14: Power In AC Circuits
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 Summary14: Power in AC CircuitsE1.1 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 SummaryE1.1 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 SummaryIntantaneous Power dissipated in R: p(t) E1.1 Analysis of Circuits (2017-10213)v 2 (t)RAC 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 SummaryIntantaneous Power dissipated in R: p(t) E1.1 Analysis of Circuits (2017-10213)v 2 (t)RAC 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 SummaryIntantaneous Power dissipated in R: p(t) v 2 (t)RAverage Power dissipated in R:P E1.1 Analysis of Circuits (2017-10213)1TRT0p(t)dtAC 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 SummaryIntantaneous Power dissipated in R: p(t) Average Power dissipated in R:P E1.1 Analysis of Circuits (2017-10213)1TRT0p(t)dt 1R 1TRT0v 2 (t)Rv 2 (t)dtAC 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 SummaryIntantaneous Power dissipated in R: p(t) Average Power dissipated in R:P E1.1 Analysis of Circuits (2017-10213)1TRT0p(t)dt 1R 1TRT02v 2 (t)Rv (t)dt hv2 (t)iRAC 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 SummaryIntantaneous Power dissipated in R: p(t) Average Power dissipated in R:1TRT1R1TRT2v 2 (t)Rhv2 (t)iP p(t)dt v (t)dt R00v 2 (t) is the value of v 2 (t) averaged over timeE1.1 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 SummaryIntantaneous Power dissipated in R: p(t) Average Power dissipated in R:1TRT1R1TRT2v 2 (t)Rhv2 (t)iP p(t)dt v (t)dt R00v 2 (t) is the value of v 2 (t) averaged over timeWe define the RMS Voltage (Root Mean Square): VrmsE1.1 Analysis of Circuits (2017-10213)p, hv 2 (t)iAC 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 SummaryIntantaneous Power dissipated in R: p(t) Average Power dissipated in R:1TRT1R1TRT2v 2 (t)Rhv2 (t)iP p(t)dt v (t)dt R00v 2 (t) is the value of v 2 (t) averaged over timeWe define the RMS Voltage (Root Mean Square): Vrmshv2 (t)ip, hv 2 (t)i(Vrms )2R The average power dissipated in R is P RVrms is the DC voltage that would cause R to dissipate the same power.E1.1 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 SummaryIntantaneous Power dissipated in R: p(t) Average Power dissipated in R:1TRT1R1TRT2v 2 (t)Rhv2 (t)iP p(t)dt v (t)dt R00v 2 (t) is the value of v 2 (t) averaged over timeWe define the RMS Voltage (Root Mean Square): Vrmshv2 (t)ip, hv 2 (t)i(Vrms )2R The average power dissipated in R is P RVrms is the DC voltage that would cause R to dissipate the same power.We use small letters for time-varying voltages and capital letters fortime-invariant values.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 2 / 11
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 SummaryCosine Wave: v(t) 5 cos ωt. Amplitude is V 5 V.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 3 / 11
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 SummaryCosine Wave: v(t) 5 cos ωt. Amplitude is V 5 V.222Squared Voltage: v (t) V cos ωt VE1.1 Analysis of Circuits (2017-10213)212 12cos 2ωt AC Power: 14 – 3 / 11
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 SummaryCosine Wave: v(t) 5 cos ωt. Amplitude is V 5 V.222Squared Voltage: v (t) V cos ωt V212 12cos 2ωt 2Mean Square Voltage: v 2 V2 since cos 2ωt averages to zero.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 3 / 11
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 SummaryCosine Wave: v(t) 5 cos ωt. Amplitude is V 5 V.222Squared Voltage: v (t) V cos ωt V212 12cos 2ωt 2Mean Square Voltage: v 2 V2 since cos 2ωt averages to zero.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 3 / 11
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 SummaryCosine Wave: v(t) 5 cos ωt. Amplitude is V 5 V.222Squared Voltage: v (t) V cos ωt V212 12cos 2ωt 2Mean Square Voltage: v 2 V2 since cos 2ωt averages to zero.RMS Voltage: VrmsE1.1 Analysis of Circuits (2017-10213)p hv 2 i 1 V2 3.54 VAC Power: 14 – 3 / 11
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 SummaryCosine Wave: v(t) 5 cos ωt. Amplitude is V 5 V.222Squared Voltage: v (t) V cos ωt V212 12cos 2ωt 2Mean Square Voltage: v 2 V2 since cos 2ωt averages to zero.RMS Voltage: Vrmsp hv 2 i 1 V2 3.54 VNote: Power engineers always use RMS voltages and currents exclusivelyand omit the “rms” subscript.For example UK Mains voltage 230 V rms 325 V peak.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 3 / 11
