14: Power in AC Circuits

1 14: Power in AC Circuits14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 1 / 11 Average Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Average Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Powerdissipated inR:P=1 TRT0p(t)dtAverage Power14.

2 Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Powerdissipated inR:P=1 TRT0p(t)dt=1R 1 TRT0v2(t)dtAverage Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Powerdissipated inR:P=1 TRT0p(t)dt=1R 1 TRT0v2(t)dt=hv2(t)iRAverage Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Powerdissipated inR:P=1 TRT0p(t)dt=1R 1 TRT0v2(t)dt=hv2(t)iR v2(t) is the value ofv2(t)averaged over timeAverage Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Powerdissipated inR.

3 P=1 TRT0p(t)dt=1R 1 TRT0v2(t)dt=hv2(t)iR v2(t) is the value ofv2(t)averaged over timeWe define theRMS Voltage(Root Mean Square):Vrms,phv2(t)iAverage Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Powerdissipated inR:P=1 TRT0p(t)dt=1R 1 TRT0v2(t)dt=hv2(t)iR v2(t) is the value ofv2(t)averaged over timeWe define theRMS Voltage(Root Mean Square):Vrms,phv2(t)iThe average Power dissipated inRisP=hv2(t)iR=(Vrms)2 RVrmsis the DC voltage that would causeRto dissipate the same Power14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 2 / 11 Intantaneous Powerdissipated inR:p(t) =v2(t)RAverage Powerdissipated inR:P=1 TRT0p(t)dt=1R 1 TRT0v2(t)dt=hv2(t)iR v2(t) is the value ofv2(t)averaged over timeWe define theRMS Voltage(Root Mean Square):Vrms,phv2(t)iThe average Power dissipated inRisP=hv2(t)iR=(Vrms)2 RVrmsis the DC voltage that would causeRto dissipate the same usesmall lettersfor time-varying voltages andcapital lettersfortime-invariant Wave RMS14.

4 Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 3 / 11 Cosine Wave:v(t) = 5 cos t. Amplitude isV= Wave RMS14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 3 / 11 Cosine Wave:v(t) = 5 cos t. Amplitude isV= Voltage:v2(t) =V2cos2 t=V2 12+12cos 2 t Cosine Wave RMS14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 3 / 11 Cosine Wave:v(t) = 5 cos t.

5 Amplitude isV= Voltage:v2(t) =V2cos2 t=V2 12+12cos 2 t Mean Square Voltage: v2 =V22sincecos 2 taverages to Wave RMS14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 3 / 11 Cosine Wave:v(t) = 5 cos t. Amplitude isV= Voltage:v2(t) =V2cos2 t=V2 12+12cos 2 t Mean Square Voltage: v2 =V22sincecos 2 taverages to Wave RMS14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 3 / 11 Cosine Wave:v(t) = 5 cos t. Amplitude isV= Voltage:v2(t) =V2cos2 t=V2 12+12cos 2 t Mean Square Voltage: v2 =V22sincecos 2 taverages to Voltage:Vrms=phv2i=1 2V= Wave RMS14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 3 / 11 Cosine Wave:v(t) = 5 cos t.

6 Amplitude isV= Voltage:v2(t) =V2cos2 t=V2 12+12cos 2 t Mean Square Voltage: v2 =V22sincecos 2 taverages to Voltage:Vrms=phv2i=1 2V= : Power engineersalwaysuse RMS voltages and currents exclusivelyand omit the rms example UK Mains voltage = 230 V rms = 325 V Wave RMS14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 3 / 11 Cosine Wave:v(t) = 5 cos t. Amplitude isV= Voltage:v2(t) =V2cos2 t=V2 12+12cos 2 t Mean Square Voltage: v2 =V22sincecos 2 taverages to Voltage:Vrms=phv2i=1 2V= : Power engineersalwaysuse RMS voltages and currents exclusivelyand omit the rms example UK Mains voltage = 230 V rms = 325 V , a ~ overbar means 2: thuseV=1 Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej VI=|I|ej IPower Factor+14.

7 Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power dissipated in loadZisp(t) =v(t)i(t) =|V||I|cos ( t+ V) cos ( t+ I) Power Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are.

8 V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power dissipated in loadZisp(t) =v(t)i(t) =|V||I|cos ( t+ V) cos ( t+ I)=|V||I| 12cos (2 t+ V+ I) +12cos ( V I) Power Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power dissipated in loadZisp(t) =v(t)i(t) =|V||I|cos ( t+ V) cos ( t+ I)=|V||I| 12cos (2 t+ V+ I) +12cos ( V I) =12|V||I|cos ( V I) +12|V||I|cos (2 t+ V+ I) Power Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power dissipated in loadZisp(t) =v(t)i(t) =|V||I|cos ( t+ V) cos ( t+ I)=|V||I| 12cos (2 t+ V+ I) +12cos ( V I) =12|V||I|cos ( V I) +12|V||I|cos (2 t+ V+ I) Power Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are.

9 V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power dissipated in loadZisp(t) =v(t)i(t) =|V||I|cos ( t+ V) cos ( t+ I)=|V||I| 12cos (2 t+ V+ I) +12cos ( V I) =12|V||I|cos ( V I) +12|V||I|cos (2 t+ V+ I)Average Power :P=12|V||I|cos ( )where = V IPower Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power dissipated in loadZisp(t) =v(t)i(t) =|V||I|cos ( t+ V) cos ( t+ I)=|V||I| 12cos (2 t+ V+ I) +12cos ( V I) =12|V||I|cos ( V I) +12|V||I|cos (2 t+ V+ I)Average Power :P=12|V||I|cos ( )where = V I= eV eI cos ( ) Power Factor+14: Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analysis of Circuits (2017-10213)AC Power : 14 4 / 11 Suppose voltage and current phasors are:V=|V|ej V v(t) =|V|cos ( t+ V)I=|I|ej I i(t) =|I|cos ( t+ I) Power dissipated in loadZisp(t) =v(t)i(t) =|V||I|cos ( t+ V) cos ( t+ I)=|V||I| 12cos (2 t+ V+ I) +12cos ( V I) =12|V||I|cos ( V I) +12|V||I|cos (2 t+ V+ I)Average Power :P=12|V||I|cos ( )where = V I= eV eI cos ( )cos is thepower factorPower Factor+14.

10 Power in AC Circuits Average Power Cosine Wave RMS Power Factor+ Complex Power Power in R, L, C Tellegen s Theorem Power Factor Correction Ideal Transformer Transformer Applications Analy