Electrical Machines I. Week 5-6: Singly Excited Systems

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1 Elctrical Machins I Wk 5-6: Singly Excitd Systms

2 RECALL REMEMBER.!! Enrgy stord in linar and non linar systms Rlationship btwn forc and stord nrgy in magntic systms Enrgy and Co-nrgy

3 Singly Excitd Magntic Fild Systm: Forc and torqu Linar systm involvs an objct moving from on point to anothr in a straight lin Rotational systm involvs an objct rotating about an axis Forc: Torqu: Distanc (displacmnt): Distanc (displacmnt): Vlocity: Mass: Angular vlocity: Momnt of inrtia:

4 Singly Excitd Magntic Fild Systm: Forc and torqu Linar systm Powr: Nwton s law of motion: Rotational systm Powr: Nwton s law of motion: Opposing forc du to spring and friction for xampl Opposing forc du to load torqu and friction for xampl

5 Singly Excitd Magntic Fild Systm: Prviously w hav calculatd th forc acting on a plungr as a function of th systm variabls as follows: 1 F i dl dx F W x m It is our objctiv today is to calculat th forc or torqu acting on a singl xcitd rluctanc machin in trms of systm variabls A rluctanc motor is an ac synchronous motor whos rluctanc changs as a function of angular displacmnt. Owing to its constant spd opration, it is commonly usd in lctric clocks, rcord playrs, and othr prcis timing dvics Th word synchronous actually man xisting or occurring at th sam tim and in lctric machins man that th motor rotats with a spd that is rlatd to th supply frquncy

6 Singly Excitd Magntic Fild Systm: Rluctanc motor Excitation(fild) winding I V s Rotor axis What is th rquird torqu to b abl to mak this motor rotat in trms of systm variabls? ω m Stator magntic axis Rotor Singl phas, pol rluctanc motor : Mchanical spd and is NOT EQUAL to π

7 Singly Excitd Magntic Fild Systm: Rluctanc motor Q F W x m T Wm θ QWm 1 Li W will study th rotor movmnt from 0 to 360 and startd to plot th inductanc and rluctanc variation with tim ϕ L, (a) (b) (c) (d), R min ϕ,l, R ϕ min Lmin,, R ϕ L,, R min

8 Singly Excitd Magntic Fild Systm: Rluctanc motor Whn th magntic axs of th rotor and th stator ar at right angls to ach othr (th quadratur or q- axis position), th rluctanc is imum, lading to a minimum inductanc. As th rotor rotats with a uniform mchanical spd th inductanc gos through ima and minima Maximum inductanc Minimum rluctanc sinc this is a two pol machin Minimum inductanc Maximum rluctanc On lctrical cycl of half on mchanical cycl of

9 Singly Excitd Magntic Fild Systm: Rluctanc motor If w try to writ th gnral quation of this inductanc as a function of thta it will b in th form of: Offstshift Pak inductanc valu θ is bcaus complt rotations ar mad in th 360..

10 Singly Excitd Magntic Fild Systm: Rluctanc motor.. Q QT W m 1 W θ m L ( θ) i T 1 L T i θ 1 i d q 1 This is a mchanical angl ( ) ( ) L L *sinθ Rplac th variabl Whr is th initial position of th rotor s magntic axis with rspct to th stator s magntic axis.

11 Singly Excitd Magntic Fild Systm: Rluctanc motor 1 ( L L ) sin ( ω t+δ) T i d q m Sinc th currnt is sinusoidal and can b xprssd as: T 1 I 1 Qcos a Qsinacosb i I cosωt ( ) L L cos ( ωt) ( ω t+ δ) d ( 1+ cosa) sin q m ( a+ b) + sin( a b) s s If you assum it as a sin function its totally corrct as wll. It is prfrrd that w assum that th supply voltag is sinusoidal and sinc th coil is inductiv, currnt wavform will lag th sinusoidal voltag by 90. This mans that th currnt will b a cosin function

12 Singly Excitd Magntic Fild Systm: Rluctanc motor T 1 I 4 ( L L ) d q sin sin ( ω ) ( ( ) ) mt+ δ 0.5 sin ωst ωmt δ ( ( ωt+ ω t) + δ) s m This big quation implis that for this rluctanc machin to b abl to rotat th avrag valu of th torqu trms of this quation MUST NOT qual to zro 1. First trm: avrag is zro as it is priodical so no torqu is producd by this trm. Scond trm: if w ar abl to mak, this trm will b r-writtn as This condition will produc an avrag torqu 0.5sin ( δ)

13 Singly Excitd Magntic Fild Systm: Rluctanc motor T 1 I 4 ( L L ) d q sin sin ( ω ) ( ( ) ) mt+ δ 0.5 sin ωst ωmt δ ( ( ωt+ ω t) + δ) s m 3. Third trm: if w ar abl to mak, this trm is discard (sam as th scond trm but with a ngativ sign which dos not imply) Thus th condition which will succssfully driv this motor will b rprsntd as T 1 I 8 ( L L ) sin( δ) d q [ ] Givn that th machin Maximum torqu

14 Singly Excitd Magntic Fild Systm: Rluctanc motor T 1 I 8 ( L L ) sin( δ) d q [ ] Givn that th machin Th torqu sign dpnds on th valu of δ, if δ is (+v), T will b (-v) in this cas, th torqu is opposit to th dirction of rotation. i.. th machin acts as gnrator. if δ is (-v), T will b (+v) in this cas, th torqu is sam dirction of rotation. i.. th machin acts as motor. Avrag torqu will b qual to zro if This condition happns if th rotor is cylindrical (not salint) Maximum torqu occurs whn,

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