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 SummaryCosine Wave: v(t) 5 cos ωt. Amplitude is V 5 V.222Squared Voltage: v (t) V cos ωt V212 12cos 2ωt 2Mean Square Voltage: v 2 V2 since cos 2ωt averages to zero.RMS Voltage: Vrmsp hv 2 i 1 V2 3.54 V VeNote: Power engineers always use RMS voltages and currents exclusivelyand omit the “rms” subscript.For example UK Mains voltage 230 V rms 325 V peak. e In this lecture course only, a overbar means 2: thus VE1.1 Analysis of Circuits (2017-10213) 1 V2.AC Power: 14 – 3 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθIE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθIE1.1 Analysis of Circuits (2017-10213) v(t) V cos (ωt θV )i(t) I cos (ωt θI )AC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθIE1.1 Analysis of Circuits (2017-10213) v(t) V cos (ωt θV )i(t) I cos (ωt θI )AC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI )E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI ) V I E1.1 Analysis of Circuits (2017-10213)12cos (2ωt θV θI ) 12cos (θV θI ) AC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI )1212 V I cos (2ωt θV θI ) cos (θV θI ) 21 V I cos (θV θI ) 12 V I cos (2ωt θV θI )E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI )1212 V I cos (2ωt θV θI ) cos (θV θI ) 21 V I cos (θV θI ) 12 V I cos (2ωt θV θI )E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI )1212 V I cos (2ωt θV θI ) cos (θV θI ) 21 V I cos (θV θI ) 12 V I cos (2ωt θV θI )Average power: P 12 V I cos (φ)E1.1 Analysis of Circuits (2017-10213)whereφ θV θIAC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI )1212 V I cos (2ωt θV θI ) cos (θV θI ) 21 V I cos (θV θI ) 12 V I cos (2ωt θV θI )Average power: P 12 V I cos (φ) Ve Ie cos (φ)E1.1 Analysis of Circuits (2017-10213)whereφ θV θIAC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI )1212 V I cos (2ωt θV θI ) cos (θV θI ) 21 V I cos (θV θI ) 12 V I cos (2ωt θV θI )Average power: P 12 V I cos (φ) Ve Ie cos (φ)E1.1 Analysis of Circuits (2017-10213)whereφ θV θIcos φ is the power factorAC Power: 14 – 4 / 11
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 SummarySuppose voltage and current phasors are:V V ejθVI I ejθI v(t) V cos (ωt θV )i(t) I cos (ωt θI )Power dissipated in load Z isp(t) v(t)i(t) V I cos (ωt θV ) cos (ωt θI )1212 V I cos (2ωt θV θI ) cos (θV θI ) 21 V I cos (θV θI ) 12 V I cos (2ωt θV θI )Average power: P 12 V I cos (φ) Ve Ie cos (φ)whereφ θV θIcos φ is the power factorφ 0 a lagging power factor (normal case: Current lags Voltage)φ 0 a leading power factor (rare case: Current leads Voltage)E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 4 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθIE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI )E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQComplex Power:E1.1 Analysis of Circuits (2017-10213)S , Ve Ie P jQ measured in Volt-Amps (VA)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQS , Ve Ie P jQ measured in Volt-Amps (VA)e Ie measured in Volt-Amps (VA)Apparent Power: S , VComplex Power:E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQS , Ve Ie P jQ measured in Volt-Amps (VA)e Ie measured in Volt-Amps (VA)Apparent Power: S , VComplex Power:Average Power:E1.1 Analysis of Circuits (2017-10213)P , ℜ (S) measured in Watts (W)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQS , Ve Ie P jQ measured in Volt-Amps (VA)e Ie measured in Volt-Amps (VA)Apparent Power: S , VComplex Power:Average Power: P , ℜ (S) measured in Watts (W)Reactive Power: Q , ℑ (S) Measured in Volt-Amps Reactive (VAR)E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQS , Ve Ie P jQ measured in Volt-Amps (VA)e Ie measured in Volt-Amps (VA)Apparent Power: S , VComplex Power:Average Power: P , ℜ (S) measured in Watts (W)Reactive Power: Q , ℑ (S) Measured in Volt-Amps Reactive (VAR)e Ie PPower Factor: cos φ , cos V S E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQS , Ve Ie P jQ measured in Volt-Amps (VA)e Ie measured in Volt-Amps (VA)Apparent Power: S , VComplex Power:Average Power: P , ℜ (S) measured in Watts (W)Reactive Power: Q , ℑ (S) Measured in Volt-Amps Reactive (VAR)e Ie PPower Factor: cos φ , cos V S Machines and transformers have capacity limits and power losses that areindependent of cos φ; their ratings are always given in apparent power.