Introduction to AC Power, RMS RMS. ECE 2210 AC Power p1. Use RMS in power calculations. AC Power P =? DC Power P =. V I = R =. I 2 R. V p.

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1 ECE MS I DC Power P I = Inroducion o AC Power, MS I AC Power P =? A Solp //9, // // correced p4 '4 v( ) = p cos( ω ) v( ) p( ) Couldn' we define an "effecive" volage ha would allow us o use he same relaionships for AC power as used for DC power? P ave = = = eff = MS = = r oo of he Mean of he Square Use MS in power calculaions v( ) Square / ( v( ) ) d \ Mean (average) oo average or "effecive" power = p average = Sinusoids r ( v( ) ) d p cos( ω ) d p cos( ω ) d ( ) d cos( ω ) d = ECE AC Power p

2 Common household power f = 6 Hz Neural, N Line, L whie black, ω = 77 rad (also ground) sec = 667 Ground, G, green Wha abou oher wave shapes?? v( ) riangular v( ) ECE AC Power p r = r = 7 Square v( ) v( ) average = average = r = Works for all ypes of riangular and sawooh wavefor r = Same for DC How abou AC + DC? r ( v( ) ) d / \ p \ / v( ) DC p cos( ω ) DC d p cos( ω ) p cos( ω ) DC DC d p cos( ω ) d p cos( ω ) d DC zero over one period DC d = rac DC = rac DC ECE AC Power p

3 ECE AC Power p sinusoid: r = I r = I p recified average ra ra π v( ) I ra = I p π d riangular: r = I r = I p ra p I ra I p square: r = I r = I p ra = r = I ra = I r = I p waveform + DC r = rac DC Mos AC meers don' measure rue MS Insead, hey measure ra, display ra, and call i MS ha works for sine waves bu no for any oher waveform Some wavefor don' fall ino hese for, hen you have o perform he mah from scrach For waveform shown v() he average DC ( DC ) value (vols) 4 ( 4 ) ( 5 )( ) ime 6 () 4 he MS (effecive) value Graphical way 4 ( 4 ) 5 ( ) 6 = v() (vols ) MS MS ime () O MS ( v( ) ) d ( ) d ( 5 ) d 4 ( ) ( 5 ) 6 he volage is hooked o a resisor, as shown, for 6 seconds he energy is ransfered o he resisor during ha 6 seconds: MS P L P L W L W L P L 6 sec W L joule All convered o hea L 5 Ω ECE AC Power p

4 Use MS in power calculaions P = I r r = for esisors ONLY!! ECE AC Power p4 Capaciors and Inducors i C i L v L + + v C - - i C ( ) v C ( ) v L ( ) i L ( ) p( ) p( ) Average power is ZEO P = Average power is ZEO P = Capaciors and Inducors DO NO dissipae (real) average power eacive power is negaive eacive power is posiive Q C I Cr Cr Q L I Lr Lr = I Cr ω C = Cr ω C = I Lr Lr ω L = ω L If curren and volage are no in phase, only he in-phase par of he curren maers for he power-- DO PODUC I cos( θ) I I I cos( θ) "Leading" Power "Lagging" power Inducor dominaes Capacior dominaes ECE AC Power p4

5 eal Power P = I r r = oher wise for resisors P = r I r cos( θ ) = I r Z cos( ) r θ Z cos( θ) P = "eal" Power (average) = r I r pf = I r r Z pf pf Z eacive Power oher wise capaciors -> - Q Q C I Cr X C = inducors -> + Q Q L I Lr X L = Cr X C X C = ECE AC Power p5 unis: was, kw, MW, ec ω C Lr X L = ω L X L and is a negaive number and is a posiive number Q = eacive "power" = r I r sin( θ) unis: A, ka, ec "vol-amp-reacive" Complex and Apparen Power complex congugae / S = Complex "power" = r I r = P jq = r I r /θ unis: A, ka, ec "vol-amp" NO I r Z NO r Z S = Apparen "power" = S r I r = P Q unis: A, ka, ec "vol-amp" Power facor pf = cos( θ ) = power facor (someimes expressed in %) < pf < θ is he phase angle beween he volage and he curren or he phase angle of he impedance θ = θ Z θ < Load is "Capaciive", power facor is "leading" his condiion is very rare θ > Load is "Inducive", power facor is "lagging" his condiion is so common you can assume any power facor given is lagging unless specified oherwise ransformers and moors make mos loads inducive Indusrial users are charged for he reacive power ha hey use, so power facor < is a bad hing Power facor < is also bad for he power company o deliver he same power o he load, hey have more line curren (and hus more line losses) Power facors are "correced" by adding capaciors (or capacive loads) in parallel wih he inducive loads which cause he proble (In he rare case ha he load is capaciive, he pf would be correced by an inducor) S (A) P (W) Q (A) Q (A) "Lagging" power P (W) ECE AC Power p5 "Leading" Power S (A)

