EE 3410 Electric Power Fall 2003 Instructor: Ernest Mendrela. Electromechanical Energy Conversion Introduction to Electric Machines

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1 EE 34 Electrc Power Fall 3 Instructor: Ernest Mendrela Electroechancal Energy Conerson Introducton to Electrc Machnes. The ery rst experence wth electrc (lnear) otors An operaton o any electroechancal dece, n that nuber electrc achnes, t s electrc otors and generators, can be seen as: a) An pact o agnetc eld on a pece o erroagnetc ateral, b) An nteracton o two agnetc elds: agnetc eld o prary part and agnetc eld o secondary part; the secondary agnetc eld can be generated: - ether by a current lowng n a wndng (conductor), - or by peranent agnet. The achnes that belong to the group (a) are called the reluctance otors, and the achnes that operates on the bass o the ode (b) are the rest o the ajorty o electrc achnes, t s generators and otors. We wll dscuss these arous electrc achnes startng wth our ery rst experence wth peranent agnets. In electrc achnes the basc requreent or the prary agnetc eld s: t has to be n oton. It can oe wth: - translaton oton we call t a agnetc traellng eld, whch exsts n lnear achnes - rotary oton a rotatng agnetc eld, whch s n rotatng achnes - coplex oton - descrbed by two or three space co-ordnates, whch s generated n electrc otors wth two degrees o echancal reedo, e.g.: rotary-lnear otors wth helcal oton o the rotor, otors wth sphercal rotor X-Y lnear otors. In the ollowng sectons we wll concentrate on the lnear achnes. The rotatng achnes wll be seen as a geoetrcal conerson o the lat lnear structure nto the cylndrcal or dsc structure... Reluctance (lnear) synchronous otor Iagne, we hae the bar peranent agnet and we slde t under the table-top as shown n Fg.. Aboe s the ron bar, whch s pulled by the agnetc eld o peranent agnet. In ths pertcular case the agnetc traellng eld s obtaned by the peranent agnet, called the prary part or stator, whch s dren by our hand wth a speed. The, so-called, secondary part or rotor s ong slghtly behnd the agnetc traelng eld o the prary part but wth the sae speed. Snce t oes synchronously wth the agnetc traellng eld we call ths otor a synchronous otor.

2 The nae reluctance eans, that the reluctance o the agnetc crcut, whch the agnetc lux s closed through, wll change as the lux s ong on, because o nte length o a secondary part. Due to ths change the agnetc orce s produced that acts on the rotor. I the secondary part would be ery (nntely) long there would be no agnetc orce F x that dre the rotor. There wll be only attracte orce F y (see Fg.). Rotor (Fe) x Φ (agnetc lux) S (speed) Stator (peranent agnet) Fg. Lnear reluctance otor ored by ong peranent agnet and a pece o sold ron.. Peranent agnet (lnear) synchronous otor The secondary part can be n a or o peranent agnet as shown n Fg.. In ths case the agnetc orce actng on the rotor s an eect o nteracton o two agnetc elds: one produced by the stator and another one by the rotor. The rotor oes here synchronously wth the agnetc eld o the stator. Because o that, and, snce the secondary part s n or o peranent agnet the otor s called a peranent agnet synchronous otor. Rotor (peranent agnet) S x S Stator (peranent agnet) Fg. Lnear peranent agnet synchronous otor ored by the ong stator peranent agnet that s pullng the rotor peranent agnet So ar we analyzed the otors where the prary agnetc eld has been produced by the peranent agnet. Ths can be replaced by the electroagnet. Its col ay be suppled ro the dc oltage source as t s shown n Fg.3. The operaton o the otor