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Complex 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 Summarye 1 V ejθV andIf V2I 12 I ejθIe Ie The complex power absorbed by Z is S , Vwhere * means complex conjugate.Ve Ie Ve ejθV Ie e jθI Ve Ie ej(θV θI ) Ve Ie ejφ Ve Ie cos φ j Ve Ie sin φ P jQS , Ve Ie P jQ measured in Volt-Amps (VA)e Ie measured in Volt-Amps (VA)Apparent Power: S , VComplex Power:Average Power: P , ℜ (S) measured in Watts (W)Reactive Power: Q , ℑ (S) Measured in Volt-Amps Reactive (VAR)e Ie PPower Factor: cos φ , cos V S Machines and transformers have capacity limits and power losses that areindependent of cos φ; their ratings are always given in apparent power.Power Company: Costs apparent power, Revenue average power.E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 5 / 11
Power in R, L, C14: 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 Summarye Ie P jQFor any impedance, Z , complex power absorbed: S VE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 6 / 11
Power in R, L, C14: 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 Summarye Ie P jQFor any impedance, Z , complex power absorbed: S Ve IZe (b) Ie Ie IeUsing (a) VE1.1 Analysis of Circuits (2017-10213)22 Ve 2we get S Ie Z Z AC Power: 14 – 6 / 11
Power in R, L, C14: 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 Summarye Ie P jQFor any impedance, Z , complex power absorbed: S Ve IZe (b) Ie Ie IeUsing (a) V2Resistor: S Ie R E1.1 Analysis of Circuits (2017-10213) Ve R22 Ve 2we get S Ie Z Z 2φ 0Absorbs average power, no VARs (Q 0)AC Power: 14 – 6 / 11
Power in R, L, C14: 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 Summarye Ie P jQFor any impedance, Z , complex power absorbed: S Ve IZe (b) Ie Ie IeUsing (a) V2Resistor: S Ie R Ve 22 Ve 2we get S Ie Z Z 2φ 0RAbsorbs average power, no VARs (Q 0)2 Ve 2Inductor: S j Ie ωL j ωLφ 90 No average power, Absorbs VARs (Q 0)E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 6 / 11
Power in R, L, C14: 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 Summarye Ie P jQFor any impedance, Z , complex power absorbed: S Ve IZe (b) Ie Ie IeUsing (a) V2Resistor: S Ie R Ve 22 Ve 2we get S Ie Z Z 2φ 0RAbsorbs average power, no VARs (Q 0)2 Ve 2Inductor: S j Ie ωL j ωLφ 90 No average power, Absorbs VARs (Q 0) Ie 2Capacitor: S j ωC j Ve2ωCφ 90 No average power, Generates VARs (Q 0)E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 6 / 11
Power in R, L, C14: 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 Summarye Ie P jQFor any impedance, Z , complex power absorbed: S Ve IZe (b) Ie Ie IeUsing (a) V2Resistor: S Ie R Ve 22 Ve 2we get S Ie Z Z 2φ 0RAbsorbs average power, no VARs (Q 0)2 Ve 2Inductor: S j Ie ωL j ωLφ 90 No average power, Absorbs VARs (Q 0) Ie 2Capacitor: S j ωC j Ve2ωCφ 90 No average power, Generates VARs (Q 0)VARs are generated by capacitors and absorbed by inductorsE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 6 / 11
Power in R, L, C14: 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 Summarye Ie P jQFor any impedance, Z , complex power absorbed: S Ve IZe (b) Ie Ie IeUsing (a) V2Resistor: S Ie R Ve 22 Ve 2we get S Ie Z Z 2φ 0RAbsorbs average power, no VARs (Q 0)2 Ve 2Inductor: S j Ie ωL j ωLφ 90 No average power, Absorbs VARs (Q 0) Ie 2Capacitor: S j ωC j Ve2ωCφ 90 No average power, Generates VARs (Q 0)VARs are generated by capacitors and absorbed by inductorsThe phase, φ, of the absorbed power, S , equals the phase of ZE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 6 / 11
Tellegen’s Theorem14: 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 SummaryTellegen’s Theorem: The complex power, S , dissipated in any circuit’scomponents sums to zero.xn voltage at node nVb , Ib voltage/current in branch bE1.1 Analysis of Circuits (2017-10213)(obeying passive sign convention)AC Power: 14 – 7 / 11
Tellegen’s Theorem14: 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 SummaryTellegen’s Theorem: The complex power, S , dissipated in any circuit’scomponents sums to zero.xn voltage at node nVb , Ib voltage/current in branch b(obeying passive sign convention)abn 1 if Vb starts from node n, 1 if Vb ends at node n 0elseE1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 7 / 11
