1967, No. 9 271
The skin effect
H. B. G. Casimir and J. Ubbink
1. Introduction; the current distribution for various configurations IT. The skin effect at high frequencies Ill. The skin effect in superconductors
"It was discovered by mathematical reasoning that when an electric current is started in a wire, it begins entirely upon its skin, infact upon the outside ofits skin; and that, in conse- quence, sufficiently rapidly impressed fluctuations of the current keep to the skin of the wire, and do not sensibly penetrate its interior. Now very few (if any) unmathematical electricians can understand this fact; many of them neither understand it nor believe it. Even many who do believe it do so, I believe, simply because they are told so, and not because they can in the leastfeel positive about its truth of their ownknowledge. As an eminent practician remarked, after prolonged seep- ticism, 'When Sir W. Thompson says so, who can doubtit? " These were the wordsof Heaviside in apleafor the use of mathematical methods in 1891. Now, seventy-five years later, the skin effect is such common knowledge that one sometimes thinks one understands it even without "mathematical reasoning". The expression for this effect put forward in 1886 by Rayleigh and now often used as a matter of course, does however have its limitations. For example, its application to pure metals at very highfrequencies leads to incorrect results, as noted by H. London in 1940. Superconductors are another special case, where theformula predicts an infinitely thin skin layer. These are afew of the problems which will be dealt with in this article, showing as far as possible their inter-relationship. The article is divided into three parts, the first of whichfollows here.
I. Introduction; the current distribution for various conûgurations
If a direct current flows in a conducting wire, it will that the charges in the wall of the cage are mobile. be distributed uniformly over the cross-section. With The tendency of the current to flow at the surface is alternating current, however, the current distribution is closely connected with the stable character of the not homogeneous and, if the frequency, conductivity electromagnetic phenomena. We see this in the follow- and dimensions of the conductor satisfy certain con- ing way. Let us assume that inside the metal there is a ditions, to be dealt with later, the current flows mainly filamentary current I which is increasing in strength in a thin layer at the surface of the conductor. This (fig. 1). This current is associated with a rotational phenomenon is called the skin effect. It is an electro- magnetic field H around it which is also increasing. A dynamic effect, that is to say, it is a result of the way in changing magnetic field induces a rotational electric which time-varying electric and magnetic fields and field E which, in turn, induces a current in the metal. electriccurrents areinterrelated. The skin effect phenom- According to Lenz's law, the direction of E is such that enori is quite different from the action of a Faraday it opposes the increase of I, thus keeping the situation cage, for instance, which acts as a barrier to a static stable. The figure shows that simultaneously at some electric field purely and simply as a result of the fact distance from I a current is generated parallel to I. The net result therefore is that the current is forced out- Prof. Dr. H. B. G. Casimir is a member of the Board of Manage- ment of N. V. Philips' Gloeilampenfabrieken; Dr. J. Ubbink is with wards. Furthermore, we see that this effect increases Philips Research Laboratories, Eindhoven. with frequency (E is larger the more rapid the change
L.._ ~ ~~__ ~_ PHILIPS TECHNICAL REVIEW VOLUME28
subject to many cycles of the alternating field between two collisions and within the mean time that it spends t in the skin layer. Broadly speaking, the field then "sees" in effect a layer of free electrons. Finally, at still higher frequencies, the "plasma fre- quency" of the metal will be reached above which the metal becomes transparent to the radiation. These skin effect complications at high frequencies form the subject of part IL Fig. 1. A filamentary current J is accompanied by a magnetic Metals in the superconducting state form a special field H. As J and H increase, an electric field E is produced class of conductors. These will be discussed in part "rIL whose direction near J is such' astto oppose the increase in J, whereas further away from J the induced field E produces a We consider the two-fluid model, in which the electrons current in the conducting medium parallel to J. are divided into two types, normal and superconduct- ing. The superconducting electrons, although they dissi- pate no energy, do have screening properties (even at ofH) and with conductivity (the larger the conductivi- cu = 0). They-therefore cause a skin effect: fields can ty, the larger the current caused by E). penetrate only to the London penetration depth, In this :first article, we shall consider in some detail which is independent ofthe frequency, and the "super- the configuration of the current for one or two special current" in this layer does not cause energy losses. The situations. normal electrons within this layer do, however, absorb The skin effect can be a disadvantage in transporting electromagnetic enemy for cu=!