3 3 does not der ro the preous otor and to get the agnetc lux ong we ust push the electroagnet. The otor stll can be called peranent agnet lnear synchronous otor. rotor S S col suppled wth dc current stator Fg.3 Peranent agnet (lnear) synchronous otor: stator agnetc eld s produced by the col suppled ro the dc source.3 Synchronous (lnear) otor The peranent agnet o the secondary part can be replaced by the electroagnet as shown n Fg.4. In ths case the otor s called synchronous lnear otor. rotor F S g S stator Fg.4 Lnear synchronous otor: stator and rotor are electroagnets The rotatng counterpart o the lnear oton structure shown n Fg.4 s the achne (n ths partcular case otor) shown n Fg.5. The stator (prary part) electroagnets, suppled by the current represent a rotatng agnetc eld that rotates wth angular speed ω. The rotor electroagnet suppled by the current s dren by the torque T, slghtly behnd the stator agnetc eld by an angle δ but wth the sae speed ω. The agnetc orce, whch acts on the secondary agnet, depends on the utual dsplaceent o both parts. I ths dsplaceent s expresses n ters o the angle δ as shown n Fg.6, then the orce F changes, practcally, snusodally as shown n the graph n Fg.7.

4 4 stator ω T S δ rotor S Fg.5 Rotary synchronous otor The angle δ s counted ro the prary part to the secondary part. Thereore t s negate. The axu negate alue the orce reaches at angle (-9 ). The orce can be expressed by the uncton: ( δ ) or expressed n ters o Cartesan s coordnate: F = F ax sn () F π = F sn x τ ax () where τ s the agnetc pole-ptch (see Fg.4). F (T) S rotor stator δ δ δ 8 τ (pole ptch) S S 36 δ (x) - F - F ax (- T ) ax Fg.6 Lnear synchronous otor Fg.7 Force (torque) power angle characterstc o lnear (rotary) synchronous otor In case o cylndrcal structure shown n Fg.5 the rotor rotates synchronously wth the stator agnetc lux due to the torque T actng on the rotor. Its alue depends snusodaly on the angle δ as shown n Fg.7.

5 5. Force and torque n electrc achnes In electrc otors the electrc energy s conerted nto echancal energy. In electrc generator the process o energy conerson s reersed: a echancal energy s conerted nto electrcal energy. In both cases the agnetc eld (agnetc lux Φ - see Fg.8) s the edu n the electroechancal conerson process. rotor F g stator Φ Fg.8 Illustraton to electroechancal energy conerson Look at Fg.8. In the lnear synchronous otor the electrc energy s delered to the syste through the stator and rotor wndng ternals called electrcal ports. Ths energy s conerted to the energy o agnetc eld, whch s next conerted nto echancal energy. The entre process o energy conerson s shown scheatcally n Fg.9. For the generator the process s reersed: the echancal energy s delered to the rotor through a rotor shat and due to the agnetc lux generated by the rotor current t s conerted nto electrcal energy leang the syste through the stator wndng ternals (Fg.). W e e W Motor W W W e e Stator and rotor wndngs Magnetc lux Machne agnetc crcut Magnetc lux Machne echancal syste Rotor shat T ω Generator Fg.9 Dagra o electroechancal energy conerson; no power losses are ncluded Durng the process o energy conerson the power losses dsspate n the syste. In the rotor and stator wndngs a part o electrcal energy s conerted nto heat due to Ohc power losses n the wndng resstance. In the rotor and stator cores, part o eld energy s lost. In the echancal part o the syste part o echancal energy s lost as

6 6 heat n bearngs and due to the rcton between the rotatng rotor and the ar (wndage losses). The process o energy conerson wth ncluson o power losses s shown n Fg.. These power losses are conerted nto the heat energy. S Φ δ ω S T Fg. Illustraton to electroechancal energy conerson n rotary synchronous generator W Wndng losses W e R Wndng losses R e W W W W W e e Stator and rotor wndngs Magnetc lux Machne agnetc crcut Magnetc lux Machne echancal syste Rotor shat T ω Core losses Mechancal losses Fg. Dagra o electroechancal energy conerson wth ncluson o power losses. Feld energy In both: otor and generator the eld energy s conerted ether nto electrc or echancal energy. In peranent agnet achne the agnetc lux s generated by the agnet and n case o electroagnet the agnetc eld s generated by the current. To deterne the agnetc eld energy stored n the otor let us consder the electroagnetc structure shown n Fg. consstng o prary part, whch does not oe and the secondary part, that can oe and whch does not hae the wndng. Let assue that at present the secondary part does not oe. Suppose we ncrease now the

