jsf the magnetic induction Β produced by On Eddy Currents in a Rotating Disk ί at the surface of the sheet. The eddy currents are generated not only by the changes in the magnetic in- W. R. SMYTH Ε "duction B' of the external field, but also NONMEMBER AIEE by the changes of the magnetic induction -B of eddy currents elsewhere in the sheet. DEVICE which often occurs in should know its derivation which, as given One of Maxwell's equations combined A electric machines and instruments by Maxwell, is difficult for modern stu- with Ohm's law gives the induced current consists of a relatively thin conducting dents to follow. A simplified proof which to be disk rotating between the pole pieces of a brings out the points essential for our permanent magnet or electromagnet. problem is given below. VXE=Vx- = -~ {B'+B) (3) The author has received inquiries as to The object is to calculate the magnetic y àt the method of calculating the paths ol the induction Β produced by the eddy cur- Writing out the ζ component of this equa- eddy currents and the torque in such cents of density ί induced in a thin plane tion and using equation 2 give cases. The following rather simple sheet of thickness b, unit permeability and

l/d% àix\ method, which is quite accurate for a conductivity y lying in the xy plane by a 1 ÎàBx àBj permanent magnet, seems not to be de- fluctuating magnetic field of induction y\àx by / 2irby\ àx ày scribed in the literature. It assumes that Bf Evidently the only components of (4) the disk is so thin that the skin effect can ί effective in producing magnetic effects be neglected. This is true for all fre- parallel its surface. Let the eddy cur- Another of Maxwell's equations states quencies that can be produced mechani- rents be confined to a finite region of the that cally. To facilitate calculation in the sheet which may or may not extend to in- special case of circular poles it is also as- finity, and let us define the stream func- (5) tion U(xy y) at any point in the sheet to àx ày àz be the current flowing through any cross section of the sheet extending from Ρ to Combining equations 4 and 5 gives

its edge. The line integral of Β or Η over J{BZ'+B2)_ 1 àBz (6) the closed path that bounds this section àt 2wby àz equals 4TTU, From symmetry the coa tribution from the upper and lower halves When àBg/àt is known, this equation of the path is the same so we may write gives the boundary condition on Bz in the plane of the sheet. This, combined with the equations V Χ Β = 0 and V* Β=0 which hold outside the sheet, and the fact that Figure 1. Geometrical relations for deriva- Β vanishes at infinity serves to determine tion of formulas for stream function (1) Β everywhere. By equations 1 and 2 the current density and stream function any- sumed that 2woxiby=ea is much less than where the choice of sign depends on the where in the sheet can be found. eue where ω is the angular frequency of side of the sheet chosen for the integra- The explicit expression for Β in terms rotation in radians per second, a the pole- tion. Differentiating this equation gives piece radius, b the disk thickness, and y

àU By^JL^. àU Bx the electric conductivity, all in centi- HI X (2) meter-gram-second electromagnetic units. — =±— bly= =^— ày 2ττ * v àx 2ττ This produces a fractional error of less These equations connect the eddy current than ea in the eddy current densities and density with the tangential components of less than (ea)2 in the torque. In the case of the electromagnet the situation is complicated by the presence of the per- meable pole pieces in the magnetic field of the eddy currents. This may send a iarge demagnetizing flux through the electromagnet. An approximate solu- tion for this case will be considered.

Maxwell's Formula Figure 3. Lines of flow of eddy currents in- This calculation starts from a formula duced in rotating disk by two circular magnet given by Maxwell in 1873,1 but apparently poles little known to engineers. To apply it one

Paper42-140, recommended by the AIEE committee on basic sciences for presentation at the AIEE summer convention, Chicago, 111., June 22-26, 1942. Manuscript submitted December 12, 1940; made available for printing May 20, 1942. Figure 2. Lines of flow of eddy currents in- W. R. SMYTHB is professor in the department of physics, California Institute of Technology, Pasa- duced in rotating disk by single circular magnet Figure 4. Geometrical relations for calcula- dena, Calif. pole tion of demagnetizing flux

