Chapter 31 Alternating Current Circuits

1 Chapter 31 Alternating Current CircuitsMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/20122 Alternating Current Circuits Alternating Current - Generator Wave Nomenclature & RMS AC Circuits : Resistor; Inductor; Capacitor Transformers - not the movie LC and RLC Circuits - No generator Driven RLC Circuits - Series Impedance and Power RC and RL Circuits - Low & High Frequency RLC circuit - Solution via Complex Numbers RLC circuit - Example ResonanceMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/20123 GeneratorsBy turning the coils in the magnetic field an emf is generated in the coils thus turning mechanical energy into Alternating (AC) 2426 Chap31-AC Circuits -Revised: 6/24/20124 GeneratorsRotating the Coil in a Magnetic Field Generates an Emf Examples.

2 Gasoline generator Manually turning the crank Hydroelectric powerMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/20125 Generatorsmmmpeakpeak = NBAcos = t = NBAcos td= - = NBA sin tdt=sin t; = NBA MFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/20126 Wave Nomenclature and RMS ValuesMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/20127 Wave NomenclatureApeak-peak = Ap-p = 2 Apeak = 2Ap; Ap = Ap-p /2 MFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/20128 sinx = A t -cos{} sintx = A2 -cosT()() 2 22x = A sin t -x = A sin t cos - sin cos tx = A sin t (0) - (1)cos tx = -Acos tThe minus sign means that the phase is shifted to the right.

3 A plus sign indicated the phase is shifted to the leftShifting Trig FunctionsMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/20129 Shifting Trig Functions t -= 02 t =2 11Tt = ; =2 2 TTt ==2 2 4 sin t -= 02 Shifted T rig ( t)sin( t- )MFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201210 Root Mean SquaredProcedure Square it (make the negative values positive) Take the average (mean) Take the square root (undo the squaring operation)The root mean squared (rms) method of averaging is used when a variable will average to zero but its effect will not average to 2426 Chap31-AC Circuits -Revised: 6/24/201211 Sine (Ra dia ns)SIN(Theta)SIN2(Theta)RMS ValueRoot Mean Squared AverageMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised.

4 6/24/201212 Average of a Periodic Function Tavgpocos( T)T Tppavgpo0cos(0)pavg1V = V=V(t)dt; V(t) = V sin tTVV1V = V sin tdt = sinxdx = - d(cosx)T T TVV= -(1 - 1) = 0 TMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201213 Root Mean Squared()()()() T222pavgo222 Tppp2 2avgo2p2avg2 RMSppavg1V= V=V (t)dt; V(t) = V sin tTVVVV = sin tdt = = T T2VV=21 VV=V = 2426 Chap31-AC Circuits -Revised: 6/24/201214 Root Mean Squared() 2 RMSppavg1 VV=V = RMS voltage (VRMS )is the DC voltage that has the same effect as the actual AC 2426 Chap31-AC Circuits -Revised: 6/24/201215 RMS PowerThe average AC power is the product of the DC equivalent voltage and Current .

5 ()()avgp pppRMSRMSavgRMSRMSavgRMS RMS1P= V I2 VIsince V= and I=221P=2 V2 I2P= VIMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201216 Resistor in an AC CircuitMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201217 Resistor in an AC CircuitFor the case of a resistor in an AC circuit the VR across the resistor is in phase with the Current I through the phase means that both waveforms peak at the same 2426 Chap31-AC Circuits -Revised: 6/24/201218 Resistor in an AC circuit ()22p22pP(t) = I (t)R = I cos tRP(t) = I Rcos tThe instantaneous power is a function of time.

6 However, the average power per cycle is of more 2426 Chap31-AC Circuits -Revised: 6/24/201219 Inductors in an AC CircuitMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201220 Coils & Caps in an AC CircuitMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201221 Inductors in an AC CircuitFor the case of an inductor in an AC circuit the VL across the inductor is 900 ahead of the Current I through the 2426 Chap31-AC Circuits -Revised: 6/24/201222 Inductors in an AC circuit ()L peakpL peakL peakpLLV I = I sin t =cos t -2 LVVI == LXX = LXL is the inductive reactanceMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201223 Average Power - Inductors() () LL peakpL peakpTavgL peakp0TL peakpavg0TL peakpavg0P(t) = V I = Vcos t I sin tP(t) = VI cos t sin t1P=VI cos t sin tdtTVIP=cos t sin tdtTVIP=sin2 tdt = 02 TInductors don t dissipate energy, they store 2426 Chap31-AC Circuits -Revised.

7 6/24/201224 Average Power - InductorsInductors don t dissipate energy, they store voltage and the Current are out of phase by we saw with Work, energy changed only when a portion of the force was in the direction of the electrical Circuits energy is dissipated only if a portion of the voltage is in phase with the 2426 Chap31-AC Circuits -Revised: 6/24/201225 Capacitors in an AC CircuitMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201226 Capacitors in an AC circuit ()CpC pCC ppppppV =cos t = V cos tQ = V C = V Ccos t = Q cos tdQI == - Q sin t = -I sin tdt I = - Q sin t = I cos t +2 For the case of a capacitor in an AC circuit the VC across the capacitor is 900 behind the Current I on the 2426 Chap31-AC Circuits -Revised: 6/24/201227 Capacitors in an AC CircuitCpCpppCpCCVVI = Q = CV ==1X C1X = CXC is the capacitive 2426 Chap31-AC Circuits -Revised: 6/24/201228 Electrical TransformersMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised.

8 6/24/201229 Electrical TransformersMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201230 Electrical TransformersMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201231 MFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201232 Electrical TransformersMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201233 Electrical TransformersBoth coils see the same magnetic flux and the cross sectional areas are the same00 1 10 2 21 12 2121221221211B = nI n I = n In I = n InI =InNInNL===NInNLMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201234 Electrical TransformersConservation of EnergyPrimary Power = Secondary Powerin 1out 2out12in212outin1V I = V IVIN==VINNV=VNInduced voltage/loopMore loops => more voltageVoltage steps up but the Current steps down.

9 MFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201235LC and RLC Circuits Without a GeneratorMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201236LC circuit - No GeneratorTo start this circuit some energy must be placed in it since there is no battery to drive the circuit . We will do that by placing a charge on the capacitorSince there is no resistor in the circuit and the resistance of the coil is assumed to be zero there will not be any 2426 Chap31-AC Circuits -Revised: 6/24/201237LC circuit - No GeneratorApply Kirchhoff s rule2222 RdIQL+= 0dtCdQSince I =dtd QQL+= 0dtCd Q1= -QdtLC1 =LCThis is the harmonic oscillator equationMFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201238LC circuit - No Generator()pppQ(t) = Q cos tdQI(t) == - Q sin tdt I(t) = - Q cos t +2 The circuit will oscillate at the frequency R.

10 Energy will flow back and forth from the capacitor (electric energy) to the inductor (magnetic energy).MFMcGraw-PHY 2426 Chap31-AC Circuits -Revised: 6/24/201239 RLC circuit - No GeneratorLike the LC circuit some energy must initially be placed in this circuit since there is no battery to drive the circuit . Again we will do this by placing a charge on the capacitorSince there is a resistor in the circuit now there will be losses as the energy passes through the 2426 Chap31-AC Circuits -Revised: 6/24/201240 RLC circuit - No GeneratorApply Kirchhoff s rule22dIQdQL+ IR += 0.