Philipse Technical Review

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1 , _,) OL. 5., N.2 FERUARY 1940 Philipse Technical Review RELATNG DEALNG WTH TECHNCAL PROLEMS TO THE PRODUCTS, PROCESSES AND NESTGATQNS OF N.. PHLPS', GLOELAMPENFAREKEN EDTED Y THE RESEARCH LAORATORY OF N.. PHLPS' GLOELAMPENFAREKEN,. EÎNDHOEN, HOLLAND.. THE USE OF, MODERN STEELS FOR PERMANENT MAGNETS by A. TH. van URK.. n. rder t btain a certain magnetic field strength in a given air gap with as little magnet steel as pssible, the cnstructin f the magnet must be s chsen that the prduct f inductin and field strength in the interir f the magnet steel is as large as pssible. n the case f mdern magnet steels with very high cercive frce and relatively lw residual magnetism, this nditin leads t quite different cnstructins than in the case f the lder kinds f steel with lw cercive frce and high residual magnètism, t is' explained in this article hw ne sets abut the cnstructin f a magnet. n 'the first part spreading is neglected: in the 'secnd part a semi-empirical methd is given f taking' spreading int accunt. " nj.: '... ntrduetin, f we 'wish t imprve upn a magnet made f the tungsten steel ) frmerly used, it wuld seem bvius simply t substitute fr the tungsten 'steel a piece f mdern magnet steel f the same shape. The result wuld," hwever, usually be fund very disappinting: the magnet is scarcely, imprved at all. The mdern magnet steel may even prduce a weaker field than tungsten steel. On the ther hànd it has been fund that upn judicius cnstructin f the whle magnet' mdern magnet steels d give. very much better results than the steels frmerly used, and that with the magnet steel "Ticnal 2A" fr instance a field can be excited in a given air gap which culd nly be btained with -ten times as much' tungsten steel. t fllws frm this that the disappintment mentined abve must nt be ascribed t the steel itself, but t the fact that a design intended fr tungsten steel is 'nt suitable fr making the mst fthe specific qualities f-the mderil magnet steels.' The aim f this article is t find ut the reasn fr this, and t learn hw a magnet shuld be designed in rder t use a given kind f magnet steel t the greatest advantage. 1) Tungsten steel may be cnsidered as the best representative f the lder type f magnet steel. Mdern magnet steels differ in physical respects frm the steels used earlier mainly in the nature f the rearrangement prcesses in the material by which the magnetic hardness is btained; see in this cnnectin: J. L. Snek, Philips techu. Rev. 2, 233, Cmparisn f mdern and lder kinds "f steel The prperties f a magnet stel are deterniind by its magnetizatin curve. The shape f this curve can be established apprximàtely by the" values f residual magnetism and cercive frce (see fig. 1). The reidual magnetism r remanece is the maximum inductin which remains in the steel when the magnetizing field is reduced t zer after the steel has been magnetized t saturatin. The cercive frce is thé magnetic field which must be applied in a directin ppsite t the residual induc-, tin in rder t reduce the latter t zer. Hi Fig. 1. nductin f a magnet steel as a functin f the internal field Hi. The remanence is r. the cercive frce He. Fig. 2 gives the'magnetizatin curve f a mdern nickel- aluminium steel, such fr example as the previusly.mentined "Ticnal 2A", cmpared with that f a tungsten steel. t may be- seen that an increase f the cercive frce f nearly tenfld is btained by the sacrifice f part f the remanence, which has fallen back t nearly ne half. n spite

