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AC Power

AC Power. Dr. Holbert April 9, 2008. Constant Term. Wave of Twice Original Frequency. Instantaneous Power: p ( t ). For AC circuits, the voltage and current are v ( t ) = V M cos(  t + v ) i ( t ) = I M cos(  t + i ) The instantaneous power is simply their product

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AC Power

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  1. AC Power Dr. Holbert April 9, 2008 EEE 202

  2. Constant Term Wave of Twice Original Frequency 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(2t+v +i)] EEE 202

  3. Average Power (P) • Calculate average power (integrate power over one cycle and divide by period) • Recall that passive sign convention says: P > 0, power is being absorbed P < 0, power is being supplied EEE 202

  4. Average Power: Special Cases • Purely resistive circuit: P = ½ VM IM The power dissipated in a resistor is • 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 EEE 202

  5. Average Power Summary Does the expression for the resistor power look identical to that for DC circuits? EEE 202

  6. Effective or RMS Values • Root-mean-square value (formula reads like the name: rms) • For a sinusoid: Irms = IM/2 • For example, AC household outlets are around 120 Volts-rms EEE 202

  7. 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 EEE 202

  8. 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? EEE 202

  9. Class Examples • Drill Problems P8-10, P8-11, P8-12 EEE 202

  10. Extra Slides EEE 202

  11. 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* EEE 202

  12. Maximum Average Power Transfer ZTh ZL = ZTh* + Voc ZL – • Note that ONLY the resistive component of the load dissipates power EEE 202

  13. Max Power Xfer: Cases EEE 202

  14. Power Factor (pf) • Derivation of power factor (0  pf  1) • 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 EEE 202

  15. 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 EEE 202

  16. From a Load Perspective • Recall phasor relationships between current, voltage, and load impedance • The load impedance also has several alternate expressions EEE 202

  17. Im S=I2rmsZ Q=I2rmsIm(Z) v- i Re P=I2rmsRe(Z) Power Triangle • The power triangle relates pf angle to P and Q • 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) power EEE 202

  18. Summarizing Complex Power (S) Complex power (like energy) is conserved, that is, the total complex power supplied equals the total complex power absorbed, Si=0 EEE 202

  19. 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 EEE 202

  20. Complex Power (S) • Definition of complex power, S • 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 EEE 202

  21. 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 EEE 202

  22. Real Power (P) • Alternate expressions for the real or average power (P) EEE 202

  23. Reactive Power (Q) • Alternate expressions for the reactive or quadrature power (Q) EEE 202

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