= 0, givmg some alternating current energy along a wire or cable. To (small) high frequency losses. There is, however, a keep the resistance low, the cross-section of the con- frequency limit above which the superconducting ductor should be as large as is practicable but the effect electrons also absorb energy. This absorption is due of increasing the diameter is far less than with direct to the transfer of electrons from the superconducting current. For alternating currents it is advantageous to to the normal state by the radiation, via a quantum use hollow cables (in power engineering), or braided process. A superconductor differs very little from an cable (in radio engineering). At microwave frequencies ordinary metal at frequencies above this frequency the skin effect can be put to good use: the effect makes limit (which lies in the microwave range). it possible to transport and store electromagnetic Since the skin effect is based entirely upon the dy- energy without radiation losses by using closed wave- namic properties of electromagnetic fields and currents guides and resonant cavities. as given by Maxwell's equations, it will be useful to At high frequencies the skin layer may be regarded as set down the four equations here: a layer screening electromagnetic radiation incident upon the metal: as a result ofthe conducting properties curl H = oD/of + J, (1) of the metal the radiation penetrates into the metal no curl E = -oB/Ot, (2) further than the depth of the skin layer. Even a single electron, whether bound or free, possesses screening div B = 0, (3) properties to a certain extent: incident radiation is div D = (J. (4) scattered by the electron so that the power travelling straight on is less than the incident power. The following points should be noted:
As cu(the angular frequency) or (J (the conductivity) a) In what follows we shall in general regard the mat- increases, the penetration depth ~ decreases. In simple erial as a medium with a given relative dielectric con-
electron theory, (J is proportional to the mean free stant and permeability er and pr (so that in the material path I of the conduction electrons. As (J or cuincreases, D = eE, B = pH, with e = ereO,p = prpo), in which there will therefore be an instant at which ~ becomes the free electrons carry the current. er and pr are gener- smaller than I. The current density at a given point ally of the order of unity for non-ferromagnetic will then no longer be dètermined simply by the materials. Interesting complications which may arise local field intensity and the static conductivity, and the when pr becomes much greater than unity (ferromag- simple theory of the skin effect will no longer apply. netism) will be discussed in the last section of part 1. This situation is referred to as the "anomalous skin b) Over a very wide-frequency range, the term oD/Ot effect". (the ·"displacement. current") in the metal is negligible . As the frequency increases, other effects may become with respect to the current density J and may therefore
significant, namely, relaxation effects: the electron is be ignored. When J can be represented simply as (JE, 1967, No. 9 SKIN EFFECT, I 273
this amounts to taking w as negligible with respect tb the wires are thin and long compared with their sep- a/e. For copper at room temperature for instance, aration: rw« a e; L (rw is the radius of the wires, 8 a F::::i 10 (Qm)-l and a/e F::::i 1019 S-l, which is very much L their length and a is the spacing). It follows from (1), higher than the frequencies with which we shall be using Stokes's theorem, that if H is thè.field. at-a dis-, dealing in this article. 'öD/'öt can become comparable tance x from a wire carrying a current I: to J only in the relaxation range, where J becomes smaller than (jE; even then 'öD/'öt begins to become 2nxH = I, hence H = I/2nx, really significant only at frequencies near the plasma and the total flux through a surface bounded by two frequency. values of x, x = p and x = q, is: Parallel wires The case of a number of parallel wires lying in a plane and connected in parallel may be used as a simple illustration of the essential features of the skin effect. The total flux between 1 and 2 is therefore: The currents in. the wires affect one another by in- duction so that the current in the innermost wires is poL (a a 2a) - h In - - 12In - - Ia In - = less than in the outer wires. 2n rw' rw a Let us consider three equidistant wires, 1, 2 and 3, connected together at their ends and connected to an = poL- [(h - 12) In -a - Ia In 21 2n rw . a.c. source of angular frequency w (fig; 2). If the frequency is so low that induction effects can Let us assume that the fields and currents vary with be ignored, the currents h, 12and Is through 1, 2 and 3 time as exp (jwt). It then follows from (2) and Stokes's are all equal in phase and amplitude. This is no longer the case at higher frequencies. Consider the circuit (1, 2) formed by wires 1 and 2. There is a flux CP3, produced by la, through this circuit. An increase in Is induces an e.m.f. in (1, 2) opposing 12 just as it --i I does h since 12and Is are on the same side of (1, 2). I An increase in la therefore tends to oppose 12 and to ! _ ------_
- '-" reinforce h; a decrease in la tends to reinforce 12; this I (--I..... _ has the result that 12 lags slightly in phase behind I: I ---- and Is. Once there is a difference between hand 12, [ I ---- I h - 12 represents a circulating current in (1,2) pro- I
9
9 Q
1.0 - -
Q Q I o 9 i 9
0.5