2 30 PHLPS TECHNCAL REEW l. 5, N. 2 f its lwer remanence, hwever, the magnet steel "Ticnal" must in general be cnsidered very much better than tungsten steel. n rder t shw why this is s, we shall examine the results btained in slutin f the prblem f btaining a certain field strength in a given air gap with as little magnet steel as psle. J.! /' Hi erst 1:0') -100D gauss w/ J4467 Fig. 2. nductin f a tungsten steel (W) and f the magnet. steel "Ticnal 2A", as functins f the cinternal field Hi which is ppsite in directin t the inductin in the part f the curve given. Simplified calculatin f a magnet We shall begin with a magnet cnstructed as, indicated in fig. 3, namely a magnet f length L and crss sectin S, upn 'which tw pieces f sft irn -are fastened, which tgether bund an air g'ap f length Z and crss sectin s. Fr the' sake f simplificatin we shall assume fr the present that the lines f frce f the magnet must all pass thrugh the air gap in rder t pass frm the nrth t the suth ple. Thus spreading is neglected. Furthermre, the, fields in the magnet steel and in the air. gap will be cnsidered hmgeneus, and we shall als assume that the sft irn is abslutely cnductive fr magnetic flux, i.e. at any arbitrary value f the inductin 13 the field Hi in the sft irn is zer. The magnetic cnditin f the mdel thus sim- 'plified can he described by three quantities: the external field strength H in the air gap, the internal field strength Hi and th inductin f the magnet Fig. 3.' Mdel fr-simpliûed calculatin f a magnet. The air gap has 'a length 1 and a crss-sectin area f S; the magnet steel has a length L and a crss-sectin area S. steel. Nw three equatins are sufficient fr the calculatin f a magnet in the first apprximatin. The first equatin is derived frm the prpsitin that the ttal magnetic flux is,the same thrugh every crss sectin f the system, and therefre als thrugh a crss sectin f the magnet and f the air gap. The equatin is therefre as fllws: S = Hs. '() The secnd equatin fllws frm the prpsitin that the line integral, f the magnetic field strength is zer alng every clsed path. Since the field strength is assumed t be zer in the sft irn, this means that: HZ+ HiL = 0, HiL = -Hl. (2) (2a) Hi is therefre ppsite in sign (ppsite in directin) t H., Finally as third equatin we have the relatin ' between and Hi which is given by the magnetizatin curve f the magnet steel: We are practically cncerned with the part f the curve in which the directin f the field strength is ppsite t that f the inductin (cf. fig. 2), s that the field has a demagnetizing actin. efre we make use f this curve, we shall hwever first draw a general cnclusin which becmes bvius when we multiply equatins (1) and (2a) byeach therr The result f this peratin: expresses the fact that the vlume Zs f the air gap, multiplied by the square f the field strength, is equal t -the vlume LS f the magnet, multiplied by the abslute value f the prduct Hi. Since the prblem is, with a given air gap and field strength in the air gap, i.e. with a given secnd term, t keep the vlume LS f the, magnet as, small as pssible, it fllws frm (4) that we must chse magnet steel r a design' f cnstructin whereby the prduct Hi is as large as pssible. Nw fr a given magnet steel Hi is nt a cnstant; bt depends upn the value f Hi. That is bvius, because the prduct disappears fr Hi = '0 ( is equal t the remanence}, as well as fr = 0 (Hi is equal t the cercive frce), while at intermediate values it is nt equal t zer. With a given value f Hi, therefre, the prduct will have a maximum (see fig. 4). (3) (4)

3 -...-=' _-.' FERUARY 1940 PERMANENT MAGNETS 31 The height f-the maximum f Hi nw prvides us with a measure f the quality f a magnet steel, while the crrespnding vallie f Hi determines' the mst favurable design f the magnet. Fr tungsten steel the maximum lies at 30 ersted and amunts t 2 X 10 5 gauss-ersted; fr "Ticnal 2A" the maximum lie at 400 ersted and amunts t 1.8 X log gauss-ersted. Accrding t equatin (4.), therefre, nine times less "Ticnal" steel than tungsten steel is necessary t prduce the same field in a given air gap. v J ',,- -, / \. '\ ffll (gauss ersted) t6 1. '2-1/ \ f \ / \ / '\, H (erste 1 w0-9cxj aa Fig. 4. Abslute value f the prduct EHi (inductin times internal field) as a functin f H, fr a tungsten steel (W)' and fr the magnet steel "Ticnal 2A". 