SEPTEMBER 1942, VOL. 61 Smythe—Eddy Currents in Rotating Disk TRANSACTIONS 681 of Β' which was given by Maxwell1 can The initial field of the eddy currents unprimed or primed according as they be obtained as follows. The right side of formed in that interval must be equal and refer to the axis of rotation or to the pole- equation 6 is finite at all times which opposite to this and must die out as if piece axis. The scalar magnetic poten-

means that if At->0 then A(BZ'+BZ)—>0. their source moved away with a uniform ίΐβΙΏ' of its lower face, of area 5, at the" Thus an abrupt change in Β' instan- speed l/(2irby). Thus the eddy currents point Ρ is seen from Figure 1 to be taneously induces eddy currents such as at a time t due to a change in the interval will maintain B'+B unchanged in the dr at a time r before t is given by 2 sheet. Therefore, for a specified change (2*o)*J8 R2 ((2ττα) in Β' the initial value of Β is known, and, dB= Fit — r, x, y, s± —^—\ir (12) fxdrx'dJà' àt \ ' 2Tcby) if no further changes in Β' occur its sub- 2 2 Jo Jo Vh*+rl' +r' -2rl'r' cos θ' sequent values as the eddy currents decay This is Maxwell's formula. It has many are found by putting ö£'/Ö/=0 in equa- applications.2 When the field is pro- where r/2 = r2+c2-2ri: cos (ω/+0). This tion 4 and solving. A second abrupt duced by moving permanent magnets, it combined with equations 12 and 13 give f change in B produces a second set of is convenient to express U in terms of the the stream function to be eddy currents, and so forth. At any in- scalar magnetic potential Ω. Since we stant the actual field of the eddy currents have unit permeability we may write U -X is a superposition of these. As the mag- 'W Jo Jo ^ nitudes of the discontinuous changes in ί7=— / Brdr = —- Ι —dr = — 13) the external field become smaller, and the 2ττ J 2τ J ôr 2TT I n'dn'dB'dr \ intervals between them shorter, we ap- \V hT*+rl'*+RT'*-2rl'Rr' cos θ') proach as a limit a continuously changing 2 2 2 Application to Magnet Moving in a where jRr' = r +c -2rc cos (ω(ί-τ)+0) magnetic field. Circle anahT = h+r/(2Trby) = h+u. Let us now Suppose that the sources of the inducing bring the pole piece down close to the We now take the case of a magnetic field lie above the xy plane where z>0. plate so that hT=r/(2Tby)t and bring up a At /=0 the source changes abruptly the field produced by a long right circular similar pole of opposite sign from the induction being #i' = Fi(x, y, z) when cylinder of radius a, uniformly and per- other side, so that the eddy current den- manently magnetized parallel to its axis, — oo

sin iß-\-eu){Ru' — ri cose')ri'dri'de'du Bt-e-Bi'-A'-FiOc, y, z)-F2(x, y, z) (7) (15) τ(2ττα) 2 2 2 •ΙΤί iV(« +V -fiV --2i?wVi' cos θ') Since B2 is not a function of t, equation 6. 2 2 2 reduces to where € = 2ττω&τ and Ru' = r +c -2rcX being c centimeters from the ζ axis, and cos (θ+eu). For 3,000 rpm with a copper its lower end h centimeters above the 1_ sheet 0.25 millimeter thick €«0.01 so (8) z = 0 plane in which lies an infinite plane 2 2wby àz that u in the denominator has reached sheet of thickness b and conductivity 7. the value 100 when eu Teaches 0.1. In piese equations, 7 and 8, are satisfied by Its upper end is too remote for considera- calculating such a quantity as the torque tion. Polar co-ordinates will be written where the current density is integrated f = Fi ( x, y, ζ =*= —4—J — \ 2irby/ over the pole piece, the neglect of e pro duces a fractional error less than (e/a)2,

