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Chapter 31 Alternating Current PowerPoint® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Scott Hildreth – Chabot College Questions about AC Circuits • How do AC circuits work, compared with DC? • Advantages? Disadvantages? • Westinghouse vs. Edison? • What roles do inductors, capacitors, and resistors play in AC circuits? • How can we mathematically model AC circuits and the complex relationships of voltage and current through all components? Copyright © 2012 Pearson Education Inc. Goals for Chapter 31 • To use phasors to describe sinusoidally varying quantities • To use reactance to describe voltage in a circuit • To analyze an L-R-C series circuit • To determine power in ac circuits • To see how an L-R-C circuit responds to frequency • To learn how transformers work Copyright © 2012 Pearson Education Inc. Introduction • How does a radio tune to a particular station? • Use a variable capacitor in concert with inductors and resistors! Copyright © 2012 Pearson Education Inc. Alternating currents • Voltage (supply) is a sinusoidal function of time V(t) = Vmaxcos wt Copyright © 2012 Pearson Education Inc. Alternating currents • Voltage (supply) is a sinusoidal function of time V(t) = Vmaxcos wt • Resulting current is ALSO a sinusoidal function in time i(t) = imax cos wt • But … phases of these are not necessarily the same through the circuit! • When Voltage is maximum, Current may not be! • V(t) = Vmaxcos (wt +/-f) but i(t) = imax cos (wt) • If f = 0 , Voltage and Current are described as “in phase” • If f 0 , Voltage and Current are described as “out of phase” Copyright © 2012 Pearson Education Inc. Alternating currents across a resistor… How do Resistors affect an AC circuit? Copyright © 2012 Pearson Education Inc. Alternating current across a resistor… • Current and Voltage are in phase across resistors • VR(t) = Vmaxcos wt Copyright © 2012 Pearson Education Inc. & iR (t) = imax cos wt Alternating currents across a resistor… • Current and Voltage are in phase across resistors • VR(t) = Vmaxcos wt Copyright © 2012 Pearson Education Inc. & iR (t) = imax cos wt Alternating currents across a capacitor… How do CAPACITORS affect an AC circuit? Copyright © 2012 Pearson Education Inc. Alternating currents across a capacitor… • Current and Voltage are out of phase across capacitors • VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt • Capacitors take time to reach maximum voltage • Voltage across capacitor LAGS behind current! Copyright © 2012 Pearson Education Inc. Alternating currents across a capacitor… CAPACITORS • VOLTAGE lags CURRENT • CURRENT leads Voltage I C E Copyright © 2012 Pearson Education Inc. Alternating currents across a capacitor… • Current and Voltage are out of phase across capacitors • VC(t) = Vmaxcos (wt - f) & Copyright © 2012 Pearson Education Inc. iC (t) = imax cos wt Alternating currents across a capacitor… • Current and Voltage are out of phase across capacitors • VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt • Note current is max at time t = 0 • But charge on capacitor is not yet built up to a maximum! • Charge on plates max AFTER current already decreasing (but still positive) Copyright © 2012 Pearson Education Inc. Alternating currents across a capacitor… • Current and Voltage are out of phase across capacitors • VC(t) = Vmaxcos (wt - f) & iC (t) = imax cos wt • Note current is max at time t = 0 • Voltage isn’t maximum until some time t = + f/w later! • Voltage E will “lag” current I across a capacitor C • Remember “I – C – E” Copyright © 2012 Pearson Education Inc. Alternating currents across a capacitor… • Current and Voltage are out of phase across capacitors • VC(t) = Vmaxcos (wt - f) & Copyright © 2012 Pearson Education Inc. iC (t) = imax cos wt Alternating currents across a inductor… How do INDUCTORS affect an AC circuit? Copyright © 2012 Pearson Education Inc. Alternating currents across an inductor… • Current &Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt • Inductors “fight” current change, and push hardest in the opposite direction when current changes from – to + or + to - Copyright © 2012 Pearson Education Inc. Alternating currents across an inductor… • Current &Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt • So voltage across the inductor will reach maximum BEFORE the current through it builds to max… Copyright © 2012 Pearson Education Inc. Alternating currents across an inductor… INDUCTORS • CURRENT lags VOLTAGE • Voltage leads Current E L I Copyright © 2012 Pearson Education Inc. Alternating currents across an inductor… • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & Copyright © 2012 Pearson Education Inc. iL (t) = imax cos wt Alternating currents across an