14 D field. On the ther hand the inductin fr ptimum field strength in the case f the magnet steel "Ticnal" is slightly lwer than with a tungsten steel; accrding t 'equatin (6), therefre, the crss sectin S must be taken slightly larger. We are' nw als able t understand why the replacing f a tungsten steel by the magnet steel "Ticnaf" (withut alteratin in design) des nt always lead t a better result. T shw this we divide.equatin (1) hy, (2a): L s H'=-SZ', Thus if we have a magnet and an air gap f knwn dimensins, the rati /Hi is knwn. n the diagram f the magnetizatin curve this gives us a straight line thrugh the rigin which,cuts the magnetizatin curve at a pint which gives us the values f and Hi. Nw it.is quite pssible (see fig. 5). fr this line t cut the curve f the better steel at a pint which gives a lwer prduct f 'H; than the pint at which it cuts the curve fr the prer steel. t is, hwever, clear that in this case the magnet f the better steel is very badly prprtined. y changing the design it is certainly pssible t btain a cn" siderably better result with the same quantity f magnet steel. (7) After we have calculated the vlume f magnet steel needed, we can calculate the required length f the magnet with the help f equatin (2). We find H L= - -l... :-.. '(5) Hi and we must substitute here fr Hi the ptimum value (with "Ticnal" 400 ersted). The crss sectin f the magnet is nw f curse als knwn. The fllwing relatin is valid, accrding t equatin (1):. Hs s=- ' (6) where is the inductin which, accrding t equatin (3), crrespnds t the ptimum value f the demagnetizing field Hi. Frm equatins (5) and (6) it is clear hw the design f a magnet must be changed when a tungsten steel is replaced by "Ticnal". The ptimum field Hi ha-s increased thirteen times and accrding t equatin (5) this means that the length f a "Ticnal" magnet must be nly 1/ 13 f that f a tungsten steel magnet which prduces the same. Fig. 5. The rati between the inductin E and the internal field Hi with a given design is independent f the prperties f the magnet steel. With a sufficiently high value f this rati (slpe f the straight line EfHi = cnstant) the prduct EHi is.greater fr the magnet steel 1 than fr the steel 2, althugh the maximum value f the prduct is smaller fr steel 1 than fr steel 2. The spreading f lines f frce Until nw' we have failed t take int accunt the spreading f the lines f frce.. We reached the cnclusin that the quality f a agnt steel is

4 32 PHLPS TECHNCAL REEW vei. 5, N. 2 determined slely by the maximum value f the prduct Hi. A lwering f the remanence f the magnet steel therefre meets with n bjectins, if nly' it is accmpanied by a crrespnding in-, crease. in the cercive frce. Such a change in the magnetic prperties has; hwever, the result that the design must be altered, and the new design may have cnsiderably mre spreading than the ld. An example will serve t illustrate this pint Fig. 6. a) Diagram f a crss sectin f a lud speaker magnet, cnstructed f the magnet steel "Ticnal 2A". b) Crss sectin f the same magnet, if the cercive frce f the steel were three times as great, and the remanence ne third. The shape changes in such a wily that the spreading becmes cnsiderably greater. ' Fig. 6a shws the crss sectin f a magnet such as is used in lud speakers. The' magnet steel is shaded, while the sft irn f the ple pieces is left white. The' relative dimensins. f this magnet are adapted t the -jl curve f the magnet steel."ticnal 2A". f a tungsten steel were used, the height f the magnet wuld have t be cnsiderably greater, while the diameter culd be taken slightly smaller. Let us nw cnsider the imaginary -case where, while the prduct Hi is kept cnstant, the cercive frce f the steel is three times as great as that f "Ticnal", while the remanence is nly ne third f that f "Ticnal". The magnet wuld then have the shape given in fig. 6b. As far as spreading is cncerned, the design is much less satisfactry. n the first place, due t the increase in diameter,. the