Is Vs. Electrical Machinery Part 4 Some Ideal Machines. Transformers. fig 4.1. Vp t. fig 4.2

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1 Elecrical Machinery Par 4 Some Ideal Machines Transformers Le us now add a second winding o an inducor o make a ransformer (fig 4.1) he primary winding has p urns and he secondary winding has s urns. The baery volage is Vp and when i is conneced o he primary winding, i produces a magneic flux Φ in he core: Φ ( Vp ) / p The flux rises a a rae ha induces a volage ℰ in every urn of boh windings such ha he oal volage induced in he primary winding is equal o he baery volage Vp and opposing i: Ip Vp ℰ p Similarly, he oal volage Vs induced in he secondary winding is: R Vp p Is Vs s Vs ℰ s Vp ( s / p ) If we now connec a load resisor R o he secondary a curren Is flows in accordance wih Ohm s Law: Φ fig 4.1 Is Vs / R Energy is now flowing from he ransformer o he load bu where is i coming from? When he load R is conneced, he flux Φ mus coninue o rise a is original rae o mainain he induced volage ha balances he supply volage Vp hus he oal curren encircling he flux in he core mus also coninue o rise a is original rae. Before he load is conneced, he only curren flowing is ha in he primary winding so he oal curren encircling he flux is: Ip p Afer he load is conneced, here is also a curren flowing in he secondary he oal curren encircling he flux due o his is: Is s The primary curren auomaically adjuss iself o leave he oal curren encircling he flux in he core unaffeced by he curren in he secondary winding. The primary and secondary currens encircle he core in opposie direcions hus he primary curren Ip has o increase by an amoun: Is ( s / p ) Ip Load conneced Energy ransferred o load Fig 4.2 shows graphically how he primary curren increases. We can herefore say: The primary volage conrols he secondary volage. The secondary curren conrols he primary curren. Energy sored in inducor fig 4.2 Vp

2 Are we saying ha i is possible o connec an unchanging volage source o he primary winding and obain an unchanging volage from he secondary winding? The imporan hing o noe is ha he acion of ransferring volage, curren, and energy from he primary circui o he secondary circui does no depend on he inpu volage changing ye alone be alernaing. Bu as illusraed in fig 4.2 he primary curren and flux in he core will evenually rise o an impracical level. Acual limiaions will be considered in Par 5 bu for now we will jus acknowledge ha here will be some limi. There are various mehods ha can be used o cause he curren and flux o reurn o zero bu he simples, and mos relevan o he presen discussion, is o reverse he supply volage. This is illusraed in fig 4.3 for he primary winding of a ransformer wih no load. Pp Vp Ip fig 4.3 As before, he curren Ip sars o rise when he volage Vp is applied. When he curren reaches he desired maximum value, he volage is invered and he curren falls o zero; i hen coninues in a negaive direcion unil he volage is again invered. The dashed line Pp shows he produc of Ip and Vp. The area under his line represens he energy Ep sored in he inducor. When he polariy of Vp changes, Pp becomes negaive, ha is, power sars o flow ou of he inducor, hus reducing he sored energy. When he sored energy is released, i is reurned o he source and becomes available for reuse. Thus, on average, no energy is aken. The curren Ip is called he magneising curren and examinaion of he waveforms show ha his curren lags he volage by a quarer cycle and hus he curren is flowing in he reverse direcion o he applied volage half he ime. Any aemp o calculae power using separae volage and curren meers or of aemping o apply Ohm s Law o he circui, is doomed o failure. Measuremens need o be made wih a proper wa meer or joule meer. In Par 3 we saw ha, by increasing he number of urns on an inducor by a facor of, he flux Φ is decreased by a facor of while he curren is decreased by a facor of 2. In realiy, here will be a minimum number of primary urns ha are required o preven he flux, and he magneising curren, rising oo quickly for a given maximum supply volage and minimum supply frequency. Thus he number of urns on he primary winding of a ransformer is by no means arbirary. Thus, even for a perfecly lossless ransformer, we are only jusified in saying: primary urns primary volage secondary curren secondary urns secondary volage primary curren if we ignore he magneising curren. Bu, as implied above, he facors ha deermine he magneising curren play an essenial par in he design of he whole ransformer.