• • • • . . . . •1 2• 3• 1 2 3 4 2 3 4 5
Fig. 4. The distribution of an alternating current over three, four or five wires, with a = 3. The wires are shown below in cross-section, distributed over the same width in the three cases. The current in each wire is shown vertically. The current is normalized to give an average "current density" of 1 (the average current per wire). The fact that a is real implies high fre- quencies and the value 3 corresponds to a wire spacing/wire diameter ratio of about 10. 1967, No. 9 SKIN EFFECT, I 275
be realized at very high frequencies where the imagi- c5k is a quantity with the dimension of a length, and is nary part becomes insignificant (there are therefore no called the (classical) "skin depth". Equation (9) cannot phase differences between the currents). The value 3 for be solved by elementary means. However, if a dimen- the real term corresponds to a value of about 20 for sionless complex variable a/rw. In each case the wires are distributed over the • l' same width and the total current is chosen such that X = (I-J)- . . . . (11) the average current per wire (a kind of current density) c5k is unity. is introduced the equation reduces to: Although, as we have seen, it is possible to calculate d2J dJ the "skin effect" for a small number of parallel wires, x2 - X - x2J = 0, . . . (12) 2 + + there is little sense in performing this type of calculation dx dx for an increasing number of wires in order to arrive, in a Bessel differential equation of order zero. The only the limit, at the effect in a solid conductor. The anal- solution which remains finite at x = 0 is the' Bessel ysis of this case is much better dealt with by considering function of the first kind. The absolute value and argu- the conductor from the start as a continuum with con-
tinuously distributed fields and currents, as below. J_ ! Wire of circular cross-section I The distribution of an alternating current over the I cross-section of a cylindrical wire is the standard I r-- example in discussing the skin effect. As the problem E, E2 has circular symmetry, we look for a circularly-symme- I tric solution in which the electric field E and the current 1 dr_ Ï" r- density J are parallel to the axis of the wire and the I - magnetic field H is perpendicular to a plane passing b ~ through the axis of the wire (fig. 5). E, Hand Jare I simply functions of the distance r from the axis of the I cylinder, with time dependence exp (jWI). Applying j_ Stokes's theorem to an area with a contour a of radius 1', it follows from (1) that for H in the wire:
r 2nrH_ = f 2nr.J dr . o
Differentiating with respect to 1': dH H+r- =Jr. (7) dr
Ifwe now apply equation (2) and Stokes's theorem to a surface with a contour b, we find: dE El - E2 = - -dr = -jwf1Bdr . Fig. 5. Diagram relating to the calculation of the current distri- dr bution in a cylindrical wire. Above: longitudinal cross-section. I Below: transverse cross-section. With J = aE, it follows that: . 1 dJ H=-j- (8) «uur dr ment of this Bessel function are tabulated, e.g. in the Jahnke-Emde tables, for a variable of the complex Differentiating (8) and substituting for Hand dH/dr in form (11). The amplitude of J (the modulus of the (7) gives a differential equation for J: Bessel function) is shown in fig. 6 as a function of d2J 1 dJ 2j r/c5k for 0 < r/c5k < 10; the amplitude multiplied by - +---J=O (9) the cosine of the phase (the argument of the Bessel dr2 r dr c5k2 ' function) is shown by the dashed curves. This last curve where ..•.. ,(10) give's an idea of the distribution of the current at a given time. It can be seen from the figure that no 276 PHILlPS TECHNICAL REVIEW VOLUME28
checked with (10» may be used to give an idea of the skin depth at various frequencies. For copper at room
J J temperature (0' = 0.6 X lOs Q-lm-l, fir = 1, . so that . p, = f-lO = 1.26 X 10-6 Hjm) the penetration depth is i i a) 15k= 1 cm (or, more accurately, 0.9 cm) at 50 Hz; b) 15k= 1micron at a wavelength of7 cm (microwave region). 100 The skin depth in the microwave region is thus very small. Experimental verification of equation (10) in the microwave region therefore requires very careful atten- tion to the surface quality. Equation (10) èan, for I example, be tested by measuring the Q (quality factor) of I a resonant cavity. Ij Q is a direct measure of th~ power I dissipation in the wall and the dissipation is directly 50 <, I '\ related to the skin depth. The measured Q of a reso- \ I nant cavity is generally considerably lower than tl.lat \ I I predicted by (10), partly because the grooves produced \ during machining aré deeper than the skin depth, so \ I \ / that the surface becomes effectively much larger. \ / Little improvement is achieved by polishing such a / surface. It is however possible to approach the theoret- 0 00 2 ical value of Q very closely by taking the greatest I possible care in machining the surface. Gevers [1] Fig. 6. The current distribution in a cylindrical wire. The ampli- obtained 98 % of the theoretical Q for a resonant cavi- tude of the current density is plotted vertically and r/(jk horizon- ty machined with a feed much smaller than the radius tally. The dashed line shows the current distribution at an arbi- trary instant. Both curves are also given for small r/(jJ, with the of curvature of the point of the tool: the tool used had vertical scale enlarged 50 times. . a radius of about 100 fLm, and the feed was about 1 fLm.
