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 </<0 and B2' = F2(x, y, z) whea sity is doubled. We now carry out the 0< t< co, As just shown the eddy cur- so as to give a total flux Φ. The magnetic differentiation with respect to t and set rents generated at t = 0 initially keep the pole density in the face is therefore t±= 0 so that the 0 = 0 line bisects the pole 2 field on the negative side of an infinite Φ/(27τα) . This magnet moves in a circle piece when t = Q. The integral then be- sheet unchanged. When s<0 we have with a uniform angular velocity ω its axis comes therefore 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.