Chaper 7 AC ower and Three-hae Crcu
Chaper 7: Oulne eance eacance eal power eacve power
ower n AC Crcu
ower and Energy Gven nananeou power p, he oal energy w ranferred o a load beween and : w p d The average power he average rae of energy ranfer defned a: w. p d
Average ower Le T and for any neger ulple of he perod of p, he average power over T he ae a he long er average by leng. The long er average w T p d T T
Average ower of a erodc Funcon Suppoe p con of a conan coponen and a perodc funcon, long er average equal o he conan coponen. ax n
Average Dpaed ower The average power dpaed by a reor wh a peak curren and a peak volage : /, co co φ ω φ ω
Average Dpaed ower Wh an arbrary load: φ coθ Source Supple θ θ φ coθ Load eurn
Average Dpaed ower The average power dpaed by he load :, co / co / θ ω ω ω θ [ ] co co co co co co v v p v φ θ ω θ φ ω φ θ ω φ ω φ θ ω
Exaple 7. AC ower Calculaon 40 80 6.4 4.8 k j k C 5 40 5 40 60 5 40, 80 40 80 40 0 4.8 W W W k ω
For a conan curren hrough a reance: T p d T T T d
oo Mean Square alue An effecve conan curren o wh repec o power dpaon ha he followng for: T d T For a perodc curren, he effecve curren equal o he roo ean quare value. The roo-ean-quare defned a T d T Lkewe, he value of a perodc volage T v d T Snuodal Snuodal
oo Mean Square alue ω ω.
Exaple 7.: MS alue of a Half- ecfed Wave n ω for 0 T 0 for < < T T T / T 0 /, n π d T 4 / 4
Maxu ower Tranfer v. Maxu Effcency Eff jx jx,, S X X X X >, 0 For axu power ranfer and
Maxu ower Tranfer When jx he pedance are ached for axu power ranfer. * Maxu avalable power 50% effcency: ax / 4
Maxu ower Tranfer f he rao X/ fxed bu can be adjued, he axu value of. coθ, X nθ Le d d coθ coθ 0, we have X nθ
Exaple 7.3: ower Tranfer fro an Ocllaor., 6 j8 0kΩ 53. 0 Cae : Mached load pedance ax * 6 j8kω. 60µ W, Eff 50% 4 6 Cae : X 4 5.µ W,, 7 4 j7 c 5c 6.3 0kΩ 6.3 Eff 0 6.% 0
ower Sye
Load n a AC ower Sye For ac power ye operang a a fxed frequency e.g., p 60Hz, frequency-dependen effec can be gnored. n h cae, any load pedance can be wren a: θ jx coθ j nθ
haor n MS φ v For Snuod: φ φ v θ
eal ower and eacve ower nananeou power : [ ] p co co θ ω φ θ v p p [ co ω φ v ] Qn ω X eal power average aborbed power n W: co θ φ v eacve power rae of energy exchange n Ar: Q n θ X olage-apere reacve
eal ower and eacve ower p p p X [ co ω φ v ] Qn ω φ v
nducve v. Capacve For a ngle nducor : For a ngle capacor : Q L Q C ωl L C ωc L ωl ωc C A load wh nducve reacance X>0, Q>0 A load wh capacve reacance X<0, Q<0 eacve power ncreae he curren o acheve he ae average power wang power
Exaple 7.4: ower-tranfer Effcency 4-j8 6-j8 kw Eff kar Q kw A L L 3.6 67%, 3 7. 30 8 5.4 30 6 30 300 9% 0 3.7, 37 3.6 0 300 no capacor, f kw W A L
Coplex ower Apparen power n A: Q Coplex power: S * θ jq Magnude of he coplex power apparen power. S The coplex power obey he conervaon law. n oher word, when everal load are conneced o he ae ource, he oal coplex power fro he ource he ae a he u of he coplex power of he load.
ower Facor ower facor : pf / S coθ for a pave load, 0,0 pf X S / pf >.
ower Facor A load wh pf draw nu ource curren. f he load nducve, Q>0 and θ>0, a laggng power facor curren phaor lag he volage phaor. f he load capacve, Q<0 and θ<0, a leadng power facor curren phaor lead he volage phaor. Connecng a capacor n parallel wh an nducve load can ake pf power-facor correcon. Q ± S ± S pf
Table 7.
Table 7.
Exaple 7.5: Degnng ower-facor Correcon A S ka Q S Q Q Q A k kar Q ka k S A k kar Q ka k S 84 500, 9.8, 50 500 5, 7, 5 0.96 4 60 500 80, 64, 80 0.6 48
Exaple 7.5: Con. ower facor correcon: c 0, Q c Qc C ω c c 57kAr 605µ F 7kW, Q 7kW 44A 500 0
Exaple 7.6: provng ower-tranfer Effcency only correc he load 4800 80A 4 j55 0 j40 3580 S 4 j55 L 0 83% 4 0 54k j35kar
Exaple 7.6: Con. 96% 00 4 00 3 7 5 04 4570 00, 45.7 5 04 4800 5 04 5 4 00 00 0 000 afer correcon 000 40 000 0 40 0 Add Ω kar j kw j S A j j j C j j C j Y C j L eq eq ω ω ω
Waeer v M v M v T v M v d T K φ ω φ ω φ ω φ ω γ >> co, co Le co, co angle Seady ae deflecon
Waeer e [ ] * γ KM T K M M M co T co ω φ co ω φ K M φ φ v v M d
Chaper 7: roble Se 4, 7,, 6, 9,, 6, 3, 34, 40