4. AC Circuit Analysis

Transcription

1 4. A icui Analysis J B A Signals sofquncy Quaniis Sinusoidal quaniy A " # a() A cos (# + " ) ampliud : maximal valu of a() and is a al posiiv numb; adian [/s]: al posiiv numb; phas []: fquncy al numb; a() A cos (" + ) A A cos (" + ) $ T # # f T $ piod [s]; naual fquncy [Hz]; -θ/ω Tπ/ω/f A T + T A cos (# + " ) d A 0,707 A Effciv valu - ms valu (oo-man-squa valu)

2 A Signals sofquncy Quaniis a() A cos (ω + θ a ) b() B cos (ω + θ b ) ϕ θ a - θ b phas shif angl (a and b a ϕ ou of phas) 0 a() and b() a in phas; > 0 a() lads b() by (advancd); < 0 a() lags b() by (dlayd). a() A cos ω b() B cos (ω ϕ) b() a() ϕ θ b 0 /ω (" a # " b ) /ω a() lads b() by b() a() 0 a() and b() in phas a() b() ±π a() and b() in opposiion of phas b() a() ±π / a() and b() 90 ou of phas A Signals: Sinusoids and Phasos Eul dniy : j cos + jsin A j(ω+θ ) A cos (ω + θ) + j A sin (ω + θ) a() A cos (ω + θ) A j(ω+θ ) A jω wh : A A jθ A jθ A cos θ + j sin θ ( ) A M A a() ( ω + θ ) m

3 A Signals: Sinusoids and Phasos j( " + ) j j" j" [ A ] [ A ] [ A ] a() A cos (" + ) wh h phaso is a cun o a volag givn by h complx A : j A A A ( cos + jsin ) A Th is a biuniqu cospondnc bwn sinusoids a h sam fquncy and phasos. $ a() (" + ) #m A Sinusoid (im domain) A cos( + ") ( A cos" ) cos + (- A sin " ) sin Phaso (fquncy domain) cangula fom Exponnial fom Pola fom A A cos + j A sin A j A A A A Signals: Sinusoids and Phasos omplx plan psnaion A A wh a b A j' A cos A sin ' ' a A a + + b ( cos' + jsin ' ) j b -& b # ' an $, wh % a " a > 0 m b A a j A A 3

4 A Signals: Sinusoids and Phasos a() A cos ( + " ) b() B cos ( + " ) a b j [ A ] [ B ] j Uniqunss: Two sinusoids a h sam fquncy a qual if and only if hy a psnd by h sam phaso a() b() A B As jω is a complx numb which oas on h complx plan whn incass, a() is qual o b() a any, if boh h al and h immaginay pas of A and B A a + jb a qual : ja ja - b a() b() A jω B jω ( cosω + j sinω) A ( cosω + j sinω) B ja m A cosω A + j sinω A cosω B + j sinω B cosω A -sinω m A cosω B -sinω m B a() b() A B and m A m B. A Signals: Sinusoids and Phasos a() A cos ( + " ) b() B cos ( + " ) a b j [ A ] [ B ] j inaiy: Th lina combinaion (wih consan and al cofficins) of phasos psns h sam lina combinaion of sinusoids a h sam fquncy c a() + c b() ca + cb suls c fom h following laion : j j [ A ] c [ B ] [( c A c B) ] j + + 4

5 A Signals: Sinusoids and Phasos a() A cos ( + " ) Divaiv: f A is h phaso of a() Acos( + (ω), ") h faso of h im divaiv of a(), da() d (ω) d d [Acos(+")], is givn by jωa suls d d fom h da() d a jω A; [ ] j A d a() jω ( jωa ) - ω A d following laion : ja j d j( ) d { [ A + " ]} { [ A ]} [ A cos( + " )] d d - A sin( ) [ j A ] [ ] j( + " ) j A j + " m A A icui Analysis n h cicui of h figu fom h lmn quaion i suls : v() i() v() di d + n a lina cicui a sinusoidal cun i() i(')d' + i() cosponds o a sinusoidal volag v() a h sam fquncy and diffn phas. dv d i d d + d i + d i v() V cos(ω +θ v ) V jω i() cos(ω +θ i ) jω jωv jω (jω ) jω + + jω jω + jω wh : jθ V V V ; jθ. jωv jω -ω jω + jω jω + jω jωv jω -ω + jω + jω jωv jω -ω + jω + jω jωv -ω + jω + 5