surface frm which the spread field riginates is increased, and in the secnd place, due t the decrease in height, the resistance f the air path fr the lines f frce, f the spread' field is cnsiderably decreased. Althugh when spreading is disregarded th secnd magnet steel is just as gd as the first, it will nt be pssible t btain as gd results with it when it is' used as a lud speaker magnet. t may be stated as a general cnclusin, that the questin as t which magnet steel is the mst suitable fr exciting a field f a certain strength in an air gap f a given frm, cannt be answered by nting exclusively the maximum value f Hi, but that and Hi themselves als play a part. n particular 'the remanence f the magnet steel "Ticnal 2A" is fund t be t lw fr certain applicatins. A new magnet steel has therefre been develped which has a slightly lwer cercive frce than "Ticnal 2A", but practically twice its remanence, namely gauss. The maximum value f Hi in the case f this steel is als twice as great. n jig. 7 may be seen the magnetizatin curve f this magnet steel which has been called "Ticnal 3.8". f we nw return t the questin f hw a magnet must be cnstructed in rder t use the magnet Steel tthe greatest pssible advantage, ur' qualitative cnsideratins f the influence f spreading are inadequate, and must. be supplemented by a methd f taking the spreading int accunt quantitatively in the calculatin f th magnet.., gauss ] / / f/»: /8 r! f. ' \./ 1\ / 1\ Hi ers! ) -400, -200 \ \- \ ' Fig. 7. Magnetizatin curve f the magnet steel "Ticnai 3.8", cmpared with that f "Ticnal 2A". The dtted lines refer t the examples treated in the cnclusin. Mre precise calcnlatin f a- magnet The mre elabrate calculatin f a mgnet which we shall carry ut in the fllwing differs frm the abve simplified.treatmerrt in the fact '

5 FERUARY 1940 PERMANENTMAGNETS 33 that we determine nt nly the flux in the effective air gap, but the flux between each pair f surfaces between which lines f frce pass thrugh the air. n calculating these different cmpnents f the ttal flux we can t ur advantage make use 'f the cncept f "magnetic ptential", which is defined in the fllwing way: UM - P f u, ds, P where ds represents an element f a line drawn frm a pint P, at which the magnetic ptential is cnsidered t be zer, t the pint P, at which the magnetic ptential is indicated by the integral. H s is the cmpnent f H in the directin f the line element ds. The flux fr each tw surfaces between which lines f frce pass can nw be represented by the magnetic ptential difference, multiplied.by. the magnetic cnductivity. The latter quantity is therefre defined in, exactly the same way as an electrical cnductivity. The magnetic cnductivity.f a rd f length Z, crss sectin s and permeability (.L is: g = (.LsjZ. The specific cnductivity is thus nthing else than the permeability. The cnductivity f the air path between tw surfaces is equal t the crss sectin f the bundle f lines f frce which pass between the tw surfaces. Actually bth the crss sectin and the length must be cnsidered as sme kind f average values. The way in which these average values are determined will be discussed presentlyn the basis f an example. ' f Uk is the ptential difference between every pair f,surfaces and gk is 'the crrespnding cnductivity, the ttal flux (8) The ttal flux (P is ften cmpared with the effective flux (P = U g which passes thrugh the air gap. Therati (Pj(Pis called the cefficient f spreading a. When the cefficient f spreading' is knwn" the calculatin f the magnet can be carried ut in the way discussed at the beginning f this paper. Equatin (1) must be replaced by the mre precise relatin' S=Hsa. and this, tgether with equatins (2) and (3) gives the three unknwns, Hi and H: The whle prblem is thus reduced t the meas- (9) urement r calculatin f the terms Uk gk r u,g', which appear in the calculatin f the' cefficient f spreading, a. Often there is.appreciahle spreading nly between the sft irn ple pieces, between which the air gap is situated. The different magnetic ptential differences Uk are then -practically equal t each ther and t, UM. n that case nly the cnductivities gk r g ccur in equatin (8)., The cnductivities can be calculated exactly in certain simple cnfiguratins. 