F2{xt y, so that the result should be good to one per cent for a sheet one millimeter thick. We may therefore drop the e terms se Because the eddy currents must die out, 350Ύ and their magnetic field must be sym- that Ru' becomes the r' in Figure 1 and χ integrate with respect to u giving metrical about the sheet, we take the plus Λ I sign when s is positive and the negative \ Cu cold \Cu hoi Κ sin θ sign when ζ is negative. Thus equation U= — X

r , / 1 9 shows that, in addition to B2 which (r -r1 cos e'WdrSdd (16) would exist if no sheet were present, there 1L ''W-a-Vcoee' is a decaying field due to eddy currents where we have written Κ for the coeffi- which appears, from either side of the cient of the integral in equation 15. The sheet, to be caused by a pair of images integral with respect to 0', from Dwight's receding with uniform velocity l/(2Tby). table of integrals 860.2, is zero when Suppose our inducing field has the form ι r'Oi' and π/r' when r'>ri'. Thus the B' = F(t, x, y,z) (10) upper limit for the r/ integration is a when r'>a and r' when r'

r ωrcbyΦ sin θ time interval dr is given by Figure 5. Curves showing torque versus speed r'>a (17) for large disk rotating between the four rec- 2^ àB' à tangular pole pairs of an electromagnet, o)fcby& sin 0 (11) r'

682 TRANSACTIONS Smythe—Eddy Currents in Rotating Disk ELECTRICAL ENGINEERING The next question is how to restrict the Thus, using equation 2, we have calculated from equations 19 and 20 ap- eddy currents to the interior of the disk c+a plied to the cases of equations 22 and 24 ΓτΜτΦΊ„ Φ r àU bounded by the circle r=A. We observe are shown in Figures 2 and 3 where a = that if we use equation 15 to calculate U Λ/7 cm, c = 7 cm, A = 10 cm and ωδγΦ/ for a second magnet also carrying a flux where 0i and r are connected by the rela- (2ΤΓ) = 3.5. The value U on the outer "? but with circular pole pieces of radius tion r2+c2—2rc cos 0i = i2. Substituting boundary is zero and changes by steps of 2 a"=Aa/c centered at c"=A /c, so thai for U from equation 20 and integrating one in Figure 2 and steps of two in Fig- 2 2 2 2 2 Ra" = r +(A /c) -2r(A /c) cos (0+6*0 > with respect to 0 give ure 3. and change the variables of integration 2 from n' to r\A/c, and from u to Au/c, then ^ ωώγφ τ if- Demagnetizing Effects the resultant expression is identical with =—ττ-χ equation 15, except that we have cRJ'/A 2 2 2 2 a A r sin ek So far the magnet pole pieces have been r sin 0, - 2 2 2 instead of Ru' and Ae/c instead of e. But «y c-a \ c r +A*-2A rc cos 0-y assumed to be so hard that they do not when r=A we see that cRu"'/A=RU'', so (21) short-circuit the flux of the eddy currents. that both magnets, one outside and one This is not true for the permeable pole inside the circle r=A, give the same U The integration is simplified by taking a 2 2 2 pieces of an electromagnet, whose effect new variable u so that 4acu = r — (c—a) on this circle. Furthermore by taking the may be calculated approximately by ob- which gives the limits 0 and 1. Thus we air gap in each magnet small, the fluxes serving that the current 2 U is enclosed by obtain, writing out e, are confined to the areas under the pole the rectangular path 1-2-3-4-1 in Figure 4, pieces, so that neither induces directly 2 2 2 2 ώΊΦ ο ( A a \ which lies in the upper and lower pole 1 1 2 eddy currents on the other side of the 2 2 2 2 2ττα V * (A -c ) / ογΦ /λ (22) pieces except where it cuts across the disk circle r=A. It is evident that if the and gap normally at r = Y\ and Θ— =*=0i. This formula gives the torque in dyne fluxes from the two magnets cut the sheet If the reluctance of this circuit lies en- centimeters when ω is in radians per in opposite directions, then U=0 when tirely in the air gaps, each of length g, then r-A and the currents induced by the second, Φ in maxwells, a, b, c, and A in the magnetic flux density Be due to the inner pole are kept inside the circle. This centimeters and y in electromagnetic eddy currents alone at rh =*=0i is 4π Ό I g. units. If we are given the volume resistiv- is exactly the boundary condition ΐοτ β Substituting for U from equation 20 and ity ρ of the disk in ohm-centimeters y — disk of radius A, except that the calcu- writing as before e = 2πω& gives lated system includes the currents in- »-»/p. duced in the region rAy which does not exist in the case of an undesired force on the disk axis which may be avoided by using two identical the disk. This field is proportional to This shows that when b and g are com- Φ€ which is, by hypothesis, small com- magnets on opposite sides of the axis and parable in size Be cannot be neglected equidistant from it. This approximately pared with Φ, and in addition the source is compared with the original flux density further away, so that the fractional error doubles the torque given by equation 22. 2 Φ/(7τα ). The sin 0X term shows that the The additional torque from the eddy cur- in U will be less than e. We should note radial component of the eddy currents also from the symmetry that the radial rents of one magnet flowing under the induced by Be have opposite signs under poles of the other may be found by an in- component of these secondary currents the two halves of the pole piece, so that tegral similar to equation 21 wThich is is opposite in sign on the two sides of the they contribute nothing directly to the &*=0 line, so that their effect cancels out torque, but on the other hand they form completely in calculating the torque r2+c2+2rc cos 0] closed circuits about the central portion which therefore should be accurate to and so produce a demagnetizing magneto- 2 2 2 terms in e . The contribution to U from r A sin θχ \dr (23) motive force in the electromagnet. The the outer magnet is found by putting r2c2+AA+A2rc cos 0] c2R"2/A2 for R2 inequation 17. Adding Integrating by the same substitutions as this to equations 17 and 18, we obtain for equation 21, adding to equation 22 and the stream function of the eddy currents multiplying by two give in the disk |

