U ANAYSS hapter Snusdal Alternatng Wavefrs and Phasr ncept Snusdal Alternatng Wavefrs and Phasr ncept ONNS. Snusdal Alternatng Wavefrs.. General Frat fr the Snusdal ltage & urrent.. Average alue..3 ffectve alues. Basc leents and Phasrs.. leents.. Average Pwer and Pwer Factr..3 ectangular and Plar epresentatn..4 pedance and Phasr Dagra..5 Krchhff s aws.3 Seres and Parallel rcuts.3. Seres rcuts.3. Parallel rcuts Snusdal Alternatng Wavefrs and Phasr ncept
. Snusdal Alternatng Wavefrs.. General Frat fr the Snusdal ltage & urrent A sn α A snωt Bascatheatcalfratfrthesnusdalwavefr A A α (rad) e snωt snωt snα snα Phasr elatns A sn θ θ A ( θ ) ( θ ) α A A sn θ θ A ( θ ) ( θ ) α A A sn( ω t θ ) A sn( ωt θ ) Snusdal Alternatng Wavefrs and Phasr ncept 3.. Average alue: algebrac su f the areas G (average value) length f the curve A A α (rad) G ÿ A snα he average value f a pure snusdal wavefr ver ne full cycle s zer. f the wavefr equatn s: then the average value s: a A snα A G A Snusdal Alternatng Wavefrs and Phasr ncept 4
..3 ffectve alues: e ~ Swtch Swtch ac ac surce dc dc surce he effectve value f an ac current s the equvalent dc current that delvers the sae pwer as the ac current. P ac ( sn ωt) ( sn ωt) sn ωt ( cs ωt ) ac he average pwer delvered s: P ac cs ωt P av ( ac) Snusdal Alternatng Wavefrs and Phasr ncept 5 quatng the average pwer delvered by the ac generatr t that delvered by thedcsurce, Pav( ac) Pdc dc dc. 77 Sae thng fr the vltage: dc eff. 77 eff. 77 n general, the effectve value f any quantty pltted as a functn f te can be fund by usng the fllwng equatn: eff ÿ dt area ( ) hs prcedure gves us anther desgnatn fr the effectve value, the rt-ean-square (rs) value. Snusdal Alternatng Wavefrs and Phasr ncept 6 3
. Basc leents and Phasrs.. leents espnse f the basc, and eleents t a snusdal vltage and current. esstr: v snωt snωt snωt v v ωt Fr a purely resstve eleent, the vltage acrss and the current thrugh the eleent are n phase Snusdal Alternatng Wavefrs and Phasr ncept 7 nductr: d d ω dt dt ( sn ωt) ω cs( ωt) sn( t ) v 9 ω X ω : reactance f the nductr (Ohs). nductve reactance s the ppstn t the flw f current, whch results n the cntnual nterchange f energy between the surce and the agnetc feld f the nductr. Unlke resstance, reactance des nt dsspate electrc energy. v v ωt Fr an nductr v leads by 9,r lags v by 9. Snusdal Alternatng Wavefrs and Phasr ncept 8 4