Conductor of arbitrary shape significant amplitude or phase variations occur while If we consider a conductor of any shape in which an rjch < 1, or, in other words, the skin effect is not en- alternating current is flowing, then in order to calcu- countered in wires thinner than the skin depth. late the distribution ofthe current in the conductor, we
For r » 15kweobtain a relatively large increase in the have to find solutions of Maxwell's equations in the current density. If we try J = exp (ex) as a solution of conductor and in the space around it that are compat- (12), we see that the relative value of the second term ible at the bounding surface. decreases as x increases. We can therefore consider the Once more we shall restrict ourselves to a single fre- extreme case in which the second term is neglected quency low enough to permit the displacement current and in this case the simple exponential function is, in to be neglected and again assume that the currents and fact, a solution. The current is now substantially at the fields have the time dependence exp (jwt). If we now surface of the wire, i.e. R » 15k,where R is the radius take the curl of (1) and substitute J = 0 outside the of the wire. It is therefore convenient to introduce the conductor and J = O'Eand curl E = -jwp,H (from variable z = R - r. Returning to (9) and omitting the (2» inside it, we find (since curl curl = grad div-tl second term, we find immediately as a solution: and div H = 0): J = Jo exp [-(1 + j)Zjt5k] , (13) outside the conductor: tlH = 0, (14) Jo being the current density at the surface. Bearing in inside the conductor: tlH = jWf-loH. (15) mind that this still has to be multiplied by exp (jwt), it It is not practicable to search for solutions to these will be seen that the variation of the current density is equations that are compatible at an arbitrary boundary given by an attenuated wave travelling inwards. The surface. The case of the cylindrical wire was so simple attenuation is very large: at the first point where the because the solutions for the regions inside and outside phase is opposite to that at the surface, zjt5k = n, the amplitude is exp (-n) = 0.05 times that at the surface. tn M. Gevers, Measuring the dielectric constant and loss angle The following rules of thumb (which can easily be ofsolids at 3000 Mc/s, Philips tech. Rev. 13, 61-70,1951. 1967, No. 9 SKIN EFFECT, 1 277
the wire can in fact be obtained independently of each cause the fields vary very little along the surface, other, provided that we restrict ourselves to sol utions 6.H "'" o2H/oz2, so that in the metal: with circular symmetry. The matching of the two solu- o2H tions is no problem, for the phase and amplitude for ~ =jw,uaH, vZ~ both solutions are constants at the boundary plane and only these constants have to be matched. The ex- from which it follows, with the boundary condition ternal solution is not affected by the radial current that the fields are zero in the interior, that distribution in the wire. H = Ho exp -(1 If we now consider the limiting case in which the r + j)Z/Ok] , skin layer is thin compared to the dimensions of the In conforrnity with the current distribution already conductor and the radii of curvature of the surface, we found in (13). can simplify the problem even if the conductor does not Part (b) of the problem can be formulated as a poten- have circular symmetry. tial problem with a simple boundary condition, since The problem then breaks down into two parts: curl H = 0 in the region outside the conductor. It a) How do field and current vary in a direction per- follows from this that H can be derived from a mag- pendicular to the surface? netic potential tp, i.e. H = grad ip. Since div H = 0 b) How do field and current vary along the surface? we have at once 6.cp = div grad cp = O. The boundary Let L be a length much greater than the skin depth, condition for this potential is Hn = àcp/àz = O. In but considerably smaller than the smallest radius of other words, the magnetic equipotential surfaces ter- curvature of the surface. L is then much smaller than minate at right angles to the surface of the con- the conductor itself or the apparatus exciting the mag- ductor. It should be noted that we must be prepared to netic field, such as a coil. The variation in the magnetic find a potential which is not a single-valued function of field outside the conductor over a distance L, parti- position. For every revolution around a current-carry- cularly