3 Consider a ransformer as in fig 4.1 having a secondary wih n imes he number of primary urns hus making a 1:n sep-up ransformer. The waveforms in fig 4.4 are for a ransformer wih n2. (A) Vp (B) Imag (C) Vs (D) Is (E) Ip fig 4.4 A) A volage Vp is applied o he primary winding B) A magneising curren Imag flows in he primary circui C) A volage Vs n Vp is induced in he secondary winding D) A curren Is Vs R flows in he secondary circui E) The resulan oal primary curren Ip n Is + Imag

4 Le us now apply wha we know in order o build a lossless bu oherwise realisic ransformer. Firs le us consider he primary winding of he ransformer: Le he supply frequency be 50 Hz and he supply volage be ±200 V. A maximum flux densiy of 1 T would be reasonable and will occur when 200 V has been applied for 5 ms. This will also be he ime when he magneising curren is a maximum. We will assume ha he cross-secional area of he core is 10 square cenimeres. We will also need o leave a gap in he core. Wih our ideal magneic maerial, if we did no leave a gap, he relucance of he magneic circui would be zero, which would make he equaions in Par 3 moo. A gap of 0.1 mm will do. When we ge o Par 5 Losses i will be apparen why we have chosen hese paricular values. For now hey jus need o be acceped as convenien round numbers. Le us now plug hese values ino our exising equaions: B1T A m2 V 200 V s d m Flux: ΦBA Wb Turns: V Φ Relucance: ℛ d A / Wb μa Inducance: L H ℛ Curren: I V A L Energy: E (V ) J 2L oe ha none of he above quaniies depend on he energy acually being ransferred from he inpu o he oupu of he ransformer; all hese quaniies remain consan regardless of he load. Alhough a (magneising) curren flows and associaed energy is sored in he ransformer, his does no, of iself, consiue any form of loss from he ransformer, as we have seen (fig 4.3) above. When we consider losses in Par 5, we will find ha any losses associaed wih magneising he core end o be independen of load and hus remain reasonably consan hus he erm fixed losses. In he absence of losses we are able o draw unlimied power from he ransformer. Again, when we ge o Par 5, we will see ha when we chose he cross-secional area of he core we were making a major sep owards deermining he maximum power ha can acually be ransferred o he load. This is basically limied by losses due o he resisance of he windings. These losses will vary wih he curren drawn hus he erm variable losses. We will make his a sep-down ransformer and pu in some appropriae figures: Secondary volage 10 V Primary urns : Secondary urns 20 : 1 Secondary urns 50 Secondary curren 5 A Primary curren 0.25 A + magneising curren.

5 Generaors & Moors If we sar wih a curren-carrying conducor moving in a magneic field as described in Par 3, we need o consider how o conver his unidirecional linear moion ino coninuous circular moion. Fig 4.5 shows a simple moor as found in school exbooks and on examinaion papers. As he single-urn coil roaes, he flux Φ passing hrough i varies cyclically as does he rae of change of Φ which equals he volage induced in he coil ℰ which swings posiive and negaive each revoluion. The spli-ring commuaor no only makes elecrical connecion o he roaing coil bu also auomaically reverses he connecions o he coil jus as he induced volage passes hrough zero. Alhough we can make a moor based on i and wach i whiz round merrily, i makes a very poor example. While i does illusrae he essenial pars of a moor i is ulimaely a pracical and pedagogical dead end. We will deal wih i laer. S fig 4.6 fig 4.5 In order o make i ino a useful moor we need o go back o ha curren-carrying conducor moving in a magneic field as described in Par 3. If insead of a single conducor moving in a gap in he core, we cu a pair of gaps, we can moun a coil wih is commuaor aached so ha i is free o urn in he magneic field as shown in fig 4.6 We now have a properly defined, low relucance, magneic circui. The purpose of he core inside he coil is simply o reduce he relucance of he magneic circui. The core may roae wih he coil or he coil may roae independenly. We can furher reduce he relucance of he magneic circui by winding he coil in slos in he core and hus make he gap even smaller as shown in fig 4.7 Fig 4.8 shows he alernaing volage induced in he coil. S fig 4.7 Fig 4.9 shows he volage as seen a he moor erminals afer he commuaor he significan poin is ha he volage is no consan bu dips o zero, wice per revoluion as he induced volage changes polariy. In Par 3 we found ha he velociy, baery volage, induced volage, force, and curren all played nicely ogeher o give an induced volage equal o he baery volage and consan velociy, force, and curren. In his case, he induced volage, as seen a he erminals, does no balance he baery volage he coil shor-circuis he baery wice per revoluion and he whole argumen collapses in ruins.