'Usually, hwever, 'the calculatin is yery difficult r even impssible. Since this prblem has great practical significance, fr instance fr' the cnstructin f measuring instruments, we shall explain in the' fllwing hw frmulae can be arrived at by 'a semi-empirical methd which make it pssible t calculate the field in the air gap f a magnet fr different shapes and sizes f the magnet and different kinds f magnet steel. As crss sectin f the bundle, ' f lines f frce which' pass between,,tw surfaces, we take the average size f'the tw surfaces, n rder t determine the ength f path in the air, ;e chse a simple shape fr the lines f frce, a straight line. r part f a circle, fr instance. The frmula fr the cnductivity s btained is finally cmpleted by a numerical factr which is determined experimentally by measuring n a mdel the flux which passes between these tw surfaces. t is clear that with the help f the frmulae s btained it is nly pssible t make precise éalculatins fr a mdel which is, an enlarged r reduced reprduetin f the experimental mdel. The great value f the methd, hwever, lies in the fact that the empirically fund numerical factrs apparently d nt change when the relatins between the varius dimensins f the magnet and the nature f the magnet steel in the actual practical mdel are quite different frm thse f the experimental mdel. With the help f the same frmula, theref.re, it is pssible t anticipate the prperties f very different types f magnets. A simple example will be give in the fllwing. Cmplete calculatin f a simple example As example we have chsen the design repreeented in jig. 8, which cnsists f tw cylindrical magnets M lying Î? a straight line, prvided with irn ple pieces P and clsed by an irn armature which is thick enugh t have practically n magnetic resistance. The distance a between the armature and the ple pieces is made s great that 'practically n appreciable flux, will, pass acrss this air space.,,

6 34 PHLPS 'TECHNCAL' REEW l. 5, N. 2 Let us 'assume that the left-hand ple piece is the nrth ple' and the right-hand ne the suth ple. The flux thrugh the air space can be divided as fllws: 1) the flux (/>1 frm the left-hand magnet t the right-hand magnet; 2) the flux (/>2 frm the wall f the cylinder f the left-hand ple piece t that f the right-hand ple piece; 3) the effective flux (/>a frm the flat surface f the left-hand ple piece t that f the, righthand ple piece. ep, t will be seen that in additin t the quantities' already mentined anther factr a has been intrduced. This factr is determined experimentally by measuring the flux (/>1' The frmula is in this way made t prduce the crrect result fr a given mdel. n general it is fund that the empirical crrectin factr des nt change very much when ne passes frm the experimental mdel t a design. with quite different dimensins, r, 1 1, 1 2, d; this is the basis f the whle methd f calculatin. The flux (/>2 thrugh the side wall f the sft irn ple piece is calculated in exactly the same way as the flux (/>1; the result is: './ Fig. 8, Mdel f a magnet whse effective flux (/J and spread flux (/Jl + (/J2 can he calculated by a semi-empirical methd. Crss hatched: magnet steel, white: sft irn. The flux (/>1 will be prprtinal t the cylindrical surface' 2 'TC T 1 2 ' f the magnet, and inversely próprtinal t the length f the lines f frce. F this latter length we take as average value the length,f the semicircle drawn in the figure, which is equa t where (J is again a factr which must be determined experimentally. finally we calculate the effective flux. The length f the lines f frce (d) and the crss sectin f the bundle flines f frce (:it r 2 ) are well defined with a sufficiently narrw air gap, s