the component perpendicular to the surface, is ing wire, there is a certain increase in the magnetic then very slight. Now consider a flat "box" at the potential. surface (length and width "'" L, thickness re Ok) COIll- If we now restrict ourselves to a cylindrical, but not pletely containing the skin layer (see fig. 7). The com- necessarily circularly-symmetric, conductor, question ponent perpendicular to the surface, H«, varies very (b) can be reduced to a potential problem in another little over the upper wallof the box. From (3) and way. Under these conditions the current distribution Gauss's theorem, the inward flux passing through along the sur/ace is the same as the distribution of the charge over the surface of an insulated, charged conduc- tor of the same shape. In other words, the boundary condition is that the potential is constant at the surface of the cond uctor. That this is so may be seen from the following. A magnetic field may be described by a vector potential A such that
curl A = H; (16) Fig. 7. The magnetic fields at the surface of a flat volume element enclosing the skin layer. for A we choose the gauge:
div A = O. (17) the walls of the box is the same as the outward flux, and, since no fl ux passes through the lower wall, It follows at once from (I) (with oD/Ot = 0, J = 0 and HnL2 is, at most, of the order of Ht.OkL (or even less, curl curl = grad div -6.) that because the fluxes through the side walls virtually corn- 6.A = o. pensate one another). Hn is thus at most about (ol convenient choice. Equation (16) is then equivalent so that the requirement !lzA = 0 becomes !lwA = O. to Hy = bA/~z,.l{z.= -bA/by. Let us assume that the line on which A is constant is Let us again take at some point on the surface a local Y = 0, -1 ~ X ~ +1. Now, by choosing a relation system ofaxes, with the z axis perpendicular to the between wand Z which maps this line on to a circle, surface. In our limiting case <5k « L the magnetic field we reduce our problem to a simple circular-symmetric has, as we saw, no vertical component at the surface: problem-Such a relation is Hz = 0, so that M/by = 0, which means that A is Z'= l/w). constant at the surface. The two-dimensional problem tew + thus reduces to finding a scalar function A(y, z) which The circle Iwl = 1 described by IV = exp (j-&) as -&varies satisfies !lA = 0 outside the conductor and is con- is converted into Z = cos -&,i.e. X = cos -0-, Y = 0, stant at the surface. which describes our line. Taking IV = r exp (j-&),then Flat strip for a circular-symmetrical A outside the circle, !lA = 0 is equivalent to: We shall use the above to calculate the current distribution over the width of a flat strip for the case of b2A 1 bA -+- -=0. a very small skin depth. If we consider a flat strip (width br2 r ör . B, thickness D) in which an alternating current flows The solution is A = e In r, in which e is an arbitrary in the longitudinal direction, we can distinguish be- constant. From (1) and Stokes's theorem the surface tween various frequency ranges. At very low frequen- current density Js, i.e. the current per unit width of cies (direct current) the current is uniformly distributed surface Js = J Jdz, is equal to the tangential magnetic over the cross-section. At higher frequencies there is a field, and hence: range in which the current is still uniformly distri- bA buted over the thickness (<5k »D) but no longer over J« = lim -. the width. The calculation of the distribution over the Y-+O bY width in this situation has been carried out by Belevitch, For points close to the surface, r ~ 1: let us put Gueret and Liénard [2]. At very high frequencies the r =1 +e (s « 1). Then: current flows in a skin layer that is thin compared to Y = Im Z = ter - l/r)sin -&~ e sin-& the strip thickness (<5k« D), and here, again the and distribution is not uniform over the width. It is this bA öln r e br e ös e distribution that we shall now calculate. - = e·--=- -=- -=-----. bY bY r bY r bY (1 e)sin ij. , We take <5k« D, so that the above considerations + apply, and D « B. With this latter condition we can e e idealize the strip as infinitely thin so that its cross hence Js=--=. . sin -& Vl-X2 section becomes a line and our boundary condition becomes: A is constant over the line. The two-dimen- The current along one face of the strip is: sional potential problem .can now be solved by con- +1 formal mapping. We take co-ordinates X, Y in the . edX 10 = = en plane of our problem. Consider the complex