Revision: December 13, E Main Suite D Pullman, WA (509) Voice and Fax
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1 .9.1: AC power analyss Reson: Deceber 13, E Man Sute D Pullan, WA ( Voce and Fax Oerew n chapter.9.0, we ntroduced soe basc quanttes relate to delery of power usng snusodal sgnals. We saw that power dsspated by a load (essentally, any energy whch s conerted to nonelectrcal energy such as heat or work s the aerage power. Reacte power results fro energy whch s stored by capactors and nductors n the load and s then returned to the source wthout dsspaton. n ths chapter, we contnue our study of AC power analyss. We wll ntroduce the concept of the root-ean-square ( alue of a sgnal as a way to represent the power of a te-aryng sgnal. We wll also ntroduce coplex power as a way to conenently represent both aerage and reacte power as a sngle coplex nuber. We also ntroduce power factor as a way to represent the effcency of the transfer of power to a load. Before begnnng ths odule, you should be able to: Defne power n ters of oltage and current (Chapter 1.1 Express snusodal sgnals n phasor for (Chapter.7.1 Calculate pedances for resstors, capactors, and nductors (Chapter.7.3 Use pedances for electrcal crcut analyss (Chapter.7.4 Defne nstantaneous, aerage, and reacte power. After copletng ths odule, you should be able to: Defne real power, reacte power, and coplex power Defne sgnal alues and calculate the alue of a gen snusodal sgnal State, fro eory, the defnton of power factor and calculate the power factor fro a gen cobnaton of oltage and current snusods Draw a power trangle Ths odule requres: N/A Doc: XXX-YYY page 1 of 6 Copyrght Dglent, nc. All rghts resered. Other product and copany naes entoned ay be tradearks of ther respecte owners.
2 .9.1: AC power analyss alues: t s often desrable to copare dfferent types of te-aryng sgnals (for exaple, square waes s. trangular waes s. snusodal waes usng a ery sple etrc. Dfferent types of sgnals are often copared by ther (root-ean-squared alues. The general dea behnd the alue of a te-aryng sgnal s that we wsh to deterne a constant alue whch delers the sae aerage power to a resste load. The aerage alue, P, of an nstantaneous power p(t s defned to be: The power delered to a resste load by a constant oltage or current source s, fro chapter 1.1, (1 eff and V eff are the effecte (or constant current and oltage, respectely, appled to the resste load, R. t s our goal to equate equatons (1 and ( to deterne the effecte oltage or current alues whch deler the sae aerage power to a resste load as soe te-aryng waefor. Assung that a current s appled to a resste load, the nstantaneous power s p( t R ( t. Substtutng ths nto equaton (1 and equatng to equaton ( results n: ( R + T 1 eff T t 0 R ( t dt (3 Solng ths for eff results n: eff 1 T + T ( t dt (4 and the effecte current s the square root of the ean of the square of the te-aryng current. Ths s also called the (or root-ean-square alue, for rather obous reasons. A slar process can be appled to the oltage across a resste load, so that Equatng ths expresson to equaton (1 results n: p( t ( t. R V eff V 1 T + T ( t dt (5 So that the defnton of an oltage s equalent to the defnton of an current. page of 6 Copyrght Dglent, nc. All rghts resered. Other product and copany naes entoned ay be tradearks of ther respecte owners.
3 .9.1: AC power analyss Equatons (4 and (5 are applcable to any te-aryng waefor; the waefors of nterest to us are snusods, wth zero aerage alues (per equatons ( and (3. n ths partcular case, the alues can be calculated to be: and V V eff V (6 eff (7 where V and are the peak (or axu alues of the oltage and current waefors, per equatons ( and (3. Please note that equatons (6 and (7 are applcable only to snusodal sgnals wth zero aerage alues. The aerage and reacte powers gen by equatons (7 and (8 can be wrtten n ters of the alues of oltage and current as follows: and P V (8 Q V sn( θ θ (9 Apparent power and power factor n the preous subsectons, we hae seen that aerage power can be represented n ters of ether the agntudes of the oltage and current or the alues of the oltage and current and a ultplcate factor consstng of the cosne of the dfference between the oltage phase and the current phase: V P V t s soetes conenent to thnk of the aerage power as beng the product of apparent power and a power factor (abbreated pf. These are defned below: The apparent power s defned as ether V or V. (The two ters are, of course, equalent. Unts of apparent power are desgnated as olt-aperes (abbreated VA to dfferentate apparent power fro ether aerage power or reacte power. The power factor s defned as cos( θ θ. Snce cosne s an een functon (the sgn of page 3 of 6 Copyrght Dglent, nc. All rghts resered. Other product and copany naes entoned ay be tradearks of ther respecte owners. the functon s ndependent of the sgn of the arguent, the power factor does not ndcate whether the oltage s leadng or laggng the current. Thus, power factor s sad to be ether leadng (f current leads oltage or laggng (f current lags oltage.