2 2 2 2 2 2 ωάΎΦ € /±€ +α 2a A (A*+ T = 2 ωτώ^Φ sin θ l 7TÛ 4c (A* R>a U> y ) 2 2 2 \r +c -2rc cos 0 = ωτΦ £>2' (24) This holds when the two magnet fields are 2 2 A 2 (19) c r +A -2rcA cos 0 antiparallel. If we subtract the integral of equation 23 from equation 22 and mul- ωrcbyΦ sin θί R

SEPTEMBER 1942, VOL. 61 Smythe—Eddy Currents in Rotating Disk TRANSACTIONS 683 stream function Ue of these eddy currents curves shown in Figure 5 which give hot kfgh speeds, it is interesting to see what is calculated as U is, but to simplify mat- and cold copper disks the same Tm for results they give for a comparable case. ters we carry out the operations from different ω^. Let us take δ = 0.4 cm, A = 25 cm, a = 4 equations 14 to 16 for a single element of These calculations of demagnetizing cm, c = 21 cm, g = 0.6 cm, 5 = 2,000 gauss the pole face, along with its image ele- effects have been worked out for a single and assume the reluctance entirely in the ment outside the circle r—A. We then pole. For an even number of poles with air gap. In equation 22, Pi =1.23, in give each element the strength indicated alternating signs, we have seen that the equation 27, 0 = 3.85 and in equation 28 in equation 26 and set up a definite in- torque per pole is increased, but the de- (ft = 0.012. The angular velocity for maxi- tegral for U over the pole-piece area. magnetizing forces are also increased so mum torque for copper (γ = 1/1,700) is This method is less exact than setting up that the torque obtained by multiplying given by equation 30 to be 27.9 radians equation 14 for the whole face, because it equation 29 by the number of poles will per second or 267 rpm. Tm is 1.15 Χ1Θ* ignores that part of the flux threading dS probably not be far wrong. The speed for dyne cm or 1.17 kilogram-meters for this from the current induced by Be outside maximum torque given by equation 30 single pole and roughly four times this for this area, which is of the order eB^S. wiH certainly be decreased, perhaps con- four poles. Expressing T' in kilogram-

The eddy currents Ue are evidently siderably, because of the increase in β. meters and ω' in rpm, equation 29 be- equivalent to a magnetic shell of variable The only formula we can find for this comes