6 ℰ ℰ fig 4.8 fig 4.9 We can solve his problem by having muliple coils arranged in slos around he roor core and a muli-segmen commuaor. Fig 4.10 shows an acual roor from a small moor. There are numerous ways of arranging hese coils in he slos and arranging heir connecions o he commuaor. We will choose a simple example wih 12 slos, 12 coils, and 12 commuaor segmens. fig 4.10 S fig 4.11 fig 4.12 Le us use he generic erms: roor for he par ha roaes, and saor for he par ha remains saionary. Fig 4.11 shows he roor and saor of such a moor rolled ou fla. The coils and commuaor move in he direcion shown while he brushes and magneic poles remain fixed. Saring a commuaor segmen 1, here is a single urn coil, shown in red, which passes hrough a slo in he roor and reurns hrough a diamerically opposie slo and finishes on segmen 2. Then, from his segmen here is anoher coil which finishes a segmen 3. oe ha he coils ha sar on segmens 2 o 6, go away from he commuaor (shown by a solid line) over a souh pole and reurn o he commuaor (shown by a dashed line) over a norh pole; as he roor urns, he volages induced in each of hese coils add up. We herefore place a brush in conac wih each end of hese five coils as shown. The fac ha hese brushes are also each shoring ou a coil (he red one and he blue one) is of no consequence because here is no volage being induced in eiher of hese coils. As we go on, he coils saring on segmens 7 o 12, go ou over a norh pole and reurn over a souh pole; as he roor urns, he volages induced in each of hese coils add up bu he volages are acing in he opposie direcion o hose induced in he firs se. When he end of he las coil is finally conneced o segmen 1, hus making a complee circui, he volages around he roor have cancelled ou and he shor circui is of no consequence. Also, he wo exising brushes are in jus he righ place o make conac wih he correc commuaor segmens conneced o hese coils. Fig 4.12 is he equivalen circui of he roor wih he volage induced in each coil shown by a cell.

7 Fig 4.13 shows he ubiquious Wesminser Moor Ki assembled. I would be useful o invesigae why i works so badly and indeed why i works a all. This represens a pracical realisaion of fig 4.1 bu wih wo significan differences: The impossible pair of magneic monopoles floaing in space are replaced by a pair of slab magnes adhering o a mild seel yoke. The spli-ring commuaor has been reduced o a pair of bare wire ends ha only make conac wih he brushes momenarily wice per revoluion. fig 4.13 Wihou he yoke he flux from each slab magne naurally flows from he norh o he souh pole of he same magne. Having a facing pair only slighly modifies ha. Adding he seel yoke provides a low relucance pah joining he magnes and hus making hem ino a single magne. There is now a small, bu reasonably well defined, flux passing across he gap in which he roor urns. The field is weak because he gap is wide bu a leas i is where i is needed. The original commuaor allowed he brushes o shor he commuaor segmens and hence he baery; he modified design avoids his. I also has he effec of confining he region where curren can flow o a small angle eiher side of he horizonal posiion of he roor. The force acing on he roor and he induced volage in he roor are boh a maximum in his posiion; boh fall o zero when he roor is verical. Fig 4.14 shows he roor curren (A) v ime (ms). The baery only supplies curren when he roor is in is mos favourable posiion. The moor progresses a abou 1600 revoluions/minue in a series of kicks wo per revoluion. We have already esablished ha, for a single conducor of lengh l carrying curren I moving a velociy v in a field of flux densiy B, he induced volage ℰ and force F are: ℰ Blv V and F BIl For a recangular coil 5 cm x 3 cm having 5 urns, roaing a 1600 rpm, and a 1.5 V baery: B 1.2 T and F 2.5 These resuls are absurd. A flux densiy of 1.2 T over such a large volume would require a very large, very heavy, and very expensive magne. A force of 2.5 requires ha here mus be an equal mechanical load for i o ac agains bu here is no such load, jus windage and fricion. Fig 4.15 shows he curren drawn by he same moor when he roor is forcibly held saionary. This is no much higher han wih he moor urning and indicaes ha he vas majoriy of he baery volage is los in he resisance of he roor and oher conducors. All his energy is wased as hea while he roor is jus lef beaing he air and doing no useful work.

