Inductor stores energy when it produces opposition to current growth during two ¼ cycle segments (the source does work). When the current in the circuit begins to decrease, the energy is returned to the circuit

AC Power Dissipation in a Resistor Instantaneous power Power is dissipated in R, not in L or C cos2(x) is always positive, so Pinst is always positive. But, it is not constant. Power pattern repeats every p radians (t/2) The RMS power is an AC equivalent to DC power Integrate Pinst in resistor over t: Integral = 1/2 RMS means “Root Mean Square” Square a quantity (positive) Average over a whole cycle Compute square root. COMPUTING RMS QUANTITIES: For any RMS quantity divide peak value such as Im or Em by sqrt(2) For any R, L, or C Household power example: 120 volts RMS  170 volts peak

Power factor for an AC LCR Circuit The PHASE ANGLE F determines the average RMS power actually absorbed due to the RMS current and applied voltage in the circuit. Claim (proven below): |Z| Irms Erms F wDt XL-XC R Proof: Start with instantaneous power (not very useful): Change variables: Average it over one full period t: Use trig identity:

Power factor for AC Circuits - continued Odd integrand Even integrand Recall: RMS values = Peak values divided by sqrt(2) Alternate form: If R=0 (pure LC circuit) F  +/- p/2 and Pav = Prms = 0 Also note:

Example 2 continued with RMS quantities: R = 3000 W L = 0.33 H C = 0.10 mF Em = 100 V. R L C E VR VC VL fD = 200 Hz Find Erms: Find Irms at 200 Hz: Find the power factor: Find the phase angle F directly: Find the average power: or Recall: do not use arc-cos to find F

Example 3 – Use RMS values inSeries LCR circuit analysis A 240 V (RMS), 60 Hz voltage source is applied to a series LCR circuit consisting of a 50-ohm resistor, a 0.5 H inductor and a 20 mF capacitor. Find the capacitive reactance of the circuit: Find the inductive reactance of the circuit: The impedance of the circuit is: The phase angle for the circuit is: The RMS current in the circuit is: The average power consumed in this circuit is: If the inductance could be changed to maximize the current through the circuit, what would the new inductance L’ be? How much RMS current would flow in that case? F is positive since XL>XC (inductive)

Transformers power transformer Devices used to change AC voltages. They have: Primary winding Secondary winding Power ratings power transformer iron core circuit symbol

Transformers Ideal Transformer Assume zero internal resistances, iron core zero resistance in coils no hysteresis losses in iron core all field lines are inside core Assume zero internal resistances, EMFs Ep, Es = terminal voltages Vp, Vs Faradays Law for primary and secondary: Assume: The same amount of flux FB cuts each turn in both primary and secondary windings in ideal transformer (counting self- and mutual-induction) Assuming no losses: energy and power are conserved Turns ratio fixes the step up or step down voltage ratio Vp, Vs are instantaneous (time varying) or RMS averages, as can be the power and current.

Example: A dimmer for lights using a variable inductance Light bulb R=50 W Erms=30 V L f =60 Hz w = 377 rad/sec Without Inductor: a) What value of the inductance would dim the lights to 5 Watts? Recall: b) What would be the change in the RMS current? Without inductor: P0,rms = 18 W. With inductor: Prms = 5 W.