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AC Power Dr. Holbert April 9, 2008 Lect19 EEE 202 1 Instantaneous Power: p(t) For AC circuits, the voltage and current are v(t) = VM cos(t+v) i(t) = IM cos(t+i) The instantaneous power is simply their product p(t) = v(t) i(t) = VM IM cos(t+v) cos(t+i) = ½VM IM [cos(v- i) + cos(2t+v +i)] Constant Term Lect19 EEE 202 Wave of Twice Original Frequency 2 Average Power (P) • Calculate average power (integrate power over one cycle and divide by period) 1 P= T t 0 T t0 1 p(t) dt = T t 0 T V M cost v I M cost + i dt t0 1 = VM I M cos v - i 2 • Recall that passive sign convention says: P > 0, power is being absorbed P < 0, power is being supplied Lect19 EEE 202 3 Average Power: Special Cases • Purely resistive circuit: P = ½ VM IM The power dissipated in a resistor is VM2 1 2 1 P = VM I M = = IM R 2 2 R 2 • Purely reactive circuit: P = 0 – Capacitors and inductors are lossless elements and absorb no average power – A purely reactive network operates in a mode in which it stores energy over one part of the period and releases it over another part Lect19 EEE 202 4 Average Power Summary Circuit Element Average Power V or I source Resistor P = ½ VM IM cos(v- i) P = ½ VM IM = ½ IM2 R Capacitor or Inductor P=0 Does the expression for the resistor power look identical to that for DC circuits? Lect19 EEE 202 5 Effective or RMS Values • Root-mean-square value (formula reads like the name: rms) I rms 1 T t 0 T 2 i (t ) dt and t0 • For a sinusoid: Vrms 1 T t 0 T v 2 (t ) dt t0 Irms = IM/2 – For example, AC household outlets are around 120 Volts-rms Lect19 EEE 202 6 Why RMS Values? • The effective/rms current allows us to write average power expressions like those used in dc circuits (i.e., P=I²R), and that relation is really the basis for defining the rms value • The average power (P) is 1 Psource VM I M cos v i Vrms I rms cos v i 2 2 Vrms 1 2 Presistor VM I M Vrms I rms I rms R 2 R Lect19 EEE 202 7 RMS in Everyday Life • When we buy consumer electronics, the faceplate specifications provide the rms voltage and current values • For example, what is the rms current for a 1200 Watt hairdryer (although there is a small fan in a hairdryer, most of the power goes to a resistive heating element)? • What happens when two hairdryers are turned on at the same time in the bathroom? • How can I determine which uses more electricity---a plasma or an LCD HDTV? Lect19 EEE 202 8 Class Examples • Drill Problems P8-10, P8-11, P8-12 Lect19 EEE 202 9 Extra Slides Lect19 EEE 202 10 Maximum Average Power Transfer • To obtain the maximum average power transfer to a load, the load impedance (ZL) should be chosen equal to the complex conjugate of the Thevenin equivalent impedance representing the remainder of the network ZL = RL + j XL = RTh - j XTh = ZTh* Lect19 EEE 202 11 Maximum Average Power Transfer ZTh Voc + – ZL ZL = ZTh* • Note that ONLY the resistive component of the load dissipates power Lect19 EEE 202 12 Max Power Xfer: Cases Load Load Equivalent Characteristic Complex ZL = ZTh* = RTh - j XTh Purely Resistive Z R = R 2 + X 2 L L Th Th (i.e., XL=0) Further reduces to ZL= RL=RTh for XTh=0 (old DC way) Purely Reactive No Average power transfer to load; Not really a case (i.e., RL=0) Lect19 EEE 202 13 Power Factor (pf) • Derivation of power factor (0 pf 1) average power P pf = = apparent power V rms I rms V rms I rms cos v i = = cos v i = cos Z L V rms I rms • A low power factor requires more rms current for the same load power which results in greater utility transmission losses in the power lines, therefore utilities penalize customers with a low pf Lect19 EEE 202 14 Power Factor Angle (ZL) • power factor angle is v- i = ZL (the phase angle of the load impedance) • power factor (pf) special cases – purely resistive load: ZL = 0° pf=1 – purely reactive load: ZL = ±90° pf=0 Power Factor Angle Lect19 I/V Lag/Lead Load Equivalent -90 < θZL < 0 Leading Equivalent RC 0 < θZL < 90 Lagging Equivalent RL EEE 202 15 From a Load Perspective • Recall phasor relationships between current, voltage, and load impedance VIZ V VM v I M i Z Z rms 2 v I rms 2 i Z Z Vrms I rms Z • The load impedance also has several alternate expressions Z Re( Z) j Im( Z) Z cos Z j sin Z Lect19 EEE 202 16 Power Triangle • The power triangle relates pf angle to P and Q Q reactive/quadrature power tan v i P real/average power – the phasor current that is in phase with the phasor voltage produces the real (average) power – the phasor current that is out of phase with the phasor voltage produces the reactive (quadrature) Lect19power EEE 202 Im Q=I2rms Im(Z) v- i P=I2rms Re(Z) Re 17 Summarizing Complex Power (S) S P jQ I 2 rms Re Z jI 2 rms 2 Im Z I rms Z Complex power (like energy) is conserved, that is, the total complex power supplied equals the total complex power absorbed, Si=0 Lect19 Reactive Power Load Power Factor Complex Power Q is positive Inductive Lagging First quadrant Q is zero Resistive pf = 1 Real valued Q is negative Capacitive Leading Fourth quadrant EEE 202 18 More Power Terminology • average power, P = Vrms Irms cos(v- i) • apparent power = Vrms Irms – apparent power is expressed in volt-amperes (VA) or kilovolt-amperes (kVA) to distinguish it from average power Lect19 EEE 202 19 Complex Power (S) • Definition of complex power, S S Vrms I *rms Vrms v I rms i Vrms I rms v i Vrms I rms cos v i j Vrms I rms sin v i P jQ – P is the real or average power – Q is the reactive or quadrature power, which indicates temporary energy storage rather than any real power loss in the element; and Q is measured in units of volt-amperes reactive, or var Lect19 EEE 202 20 Complex Power (S) • This is really a return to phasor use of voltage and current rather than just the recent use of magnitude and rms values • Complex power is expressed in units of volt-amperes like apparent power • Complex power has no physical significance; it is a purely mathematical concept • Note relationships to apparent power and power factor of last section |S| = Vrms Irms = apparent power S = (v- i) = ZL = power factor angle Lect19 EEE 202 21 Real Power (P) • Alternate expressions for the real or average power (P) P Re S Vrms I rms cos v i Re Z S cos Z I rms Z I rms Z 2 I rms Re Z Lect19 EEE 202 22 Reactive Power (Q) • Alternate expressions for the reactive or quadrature power (Q) Q ImS Vrms I rms sin v i ImZ S sin Z I rms Z I rms Z 2 I rms ImZ Lect19 EEE 202 23