7 7 current n the prary wndng ro to. The agnetc lux wll rse ro to Φ as shown n Fg.3. We can express the agnetc lux as the lux lnkage λ = Φ, whch s the product o a nuber o wndng turns and the agnetc lux. In case o the real agnetc crcut the λ- cure s not lnear due to the saturaton o the ron core. For the lnear agnetc crcut the λ- characterstc s a straght lne as shown n Fg.3.b. Ths straght lne s descrbed by the equaton: λ = L (3) where L s a current coecent known as wndng nductance. I we derentate both sdes o the aboe equaton assung L = const, we wll obtan the equaton or the oltage e nduced n the wndng: d d e = λ = L (4) dt dt Prary part (statonary) g Secondary part (oable) Φ R R e e F Fg. Illustraton to deraton o orula or eld energy (a) λ (b) λ Φ L = slope Fg.3 Magnetc lnkage-current characterstc or: (a) nonlnear syste, (b) or lnear syste

8 8 The electrc power s equal: d pe = e = L (5) dt Snce the relaton between the power and energy s The ncreent o electrc energy: e e dwe pe dt = (6) dw = p dt = e dt = L d (7) In ths partcular case ths energy s a part o the total electrc energy delered to the wndng (see Fg.): where: dw = p dt (8) p R e = = + (9) Thus W e s equal to the agnetc eld energy stored n the agnetc lux: W e = W () I the power losses n all eleents o the syste are gnored and the secondary part s ong, then, durng the derental te nteral dt the ncreent o electrcal energy dw e s equal to the su: dwe = dw + dw () where dw s the ncreent o echancal energy equal to echancal work done durng the te dt by the ong secondary part. I the losses cannot be neglected they can be dealt wth separately. They do not contrbute to the energy conerson process. When the lux lnkage s ncreased ro zero to λ by eans o ncrease o current ro to, the energy stored n the eld s (Fg.4): W λ = dλ ()

9 9 λ W λ dw Fg.4 Feld energy on λ- characterstc Suppose the ar gap o the syste n Fg. ncreases. The λ- characterstc wll becoe ore lat and straght (see Fg.5). To antan the sae agnetc lux (lux lnkage) greater current should low n the wndng and consequently greater energy s stored n the agnetc crcut (Fg.6). Snce the olue o agnetc core reaned unchanged the ncrease o eld energy occurred n the ar-gap. λ λ λ W g g ar-gap ncrease W Fg.5 λ-i characterstcs or arous argaps n the achne Fg.6 Feld energy n the achne wth derent ar-gap The energy stored n the eld can be expressed n ters o other quanttes, e.g. agnetc lux densty B n the ar gap g. To nd lux densty B or a gen current n the wndng we wll use the equalent agnetc crcut o the syste shown n Fg.7. Ths crcut does not der ro electrc crcut or a dc current and the analogy between the electrc and agnetc quanttes are shown n Table. Table Electrc crcut Electrooe orce (e) E [V] Current I [A] Magnetc crcut Magnetoote () F = I [A turns] Magnetc lux Φ [Wb]

10 Resstance o conductor lw R = [Ω] A w γ where: l w length o wre [], A w cross-secton area o the wre [ ], γ conductty [..] Oh s law: E = R Magnetc resstance (reluctance) o agnetc crcut l R = [/H] (3) A µ where: l length o agnetc crcut [], A cross-secton area o agnetc crcut [ ] µ agnetc pereablty [H/] Oh s law: F Φ= (4) R Φ H c l c R c F H g l g R g Fg.7 Equalent agnetc crcut o the electroagnetc syste shown n Fg. Magnetc lux Φ s a product: Φ = B A (5) o B [T] agnetc lux densty, and A [ ]. For the lnear agnetc crcut a lux densty s equal B= H µ (6) where H [A/] s the agnetc eld ntensty n the agnetc crcut. The Oh s law or agnetc crcut wrtten n other or s: F = Φ R (7) Insertng (3), (5) and (6) nto (7) we obtan:

11 I = l B A A = H l µ µ µ (8) = H l where H l s the agnetc oltage drop across reluctance o the agnetc crcut. Consder now the electroagnetc syste shown n Fg. wth ts agnetc equalent crcut n Fg.7. Let H c - agnetc ntensty n the core H g - agnetc ntensty n the ar gap l c total length o the agnetc core l g length o the ar-gaps Then The lux lnkage: = Hclc+ Hglg (9) λ = Φ = A B () Fro Eqs., 9 and : Hl + Hl () c c g g W = A dl For the ar-gap H g B = () µ where µ s the agnetc pereablty o the acuu (ar-gap) equal to Fro Eqs. and 4 7 π [H/] B W = Hclc + lg A db µ B = H db A l + db A l c c g µ B = HdBV + V (3) c c g µ = w V + w V c c g g = W + W c g

12 where: w c = H db - s the energy densty n the agnetc core c B wg = - s the energy densty n the ar-gap µ V c s the olue o the agnetc core V g s the olue o the ar-gap W c s the energy n the agnetc core W g s the energy n the ar-gap For a lnear agnetc core: thereore H c B µ c = (4) c W B = db V c c c c µ c Bc = V µ c c (5) Lookng at Eqs. 3 and 5 we see that the eld energy s nersely proportonal to the pereablty µ and straght proportonal to the olue V. I we hae the electroechancal syste shown n Fg., n whch the sae lux densty s n both: ron cores and ar-gap (the sae lux Φ and the sae cross-secton area are or both parts), the agnetc energy s stored anly n the ar-gap, snce the core pereablty s equal to µ = µµ, and the relate pereablty or unsaturated ron s µ >. c o r r. Co-energy To calculate the attracte agnetc orce actng on the oable part we wll ntroduce the quantty called co-energy. It s dened as: W ' = λ d (6) It does not hae any physcal sgncance. Co-energy and energy o the syste s shown n Fg.8. Fro Fg.8 W + W = λ (7) '

13 3 I λ- characterstc s nonlnear: W s lnear (straght lne g ): W ' ' > W (Fg.9 cure g ), but λ- characterstc = W. I the ar-gap ncreases ro g to g and the current reans unchanged the co-energy wll decrease as shown n Fg.9. λ λ λ W λ g g W λ W W Fg.8 Feld energy W and eld co-energy W Fg.9 Feld co-energy or two derent alues o ar-gap n the syste.3 Mechancal energy and orces Let us consder the syste n Fg.. Let the secondary part oes ro one poston (x = x ) to another poston (x = x ). The λ- characterstcs o the syste or these two postons are shown n Fg.. I the secondary part has oed slowly the current, equal to = / R reans the sae at both postons n the steady state because the col resstance does not change and the oltage s set to be constant. Prary part (statonary) g Secondary part (oable) Φ R R e e F x x x Fg. Electroechancal syste wth oable and statonary parts

14 4 λ λ λ c d dw e b a x x Fg. Illustraton to the agnetc orce deraton The operaton pont has oed upward ro pont a b. Durng the oton: dw = e dt = dλ = area( abcd) e λ (8) λ the ncreent o electrc energy has been sent to the syste. The eld energy has changed by the ncreent The echancal energy dw = area(bc ad) (9) dw = dw dw e = area( abcd) + area( ad) area( bc) = area( ab) (3) s equal to the echancal work done durng the oton o the secondary part and t s represented by the shaded area n Fg.. Ths shaded area can be seen also as the ncrease n the co-energy: Snce: dw dw = dw (3) ' = dx (3) the orce that s causng derental dsplaceent s: ' W = x (, x) = const (33)

15 5.3. Force n lnear syste Let us consder agan the syste shown n Fg.. The reluctance o the agnetc core path can be gnored due to the hgh alue o µ c (see Eq.3) and the λ- characterstc s assued to be lnear. The col nductance L depends on the reluctance o the agnetc crcut. Fro Eqs.3 and 4 we obtan L F = (34) R Fro Eqs.3 and 34, ater transoraton we obtan: µ A L = (35) g That eans that nductance L depends on length o the ar-gap, so t s the uncton o x co-ordnate (see Fg.). Thus or the dealzed syste: where L(x) changes ts alue wth the gap length. Snce the eld co-energy s: λ = L( x) (36) W ' = λ d (37) ater nsertng Eq.36 or λ we obtan: W ' ( ) = L x d = ( ) L x (38) The agnetc orce actng on the secondary part we obtan ro Eqs.33 and 38: L x ( ) = x = const = ( ) dl x dx (39) For a lnear syste the eld energy s equal to the co-energy (Fg.9 lne g ), thus:

16 6 ' W = W = L( x) (4) Force can be expressed also n ters o agnetc lux densty n the ar-gap B g. I we assue that H c s neglgble (due to hgh pereablty µ c o the core), then or echancal syste n Fg. we obtan ro Eq.9: and Bg = H g g = g (4) µ B g = g (4) µ Fro Eqs.35, 38 and 4 we obtan: B ' g W = Ag g (43) µ The aboe expresson can be obtaned also ro the eld energy. For the lnear agnetc ' crcut W = W, thereore ro Eq.3 (or neglgble agnetc energy stored n the core): W ' Bg = V µ Bg = Ag g µ g (44) where A g s the cross-secton area o the ar-gap. Fro Eqs.33 and 43 the orce actng on the secondary part s: B = A g g g g µ Bg = A µ g (45) It eans that the agnetc orce s proportonal to the agnetc lux densty n square. The agnetc pressure F that s oten used s calculated as the orce per unt area o ar-gap. Thus ro Eq.45 we hae (the cross-secton area o the ar-gap s A g ):

17 7 F Bg = (46) µ The expresson 45 can be used to calculate the orce n all electroechancal deces where the agnetc lux densty n the ar-gap s known. So t can also be appled to deterne the drng orce n all these prte lnear otors shown n Fgs Let us consder the reluctance lnear otor shown n Fg.. The agnetc orce actng on the rotor (and the sae orce, but n opposte drecton, acts on the stator ) s expressed by Eq.45. The drng orce x actng on the rotor s the tangental (to x axs) coponent that can be deterned ro we know the angle β or the utual dsplaceent x between two otor parts. I we hae the reluctance otor wth the prary part as an electroagnet (Fg.3) we can use orula 39 to deterne the lnear orce x actng on the secondary part. The data that we hae to know s the current lowng n the stator wndng and the nductance L(x) expressed as the uncton o x coordnate. I the rotor s nntely long n x drecton as n Fg.4, then L = const. wth respect to x drecton and accordng to Eq.3 the orce x = despte ery strong attracte orce y actng on the rotor. rotor (Fe) B S y x y δ x x stator Fg. Force coponents n lnear reluctance otor rotor x x S y stator Fg.3 Force coponent F x produced n the lnear reluctance otor The Eqns.39 and 45 or orce was dered or the syste wth the col wound on the prary part only. We dere now the orce equaton or the lnear syste wth the col wound also on the secondary part (Fg.) as t s n synchronous lnear otors shown n Fg.8. To do ths let us assue agan that the secondary part does not oe and the eld energy s equal to the electrc energy:

18 8 dw = dw = e dt + e dt e = dλ + dλ (47) rotor x y S y stator Fg.4 o drng orce n the otor wth nntely (ery) long rotor In the lnear syste (the λ- characterstc s represented by the straght lne; see Fg3.b) the lux lnkages can be expressed n ters o nductances that are constant: λ = L + L λ = L + L (48) where: L s the sel nductance o the exctaton wndng L s the sel nductance o the oeable part wndng L and L are the utual nductances between two wndngs Fro Eqs.33 and 34: ( ) ( ) ( ) dw = d L + L + d L + L = L d + L d + L d (49) The eld energy s equal to the eld co-energy or the lnear systes. Thus we hae: ' ( ) W = W = L d + L d + L d = L + L + L (5) In the syste analyzed here the nductances depend on the alue o an ar-gap. It eans, that they are unctons o poston x o secondary part. Accordng to Eq.33 the orce deeloped by the lnear syste:

19 9 ' W = x (, x) = const dl dl dl = + + dx dx dx (5) The rst two ters are the two orce coponents known as reluctance orces. The thrd one s called an electroagnetc orce. Ths orce exsts een sel nductances do not depend on x co-ordnate, t s there are no two rst coponents. Ths type o stuaton exsts n the syste shown n Fg.5, where ether two or one o the part s nntely (ery) long. When the secondary oes wth respect to the prary, sel nductances o the cols rean unchanged, and only the utual agnetc couplng (utual nductance) changes. Fg.5 Cols n the nntely long stator and rotor cores Equlbru equatons To analyze transents n the wndngs and dynac behaor o the otors we hae to wrte equlbru equatons or electrcal ports and echancal port. The equlbru equatons or electrcal ports are ternal oltage equatons or both prary and secondary wndngs wrtten on the bass o the second (oltage) Krhho s law appled to equalent crcut o both wndngs shown n Fg.6. Equaton or echancal port s the equaton o oton or echancal syste o the otor shown n Fg.7, and wrtten n accordance wth the ewton s Law o Moton. The equatons wrtten or the electroechancal syste shown n Fg. are as ollows: - or electrcal ports: - or echancal port: λ t = R λ + t (53) = R + (5)

20 where: M s ass o the oable part D rcton coecent o the oable secondary part L load orce d x dx = M D L dt + dt + (54) In the aboe oltage equatons the ternal oltages are equal to the oltage drops across the wndng resstances and the oltages e and e (see Fg.6) nduced n the wndngs by the luxes lnked wth the. In the oton equaton the electroagnetc orce deeloped by the otor s equal to the nerta orce, rcton orce and load orce. R M(x) R Φ e e Fg.6 Equalent crcut o the electroechancal syste n Fg. Secondary part F ass M F - nerta M F - load L F - rcton D D Fg.7 Equalent echancal syste o the electroechancal syste n Fg. The derates: ( ( ) ) ( ) ( ) λ L x L x = e = + t t t d dl ( x) dx d dl ( x) dx = L ( x) + + L ( x) + dt dx dt dt dx dt (55)

21 ( ( ) ) ( ) ( ) λ L x L x = e = + t t t d dl ( x) dx d dl ( x) dx = L ( x) + + L ( x) + dt dx dt dt dx dt (56) Exaples o electroechancal deces wth a lnear (oscllatng) oton: - transorer wth ong col (or secondary oltage araton) (Fg.8) Voltage equatons: R x λ = R + t e L λ = R + t R M(x) snce λ = L + L and λ = L + L and L, L and L = L = M ( x) are constant n e L te d d = R + L M ( x) dt + dt, or = R + e d d = R + L M ( x) dt + dt, or = R + e Fg.8 Equalent crcut o transorer Changng the dstance between the cols we nluence M(x) and we change the oltage e and. - jupng rng (Fg.9) X Φ C X Φ Fg.9 Jupng rng

22 Idea o operaton: Magnetc lux o the col Φ col nduces the oltage E (and the current ) n the rng. The current n the rng contrbutes to the lux Φ rng. Ths lux opposes the lux Φ col accordng to phasor dagra n Fg.3.b (the nductances L and E are assued to be uch greater than the resstances R and R ). As the result the rng s repelled out o the col. (a) = Alunu rng R E L e x (b) R << L Φ I C R g M(x) E V = E L E R << L I Φ Fg.3 Equalent crcut o the jupng rng (a), and ts phasor dagra (b) When the dstance between the col and rng changes, then the resultant nductance o the col s changng too. I the capactor C has the alue that brngs the col crcut nto resonance when the rng s at lowest poston, then the heay col current contrbutes to the strong orce that repels the rng upwards. When the rng s ar ro the col the resonance dsappears, consequently the ltng orce goes to zero and the rng alls down. There the resonance occurs agan and the rng s repelled agan. The resultant eect s the jupng rng. The equalent crcut o the dece wth jupng rng s shown n Fg.3.a. - nducton lnear oscllator (Fg.3) The phenoenon descrbed aboe s appled n nducton oscllaton otor shown scheatcally n Fg.3. Two cols are placed at the ends o the erroagnetc bar. Both C C Fg.3 Schee o nducton lnear oscllator