3 strength Ue in the sheet and to get ïïe the torque is one derived by Rüdenberg. demagnetizing magnetomotive force we This formula is written as a double infinite 0 . 00785a/ ,2 2 kg m must find the equivalent uniform shell. series and is derived by considering a thin (ί +o.ooœo47w ) ' ~ Thus we have conducting strip bounded by straight lines which moves lengthwise in the nar- This formula is plotted in Figure 6. A UedS row gap between magnetic poles with comparison of Figure 5 with Figure 6 rectangular faces. The fields of adjacent indicates that our formula gives too rapid a falling off in torque at high speeds. It where 5 is the area of the pole face. We poles are antiparallel, so that the inducing should be pointed out that other condi- now have a complicated quadruple inte- fields can be expanded in a double series of odd harmonics. This formula was tions, such as the degree of saturation of gral involving the variables r'f R't θ' and 4 the iron in the magnet will upset the as- θχ whose evaluation can be simplified checked qualitatively by Zimmermann, sumed relation between magnetomotive somewhat by integrating in the proper but could not be verified quantitatively, force and Φ and may modify equations 38, order. The result is as the theoretical and experimental boundary conditions did not agree. Lentz 29, 30, and 31 considerably. 2 2 2a2A2(A2-\-c2) r

2 2 A* (A -c 2 2 and dropped the rest. His experimental method used for two and to other than loge : β Ύ ω*Φ (27) (A2-c2)' -c2a2)~ brake had the center of the disk removed circular faces. Several such calculations to simulate a ring whose width roughly have been carried out, but it is doubtful If the flux penetrating the sheet at rest is equaled that of the postulated strip. His if the additional theoretical accuracy jus-

Φ0, then when in motion we have, if (R is four poles were so far apart that their tifies publishing them. The difference the reluctance of the electromagnet, Φ<= action was nearly independent. We have 2 2 2 between the ideal boundary conditions Φ0- 0 7 ω Φ/Φ, so that redrawn in Figure 5, his experimental used here and those found in apparatus ΟΙΦο curves giving the torque in meter kilo- is such that we recommend that the (28) Φ= • grams against angular velocity in revolu- torque for one pole be calculated by equa- tions per second. The ring had inner and tion 22 for permanent magnets or by The expressions for the torque now be- outer radii of 5 centimeters and 25 centi- equation 29 for electromagnets, and the come meters and was 0.4 centimeter thick. result multiplied by the number of poles

2 œy(R^0 D The air space was 1.2 centimeters, and to give the total torque. In power ap- 7 = (29) ((R+/3Vo>2)2 the centers of the rectangular pole pieces paratus the heating of the disk will change were 20.75 centimeters from the rotation its resistivity and may cause it to expand where D has the values given in equations axis and were 6 centimeters (radial) by 8 and buckle and otherwise upset the cal- 22, 24, or 25, according to the pole ar- centimeters (tangential). The inducing culations. rangement. There is now a definite speed field was 2,150 gauss at rest. The figures for maximum torque which is found by on the hot copper curve show the esti- settingör/Οω = 0 to be mated stable mean temperature for that References speed. (30) 1. ELECTRICITY AND MAGNETISM, Maxwell. Ox- ßy A direct quantitative comparison of our ford University Press, 1892, volume 2, page 297.

formula with Lentz's data is difficult, be- 2. STATIC AND DYNAMIC ELECTRICITY, Smythe. Putting this in equation 29 gives cause he used rectangular poles, his air McGraw-Hill Book Company, New York, Ν. Y., 1939. Chapter XI. gap was so large as to spread the inducing 2 τ _3λ/30ΪΦΟ ^ 3. SAMMLUNG ELEKTROTECHNISCHE VORTRÄGE. (31) field over an unknown area, the center of 16/3 Verlag Enke, Stuttgart 1907. Bd 10. page 269 and his disk was cut away, and we do not following. This is independent of the conductivity know where his flux density was meas- 4. Elektrotechnik und Maschinenbau, Bd 40, 1922. page 11. which is surprising, although there is some ured. Although our formulas are inac- δ. Elektrotechnik und Maschinenbau, Bd 52, 1934, evidence for it in Lentz's experimental curate for such large dimensions at the pages 99-102.

684 TRANSACTIONS Smythe—Eddy Currents in Rotating Disk ELECTRICAL ENGINEERING