8 Le us now analyse he machine represened by fig 4.11 As he roor urns, he flux ΦISIDE passing hrough any single-urn coil varies cyclically as does he rae of change of ΦISIDE which equals he volage induced in he coil ℰ If we consider he BLUE coil ha is conneced o commuaor segmens 7 & 8 which are currenly under he EGATIVE brush: he flux ΦISIDE is equal o he oal flux Φ As he roor urns, ΦISIDE decreases o zero and, afer half a revoluion, he BLUE coil will be in he posiion originally occupied by he coil shown in RED and is commuaor segmens 7 & 8 are under he POSITIVE brush: he flux ΦISIDE is now equal o -Φ because i is passing hrough he coil in he opposie direcion. Therefore he oal change of he flux ΦISIDE is equal o 2Φ in half a revoluion. If he roor is urning a a speed of f complee revoluions per second, he average volage ℰ induced in his single-urn coil is equal o 4Φf vols. Le us now consider he res of he coils currenly beween he POSITIVE brush and he EGATIVE brush. These are all conneced in series wih each oher and wih he BLUE coil ha we were originally considering. If here are n coils on he roor here will ½ n coils beween he brushes. These coils will have he same average volage ℰ induced in each. Thus here will be an induced volage beween he brushes of 2Φfn vols. As here are likely o be more han a single urn on each coil, we can muliply his volage by he number of urns on each coil. Thus if here are urns on he roor in oal we can say ha he induced volage beween he brushes V 2Φf vols. The remaining ½ n coils on he roor are also conneced in series beween he brushes and also generae 2Φf vols bu, as shown in fig 4.12 above, hese form a parallel pah and have no effec on he overall induced volage, hough i does resul in he wire forming he coils only having o carry half he curren. Why have we gone o so much rouble o examine a moor ha is boh complicaed and which will never appear on an examinaion paper? We mus sar from wha we have in his case i is he moor found in GCSE exbooks where we are presened wih a number of fallacious saemens abou a moor ha barely resembles a real moor and which barely works a all. We are also asked o accep ha hese saemens apply equally o a real moor in pracical use. By a series of raional seps, from he original crude design, we have derived a moor, which could equally well be a generaor, of a ype in common use and which is reasonably efficien. If we disregard he losses, which we will deal wih in Par 5, we can use i as he basis of an ideal machine. In Par 3 we derived an equaion for he volage induced in a conducor moving in a uniform magneic field a consan velociy: ℰ Blv vols In he case of any of he roary machines discussed, boh B and v are vecors having values which vary in magniude and direcion and herefore he above assumpions do no apply. However he analysis of fig 4.11 shows ha he muliple coils effecively average ou he variaions in B and by considering he angular velociy we avoid he variaions in v and he commuaor deals wih he polariy changes. Then, by applying exacly he same principles ha we used in Par 3, we have obained he equivalen equaion for our ideal, bu realisic, machine: V 2Φf vols We will have o pu up wih he complexiy of he machine for he simpliciy of his resul.

