Notes_ACPwr_S16 (1).pdf - ECE 2210 RMS Introduction to AC Power RMS AC Power DC Power I V P = V I = 2 P = 2 = I.R A Stolp corrected p4'14 R v t = V p

Notes_ACPwr_S16 (1).pdf - ECE 2210 RMS Introduction to AC...

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A. Stolp 3/31/09, 2/20/10 11/23/13 corrected p4 '14 ECE 2210 Introduction to AC Power, RMS RMS AC Power I DC Power P = ? P = . V I = V 2 R = . I 2 R v ( ) t = . V p cos ( ) . ϖ t R V R v ( ) t V p t p ( ) t V p 2 R Couldn't we define an "effective" voltage that would allow us to use the same relationships for AC power as used for DC power? average or "effective" power = V p 2 R 2 t P ave = = = V p 2 R 2 V p 2 2 R V p 2 2 R v ( ) t 2 V p 2 S quare / V eff = V p 2 2 = V p 2 = V rms = . 1 T d 0 T t ( ) v ( ) t 2 average = V p 2 2 | \ M ean (average) R oot t RMS R oot of the M ean of the S quare Use RMS in power calculations Sinusoids V rms = . 1 T d 0 T t ( ) v ( ) t 2 = . 1 T d 0 T t . V p cos ( ) . ϖ t 2 = . 1 T d 0 T t . V p 2 1 2 . 1 2 cos ( ) . . 2 ϖ t = . V p 2 . 1 T d 0 T t ( ) 1 . 1 T d 0 T t cos ( ) . . 2 ϖ t = . V p 2 1 0 ECE 2210 AC Power p1
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Common household power ECE 2210 AC Power p2 f = . 60 Hz Neutral, N Line, L black, 120V V rms . 120 V white ϖ = . 377 rad sec (also ground) V p = = . V rms 2 170 V T = . 16.67 ms Ground, G, green What about other wave shapes?? v ( ) t Triangular v ( ) t Square V p V p t t v ( ) t 2 V p 2 v ( ) t 2 V p 2 average = V p 2 average = V p 2 3 t t V rms = V p 3 V rms = V p Works for all types of triangular and sawtooth waveforms Same for DC v ( ) t How about AC + DC ? . /|\ Vp \|/ V rms = . 1 T d 0 T t ( ) v ( ) t 2 V DC t = . 1 T d 0 T t . V p cos ( ) . ϖ t V DC 2 = . 1 T d 0 T t . V p cos ( ) . ϖ t 2 . . 2 . V p cos ( ) . ϖ t V DC V DC 2 = . 1 T d 0 T t . V p cos ( ) . ϖ t 2 . 1 T d 0 T t . . 2 . V p cos ( ) . ϖ t V DC . 1 T d 0 T t V DC 2 - - - zero over one period - - - = V rmsAC 2 0 V DC 2 = V rmsAC 2 V DC 2 ECE 2210 AC Power p2
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ECE 2210 AC Power p3 rectified average V ra = . 1 T d 0 T t v ( ) t sinusoid: V rms = V p 2 I rms = I p 2 V ra = . 2 π V p I ra = . 2 π I p triangular: V rms = V p 3 I rms = I p 3 V ra = . 1 2 V p I ra = . 1 2 I p square: V rms = V p I rms = I p V ra = V rms = V p I ra = I rms = I p Most AC meters don't measure true RMS. Instead, they measure V ra , display 1.11V ra , and call it RMS. That works for sine waves but not for any other waveform. waveform + DC V rms = V rmsAC 2 V DC 2 Some waveforms don't fall into these forms, then you have to perform the math from scratch 0 1 2 3 4 5 6 7 8 9 10 11 12 6 4 2 2 4 For waveform shown v(t) (volts) The average DC ( V DC ) value = . . 2 V ( ) . 4 ms . ( ) . 5 V ( ) . 2 ms . 6 ms 0.333 V time (ms) The RMS (effective) value 0 1 2 3 4 5 6 7 8 9 10 11 12 5 5 10 15 20 25 v(t) 2 (volts 2 ) Graphical way = . .
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