Lect19EEE 2021 AC Power Dr. Holbert April 9, 2008

Lect19EEE 2022 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 p(t) = v(t) i(t) = V M I M cos(  t+  v ) cos(  t+  i ) = ½V M I M [cos(  v -  i ) + cos(2  t+  v +  i )] Constant Term Wave of Twice Original Frequency

Lect19EEE 2023 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

Lect19EEE 2024 Average Power: Special Cases Purely resistive circuit: P = ½ V M I M 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

Lect19EEE 2025 Average Power Summary Does the expression for the resistor power look identical to that for DC circuits?

Lect19EEE 2026 Effective or RMS Values Root-mean-square value (formula reads like the name: rms) For a sinusoid:I rms = I M /  2 –For example, AC household outlets are around 120 Volts-rms

Lect19EEE 2027 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

Lect19EEE 2028 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?

Lect19EEE 2029 Class Examples Drill Problems P8-10, P8-11, P8-12

Lect19EEE Extra Slides

Lect19EEE Maximum Average Power Transfer To obtain the maximum average power transfer to a load, the load impedance (Z L ) should be chosen equal to the complex conjugate of the Thevenin equivalent impedance representing the remainder of the network Z L = R L + j XL = R Th - j X Th = Z Th *

Lect19EEE Maximum Average Power Transfer V oc + – Z Th ZLZL Z L = Z Th * Note that ONLY the resistive component of the load dissipates power

Lect19EEE Max Power Xfer: Cases

Lect19EEE 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

Lect19EEE Power Factor Angle (  Z L ) power factor angle is  v -  i =  Z L (the phase angle of the load impedance) power factor (pf) special cases –purely resistive load:  Z L = 0°  pf=1 –purely reactive load:  Z L = ±90°  pf=0

Lect19EEE From a Load Perspective Recall phasor relationships between current, voltage, and load impedance The load impedance also has several alternate expressions

Lect19EEE 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 Re Im P=I 2 rms Re(Z) Q=I 2 rms Im(Z) S=I 2 rms Z  v -  i

Lect19EEE Summarizing Complex Power (S) Complex power (like energy) is conserved, that is, the total complex power supplied equals the total complex power absorbed,  S i =0

Lect19EEE More Power Terminology average power, P = V rms I rms cos(  v -  i ) apparent power = V rms I rms –apparent power is expressed in volt-amperes (VA) or kilovolt-amperes (kVA) to distinguish it from average power

Lect19EEE 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

Lect19EEE 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| = V rms I rms = apparent power  S =  (  v -  i ) =  Z L = power factor angle

Lect19EEE Real Power (P) Alternate expressions for the real or average power (P)

Lect19EEE Reactive Power (Q) Alternate expressions for the reactive or quadrature power (Q)