9 oe ha he induced volage depends only on he flux Φ, he oal number of urns on he roor, and is speed of roaion f. oe ha i does no depend on: wheher he machine is a moor or a generaor, he number of commuaor segmens, coils, or slos, wheher he roor windings are fixed o he core or wheher hey roae independenly, wheher he coils are wound on he surface of he roor or in slos, he lengh or diameer of he roor, he curren in he roor windings, he mechanical or elecrical load on he machine. Φlv Blv vols..(i) A The wo equaions: ℰ and: V 2Φf vols... (ii) look very differen bu we can show ha boh equaions lead o he same resul. S 1 2 fig 4.16 If we consider he hypoheical case of a roor having jus wo coils (fig 4.16): 2 For a roor wih radius r: v 2πfr Hence: f v 2 πr Assume ha we have a uniform radial flux Φ ha is spread over an area A equal o half of he circumference of he roor imes is lengh l: Φ BA B(πrl) Bu: Hence: V 2Φf. (ii) V 4 B(πrl ) v 2B l v 2 πr We chose 2 because i is he smalles number of coils o which equaion (ii) applies. The wo coils are effecively in parallel so we can disregard one of hem. This leaves one coil having wo conducors, each of lengh l moving in he magneic field so as o cause he volages induced in each o add. Therefore he induced volage ℰ in one conducor is: ℰ Blv.....(i)

10 Le us now consider he effec of applying a mechanical load o he moor Fig 4.17 shows he moor circui in schemaic form we will ignore any connecions o he saor circui and jus assume ha he magneic field is supplied by a permanen magne. The moor is urning a consan speed and driving a consan mechanical load. As i is a roary machine, i will be convenien o express hese quaniies as: speed f revoluions/second orque τ newon meres [ m] The moor is supplied wih a volage V and draws a curren I. I V M fig 4.17 We have already esablished ha: V 2Φf vols f Hence: V 2Φ revoluions/second.. (1) We can also say ha: Elecrical Inpu Power Mechanical Oupu Power VI 2πτf was I Hence: πτ amperes (2) Φ oe ha he erms on he righ hand side of equaions (1) and (2) are hings over which we generally have conrol, whereas hose on he lef are a consequence of hese decisions. We can say: The elecrical inpu volage V conrols he mechanical oupu speed f The mechanical load on he oupu τ conrols he elecrical inpu curren I When we consider loses in Par 5 i will be seen ha i is desirable o keep he flux Φ as high as possible. Then, once we have chosen he operaing volage V, we can chose he number of urns o give he required speed f. If he volage is low hen only a small number of urns of hick wire will be required. Conversely, if he required operaing volage is higher, he same performance can o achieved by using more urns of hinner wire his will ake up he same space so he roor size will be unchanged. The operaing curren will be reduced proporionaely. Trading increased orque for reduced speed does require a larger roor for he same oupu power. Wha can we say abou he direcion of roaion? The erms clockwise and ani-clockwise can be ambiguous. A machine wih a shaf ha is urning clockwise, when viewed from one end; will be urning aniclockwise, when viewed from he oher. I is clearer if he roaion is shown as a vecor as in fig 4.18 where i is applied o a can moor. fig 4.18 This moor has an improved version of he Wesminser saor comprising a pair of curved magnes on he inside of a seel ube which forms he body as shown in he cuaway illusraion.

11 Swee exiss by convenion, bier by convenion, colour by convenion; aoms and Void (alone) exis in realiy... Democrius (lived abou BC) Ancilla o he Pre-Socraic Philosophers, by Kahleen Freeman, [1948] There is a resemblance beween he diagram showing mechanical roaion and ha showing elecric curren and is associaed magneic flux (fig 3.1) he essenial difference is ha he direcion of mechanical roaion represens a real movemen of aoms whereas he direcions of curren and flux are arbirary convenions. How hen have we been able o esablish a real direcion of roaion saring wih such arbirary convenions? Tradiionally, we would conceal he arbirary naure of hese convenions by applying apparenly arbirary rules ha have acually been cunningly designed o cause he righ answer o pop ou a he end. This all provides an abundance of mark fodder for examiners. Insead, we have considered energy, which always has a real direcion of flow, and he direcion of moion jus falls ou naurally wihou any hand waving. Looking a fig 3.19 we can see ha he direcion of moion indicaed by v can be reversed simply by roaing or flipping over he drawing hese operaions are exacly equivalen o reversing he baery connecions or reversing he poles of he magne. Exacly he same argumen can be applied o he roary version. Fig 4.19 shows a sylised version of our can moor. The ends of he shaf are marked using he usual do and cross convenion o indicae he direcion of he roaion vecor shown in fig 4.18 S fig 4.19a fig 4.19b Reversing he moor is equivalen o looking a i from he oher end. We can illusrae his by aking he iniial drawing (fig 4.19a) and urning i over. A horizonal flip (fig 4.19b) urns he moor around and reverses he posiive and negaive erminals and hence he curren. A verical flip (fig 4.19c) urns he moor around and reverses he direcion of he magneic field. fig 4.19c fig 4.19d A horizonal and a verical flip (fig 4.19d) reverses he curren and reverses he direcion of he magneic field while he direcion of roaion remains unchanged. A real moor migh no have he symmery of our ideal moor and i may be necessary o physically rearrange he elecric or magneic polariy o reverse i his could be by means of a simple change-over swich in he supply o he brushes or, if he permanen magnes are replaced by an elecromagne, in he supply o ha. We can use he machine as a generaor equaions (1) & (2) sill apply so we can say: The mechanical inpu speed f conrols he elecrical oupu volage V The elecrical oupu curren I conrols he mechanical load on he inpu τ