6 ransformer basics and raings A ransformer is wo coils of wire ha are magneically coupled ECE AC Power p6 ransformers are only useful for AC, which is one of he big reasons elecrical power is generaed and disribued as AC ransformer urns and urns raios are rarely given, / s is much more common where / s is he raed primary over raed secondary volages You may ake his o be he same as N /N alhough in realiy N is usually a lile bi bigger o make up for losses Also common: : s ransformers are raed in A Boh MS ransformer aing (A) = (raed ) x (raed I), on eiher side Don' allow volages over he raed, regardless of he acual curren Don' allow currens over he raed I, regardless of he acual volage Ideal ransformers Iron-core ransformer primary secondary I I rare Z L N N Ideal: P = P power in = power ou (yff, Fig7) ransformaion of volage and curren N N = = I I common urns raio urns raio as defined in Chapman ex: a = N N, same as N = N Noe: some oher exs N N define he urns raio as: N Be careful how you and ohers use his erm ransformaion of impedance I I Z I You can replace he enire ransformer and load wih (Z eq ) his "impedance ransformaion" can be very handy Z eq Z eq = N N Z Z N ransformers can be used for "impedance maching" his also works he opposie way, o move an impedance from he primary o he secondary, muliply by: ECE AC Power p6 N N

7 Oher ransformers Muli-ap ransformers: ECE AC Power p7 Many ransformers have more han wo connecions o primary and/or he secondary he exra connecions are called "aps" and may allow you o selec from several differen volages or ge more han one volage a he same ime Isolaion ransformers: Allmos all ransformers isolae he primary from he secondary An Isolaion ransformer has a : urns raio and is jus for isolaion Auo ransformers: ari-ac: LD Auo ransformers have only one winding wih aps for various volages he primary and secondary are simply pars of he same winding hese pars may overlap Any regular ransformer can be wired as an auo ransformer Auo ransformers DO NO provide isolaion A special form of auo ransformer wih an adjusable ap for an adjusable oupu volage A Linear-ariable-Differenial-ransformers has moveable core which couples he primary winding o he secondary winding(s) in such a way he he secondary volage is proporional o he posiion of he core LDs are used as posiion sensors Home power Sandard oule connecions are shown a righ he lines coming ino your house are NO -phase hey are +, Gnd, - (he wo s are 8 o ou-of-phase, allowing for 4 connecions) Neural, N whie (also ground) Ground, G, green Line, L black, -Phase Power (FYI ONLY) Single phase power pulses a Hz his is no good for moors or generaors over 5 hp hree phase power is consan as long as he hree loads are balanced hree lines are needed o ransmi -phase power If loads are balanced, ground reurn curren will be zero Wye connecion: Connec each load or generaor phase beween a line and ground Dela connecion: Connec each load or generaor phase beween wo lines Wye Dela LN = LL I L I LL LL LN I LL = I L ECE AC Power p7

8 -Phase Power (FYI ONLY) Common -phase volages: 8 φ 48 φ 77 Apparen Power: S φ LN I L LL I LL LL I L ECE AC Power p8 Power: P φ LN I L pf LL I LL pf LL I L pf = S φ pf pf = cos( θ) eacive power: Q φ LN I L sin( θ) ec = S φ P φ an = LN / ο I a = I L /α I b = I L /α - ο A AB = LL / ο B bn = LN /- ο CA = LL /- ο = LL / 5 ο BA = LL /-9 ο I c = I L /α -4 o = I L /α + ο cn = LN /-4 ο = LN / ο C lower-case leers a source end A neural (ground a some poin) I A I A A N upper-case leers a load end AN CA I B AB B AN I AB I CA I BC BN, if balanced load N C BC BN B BN I B I C C I C N CN neural is no conneced a he load AN = BN = CN = LN = LL AB = BC = CA = LL LN I A = I B = I C = I L I LL I AB = I BC = I CA = I LL = o ge equivalen line currens wih equivalen volages Z Y = Z Z Z y I L ECE AC Power p8

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