apactr: ω dv dt d dt ( sn ωt) ω cs( ωt) sn( t ) ω 9 ω X : reactance f the capactr (Ohs). ω apactve reactance s the ppstn t the flw f charge, whch results n the cntnual nterchange f energy between the surce and the electrc feld f the capactr. ke the nductr, the capactr des nt dsspate electrc energy. v v ωt Fr a capactr leads v by 9,rv lags by 9. Snusdal Alternatng Wavefrs and Phasr ncept 9 As a general rule fr crcut wth cbned eleents: f the current leads the vltage, the crcut s prednantly capactve, and f the vltage leads the current, t s prednantly nductve. f Hz(dc) ery hgh frequences ω f X ω X ω nstead f dfferentatn, t s pssble t use ntegratn dependng n the unknwn quanttes: ÿ vdt v ÿdt Snusdal Alternatng Wavefrs and Phasr ncept 5
.. Average Pwer and Pwer Factr n general the nstantaneus pwer s deterned by: p v n general, fr a snusdal vltage and current, P csθ eff eff csθ the average pwer s deterned by: Where θ s the phase angle between the vltage and the current. eff rs eff rs PF csθ ffectve vltage ffectve current Pwer Factr esstr nductr apactr P cs eff eff P cs 9 P cs9 Snusdal Alternatng Wavefrs and Phasr ncept..3 ectangular and Plar epresentatn ÿ A jb ectangular fr ÿ θ Plar fr Plar t rectangular ectangular t plar j A csθ B ÿ A B, θ tan B snθ θ xaple: v v sn sn v v v ( ωt θ) ( ωt θ ) θ θ θ θ A B A Use the calculatr drectly wth plar fr r cnvert v and v t rectangular fr, d the suatn and cnvert the result t plar fr. Snusdal Alternatng Wavefrs and Phasr ncept 6
..4 pedance and Phasr Dagra esstr: ÿ ÿÿ ÿ j r n phasr fr ÿ ÿ nductr: ÿ ÿÿ ÿ X 9 ω 9 jx r X ω n phasr fr ÿ ÿ 9 9 X apactr: ÿ ÿÿ ÿ X c 9 9 jx ω ω r X n phasr fr ÿ ÿ 9 9 X Snusdal Alternatng Wavefrs and Phasr ncept 3..5 Krchhff s aws ltage aw: the cplex (vectr) su f the ptental rses and drps arund a clsed lp (r path) s zer. Ο Ο rse Ο drps urrent aw: the cplex (vectr) su f the currents enterng and leavng a junctn s zer. Or the su f currents enterng a junctn ust equal the su f currents leavng the junctn. enterng leavng Snusdal Alternatng Wavefrs and Phasr ncept 4 7
.3 Seres and Parallel rcuts.3. Seres rcuts w eleents are n seres f they have nly ne pnt f ntersectn that s nt cnnected t ther current-carryng eleents f the netwrk. N ~ ~ θ Fr N pedances n seres the equvalent pedance s: N pedance apltude θ pedance angle Snusdal Alternatng Wavefrs and Phasr ncept 5 urrent n the crcut: θ θ (A) ltage acrss each eleent:,,, N N () Pwer delvered by each pedance : P (W) (,, N) Pwer delvered by the surce: P csθ P P P N (W) Snusdal Alternatng Wavefrs and Phasr ncept 6 8
xaple f Seres - rcut: 3Ω ~ X 4Ω 3 4 9 3 j 5 53. 3 4 ectangular fr Plar fr Seres - rcut pedance dagra: j X 4Ω 5Ω θ 53.3 3Ω Snusdal Alternatng Wavefrs and Phasr ncept 7 urrent: ltages: Plar fr: 53.3 5 53.3 53.3 3 6 53.3 () (A) 53.3 4 9 8 36.87 () ectangular fr: 36 j48 () 64 j48 () Apply Krchhff s vltage law: Ο ( 36 j48) ( 64 j48) j Phasr dagra: j 8 6 36.87 53.3 Snusdal Alternatng Wavefrs and Phasr ncept 8 9
