Lecture Electric circuit theory: AC power analysis include all of the following content: Instantaneous and average power, RMS value, apparent power and power factor, complex power, power factor improvement.
Nguyễn Công Phương Electric Circuit Theory AC Power Analysis Contents I Basic Elements Of Electrical Circuits II Basic Laws III Electrical Circuit Analysis IV Circuit Theorems V Active Circuits VI Capacitor And Inductor VII First Order Circuits VIII.Second Order Circuits IX Sinusoidal Steady State Analysis X AC Power Analysis XI Three-phase Circuits XII Magnetically Coupled Circuits XIII.Frequency Response XIV.The Laplace Transform XV Two-port Networks AC Power Analysis - sites.google.com/site/ncpdhbkhn AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn Instantaneous Power (1) p(t ) = v(t )i (t ) v(t ) = Vm sin(ωt + φv ) i (t ) = I m sin(ωt + φi ) → p (t ) = Vm I m sin(ωt + φv ) sin(ωt + φi ) Vm I m = [cos(φv − φi ) − cos(2ωt + φv + φi )] Vm I m Vm I m = cos(φv − φi ) − cos(2ωt + φv + φi ) 2 AC Power Analysis - sites.google.com/site/ncpdhbkhn Instantaneous Power (2) Vm I m Vm I m p (t ) = cos(φv − φi ) − cos(2ωt + φv + φi ) 2 p(t) Vm I m Vm I m cos(φv − φi ) t AC Power Analysis - sites.google.com/site/ncpdhbkhn Average Power (1) P= T ∫ T p(t ) dt Vm I m Vm I m p (t ) = cos(φv − φi ) − cos(2ωt + φv + φi ) 2 1 → P = Vm I m cos(φv − φi ) T ∫ T 1 dt − Vm I m T ∫ T cos(2ωt + φv + φi )dt The average of a sinusoid over its period is zero → P = Vm I m cos(φv − φi ) AC Power Analysis - sites.google.com/site/ncpdhbkhn Average Power (2) V = Vm φv I = I m φi → VI* = Vm I m φv − φi → I* = I m − φi VI* = Vm I m φv − φi = Vm I m cos(φv − φi ) + jVm I m sin(φv − φi ) P = Vm I m cos(φv − φi ) * → P = Re(VI ) AC Power Analysis - sites.google.com/site/ncpdhbkhn Average Power (3) 1 * P = Re(VI ) = Vm I m cos(φv − φi ) 2 1 P = Vm I m cos(0) = Vm I m = I m R 2 φv = φi : φv − φi = ±90 : o P = Vm I m cos(90o ) = AC Power Analysis - sites.google.com/site/ncpdhbkhn Average Power (4) • Ex.: v(t) = 150sin(314t – 30o) V i(t) = 10sin(314t + 45o) A Find P? AC Power Analysis - sites.google.com/site/ncpdhbkhn AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn 10 AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn 24 RMS Value (1) i(t) + – e(t) (AC) R →P= T ∫ T R T i Rdt = ∫ i dt T → I eff I + – E (DC) R T = i dt ∫ T → P = I 2R Veff I is the effective/RMS value of i(t) X eff = X rms = T ∫ T = T ∫ T v dt x dt AC Power Analysis - sites.google.com/site/ncpdhbkhn 25 RMS Value (2) I rms = T ∫ T i 2dt → I rms i(t ) = I m sin ωt T = i dt = ∫ T T ∫ T [ I m sin ωt ]2 dt T − cos 2ωt = Im dt ∫ T I m2 = 2T I rms Im = ∫ T Im dt = Vrms Vm = AC Power Analysis - sites.google.com/site/ncpdhbkhn 26 AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn 27 Apparent Power P = Vm I m cos(φv − φi ) Vrms Vm = I rms Im = P = Vrms I rms cos(φv − φi ) S = VrmsIrms AC Power Analysis - sites.google.com/site/ncpdhbkhn 28 Power Factor P = Vrms I rms cos(φv − φi ) S = Vrms I rms P pf = = cos(φv − φi ) S • φv − φi = → pf = → P = S = Vrms I rms • φv − φi = ±90o → pf = → P = AC Power Analysis - sites.google.com/site/ncpdhbkhn 29 AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn 30 Complex Power (1) I + Z V – * S = VI V = Vm φv I = I m φi → I* = I m − φi 1 → S = Vm I m φv − φi = Vm I m cos(φv − φi ) + j Vm I m sin(φv − φi ) 2 AC Power Analysis - sites.google.com/site/ncpdhbkhn 31 Complex Power (2) * S = VI V = ZI S = → S = ( )( ) V VV * Vm φv Vm − φv Vrms V = = = * * * Z Z Z Z 1 ZII* = Z I m φi I m − φi = ZI m2 = ZI rms 2 Z = R + jX * ( → S = ( R + jX ) I rms )( ) P = Re( S ) = RI rms 2 → = RI rms + jXI rms Q = Im(S) = XI rms AC Power Analysis - sites.google.com/site/ncpdhbkhn 32 Complex Power (3) * V S = VI = P + jQ = Vm I m φv − φi = Vrms I rms φv − φi = ZI rms = rms* 2 Z S = S = Vrms I rms = P + Q 2 P = Re(S) = S cos(φv − φi ) = RI rms Q = Im(S) = S sin(φv − φi ) = XI rms P pf = = cos(φv − φi ) S AC Power Analysis - sites.google.com/site/ncpdhbkhn 33 Complex Power (4) • Ex.: v(t) = 150sin(314t – 30o) V i(t) = 10sin(314t + 45o) A Find Vrms, Irms, S, S, P, Q, pf ? AC Power Analysis - sites.google.com/site/ncpdhbkhn 34 AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn 35 Power Factor Improvement (1) pf = cos φ R + – E pf > pf1 → φ2 < φ1 IL IC L φ1 I + – E R IL L φ2 E I C IC IL φ2 < φ1 → C = ? ( P = const) AC Power Analysis - sites.google.com/site/ncpdhbkhn 36 Power Factor Improvement (2) Q1 Q1 = P tan φ1 , Q2 = P tan φ2 ∆Q Erms ∆Q = = ωCErms → C = ω Erms XC tan φ1 − tan φ2 =P ω Erms φ φ1 Q2 P I E R + Q1 − Q2 P tan φ1 − P tan φ2 C= = 2 ω Erms ω Erms S1 – ∆Q = Q1 − Q2 ∆Q IL AC Power Analysis - sites.google.com/site/ncpdhbkhn C L IC 37 Ex Power Factor Improvement (3) A load is connected to a 220 V (rms), 50 Hz power line This load absorbs a power of 1000kW Its power factor is 0.8 Find the capacitor necessary to raise the pf to 0.9? tan φ1 − tan φ2 C=P ω Erms pf1 = 0.8 → cos φ1 = 0.8 → φ1 = 36.9o → tan φ1 = 0.75 pf = 0.9 → cos φ2 = 0.9 → φ2 = 25.8o → tan φ2 = 0.48 0.75 − 0.48 → C = 1000 × 10 = 0.0178 F 314 × (220) AC Power Analysis - sites.google.com/site/ncpdhbkhn 38 ... sites.google.com/site/ncpdhbkhn AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn... sites.google.com/site/ncpdhbkhn AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn... sites.google.com/site/ncpdhbkhn 34 AC Power Analysis Instantaneous and Average Power Maximum Average Power Transfer RMS Value Apparent Power and Power Factor Complex Power Power Factor Improvement AC Power Analysis - sites.google.com/site/ncpdhbkhn