23 3 are connected n parallel to the AC source a capactors. I the rng placed between the s close to one o the col then the resonance occur n the col and strong repulson produced orce throws the rng to the other col. When the rng appears close to ths col the resonance o ths col wll contrbutes to the repulson o the rng to the opposte col. Consequently, the rng oscllates between the cols. - Reluctance lnear oscllator (Fg.3) The erroagnetc bar loosely placed n the col s suspended by the attracte agnetc orce produced by the col agnetc lux (Fg.3.a). When the bar oes n the col the col nductance changes ts alue. I the capactor connected n seres to the col s chosen to put the crcut nto resonance when the bar s at lower poston (see Fg.3.b), then, due to the heay resonance current, a strong agnetc orce pulls the bar upwards nto the col. There the crcut s out o resonance and due to the sall current (and no agnetc orce at the center o the col) the bar alls down. When t coes at the end o the col the resonance appears agan, pullng the bar upwards. In consequence the bar wll oscllate n ertcal poston. I the col s placed n horzontal poston as n Fg.33, then the resonance current appear at both ends. The bar wll oscllate n horzontal poston. To ncrease the ecency o the oscllator end-rngs are placed at both ends (see Fg.34). At the end o the bar track the knetc energy o the ong bar s not lost but conerted nto the potental energy o the sprng and next returned to the bar. Ths reluctance oscllator s ore ecent than the nducton oscllator because there are no current losses n the ong part. (a) R x (b) C R x L L x D D g g Fg.3 Crcut dagras: (a) erroagnetc bar suspended by the agnetc eld, (b) electroagnetc dece wth jupng erroagnetc bar

24 4 L(x) (x) x x r x C Fg.33 Schee o the reluctance lnear oscllator wth the current and col nductance characterstcs C Fg.34 Reluctance lnear oscllator wth end sprngs.3. Torque n rotatng achnes Let us consder the rotatng electroagnetc syste shown n Fg.35, n whch the stator possesses wndng and the rotor wndng. The eld co-energy s expressed by Eqn.5. The torque deeloped by the otor s

25 5 T ' W = θ (, θ ) = const (57) Fro Eqs.5 and 57: dl dl dl T = dθ + dθ + dθ (58) Stator R θ R e ω e Rotor Fg.35 Rotary otor wth salent pole stator and rotor The rst two ters are reluctance torques and the thrd one s an electroagnetc torque. For the otor wth the round rotor (Fg.36) there s no torque represented by the rst ter, snce the sel nductance o the stator L does not depend on the poston o the torque (the agnetc lux generated by the rst wndng does not change as the rotor rotates). Thereore the torque o ths otor s descrbed by the ollowng equaton: dl dl T = dθ + dθ (59) e ω θ e Fg.36 Motor wth the salent pole stator and round rotor Suppose the stator has cylndrcal structure and the rotor has salent agnetc poles as shown n Fg.37. In such a otor, known as synchronous otor wth salent poles, the

26 6 sel nductance o the rotor wndng L does not change as the rotor rotates. Thereore the torque equaton takes the or: dl dl T = dθ + dθ (6) The rst ter s the reluctance torque, and the second one s known as a synchronous torque. θ Stator ω Rotor Fg.37 Motor wth round stator and salent pole rotor Machnes wth cylndrcal stator and rotor A schee o cylndrcal achne s shown n Fg.38. The sel nductances are constant and thereore no reluctance torque s produced. The torque deeloped by the otor s T dl dθ = (6) θ ω Fg.38 Motor wth cylndrcal stator and rotor Let the utual nductance changes snusodally:

27 7 L = M cosθ (6) where: M s the peak alue o utual nductance θ s the angle between the agnetc axs o the stator and rotor wndngs. Let the currents n the two wndngs be: = I cosωt (63) = I cos ω t+ α (64) ( ) The poston o the rotor wth respect to the stator depends on rotor speed and s: where: ω s the angular elocty o the rotor δ s the rotor poston at t = Fro Eqns.6, 6, 63, 64 and 56 we hae: θ = ωt + δ (65) { ( ( ) ) t ( ( ) ) t ( ( ) ) t ( ) ( ) T = I I M cosω tcos ω t+ α sn ω t+ δ I I M sn ( ω ( ω ω) ) t α δ 4 + sn ω ω + ω α + δ = sn ω + ω ω α + δ + sn ω ω ω + α + δ (66) The torque s the su o our coponents, whch ary snusodally wth te. Thereore the aerage alue o each coponent s zero unless the coecents o t are zero. Thus the aerage torque wll be nonzero : ( ) ω =± ω ± ω (67) The achne wll deelop aerage torque t rotates n ether drecton at a speed that s equal to the su or derence o the angular requences ω = π o the stator and the rotor currents ω = ω ± ω (68) There are two practcal cases: ) ω =, α =, ω = ω : (sngle-phase synchronous achne). Rotor carres dc current, stator ac current.