4 .9.1: AC power analyss t should be ephaszed agan at ths pont that the oltage and current are not ndependent quanttes; they are related by the load pedance. For the syste of Fgure 1, for exaple, the oltage and current phasors are: V Z (10 L where V s the oltage phasor across the load, s the current phasor through the load, and Z L s the load pedance. Thus, the dfference between the oltage and current phase angles s sply the phase angle of the load pedance: θ θ θ. Therefore, the load pedance sets the power factor. f the load s purely resste, θ θ, the power factor s one, and the aerage power s the ZL sae as the apparent power. f the load s purely agnary (as wth purely nducte or purely capacte loads the power factor s zero and there s no aerage or real power absorbed by the load. Coplex power and power trangles Apparent power, aerage power, reacte power, and power factor can all be represented sultaneously n a sngle paraeter called coplex power. f we defne coplex power as: * V S (11 where V s the phasor representng the oltage, s the phasor representng the current, and the superscrpt * denotes coplex conjugaton (sply changng the sgn on the agnary part of the phasor. f we substtute the agntude and phase angle representatons for the phasors n equaton (11, we obtan (snce coplex conjugaton sple changes the sgn of the phase angle of a coplex nuber: S 1 ( V θ ( θ ( θ θ V (1 So that the coplex power S s a coplex nuber wth agntude V (or, equalently, V and phase angle θ θ. t s easy to see that the agntude of the coplex power s sply the apparent power. f we represent S n rectangular coordnates, we obtan: S P + jq (13 Where P s the aerage power, V P V page 4 of 6 Copyrght Dglent, nc. All rghts resered. Other product and copany naes entoned ay be tradearks of ther respecte owners.
5 .9.1: AC power analyss as before, and Q s the reacte power, V Q sn( θ θ V sn( θ θ also as before. The coplex power, real power, reacte power, and apparent power can be represented graphcally n the coplex plane as a power trangle, as shown n Fgure 1 below. Fgure 1. Power trangle. One portant thng to note about Fgure 1 s that ths fgure dffers fro a phasor dagra the coponents shown on the power trangle are not phasors, snce they do not prode agntude and phase nforaton about snusodal sgnals. The ectors shown n Fgure 1 are sply coplex nubers. We conclude ths chapter wth an exaple. Exaple 1: For the crcut below, (a fnd the aerage power delered by the source (b fnd the power absorbed by the resstor (c fnd the apparent, real, and reacte powers delered by the source (d sketch a power trangle for the source page 5 of 6 Copyrght Dglent, nc. All rghts resered. Other product and copany naes entoned ay be tradearks of ther respecte owners.
6 .9.1: AC power analyss Exaple 1 contnued The frequency doan crcut s shown below. -j V + - The current delered by the source s therefore: A ( 10 j8 Ω (a The aerage power delered by the source s therefore: ( 100V ( 7. 8A P cos( W (b The power absorbed by the resstor s the sae as the aerage power delered by the source, so P 305W, whch s the sae as R. 10 Ω ( 100V ( 7. 8A (c The apparent power s 391VA, the reacte power s ( 100V ( 7. 8A Q sn( VAR, and the real power s sply the aerage power, P 305W. (d The power trangle s shown below: page 6 of 6 Copyrght Dglent, nc. All rghts resered. Other product and copany naes entoned ay be tradearks of ther respecte owners.
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