Physics 121 - Electricity and Magnetism Lecture 14 - AC Circuits, Resonance Y&F Chapter 31, Sec. 3 - 8 The Series RLC Circuit. Amplitude and Phase.

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Physics 121 - Electricity and Magnetism Lecture 14 - AC Circuits, Resonance Y&F Chapter 31, Sec. 3 - 8 The Series RLC Circuit. Amplitude and Phase Relations Phasor Diagrams for Voltage and Current Impedance and Phasors for Impedance Resonance Power in AC Circuits, Power Factor Examples Transformers Summaries

Relative Current/Voltage Phases in pure R, C, and L elements Apply sinusoidal current i (t) = Imcos(wDt) For pure R, L, or C loads, phase angles for voltage drops are 0, p/2, -p/2 “Reactance” means ratio of peak voltage to peak current (generalized resistances). VR& Im in phase Resistance VL leads Im by p/2 Inductive Reactance VC lags Im by p/2 Capacitive Reactance Phases of voltages in series components are referenced to the current phasor currrent Same Phase

Phasors applied to a Series LCR circuit vR vC vL Applied EMF: 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 Current: Same everywhere in the single essential branch Im Em F wDt+F wDt Phasors all rotate CCW at frequency wD Lengths of phasors are the peak values (amplitudes) The “x” components are instantaneous values measured Kirchoff Loop rule for series LRC: Im Em F wDt VL VC VR Component voltage phasors all rotate at wD with phases relative to the current phasor Im and magnitudes below : VR has same phase as Im VC lags Im by p/2 VL leads Im by p/2 along Im perpendicular to Im

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 Impedance magnitude is the ratio of peak EMF to peak current: For series LRC circuit, divide each voltage in |Em| by (same) peak current Applies to a single series branch with L, C, R Magnitude of Z: 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 the 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 should 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 across the RLC circuit. (E) Find the peak and instantaneous voltages across each circuit element. VR in phase with Im VR leads Em by |F| VL leads VR by p/2 VC lags VR by p/2 Note that: Why not? 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 flows & impedance is minimized 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

Power Dissipation in a Resistor Instantaneous power dissipation cos2(q) 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, current, voltage are useful, DC-like quantities Integrate Pinst: Integral = 1/2 Pav is an “RMS” quantity: “Root Mean Square” Square a quantity (positive) Average over a whole cycle Compute square root. For other rms quantities, 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 AC Circuits |Z| Irms Erms F wDt XL-XC R 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): Proof: start with instantaneous power (not very useful): Average it over one full period t: Change variables: Use trig identity:

Power factor for AC Circuits - continued 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: Find the average power: or Recall: do not use arc-cos to find F

Example 3 – Series LCR circuit analysis using RMS values 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 Secondary 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) but can also be regarded as 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.