that it is unnecessary t intrduce a crrectin factr. The fllwing result is btained: (12) The experiments fr the determinatin f a and {J were carried ut with a magnet f "Ticnal 3.8" having the dimensins: r 1.28 cm, d cm, 1 1 = 0.2 cm, 1 2 = 3 cm. The fllwing values were fund: The magnetic ptential then passes frm the value 0 fr the utermst line. f frce t,the value Hi fr the innermst ne; we have here taken as average value 1 2 Hi. n this way we arrive at the frmula: H = ersted; (/>1 = maxwell, (/> ", (/>0 == " ttal flux (/> " (1 - (/>/(/>0 = Frm the measured' value f (/>0' with the. help f equatin (12) we can calculate the internal field Hi, and we find a value f 373 ersted. y means. f equatin (9) and (10) the spread flux (/>1 and (/>2 can nw be calculated. 'The results are: (/>1 = a' maxwell, (/>2 = {J' " y cmparing these values with the measured values f (/>1 and (/>2 we find:

7 FERUARY 1940 PERMANENT MAGNETS 35 a fj ) ". All the quantities are nw knwn which are necessary fr setting up a frmula fr the' cefficient f spreading with the help f equatins (9)" (10), (ll) and" (12): " a = (!P 1 +!P 2 +!P)J!P =, 2 d ( 3,43 l2' 8,28l 1 ) =- +-'- +l.. :re r d d+1 1 ' Equatin (13) frms the basis fr the calculatin f a magnet f the type described abve. f the cefficient f spreading a is knwn, equatins (2), (3) and (9) giveus the therunknwns, H, Hi and. n rder t check the frmula and thus t test the whle methd, a secnd mdel was made with very different dimensins frm.' thse f the test.' mdel, while in additin a' different kind f magnet steel "Ticnal 2A" was' used instead f "Ticnal 3.8" (see fig. 7). The dimensins fthe secnd mdel are the fllwing: r cm, d 2.05 cm, 1 1 = 1.0 cm, 1 2 = 5.6 cm,." crss sectin magnet 3) 8.28 cm'', The field H in the air gap f this magnet was fnd t be 980 ersted. Let us nw find the field strength which is t -be expected n the basis. f the frmulae derived abve. Accrding t (13) we find a = 4.6. (13) 2) a and (1 are much larger than ne, which means that ur estimatin f the cnductivity befre the intrduetin f the crrectin factr was very inaccurate.. The fact that the actual cnductivity is very much higher is understandable. The area f the crss sectin f the bundle is very much greater at the middle than we have assumed. t is remarkable that this rugh methd f taking spreading int accunt gives fairly gd results in the end. 3) The magnet had a hle alng the axis s that its crsssectin area was slightly less than 17. r 2 " Accrdingt equatin (9) 8.28 = H X 9.08 X 4.6, while accrding t equatin (2) Hi = -H 2,05. 1l,2 f we eliminate H frm thes tw equatins (as was dne in the derivatin f equatin (7)), it fllws that: 9,08. 4,6. 1l,2 ' = 27,6. Hi 2,05. 8,28 Nw with the help f the magnetizatin curve we can 'als determine and Hi themselves: the line JHi = 27.6 cuts the -Hi curve f the magnet, steel used ("Ticnal 2A") at the pint = 5000 gaus;, Hi = -182 ersted. We finally find fr the field in the air gap "-112 H = 2 O Hi = 994 ersted., The discrepancy with the measured result thus amunts t nly 1 5 per cent, which must be cnsidered fairly satisfactry, "especially when it is kept in mind that the cntrl mdel had quite different dimensins frm the test mdel (the rati between the spread flux and' the effective flux is, fr example, three times a' great as in the test mdel). Having seen hw a magnet can be calculated, it remains t find ut hw a magnet can be designed inrder t prvide a given field in a given"air gap. "We begin by calculating the length and the crsssectinal area f the magnet steel accrding t the equatins (2a) and (9). Fr He.we chse a favurable value, and estimate the 'value f" the cefficient f spreading a. With the diensins f the steel btained- we design a magnet and then calculate Cf accrding t equatin (13). f the calculated value agrees with the estimated ne, the wrk is finished, therwise we repeat the calculatin with the new value f Cf and the result will be att.ained after a secnd r third attempt."