Pwer: Methd : P cs θ cs( 53.3 ) W Methd : Methd 3: P 3 W P csθ P P csθ csθ 6 cs 8 cs 9 W Pwer Factr: PF P cs θ cs53.3.6 laggng pedance angle angle between nput vltage and current When current s laggng the vltage, the pwer factr s sad t be a: laggng pwer factr. Snusdal Alternatng Wavefrs and Phasr ncept 9 xaple f Seres -- rcut: 3Ω ~ 5 X 7Ω X 3Ω X 3 j7 j3 3 j4 ectangular fr 3 5 53. 3 9 X Plar fr 9 Seres -- rcut pedance dagra: j X 7Ω X -X 4Ω 5Ω θ 53.3 3Ω X 3Ω Snusdal Alternatng Wavefrs and Phasr ncept
urrent: 5 53.3 5 53.3 (A) ltages: Plar fr: ectangular fr: 53.3 3 3 53.3 () 53.3 7 9 7 36.87 () 53.3 3 9 3 43.3 () 3 8 j4 () 56 j4 () 4 j8 () Apply Krchhff s vltage law: Ο ( 8 j4) ( 56 j4) ( 4 j8) 5 j 5 Snusdal Alternatng Wavefrs and Phasr ncept Phasr dagra: j 36.87 53.3 e dan: v v v sn 3sn 7sn 3sn ( ωt 53.3 ) ( ωt 53.3 ) ( ωt 36.87 ) ( ωt 43.3 ) Snusdal Alternatng Wavefrs and Phasr ncept
Pwer: Methd : ( 53.3 ) 3 W P cs θ 5 cs Methd : P 3 3 W Methd 3: P csθ P P P csθ csθ csθ 3 cs 7 cs 9 3 cs 9 3 3 W Pwer Factr: P PF cs θ.6 laggng pedance angle angle between nput vltage and current Snusdal Alternatng Wavefrs and Phasr ncept 3 ltage Dvder ule: N ~ xaple: 3Ω ~ 5 X 7Ω Seres -- rcut X 3Ω jx jx jx jx jx jx jx jx 3 5 5 3 53.3 3 j(7 3) 5 53.3 7 9 5 35 9 7 36.87 3 j(7 3) 5 53.3 3 9 5 5 9 3 43.3 3 j(7 3) 5 53.3 Snusdal Alternatng Wavefrs and Phasr ncept 4
.3. Parallel rcuts w eleents, branches, r netwrks are n parallel f they have tw pnts n cn. Parallel pedances: N Adttance Y Y Y YN N Y a b tal pedance f parallel pedances s always less than the value f the sallest pedance Snusdal Alternatng Wavefrs and Phasr ncept 5 Fr tw parallel pedances: Fr three parallel pedances: 3 3 3 nductance: G (seens, S) Susceptance: B (seens, S) X B X (seens, S) j B B G Adttance Dagra Snusdal Alternatng Wavefrs and Phasr ncept 6 3
xaple f Parallel -- rcut: ~ 5Ω X 8Ω 93.6.56 Parallel --rcut X Ω Y G G. Y B B 9 9.5 9 X Y B B 3 9 X 9.5 9 j B G Y Y Y Y (. J ) ( J.5) ( J.5) 3. -j. 75.36.56 B B θ.56 Y 4.68. 56 Y.36.56 B Adttance Dagra Snusdal Alternatng Wavefrs and Phasr ncept 7 urrent: G 93.6.56 B B 93.6.56 4.68.56. 8.7.56 93.6.56.5 9.7 69.44 93.6.56.5 9 4.68.56 Apply Krchhff s current law: ( j) ( 7.57 j6.574) ( 4.9.955) (.643 j4.38) j Snusdal Alternatng Wavefrs and Phasr ncept 8 4
urrent Dvder ule: ~ N N ~ xaple: 5Ω X 8Ω Parallel --rcut X Ω 4.68.56 4.68.56 5 4.68.56 8 9 4.68.56 9 8.7.56.7 69.44 4.68.56 Snusdal Alternatng Wavefrs and Phasr ncept 9 Suary: Krchhff s law apples t ac crcuts the sae way t apples t dc crcuts. Seres and parallel crcut rules apply t ac crcuts the sae way they apply t dc crcuts. nstead f resstance pedance s used. Seres Parallel N N pedance adttance Y Fr any netwrk cnfguratn(seres, parallel r seres-parallel), the angle θ by whch the appled vltage leads the surce current wll be pstve fr nductve netwrk and negatve fr capactve netwrks. Snusdal Alternatng Wavefrs and Phasr ncept 3 5