Chapter.4 MAGNETIC CIRCUIT OF A D.C. MACHINE

Chapter.4 MAGNETIC CIRCUIT OF A D.C. MACHINE The dfferent part of the dc machne manetc crcut / pole are yoke, pole, ar ap, armature teeth and armature core. Therefore, the ampere-turn /pole to etablh the requred r flux n the manetc crcut the um of the ampere-turn requred for dfferent part mentoned above. That, AT / pole AT y + AT p + AT + AT t + AT c 1. Yoke, 2. Pole, 3. Ar ap, 4. Armature teeth, 5. Armature core, 6. Leakae flux ab: Mean lenth of the flux path correpondn to one pole Manetc crcut of a 4 pole DC machne Note: 1. Leakae factor or Leakae coeffcent LC. All the flux produced by the e pole p wll not pa throuh the dered path.e., ar ap. ome of the flux produced by the pole wll be leakn away from the ar ap. The flux that pae throuh the ar ap and cut by the armature conductor the ueful flux and that flux that leak away from the dered path the leakae flux 1. Thu A leakae flux enerally around (15 to 25) % of, 1

p + (0.15 to 0.25) LC x where LC the Leakae factor or Leakae coeffcent and le between (1.15 to 1.25). 2. Mantude of flux n dfferent part of the manetc crcut a) Flux n the yoke y ( LC) /2 b) Flux n the pole p LC c) Flux n the arr ap d) Flux n the armature teeth e) Flux n the armature core / 2 3. Reluctance of the ar ap Reluctance of the ar app where l Lenth of ar ap ψ τ Wdth (pole arc) over whch the flux pan n the ar ap L Axal lenth of the armature core ψ τ L Ar ap area / pole over whch the flux pan n the ar ap Becaue of the chamfern off the pole, the lenth of ar ap under the pole vare v from l at the center of the pole to l >l at the pole tp. The lenth of ar ap to be condered for the ' calculaton of ar ap reluctance nether l nor.the lenth of ar ap at the of the pole. l l a µ µ r (τ ψ L) µ 0 0 a µ r 1.0 for arr ap ' l, but ha to be a value n tp enerally 1.5 to 2 tme the ar ap len between l and l ' th under the center Becaue of the frnn of flux, the wdth over whch the flux pae throuh the ar ap not ψ τ but t more than that. 2

The effect of varaton n ar ap lenth and frnn of flux can be nored a the former appear n the numerator and the latter n the denomnator of the expreon e for the reluctance. Whle calculatn the reluctance of the ar ap, effect of the preence of lot and duct on the armature mut alo be condered. Effect of lot on the reluctance of the ar ap Conder a mooth urface armature (A).e. havn no lot and duct. Over O a lot ptchλ, reluctance of the ar ap n the preence of mooth urface armature l A ------- (1) λ L µ 0 Over the ame lot ptch conder a lot and tooth. Becaue of the crowdn effect, the flux ntead pan only over thee tooth wdth b t, pae over ome porton of thee lot alo. Thu the wdth over whch the flux pan equal to (b t + b δ ) where δ called the Carter frnn coeffcent for lot.. It le than 1.0 and depend on the rato of lot openn to ar ap lenth and can be obtaned from the Carter frnn coeffcent curve. The reluctance of the ar ap n the preence of armature wth lot l AW ------- (2) (b t + b δ ) L µ 0 Dvdn 2 by 1, l / (b AW t + b δ ) L µ 0 l / λ L µ A 0 3

AW λ A (b + b δ ) t λ A b + b δ + b b t after addn and ubtractn bn the denomnator λ A AW λ - b (1 - δ ) K A wherek called the Carter ap expanon coeffcent for lot and reater than 1.0. It clear from the above expreon that the effect of the lot to ncreae the reluctance of the ar ap by a factor K a compared to the reluctance of the ar ap nn the preence of a mooth urface armature. Effect of ventlatn duct on the reluctance of the ar ap Conder a mooth urface armature (A).e. armature havn no lot andd duct. Reluctance of the ar ap, n the preencee of a mooth urface armature l A --------- (3)) π DLµ 0 Reluctance of the ar ap n the preence of the armature wth duct (AWD) l AWD --------- (4) π D [L - n vb v( l - δ v)] ] µ 0 where δ v the carter frnn coeffcent for duct. It le than 1.0 and a depend on the rato openn of the duct to ar ap lenth and obtaned from the Carter frnn f coeffcent curve. 4