Tellegen’s Theorem14: 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 SummaryTellegen’s Theorem: The complex power, S , dissipated in any circuit’scomponents sums to zero.xn voltage at node nVb , Ib voltage/current in branch b(obeying passive sign convention)abn 1 if Vb starts from node n, 1 if Vb ends at node n 0elsee.g. branch 4 goes from 2 to 3 a4 [0, 1, 1]E1.1 Analysis of Circuits (2017-10213)AC Power: 14 – 7 / 11
Tellegen’s Theorem14: 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 SummaryTellegen’s Theorem: The complex power, S , dissipated in any circuit’scomponents sums to zero.xn voltage at node nVb , Ib voltage/current in branch b(obeying passive sign convention)abn 1 if Vb starts from node n, 1 if Vb ends at node n 0elsee.g. branch 4 goes from 2 to 3 a4 [0, 1, 1]Branch voltages: Vb E1.1 Analysis of Circuits (2017-10213)Pnabn xn (e.g. V4 x3 x2 )AC Power: 14 – 7 / 11
Tellegen’s Theorem14: 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 SummaryTellegen’s Theorem: The complex power, S , dissipated in any circuit’scomponents sums to zero.xn voltage at node nVb , Ib voltage/current in branch b(obeying passive sign convention)abn 1 if Vb starts from node n, 1 if Vb ends at node n 0elsee.g. branch 4 goes from 2 to 3 a4 [0, 1, 1]Branch voltages: Vb Pabn xn (e.g. V4 x3 x2 )PP KCL @ node n:b abn Ib 0 b abn Ib 0E1.1 Analysis of Circuits (2017-10213)nAC Power: 14 – 7 / 11
Tellegen’s Theorem14: 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 SummaryTellegen’s Theorem: The complex power, S , dissipated in any circuit’scomponen
Cosine Wave RMS 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 Summary E1.1 Analysis of Circuits (2017-10213) AC Power: 14 - 3 / 11 Cosine Wave: v(t) 5cosωt.Amplitude is V 5V. Squared Voltage: v2(t) V2 cos2 ωt V2
Contemporary Electric Circuits, 2nd ed., Prentice-Hall, 2008 Class Notes Ch. 9 Page 1 Strangeway, Petersen, Gassert, and Lokken CHAPTER 9 Series–Parallel Analysis of AC Circuits Chapter Outline 9.1 AC Series Circuits 9.2 AC Parallel Circuits 9.3 AC Series–Parallel Circuits 9.4 Analysis of Multiple-Source AC Circuits Using Superposition 9.1 AC SERIES CIRCUITS
1 Introduction to RL and RC Circuits Objective In this exercise, the DC steady state response of simple RL and RC circuits is examined. The transient behavior of RC circuits is also tested. Theory Overview The DC steady state response of RL and RC circuits are essential opposite of each other: that is, once steady state is reached, capacitors behave as open circuits while inductors behave as .
/ Voltage Divider Circuits Voltage Divider Circuits AC Electric Circuits Question 1 Don’t just sit there! Build something!! Learning to mathematically analyze circuits requires much study and practice. Typically, students practice by working through lots of samp
circuits, all the current flows through one path. In parallel circuits, current can flow through two or more paths. Investigations for Chapter 9 In this Investigation, you will compare how two kinds of circuits work by building and observing series and parallel circuits. You will explore an application of these circuits by wiring two switches .
Lab Experiment 7 Series-Parallel Circuits and In-circuit resistance measurement Series-Parallel Circuits Most practical circuits in electronics are made up combinations of both series and parallel circuits. These circuits are made up of all sorts of components such as resistors, capacitors, inductors, diodes, transistors and integrated circuits.
2. Diesel Power Plant 3. Nuclear Power Plant 4. Hydel Power Plant 5. Steam Power Plant 6. Gas Power Plant 7. Wind Power Plant 8. Geo Thermal 9. Bio - Gas 10. M.H.D. Power Plant 2. What are the flow circuits of a thermal Power Plant? 1. Coal and ash circuits. 2. Air and Gas 3. Feed water and steam 4. Cooling and water circuits 3.
Sep 06, 2018 · SURVIVABLE FIRE ALARM CIRCUITS 40 Advancing the Science of Safety What Circuits Need Survivability? Signaling Line Circuit (SLC) When used to trigger remote NAC power supplies or booster power supplies. When used to for control modules that trigger circuits or amplifiers SURVIVABLE FIRE ALARM CIRCUITS 41 Advancing the Science of Safety
INTERNATIONAL CRIMINAL COURT FROM AMERICA’S PERSPECTIVE JOHN R. BOLTON* In the aftermaths of both World War I and World War II, the United States engaged in significant domestic political debates over its proper place in the world. President Wilson’s brainchild, the League of Nations, was the center-piece of the first debate, and the United Nations the centerpiece of the second. The .
