Self and Mutual Inductances for Fundamental Harmonic in Synchronous Machine with Round Rotor (Cont.) Double Layer Lap Winding on Stator

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Τρύφαινα Παπαντωνίου
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1 Sel nd Mutul Inductnces or Fundmentl Hrmonc n Synchronous Mchne wth Round Rotor (Cont.) Double yer p Wndng on Sttor

2 Round Rotor Feld Wndng (1) d xs s r n even r Dene S r s the number o rotor slots. Dene s r s the number o rotor slots between two poles, or the number o slots per pole. s r Sr

3 Round Rotor Feld Wndng () sr odd

4 MMF rom Round Rotor When s r s Even (1) F wth υ rom ll conductors υ turns (υ 1) γ r sr υ 1,,..., υ θ de d xs θde θd F F h 1,3,5... F cos( hθ ) F cos( h θ ) s r F h h h de h d 4 (υ 1) γ r υ sn h υ 1 Fh h Rotor wndng turns re typclly ll connected n seres.

5 Dene: MMF rom Round Rotor When s r s Even () s r υ υ 1 totl number o seres turns o rotor sr 4 1 ( 1) 4 ˆ υ γ r h Fh sn h sr υ h υ 1 h υ υ 1 where ˆ h k w, h eectve number o seres turns or h th hrmonc w, h sr 1 (υ 1) γ r sn s r υ υ 1 k h υ 1 υ wndng ctor or h th hrmonc

6 MMF rom Round Rotor When s r s Odd (1) F wth 1 rom conductors 1 turns 1 θ de F wth υ rom ll conductors υ turns ( υ 1) γ r υ,..., s r + 1 υ θ de F F where F F cos( hθ ) F cos( h θ ) h h h de h d h 1,3, ( υ 1) γ r s r Fh sn h + υ sn h h υ

7 MMF rom Round Rotor When s r s Odd () Dene: s r υ υ totl number o seres turns o rotor 4 1 Fh h h h + h where sr + 1 ˆ ( 1) 4 1 υ γ h sn sn sr + 1 υ υ 1 + υ υ ˆ h k w, h eectve number o seres turns or h th hrmonc 1 ( υ 1) γ r sr kw, h sn h + sn h sr + 1 υ υ 1 + υ υ wndng ctor or h th hrmonc

8 Mutul Inductnce between Sttor nd Rotor Feld Wndng I we pply current n rotor eld wndng, then when rotor s movng, the mgnetc eld n rgp rom rotor eld wndng s: ow, we cn clculte lux n hse A wndng rom eld current. λ B where 4 µ g d e ˆ cos( θ ) λ ˆ Φ cosθ rom eld wndng, pk Dene: B, pk Dl Φ, pk B, pk ge Dl 4 µ ˆ 8µ ˆ cosθ me g e λ 8µ Dl ˆ ˆ cosθ g e 8µ Dl ˆ ˆ where s g e me µ 4 me ˆ Dl ˆ ˆ g e θ s d ˆ k θ θ w β θ me cosθ me m cosθ θ me me θ Eectve number o turns on eld wndng. m

9 Sel Inductnce o Rotor Feld Wndng For the mgnetc eld rom rotor eld wndng s: ow, we cn clculte lux n eld wndng by ntegrtng on. pk, ˆ Φ λ Dl B pk pk,, Φ e e g Dl g Dl k ˆ 8 ˆ 4 ˆ µ µ λ ) cos( ˆ 4 d e g B θ µ e pk g B ˆ 4, µ ˆ 8 g Dl e m µ λ where θ d m l +

10 Stedy Stte Anlyss o Synchronous Mchne wth Round Rotor

11 Termnl Voltge or Round Rotor Motor + c b c b s s s c b dt d R R R R v v v v λ λ λ λ + + c b When + + c b me s me s me s me s s me s s me s s c b ) 3 cos( ) 3 cos( cos ) 3 cos( ) 3 cos( cos θ θ θ θ θ θ λ λ λ λ

12 Round Rotor Motor t Stedy Stte dλ v dt dθme At stedy stte ωe me et + r dt d v R s + s s Iωe sn ( ωet+ φr ) dt R s+ λ s + s cosθme θ ω φ d jφr + jωet R s + s + Re s Iωee e dt j et v Re e ω j et Re I e ω jφr + E I ω e et ( V φ ) ( A ) A s e V R I + jx I + E φ s A s A A X ω s e s s 8µ Dl ˆ ˆ ge

13 Round Rotor Genertor t Stedy Stte Motor V φ R I + jx I + E s A s A A Genertor V φ R I jx I + E s A s A A v dλ v R s+ dt λ ( ) + cosθ s s me + dt d jφr + jωet v R s s Re s Iωee e j et Re( e ω j et V φ ) Re( I Ae ω ) s 8µ E A Dl ˆ ˆ ge