Dvdn 4 by 3, l AWD / π D [ L - n v b (1 - ) ] µ l / π D Lµ A L A AWD Kv L - nvb v(1- δv) wherek v called the Carter ap expanon coeffcent for duct and reater than 1.0. Thu the effect of duct to o ncreae the reluctance of the ar ap by a factor K v a compared to the reluctance of the ar ap n the preence of a mooth urface armature. Combned effect of lot and duct on the reluctance of the ar ap The preence of lot and duct ncreae the reluctance of the ar ap by factor f K and K v repectvely. Toether they ncreae the reluctance by a factor K calledd the Carter ap expanon coeffcent (or extenon coeffcent or contracton coeffcent). Thu λ L K K Kv λ - b ( 1 - δ ) L - n b ( 1 - δ ) o v v 0 0 K A v v v where b o openn of the lot wdth of the lot b for open type of lot <b for em-cloed lot zero for cloed lot δ or (1 - δ ) Carter frnn coeffcent for lot and depend on the rato b o / l and can be obtaned from the carter frnn coeffcent curve. δv or (1 - δ v) Carter frnn coeffcent for duct and depend on the rato r b v / l and can be obtaned from the carter frnn coeffcent curve. 5

Calculaton of ampere-turn per pole for the manetc crcut of a DC machne m The total ampere turn / pole requred for the manetc crcut of a DC machne m to etablh flux, AT / pole um of the ampere turn requred to over come the reluctance off the yoke, pole, ar ap, armature teeth and armature core AT y + AT p + AT + AT t + AT c a) ampere turn for the yoke / pole AT y : LC / 2 Flux denty n the yoke B y tela A y Let at y be the ampere turn per metre, obtaned from the manetzaton curv the yoke materal, at B y. ve correpondn to 6

NOTE: L y Axal lenth of the yokee D Dameter of the armature d y Depth of the yoke l Lenth of ar ap A y Cro-ectonal area of yoke d y L y h p Heht of the pole b p Wdth of the pole D y Mean dameter of the yoke (D + 2l + 2h p + d y ) f Pole ptch at mean dameter of the yoke π D y / P Mean lenth of the flux path n the yoke l y abc abcde / 2 (f 2fb + 2ab) /22 π Dy 2 bp 2 dy - - / 2 P 4 2 π D y bp - - d y / 2 P 2 Total ampere-turn for the yoke / pole AT y at y l y b) ampere turn for the pole AT p : LC Flux denty n the pole B p tela A p Let at p be the ampere turn per metre, obtaned from the manetzaton curve correpondn to the pole materal, at B p. Note: L p Axal lenth of the pole L p Net ron lenth of the pole h p Heht of the pole ncludn pole hoe heht L p K L p d Dameter of the pole A p Cro-ectonal area of the pole b p L p n cae of quare or rectanular lamnated pole π d 2 /4 n cae of crcular pole 7

Mean lenth of the flux path n the pole pole heht h p Total ampere turn for the pole / pole AT p at p h p c) ampere turn for the ar ap / pole AT : nce flux mmf or AT / reluctance, ampere turn for the ar ap per pole AT x reluctance. Thouh the reluctance of thee ar ap under a pole l, t to bee multpled by the τ ψ L µ 0 Carter ap expanon coeffcent K K K v n order to account the effect of lot and duct. Therefore, AT l K l K K B ψ τ Lµ 0 4 π x 1 0-7 800,000l K B (approxmately) whereb the maxmum value of the flux denty n the ar ap alon thee center lne of the pole. That, B a v P B ψ τ L π D L ψ π D L ψ ψ P a v e r a e v a lu e o f th e f lu x d e n ty B a v f e ld f o r m f a c to r K a n d a p p r o x m a te ly e q u a l t o p o le e n c lo u r e f ψ d) ampere turn for the armature teeth / pole AT t : Flux denty n the armaturee tooth (n cae of a parallel ded lot and tapered tooth) at 1/3 heht from the root of the tooth 8

B t1/3 b L /P t 1/3 where b t 1/3 wdth of the tooth at 1/3 heht from the root of the tooth π (D - 4 / 3 h ) L Net ron lenth of the armature core K (L n v b v ) Let at t be the ampere turn per metre, obtaned from the manetzaton curve correpondn to the armature core materal, att B t 1/3. Mean lenth of the flux path n the tooth heht of the tooth h t Total ampere turn for the armature teeth / pole AT t at t h t e) ampere turn for the armature core / pole AT c : Flux denty n the armature core B c / 2 tela Let at c be the ampere turn per metre, obtaned from the manetzaton curve correpondn to the armature core materal, att B c. Note: d c Depth of the armature core A c Cro-ectonal area of the armature core d c L Mean lenth of the flux path n the armature core P Q R π (D - 2h t - d c ) l c 2 2P t - b Total ampere turn for the armature core / pole AT c at c l c Thu the total ampere-turn requred for the manetc crcut of the DC machne AT / pole AT y + AT p + ATT + AT t + AT c A c 9