inductor… • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt Consider cases: t = 0 • Note current is max, and rate of change di/dt = 0 • Voltage across inductor ONLY depends upon L di/dt! • So at that time, VL = 0! Copyright © 2012 Pearson Education Inc. Alternating currents • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & At t = 0, current max, voltage across L = 0 Copyright © 2012 Pearson Education Inc. iL (t) = imax cos wt Alternating currents across an inductor… • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt Consider cases: t >0 • Note current is positive but decreasing, and rate of change di/dt <0 • Voltage across inductor depends upon L di/dt! • Inductor reacts to decreasing current by continuing to provide EMF from a to b • So at that time, VL = Va - Vb <0! Copyright © 2012 Pearson Education Inc. Alternating currents • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & At t > 0, current +, decreasing, voltage across L <0 Copyright © 2012 Pearson Education Inc. iL (t) = imax cos wt Alternating currents across an inductor… • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & Consider cases: t = ¼ of period… • Note current is 0 at some time wt = + /2 • At that time, current is changing from + to – (large change in B field flux!) Copyright © 2012 Pearson Education Inc. iL (t) = imax cos wt Alternating currents • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & At wt = + /2 current 0, decreasing, voltage across L max negative Copyright © 2012 Pearson Education Inc. iL (t) = imax cos wt Alternating currents across an inductor… • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt • Note current is 0 and increasing at some time wt = 3/2 • At that time, current is changing from - to + (large change in B field flux!) • Inductor reacts to this change, generating E to oppose this change • VL will be largest, positive (Va > Vb) pushing the other way! Copyright © 2012 Pearson Education Inc. Alternating currents • Current and Voltage are out of phase across inductors • VL(t) = Vmaxcos (wt+f) & At t, wt = +3/2 , current 0, increasing, voltage across L max Copyright © 2012 Pearson Education Inc. iL (t) = imax cos wt How can we mathematically model AC circuits and the complex relationships of voltage and current, and power through all components? Copyright © 2012 Pearson Education Inc. How can we mathematically model AC circuits and the complex relationships of voltage and current through all components? Phasors! Copyright © 2012 Pearson Education Inc. No, not PHASERS! Copyright © 2012 Pearson Education Inc. Phasors • Graphical representation of current/voltage in AC circuits • Takes into account relative phases of different voltages • Example: current phasor graphs i (t) = imax cos wt Copyright © 2012 Pearson Education Inc. The “real” portion of a Phasor! • Projection of vector onto horizontal axis Copyright © 2012 Pearson Education Inc. The “real” portion of a Phasor! • Consider four different current phasors: IB IA w IC ID Copyright © 2012 Pearson Education Inc. The “real” portion of a Phasor! • Which phasor represents • Positive current becoming more positive? • Positive current decreasing to zero? • Negative current becoming more negative? • Negative current decreasing in magnitude? Copyright © 2012 Pearson Education Inc. I I B A w I C ID The “real” portion of a Phasor! • Which phasor represents • Positive current becoming ID more positive? • Positive current decreasing IA to zero? • Negative current becoming IB more negative? • Negative current decreasing in magnitude? IC Copyright © 2012 Pearson Education Inc. I I B A w I C ID Resistor in an ac circuit • VR = IR; VR in phase with I Copyright © 2012 Pearson Education Inc. Phasors for Voltage/Current across Resistor • VR(t) = Vmaxcos (wt) & Copyright © 2012 Pearson Education Inc. iR (t) = imax cos wt Capacitors in an ac circuit • VC(t) = Vmaxcos (wt-f) VC out of phase with I Copyright © 2012 Pearson Education Inc. Phasors for Voltage/Current across Capacitor • VC(t) = Vmaxcos (wt-f) & iC (t) = imax cos wt I - C- E: Current Leads Voltage Across Capcitor Copyright © 2012 Pearson Education Inc. Capacitance in an ac circuit • The voltage amplitude across the capacitor is VC = IXC • Xc = “capacitive reactance” = 1/wC • Xc = DECREASES as angular frequency increases • WHY? Copyright © 2012 Pearson Education Inc. Inductors in AC circuits • VL(t) = Vmaxcos (wt+f) • VL out of phase with I Copyright © 2012 Pearson Education Inc. Phasors for Voltage/Current across Inductor • VL(t) = Vmaxcos (wt+f) & iL (t) = imax cos wt E-L-I: Voltage Leads Current Across Inductor Copyright © 2012 Pearson Education Inc. Inductor in an ac circuit • The voltage amplitude across the inductor is VL = IXL • XL = “inductive reactance” = wL • XL increases as frequency increases! • WHY? Copyright © 2012 Pearson Education Inc. Comparing ac circuit elements • Table 31.1 summarizes the characteristics of a resistor, an inductor, and a capacitor in an ac circuit. Copyright © 2012 Pearson Education Inc. Root-mean-square values Copyright © 2012 Pearson Education Inc. Current in a personal computer • Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug. • What is the: • AVERAGE current, • Average of the current squared, • Current amplitude? Copyright © 2012 Pearson Education Inc. Current in a personal computer • Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug. •What is the: • AVERAGE current? 