28 8 For these condtons, ro Eqn.66 T I I M { sn ( ωt δ) snδ} = + + (69) The nstantaneous torque s pulsatng. It wll be constant or poly-phase achne. sn ω t + δ s zero. It To nd the aerage torque ro Eqn.69 we see that aerage o ( ) eans the aerage torque s: T a IIM = snδ (7) I ω = (at startng) the (sngle-phase) achne does not deelop the aerage torque. ) ω = ω - ω (asynchronous sngle-phase otor). Both stator and the rotor carry ac currents at derent requences and rotor speed ω ω and ω ω Fro Eq.66 I I M = { sn sn + + sn + + sn + ( ω α δ) ( ω α δ) T t t ( ωt ωt α δ) ( α δ)} (7) The nstantaneous torque s pulsatng (t s constant n poly-phase otor). The aerage alue o the torque s: I IM Ta = sn ( α + δ ) (7) 4 At ω = the aerage torque s zero. A sngle-phase achne should be brought to the speed derent than so t can produce an aerage torque. Ths s the prncple o operaton o nducton otor. Equlbru equatons Slar as or lnear otor the equlbru equatons or rotatng achne are as ollows: - or electrcal ports: oltage equatons (the sae as Eqns.5 and 53). λ t = R + (73)

29 9 The derates: λ t = R + (74) ( ( θ) ) ( θ) ( ) λ L L = + t t t d dl ( θ) dθ d dl ( θ) dθ = L ( θ) + + L ( θ) + dt dθ dt dt dθ dt ( ( θ) ) ( θ) ( ) λ L L = + t t t d dl ( θ) dθ d dl ( θ) dθ = L ( θ) + + L ( θ) + dt dθ dt dt dθ dt (75) (76) θ The ter = ω s the angular speed o the rotor. For the otor consdered the t utual nductances L = L = M, thus: ( θ) ( θ) λ d dl d dm e = = L( θ ) + ω + M ( θ) + ω t dt dθ dt dθ ( θ) ( θ) λ d dl d dm e = = L( θ ) + ω + M ( θ) + ω t dt dθ dt dθ (77) (78) The oltages nduced n the wndngs (see Fg.36) or two groups. The rst one contans the oltages: d d e t = L( θ) + M ( θ) dt dt (79) d d et = L( θ) + M ( θ) dt dt (8) nduced due to the araton n te o the agnetc luxes represented by the currents that generate the. These types o oltages are nduced n transorers (ndex t). The second group: ( θ) dm ( θ) dl e = r ω ω dθ + dθ (8) ( θ) dm ( θ) dl e = r ω ω dθ + dθ (8)

30 3 are the oltages nduced by the rotaton o the rotor. In that nuber the rst ters are the oltages nduced by the salency o the rotor and do not exst n the otor wth cylndrcal stator and the rotor (see Fg.38). The second ters are the oltages nduced by the utual rotaton o the two wndngs. - or echancal port: oton equaton (Fg.39) d θ dθ T = J + D + T s + Tl (83) dt dt where: J oent o nerta o the rotor D rcton coecent o the rotor T l load torque T s torsonal torque (see Fg.4 two torques act n opposte drectons and they twst the shats by angle θ), whch s due to the rotor shat elastcty and s gen by: where K s s the torsonal coecent. s s ( ) T = K θ (84) Motor T T J,ω T D T L Load Fg.39 Equalent dagra or echancal syste o the rotary otor: T electroagnetc torque, T J nerta torque, T D rcton torque, T L load torque T L Motor θ Load T Fg.4 Twst (by the angle θ) o the elastc shat caused by electroagnetc torque o the otor T and load torque T L actng n oposte drecton