Method of calculatn the ampere turn for the armature teeth: For a parallel ded lot, the e tooth tapered and therefore the flux dentyy at each and every ecton of the tooth wll be dfferent. The flux denty leat at the ar ap urface of the tooth where the flux enter the tooth and maxmum at the root of the tooth where the tooth ecton mnmum. nce the varatonn of flux denty n the tooth non-lnear becaue of aturaton of ron, the calculaton of ampere turn become dffcult. Dfferent method avalable for the calculaton of AT t are 1. Graphcal method 2. mpon method and 3. B t 1/3 method Graphcal method In th method the tooth dvded nto a number of equal part and flux denty at each tooth ecton calculated. Correpondn to each flux denty, At / m obtaned o from the manetzaton curve. Aumn lnearty between the ecton condered, ATT t calculated. Note:h t heht of the tooth or depth of the lot b t1, b t2, b t3 etc., are thee tooth wdth at dfferent ecton 1, 2, 3 etc. Flux denty at ecton l, B t1 Let the ampere turn / metre, obtaned from the manetzaton curve H 1 or at 1 at B t1. Flux denty at ecton 2, B t2 Let the ampere turn / metre, obtaned from the manetzaton curve H 2 or at 2 at B t2. Flux denty at ecton 3, B t3 Let the ampere turn / metre, obtaned from the manetzaton curve H 3 or at 3 at B t3. mlarly let H 4 be the ampere turn / metre at B t4 etc. Total ampere turn for the teeth / pole b L / P t1 b L / P t2 b L / P t3 10

AT t H 1 + H 2 2 h t + n H 2 + H 3 2 h t + n H 3 + H 4 2 where n the number of part by whch the tooth dvded. h t etc., n mpon method In th method the tooth dvded nto two equal part. The flux dentyy at each ecton calculated and the correpondn ampere turn / metre are obtaned from the manetzaton curve. Note:b t1, b t2, b t3 are the wdthh of the tooth at ecton 1, 2 and 3 Let H 1 be the AT/m correpondn to the flux denty B t1 Let H 2 be the AT/m correpondn to the flux denty B t2 Let H 3 be the AT/m correpondn to the flux denty B t3 b L /P Accordn to mpon rule,, averae ampere turn / m 1 H av ( H1 + 4H 2 + H 3 ) 6 Total ampere turn for the armature teeth / pole AT t H av h t B t 1/3 method In th method, AT t obtaned condern the flux denty at 1/3 heht from the root of the tooth. t2 t1 b L /P t3 at ecton 2. at ecton 3. b L /P at ecton 1. Flux denty n the tooth at 1/ /3 heht from the root of the tooth B t 1 /3 b t 1 /3 L /P 11

Let at t be the ampere turn perr metre, obtaned from the manetzaton curve correpondn to the armature core materal, att B t 1/3. Total ampere turn for the teeth / pole AT t at t h t. [Note: In all the above threee method, the effect of aturaton of ron nelected. n In other word all the flux under a lot ptch aumed to be pan throuh the tooth only]. Real and Apparent Flux dente When the ron not aturated, reluctance of the ron wll be le and all the flux over a lot ptch wll be pan throuhh the tooth only. However, when the ron et aturated, reluctance of the ron ncreae conderably and the flux over the lot ptch dvde telf to take both lot and tooth path. Thu the flux denty ron n the tooth, but t an apparent flux denty. The real flux denty B real wll, however, be equal to denty B app. flu x n th e to o th o r ro n p a th n e t ro n a ree a o f th e to o th A not the real or actual area of the tooth A and wll be le than flux denty the apparent flux 12

Area of ron or tooth area over whch flux pan A b t L Total area over the lot ptch λ L area of ron A + area of non-manetc path A n + n B app + A A A where B n µ 0 µ r H µ 0 H the flux denty n the non-manetc path and a K a contant equal to A n / A. The manetzn force H the ampere turn / metre to etablh B real or B n. Therefore B app B real + µ 0 H K If the lot factor K 1 + K (1 + Thu B app B real + µ H (K 1) and an equaton of traht lne. 0 A B + x B + B K A A A n n n real real n n A n ) A A + A A [Note: nce the actual value e of flux pan throuh the lot or tooth not known, k B n and B real and therefore the AT / m.e. H to etablh B n or B real are alo not known. Hence H the equaton B app B real + µ 0 H (K 1) ha two unknown B real and H. Thu the equatonn cannot be olved. However the value of B rea al and H can be found by plottn the above equaton on the manetzaton curve. The nterecton pont of the manetzaton curve and the traht lne provde the value of B real and H.] n λ L b L t then K (K 1). The co-ordnate, of the nterecton pont of manetzaton curve and traht lne provde the value of B real and H. Therefore the total ampere-turn for the armature teeth / pole AT t H h t. 13

No-load, Manetzaton or Open crcut charactertc (OCC) nce the OCC a plot of emf nduced and AT, ampere-turn for dfferent aumed voltae or flux are found out by calculatn AT y,at p, AT, AT t and AT c. Accuracy of the curve ncreae a the number of voltae condered ncreae. Informaton that can be obtaned from the open crcut charactertc, 1) The value of crtcal feld retance. 2) hunt and ere feld ampere-turn 3) Effect of armature reacton n conjuncton wth the nternal charactertc. ****************** 14