0 amps! Copyright © 2012 Pearson Education Inc. Average over 1 period = 0! Current in a personal computer • Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug. •What is the: •Average of current squared? 2.72 = 7.3 Amps2 Copyright © 2012 Pearson Education Inc. Current in a personal computer • Suppose you have a device that draws 2.7 Amps from a 120V, 60-Hz standard US power plug. •What is the: • Current amplitude? Irms = .707 I So I = 3.8 Amps Copyright © 2012 Pearson Education Inc. The L-R-C series circuit • Combine all three elements into simple series circuit • The voltage amplitude across an ac circuit is V = IZ • Overall effective resistance = Z (“impedance”) • Z = [R2 + (XL - Xc)2] ½ Copyright © 2012 Pearson Education Inc. The L-R-C series circuit • Suppose inductive reactance > capacitive reactance? • XL > XC • Inductor is dominating • Current will be out of phase with supply voltage • “E – L – I “ reminds us that current will LAG voltage. Copyright © 2012 Pearson Education Inc. The L-R-C series circuit • Suppose inductive reactance > capacitive reactance? Copyright © 2012 Pearson Education Inc. The L-R-C series circuit • Suppose capacitive reactance > inductive reactance? • X C > XL • Capacitor is dominating • Current will be out of phase with supply voltage • “I – C – E ” reminds us that current will LEAD voltage. Copyright © 2012 Pearson Education Inc. The L-R-C series circuit • Suppose capacitive reactance > inductive reactance? Copyright © 2012 Pearson Education Inc. A resistor and a capacitor in an ac circuit • 200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor VR = 1.20V cos (2500 rad/sec) x t • What is i(t)? • What is the reactance? • What is Vc(t) Copyright © 2012 Pearson Education Inc. A resistor and a capacitor in an ac circuit • 200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t Copyright © 2012 Pearson Education Inc. A resistor and a capacitor in an ac circuit • 200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t Ohm’s Law applies (that’s why it is a LAW! ) VR = IR so I = 0.006 A cos (2500 rad/sec) x t Note current is in phase with the voltage across R! Copyright © 2012 Pearson Education Inc. A resistor and a capacitor in an ac circuit • 200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t Capacitive Reactance XC = 1/wC = = Copyright © 2012 Pearson Education Inc. 1/(2500 rad/s) x 5.0 mF 80 W A resistor and a capacitor in an ac circuit • 200 Ohm Resistor in series with 5 mF capacitor. Voltage across resistor = 1.20V cos (2500 rad/sec) x t Voltage across Capacitor VC = I Xc VC = I Xc Copyright © 2012 Pearson Education Inc. = 0.006 A x 80 W = 0.48 V and = 0.48 V cos (wt - ) A useful application: the loudspeaker • The woofer (low tones) and the tweeter (high tones) are connected in parallel across the amplifier output. Copyright © 2012 Pearson Education Inc. An L-R-C series circuit • R = 300 Ohms • L = 60 mH • C = 0.50 mF • V = 50 V • w = 10,000 rad/sec • What are XL, Xc, Z, I, Phase angle f, and VR, Vc, VL? Copyright © 2012 Pearson Education Inc. An L-R-C series circuit • R = 300 Ohms • L = 60 mH • C = 0.50 mF • V = 50 V • w = 10,000 rad/sec • What are XL, Xc, Z, I, Phase angle f, and VR, Vc, VL? Copyright © 2012 Pearson Education Inc. Power in ac circuits • Power = I x V • Average Power = Irms Vrms cos f • Note that the net energy transfer over one cycle is zero for an inductor and a capacitor. Copyright © 2012 Pearson Education Inc. Resonance in ac circuits • At the resonance angular frequency w0, the inductive reactance equals the capacitive reactance and the current amplitude is greatest. (See Figure 31.18 below.) Copyright © 2012 Pearson Education Inc. Tuning a radio • RMS voltage of 1.0V; what is resonance frequency? At that frequency what are XL and XC and Z? Copyright © 2012 Pearson Education Inc. Transformers • Power is supplied to the primary and delivered from the secondary. • Terminal voltages: V2/V1 = N2/N1. • Currents in primary and secondary: V1I1 = V2I2. Copyright © 2012 Pearson Education Inc. Real transformers • Real transformers always have some power losses Copyright © 2012 Pearson Education Inc.