14 Open Crcut Voltge (1) Assume the rmture wndngs re open crcut, the mgnetc eld n the r gp comes rom the eld wndng only. Bg rom eld wndng B, pk cos( ωet θ + φr ) B, pk cos( θ ωet φr ) B Dl From otes Flux nkge n hse Wndng β ( ω e t + φr ), pk Φ, pk rom eld wndng Φ, pk cos Φ, pk ( ω t φ ) λ ˆ β ˆ cos + dλ ( ) ˆ sn( ) ˆ dφ E t Φ ω ω t+ ϕ + cos( ω t+ ϕ ) dt dt rom eld wndng, pk A, pk e e r e r dφ, pk At stedy stte, we hve dt e r E A ( t) ˆ ˆ Φ Φ, pk, pk ω sn( ω t ω cos( ω t e e e e + φ ) r + φr + ) hsor o E A (t) s: E A ω ˆ Φ e, pk e j( ϕr + )

15 Open Crcut Voltge () E A s ω e s I e j( φr + ) 8µ ˆ ˆ Dl g e B E A ω ˆ Φ Φ, pk e, pk B r, pk Dl j( φr + ) 4µ ˆ I cos( θ θ ) r me g e e We cn nd out tht they re the sme. λ ˆ Φ, pk, pk E µ Dl ˆ ˆ E Dl ˆ ˆ 1 8 I g e 8 r r ( ) ( ) j φ + j φ + A ωe I e ωλ e, pke g e A, rms e e, pk

16 Volt-Second Blnce E ˆ Φ 4.44 ˆ Φ A, rms e, pk e, pk E ˆ Φ ˆ Φ A, rms or:, pk 4.44, pk e λ, pk Exmple: 6Hz genertor s to be operted t 5 Hz, then the opertng voltge must be derted to 5/6 o ts orgnl vlue.

17 Induced hse Voltge For the net mgnetc eld, t stedy stte: Bnet Bpk cos( ω et θ + φ net ( t ) λ ˆ Φ cos ω + φ, net pk e net ) Φ pk B pk Dl The stedy stte phse voltge n rmture phse A wndng s: dλ ( ) ˆ sn( ) ˆ Vφ t Φ pkωe ωet + φnet Φ pkωe cos( ωet + φnet + ) dt j( ϕnet + ) ˆ V ω Φ e hsor ϕ e pk The rms phse voltge s, V ˆ Φ 4.44 ˆ Φ φ rms e pk e pk λ pk

18 Y Connecton (Genertor)

19 Connecton (Genertor)

20 Φ pk Exmple 1 For 3 phse, 4 pole, 4 slot, 5/6 ptch mchne wth double lyer lp wndng, the pek mgnetc led ntensty n the rgp s.45 T. There s one slot skew. The mchne sht speed s 8 rpm. The sttor nner dmeter s.5 m. The mchne length s.3 m. There re 1 turns per col. All the turns re connected n seres. The three phse cols re Y connected. (1)Wht s the rms phse voltge o the mchne? ()Wht s the rms termnl voltge o the mchne? V ˆ Φ 4.44 ˆ Φ ϕ, rms e pk e pk B Dl pk 4 ˆ k w q q S m λ pk c ACmchne1.m

21 Exmple For smple pole, 3 phse, Y connected mchne (sngle lyer wndng) shown n the gure, the pek mgnetc eld ntensty n the rgp s.t. There s no skew. The mchne sht speed s 36 rpm. The sttor nner dmeter s.5 m. The mchne length s.3 m. There re 15 turns n the col. (1)Wht s the rms phse voltge o the mchne? ()Wht s the rms termnl voltge o the mchne? V ˆ Φ 4.44 ˆ Φ ϕ, rms e pk e pk λ pk Φ pk B Dl pk For ths exmple: ˆ c (ote: Ths s sngle lyer wndng.) ACmchne.m

22 Voltge nd Speed Regulton Voltge regulton: VR V nl V V l l 1% Speed regulton: Or: SR SR n ω nl n nl ω l n ω l l l 1% 1%

23 AC Mchne Ecency out η 1% n out n loss

24 AC Mchne oss Mechnsm 1. Electrcl or copper losses (I R losses). Core losses 3. Mechncl losses 4. Stry or mscellneous losses

25 Electrcl or Copper oss Sttor Copper oss (SC): 3I R SC A s Rotor Copper oss (RC): I R RC F F

26 Core, Mechncl nd Stry osses

27 AC Genertor ower Flow power converted rom mechncl to electrcl T ω conv em m

28 AC Motor ower Flow power converted rom electrcl to mechncl T ω conv em m

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