EE2003 Circuit Theory Chapter 11 AC Power Analysis Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 AC Power Analysis Chapter 11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Instantaneous and Average Power Maximum Average Power Transfer Effective or RMS Value Apparent Power and Power Factor Complex Power Conservation of AC Power Power Factor Correction Power Measurement 2 11.1 Instantaneous and Average Power (1) • The instantaneously power, p(t) p(t ) v(t ) i(t ) Vm I m cos (w t v ) cos (w t i ) 1 1 Vm I m cos ( v i ) Vm I m cos (2w t v i ) 2 2 Constant power Sinusoidal power at 2wt p(t) > 0: power is absorbed by the circuit; p(t) < 0: power is absorbed by the source. 3 11.1 Instantaneous and Average Power (2) • The average power, P, is the average of the instantaneous power over one period. 1 P T T 0 1 p(t ) dt Vm I m cos ( v i ) 2 1. P is not time dependent. 2. When θv = θi , it is a purely resistive load case. 3. When θv– θi = ±90o, it is a purely reactive load case. 4. P = 0 means that the circuit absorbs no average power. 4 11.1 Instantaneous and Average Power (3) Example 1 Calculate the instantaneous power and average power absorbed by a passive linear network if: v(t ) 80 cos (10 t 20) i (t ) 15 sin (10 t 60) Answer: 385.7 600cos(20t 10)W, 387.5W 5 11.1 Instantaneous and Average Power (4) Example 2 A current I 10 30 flows through an impedance Z 20 22Ω . Find the average power delivered to the impedance. Answer: 927.2W 6 11.2 Maximum Average Power Transfer (1) ZTH R TH j XTH ZL R L j XL The maximum average power can be transferred to the load if XL = –XTH and RL = RTH Pmax If the load is purely real, then R L VTH 2 8 R TH 2 2 R TH XTH ZTH 7 11.2 Maximum Average Power Transfer (2) Example 3 For the circuit shown below, find the load impedance ZL that absorbs the maximum average power. Calculate that maximum average power. Answer: 3.415 – j0.7317W, 1.429W 8 11.3 Effective or RMS Value (1) The total power dissipated by R is given by: 1 P T T 0 R T 2 2 i Rdt i dt I rms R 0 T 2 Hence, Ieff is equal to: I eff 1 T T i 2 dt I rms 0 The rms value is a constant itself which depending on the shape of the function i(t). The effective of a periodic current is the dc current that delivers the same average power to a resistor as the periodic current. 9 11.3 Effective or RMS Value (2) The rms value of a sinusoid i(t) = Imcos(wt) is given by: I 2 rms Im 2 The average power can be written in terms of the rms values: I eff 1 Vm I m cos (θ v θ i ) Vrms I rms cos (θ v θ i ) 2 Note: If you express amplitude of a phasor source(s) in rms, then all the answer as a result of this phasor source(s) must also be in rms value. 10 11.4 Apparent Power and Power Factor (1) • Apparent Power, S, is the product of the r.m.s. values of voltage and current. • It is measured in volt-amperes or VA to distinguish it from the average or real power which is measured in watts. P Vrms I rms cos (θ v θi ) S cos (θ v θi ) Apparent Power, S Power Factor, pf • Power factor is the cosine of the phase difference between the voltage and current. It is also the cosine of the angle of the load impedance. 11 11.4 Apparent Power and Power Factor (2) Purely resistive θ – θ = 0, Pf = 1 v i load (R) Purely reactive θv– θi = ±90o, load (L or C) pf = 0 Resistive and reactive load (R and L/C) θv– θi > 0 θv– θi < 0 P/S = 1, all power are consumed P = 0, no real power consumption • Lagging - inductive load • Leading - capacitive load 12 11.5 Complex Power (1) Complex power S is the product of the voltage and the complex conjugate of the current: V Vm θ v I I mθi 1 V I Vrms I rms θ v θ i 2 13 11.5 Complex Power (2) 1 S V I Vrms I rms θ v θ i 2 S Vrms I rms cos (θ v θi ) j Vrms I rms sin (θ v θi ) S = P + j Q P: is the average power in watts delivered to a load and it is the only useful power. Q: is the reactive power exchange between the source and the reactive part of the load. It is measured in VAR. • Q = 0 for resistive loads (unity pf). • Q < 0 for capacitive loads (leading pf). • Q > 0 for inductive loads (lagging pf). 14 11.5 Complex Power (3) S Vrms I rms cos (θ v θi ) j Vrms I rms sin (θ v θi ) S = P + j Q Apparent Power, S = |S| = Vrms*Irms = P 2 Q 2 Real power, P = Re(S) = S cos(θv – θi) Reactive Power, Q = Im(S) = S sin(θv – θi) Power factor, pf = P/S = cos(θv – θi) 15 11.5 Complex Power (4) S Vrms I rms cos (θ v θi ) j Vrms I rms sin (θ v θi ) S = Power Triangle Impedance Triangle P + j Power Factor Q 16 11.6 Conservation of AC Power (1) The complex real, and reactive powers of the sources equal the respective sums of the complex, real, and reactive powers of the individual loads. For parallel connection: S 1 1 1 1 * V I* V (I1 I*2 ) V I1* V I*2 S1 S2 2 2 2 2 The same results can be obtained for a series connection. 17 11.7 Power Factor Correction (1) Power factor correction is the process of increasing the power factor without altering the voltage or current to the original load. Power factor correction is necessary for economic reason. 18 11.7 Power Factor Correction (2) Qc = Q1 – Q2 = P (tan θ1 - tan θ2) = ωCV2rms Q1 = S1 sin θ1 = P tan θ1 P = S1 cos θ1 Q2 = P tan θ2 C Qc P (tan θ1 tan θ 2 ) 2 2 ωVrms ω Vrms 19 11.8 Power Measurement (1) The wattmeter is the instrument for measuring the average power. The basic structure If Equivalent Circuit with load v(t ) Vm cos(wt v ) and i(t ) I m cos(wt i ) P Vrms Irms cos (θ v θi ) 1 2 Vm Im cos (θ v θi ) 20