variable .J VI X2 Z = X + jY and write down an analytical function w -1 ofZ: (the total current is 210, since both faces carry current). IV = U + j V = IV(Z). To facilitate comparison with the parallel-wire prob- lem we normalize so as to obtain an average surface .This relation maps the Z-plane on to the w-plane. current density ofunity. We therefore put e = 2/n (the The relationship tlA = 0 remains valid in the w-plane strip Y = 0, -1 ~X ~ + 1, has a width of 2). Js is for it follows from the Cauchy relations for analytical plotted in fig. 8 for e = 2/n. functions, We note that, under the conditions <5k« D« B, bu bv bv bu the current distribution over the width is indepen- bX bY' bX = - bY' dent of the frequency and the conductivity. Further, that: the solution given here forms an asymptotic approxi- bU)2 (bV)2J mation in the range of (lower) frequencies where, in tlzA = [( bX + bY: !lwA , contrast with the above conditions, the skin depth is in which large in comparison with the thickness, but small in comparison with the root of the product of width and thickness [2]. 1967, No. 9 SKIN EFFECT, I 279 Fig. 8. The distribution of H is constant. As the field must vanish for z -+ ± 00, the surface current over the width of a flat strip (width B, H = 0 everywhere outside the strips; we assume that thickness D) with Jk « D « H = Hobetween the strips. Inside each strip: B (the solid curve). The curve is normalized to give a "sur- d2H face current density" of uni- -_ =jwf.loH. ty. The points indicated by dz2 1.0 triangles, crosses and circles, taken from fig. 4, represent With the boundary conditions H = 0 for z = a + d the current distribution over &. x e three, four or five wires re- and H = Hofor z = a, we find for the field in the spectively ford = 3 (cf. fig. 4). "upper" strip (z > 0): sinh a (a + d - z) H=Ho------(18) sinh ad ' 05 in which a = (l + j)/Ok. The field in the lower strip (with H= 0 for z = -a-d, and H= Ho for z = -a) follows from this by replacing z by -r-Z, For the current density in the upper strip we find (with J = dH/dz): cosh a (a + d - z) OL_------~ J = -aHo , . . (19) sinh ad VZZZZZZ?ZZZ7IZZ7ZZZZZZZ77?J and that in the lower strip follows by replacing z by -z and multiplying the entire expression by -1. The One might be inclined to think that the problem of current per unit width in the upper strip is: the flat strip could be reduced to a one-dimensional a+d a+d problem by considering the limiting case of an infinite- ü-r d ' l'dH (20) ly wide strip (where one need consider only the co- ( Jdz =, dz dz = [Ht = -Ho. ordinate perpendicular to the plane of the strip). This a is however not possible because the contribution to- No complications arise in taking the d.c. limit of (19), wards the magnetic field from distant current strips Ok -+ 00, i.e. a -+ O. Expanding (19) and neglecting all cannot be neglected. The field due to a current strip of but first order terms in a, J = =Hejd, as is consistent width dX at a distance X (measured along the width) with (20). The system may be considered as a deformed is proportional to dX/X, and the integral of such a single-turn coil. term diverges as the boundary is extended to infinity. Thus the width cannot be made infinitely large, and the co-ordinate along the width cannot be eliminated by such means. This can however be done for the case of two in- finitely wide strips placed opposite to each other and carrying current in opposite directions. We shall deal with this briefly. A one-dimensional problem: two opposite and infinitely Fig. 9. Two strips of thickness d at a spacing 2a extending to wide strips infinity in length and width. Suppose that two strips of thickness d at a distance 2a extend to an infinite distance in both length and Ferromagnetic sphere width (see jig. 9). In this system, the current can flow In ferromagnetic metals the skin effect occurs in a lengthwise (equal but opposite currents in the two frequency range completely different from that in the strips) with the magnetic field along the width, the non-ferromagnetic metals. For a metal in which current and field depending only on the co-ordinate z flr = 5000 and 0 = gOcojJper, the value of uo is perpendicular to the plane of the strips. Even if the con- 1000 times greater than it is for copper, so that the skin dition Ok« d is not fulfilled this one-dimensional depth is only 1 mm at frequencies as low as 1 Hz. problem is very simple. Equation (1) becomes J = dH/dz. In the region [2] V. Belevitch, P. Gueret and J. C. Liénard, Le skin-effe! dans between and outside the strips, J = 0 and therefore un ruban, Rev. HF 5, 109-115, 1962. 