6 A icui Analysis i() V + j ω ω v() V Z o V Z - Elmn quaion in h fqncy domain Fom h dfiniion of h ffciv phasos i is V V and Th impdnc (complx numb) is h aio of h paso volag V and h phaso cun. wh Z + j ω ω o Z + j X sisanc X aanc, acanc, X X + X ; X ω, X - ω Th impdnc is masud in ohm [Ω]. Th cipocal of impdnc is h admianc: Y Z ( S uni : simns [S]) v() V cos(ω +θ V ) V jω i() cos(ω +θ ) jω wh V V jθ V ; jθ V Z A icui Analysis wh Z +jx o Z Z jθ Z ; Z X + X, θ Z an - V Z V Z j(θ V - θ Z ) jθ m V V Z V V + X + ω- ω X ω- θ θ V - θ Z θ V - an - θ - ω V an- 6

7 A icui Analysis Phas shif v() V cos(ω +θ V ) V jω i() cos(ω +θ ) jω wh V jθ V V ; jθ v() i() V V j(θ V -θ ) V jϕ Z Z jθ + jx Z Z V jϕ m X Z ϕ Thfo h phas shif angl ϕ θ V -θ bwn cun and volag phasos is also h angl of h load impdnc. A icui Analysis Whn assuming as h fnc : θ v 0 (his wihou any loss of gnaliy, sinc all phas angls will b fncd o h souc volag's angl) ϕ θ v - θ i - θ i v() V cos ω V jω, V V i() cos(ω ϕ) jω, -jϕ V Z wh Z + j ω ω + jx is also : Z Z jθ Z ; Z X + X ; θ Z ϕ an - V Z V Z j(θ V - θ Z ) jθ -jϕ V Z V V + X + ω- ω X ω- ϕ θ Z an - ω an- m ϕ V 7

8 A icui Analysis icui Analysis v n m i 0 v 0 (i ) ngo-diffnial Equaions f Tansfomaion fom h im domain o h fquncy domain (Sinmz ansfom) n m V 0 V 0 f ( ) Algbaic quaions Soluion and nvs Tansfomaion fom h fquncy domain o h im domain A icui Analysis Th siso: Z V v() Vcos ω V jω, V V i() cos(ω ϕ) jω, -jϕ V Z wh Z V Z V -jϕ wh V X ϕ an - 0 m v() V i() i() cos(ω ϕ) V cos ω v() i() V v() and i() in phas 8

9 A icui Analysis Th nduco: Z j ω V v() Vcos ω V jω, V V i() cos(ω ϕ) jω, -jϕ v() V Z wh Z jω i() V Z V ω -jπ V -jϕ wh ω ϕ an - i() cos(ω ϕ) v() " # ω 0 π V ω cos ω - π V ȷ ) m V i() and v() a π/ ou of phas i() lags v() by π/ (dlayd) A icui Analysis Th apacio: Z - j ω V v() Vcos ω V jω, V V i() cos(ω ϕ) jω, -jϕ v() V Z wh Z -j ω V Z V ω j π V ω -jϕ wh ϕ an - ω 0 - π i() cos(ω ϕ) V ω cos ω + π v() " & i( )d V - $% ȷ /0 m i() V i() and v() a π/ ou of phas i() lads v() by π/ (advancd) 9

10 A icui Analysis Sis sonanc v() V cos ω V jω, V V i() cos(ω ϕ) jω, -jϕ V Z -jϕ con ϕ ω ( ω ) V + ω- ω ω ω- ( ) an - A h sonanc fquncy ω 0, fo which h induciv and h capaciiv acanc in si a qual in magniud, h oal acanc is zo. Thfo i follows ha: - h impdnc is only sisiv, - h cun ampliud/ffciv valu (ω 0 ), (ω 0 ) a maximual (his is h moivaion of h m "sonanc"). - h volag and h cun a in phas. ω 0 - ω 0 0 ω 0 ϕ i() v() (ω) ω 0 ω 0 ω ω A icui Analysis Sis sonanc Z & # + j$ ' - + j( X + X ) % ' " A ' ' 0 i is X - X. V Z + j X + j X : i() v() Fo a givn " h volag phaso # is givn by h vco sum "+jx "+jx " V j X j X j X V j X j X V ω < ω 0 ω ω 0 ω > ω 0 j X 0