12 And, has hou slain he Jabberwock? Come o my arms, my beamish boy! O frabjous day! Callooh! Callay! He chorled in his joy. Lewis Carroll: Jabberwocky from Through he Looking-Glass and Wha Alice Found There (1872) We have slain he Jabberwock, is many heads have urned ou o be jus reflecions and shadows. Of he rules lised by Eric Laihwaie in Par 2, only Lenz s Law remains and wha a perfecly poinless Law i is, i can be derived a any ime from he Law of Conservaion of Energy, i remains purely as a rule of humb alongside hose oher rules ha we have shown o be unnecessary bu no necessarily un-useful. In Par 3 we finished wih a lis of hree laws and wo rules of arihmeic. We finish Par 4 having added jus a hird rule of arihmeic: Addiion. In Par 5 Losses we will mee: Subracion. We are lef wih an uneasy feeling ha, having described he operaion of boh ransformers and roary machines in erms of a very limied reperoire of physical laws, hose wo descripions seem o be irreconcilably differen surely hey should have more in common. Fig 3.12 would seem o indicae ha here should be some sor of connecion as indeed here is. A roary machine is required o have a consan flux in is magneic circui whereas a ransformer is required o have an alernaing flux in is magneic circui. The elecrical inpu and oupu of a ransformer are boh AC whereas he elecrical inpu of a moor and he elecrical oupu of a generaor are boh DC. Finally, roary machines roae whereas ransformers are saic. S fig 4.20a fig 4.20b Le us ake our moor (fig 4.20a) and hold he shaf saionary. The res of he moor is now roaing in he reverse direcion (fig 4.20b). We now have an exernal roor machine and he flux hrough he windings of our new saor due o he roaing permanen magnes is now alernaing. The commuaor provided a connecion beween he saionary and moving pars bu also served o conver beween AC and DC. The firs funcion is now redundan, as he coils concerned do no move, and we can now envisage some form of swiching arrangemen beween he moor erminals and he coils o perform he second funcion. Moors like his, no only exis bu are made in huge quaniies for use in compuer fans. Whereas he commuaor in fig 4.20a allowed he roaing windings o produce a saionary magneic field, he corresponding mechanism in fig 4.20b causes he saionary windings o produce a roaing magneic field. We can perhaps begin o see ha here is no fundamenal differences beween he differen ypes of machine and he same principles apply o hem all. There now opens up he possibiliy of building all sors of moors and generaors boh DC and AC. The machine hined a in fig 3.12 which forms a ransformer supplying is own roor curren from a roaing field in he saor is a very common one. Being able o use elecronics o swich and conrol he various currens, allows all sors of sysems ha were no pracicable wih a commuaor. This even includes allowing a ransformer o funcion wih DC. Bu we mus heed Feynman s warning in Par 1 and sop here we can leave he deails o ohers. We have shown ha even a GCSE knowledge of physics and mahemaics provides a basis for undersanding he operaion of elecrical machines which was our aim.