11 A icui Analysis Paalll sonanc: Z Z Z Z + Z, dov Z j ω, Z -j ω Z -j ω - ω ( ω ) V ω - ω A h sonanc fquncy ω 0, as h induciv and h capaciiv acanc a qual in magniud, h paalll of a capacio wih an induco has a oal acanc qual o infini. Thfo i follows ha: - h oal impdnc is infini; - h oal cun flowing hough h cicui is zo. - h cuns in h induco and h capacio a: - X V ω 0 ω ω 0 - ω 0 0 ω 0 and a ω ω j V A icui Analysis mpdanc onncions Sis of impdancs Z k q Z k Z Z Z Z n Paalll of impdancs Z q k Z k Z Z n Z Δ Z Y Wy Dla onncion Z Y Z Δ ZΔ Z Δ + Z Δ + Z Δ3 Z Δ Z Y3 ZY ZΔ3 Z Δ Z Y ZY + Z Y ZY3 + Z Y3 ZY Z Y3 3

12 Pow in A icuis p() v() i() Wih h assumpion : θ v 0 v() Vcosω; i() cos(ω ϕ) ( ) cos ϕ cos ω + sin ϕ sin ω i a () + i () i() cos ω cos ϕ + sin ω sin ϕ i a () cos ϕ cos ω; in phas cun, i () sin ϕ sin ω; aciv cun (π / ou of phas). Th isananous pow is h sum of h in phas isananous pow p a () and h isananous aciv pow p (). p() v() i() v() i a () + v() i () p a () + p () v() i() v() VMcos# i() cos( # " ) v() i() M Pow in A icuis nsananous in Phas and aciv Pow p a () v() i a ( ) Vcosω cos ϕ cos ω V cosϕ cos ω p a () is h in phas isananous pow. is always posiiv. Thby i is a pow flowing ino h cicui lmn and is dissipad by h sisanc of h lmn. This pow is consumd and hfo i is uilizd by h cicui lmn ha is h cicui load. p () v() i () Vcosω sn ϕ sn ω V sin ϕ sin ω V sin ϕ sin ω p () is h isananous aciv pow. s avag valu is qual o zo. cosponds o an ngy flowing ino h cicui lmn fo half piod [h piod of p is T/ π /(ω )] and ousid of i fo h nx half piod and hnc in and ou of h mmay lmns (inducanc and capacios). sananous in phas pow : p a () V cos cos " i () p a() v() i a() sananous aciv pow: p () V sin sin " v() p ()

13 Avag Pow Pow in A icuis Th avag pow P (said also al pow) is h avag of h isananosus pow ov on piod T π /ω. P T 0 +T p(') d' 0 T 0 +T as p a () V cos ϕ cos ω, i is : P V cos ϕ V cos ϕ p a (') + p (') d' 0 T 0 +T p a (') d' 0 wh V and a h ffciv valus of h nsion and h cun spcivly [hy a h ms valus of v() and i(). "cosϕ " is said o b h pow faco]. Th avag pow is h pow uilizd by h cicui loads, i is consumd by h cicui. n h S Sysm of Unis h avag pow is masud in wa [W]. aciv Pow Pow in A icuis Th aciv pow Q is h maximal valu of h isananous aciv pow p (): Q p () Max sign ϕ ( ) V sin ϕ V sin ϕ wh p () V sinϕ sinω V sinϕ sinω Q is h maximal valu of h pow xchangd by a cicui lmn, o gnally by a load, wih h cicui ino which h cicui lmn is insd (o wih h nwok o which h load is conncd). Q can b posiiv o ngaiv dpnding on h sign of ϕ. Fo an induciv load Q is posiiv, fo a capaciiv load Q is ngaiv. n h S Sism of Unis Q is masud in vol - amp aciv [VA]. 3

14 Pow in A icuis omplx Pow V * N * V ( ) $ N V j& V -j& j & V -& * V V N P + jq # " V j% P + jq V cos% + jv sin % Sinc i is N V * Z N P + jq V Z * % Z ( + j X ) + j X $ # P " Q X ( h avag pow dpnds on h sisanc) ( h aciv pow dpnds on h acanc ) Pow in A icuis A Pow onsvaion Whn wo loads Z and Z a in si wih a volag souc (op figu), KT ylds V V, + V, N V * * V, ( + V, ) V *, + V,, * N + N Whn h loads a in paalll wih h souc (midl figu), K ylds, +, N V * V *, + *, ( ) V *, + V *, N + N Fo a cicui of l lmns, whh h loads a conncd in si o in paalll o boh conncions a psn (boom figu), h oal pow supplid by h souc quals h amoun of pow dlivd o h cicui lmns. N V * N P + jq l V,k *,k Nk k l k l k ( P k + j Q k ) P Q l V + " k X k k k (P is du o h sisancs) k l X k k (Q is du o h acancs) k Z Z V + " V + " Z Z 4

15 Appan Pow Pow in A icuis Th appan pow is h poduc of h ffciv volag (ms valu) and h ffciv cun (ms valu). N V V n h S sism of unis N is masud in vol-amp [VA]. Fom h dfiniion of h avag pow P i is: P N cosϕ o N P cosϕ Ths laions indica h lvanc of h pow faco cosϕ. Whn h load angl ncass abov zo cosϕ dcass blow and h avag pow P, which is h pow uilizd by h load, dcass blow h appan pow N, which is h pow capaciy ndd by h soucs (fo xampl h lcical pow capaciy of h gnaos of h lcical pow saions). Pow in A icuis Pow Faco Th pow faco is h cosin of h phas diffnc bwn volag and cun. Hnc i is h cosin of h angl of h load impdnc: cosϕ cos(θ V θ ). P V cos N cos P cos V Th valu of h pow faco cosϕ angs bwn 0 and uniy. Fo a pu sisiv load h volag and cun a in phas, so ha θ V θ ϕ 0 and h pow faco cosϕ. Thfo h appan pow is qual o h avag pow. Fo a puly aciv load θ V θ ϕ ±π/ and h cosϕ 0. n his cas h avag pow is zo. n bwn hs wo xm cass h pow faco is said o b lading o lagging. ading pow faco mans ha h cun lads h volag (capaciiv load). agging pow faco mans ha h cun lags h volag (induciv load). P N 5

16 Pow in A icuis Pow Faco P V cos N cos Q V P V sn an cos P P cos V N & cos' cos $ an % - Q # P " P a fixd P and V cos$ # ", N V cos$ n h lcical sysms h pow faco has o b as big as possibl o uiliz h maximal pa of h appan pow gnad (P N) and o duc h cun a a givn pow uilizd (P) and a a givn volag. Many uiliis qus o pay h pow faco uilizd whn is valu is blow 0.9. Pow in A icuis G V ' V U ln Moivaions fo h ducion of h phas shif angl f h cun is ducd, h pow los along h lin dcass (P ln ). As consan volag V ' V is quid by h n and as V ' V + ln, in od o duc h volag losss along h lin h cun has o b as low as possibl. Th appan pow, ha has o b gnad by h lcical pow saion, dcass wih h dcas of h cun. 6

17 Shif Angl ocion P V cos ϕ Q P an ϕ Q V sin ϕ Mos loads a induciv and opa a a low lagging pow faco. n od o duc ϕ o a dmind ϕ' (usually cosϕ' 0,9 ) a capacio is s in paalll wih h load. Th capaciy is givn by h h capaciiv aciv pow which compnsas h induciv pow : Q P an ϕ Q + Q P an ϕ ' Q P an ϕ' - an ϕ ( ) wh Q is h aciv pow o b cocd and Q -ω V ( ωv) is h aciv pow of h capacio uilizd fo h load faco cocion. Hnc i is P - " # & an ϕ - an ϕ ' ( ) $% ' an + an + & # ω V$% # Pow in A icuis V V ϕ ϕ U Tminology admianc ampliud divaiv fquncydomain olag olad impdnc linaiy invsansfom naualfquncy opposiionofphas paalllsonanc phas phasshifangl phaso ouofphas ammnza ampizza divazion,divaa dominiodllfqunz ssiniadodifas ssinanicipodifas impdnza linaià asfomaa invsa fqunzaciclica, fqunza opposiziondifas aniisonanza fas angolodisfasamno faso sfasaodi 90 ouofphas sfasaodi90,in quadauadifas piod adianfquncy acanc sonancfquncy oo mansqua valu,ms valu sissonanc sinusoid sinusoidalquaniy Sinmzansfom imdomain uniqunss piodo pulsazion,fqunza angola aanza fqunzadiisonanza valofficac isonanza sinusoid gandzzasinusoidal asfomaadi Sinmz dominiodlmpo unicià Dpamn*of*Elcical,*Elconic,*and*nfomaion*Engining*(DE)*7 Univsiy*of*Bologna 34 7