280 PHILIPS TECHNICAL REVIEW VOLUME 28 In the previous sections we always thought of the magnetic flux lines (which are continuous through the alternating current in the conductor as being produced surface) is given by: by a current source connected to it. It is also possible, tan 'IjJ = !lrtan cp, however, to study the skin effect in a conducting body where in an alternating electromagnetic field, the current be- tanç = BnI/ÉtI, tan e = BnefBte. ing produced by induction. We shall now consider the case of a ferromagnetic sphere in an alternating mag- . In a static magnetic field, tan cp has a value of the netic field. An interesting complication is that the ten- order of unity (the field within the sphere is homogene- dency of the alternating field and therefore of the flux ous), so that for a ferromagnetic material tan e » 1: to be forced outwards by the skin effect is opposed the lines of flux outside the sphere terminate at right by the ferromagnetism of the sphere, which tends angles to the sphere. (For a diamagnetic material, on to concentrate the flux. Certain peculiarities in the the other hand, they are substantially tangential to the behaviour of iron particles in a high frequency field surface.) may be seen as a conflict between these opposing Let us now consider alternating fields of frequencies tendencies [3]. such that ~k« R: the fields are confined to a layer of For the sake of simplicity we assume that the mat- thickness ~k. We may now estimate tan cp to be of the erial has a permeability ftr which is independent of order of ~kfR, so that: the frequency. In reality, ftr is in general a function of !lr~k tan 'IjJ R::j -- • the frequency. Employing the usual notation for com- R plex permeabilities, !.Lr = ftr' + jpr", the ftr' tends to For a ferromagnetic material we can now distinguish become unity at high values of w, and {lr" exhibits one three cases (fig. 11): or more peaks as a function of the frequency, these a) R« ~k, "low frequency". The pattern of the flux peaks representing losses. We shall not take these lines is identical to the static case; it is homogeneous effectsinto account. inside the sphere, and outside the sphere it is character- Let us now consider a sphere of radius R in an ini- ized by flux lines perpendicular to the surface (ferro- tially uniform magnetic field (fig. 10). Using Stokes's magnetic pattern). b) bk« R «{lrbk, "medium frequency". Inside the sphere the field is concentrated in a skin layer, and outside the pattern is still ferromagnetic. c) ftr~k« R, "high frequency". Inside the sphere the field is concentrated in a skin layer, and outside the pattern is diamagnetic. If we introduce a critical frequency defined by a and R (but independent of ftr), 2 Wc=-- . (21) poaR2 ' the boundaries between regions (a), (b) and (c) are the frequencies given by: Fig. 10. Refraction of the lines of magnetic induction at the sur- W 1 face of a ferromagnetic sphere. (a) -«-, Wc ftr 1 W (b) -« -« ftr , and Gauss's theorems and equations (1) and (3), one flr Wc arrives at the well-known boundary conditions that W must apply at the surface (the suffixes nand t refer (c) !lr« - . to the normal and tangential component, i and e refer Wc to internal and external): In the case of a sphere 1mm in diameter (i.e. R = 5 X 10-4 m) and a = !acopper = 1.2 X 107 (Qm)-l, Bne = Bni, Wc R::j 5 X 105 radfs. For such a sphere, with {lr = 5000, Hte = He or Bs« = Bi: ff.lr, the frequency boundaries W = wcf ftr and W = flrWc of from which it follows that the "refraction" of the the medium-frequency region become: W = 100 and W .- 2.5 X 109 radfs, corresponding roughly to 20 Hz [3] H. B. G. Casimir, Philips Res. Repts. 2, 42-54, 1947. and 400 MHz. ]967, No. 9 SKIN EFFECT, I 281 The difference between the three cases shows up Let Jm be the amplitude of J; then (Re J)2 ~ tJIit2. particularly in the power dissipated by the sphere. Integration over the sphere gives: This depends on u, a and w in different ways in the three cases. If we borrow from magnetostatics the re- P = 6n wflHm2R3 (~)2, sult that (because of demagnetization effects) the 5 flrÖk magnetic flux density in a sphere with a high flr in an where H,« is the amplitude of H. originally homogeneous static magnetic field His B = The dissipation in cases (b) and (c) may be estimated 3floH (i.e. independent of flr), the energy dissipation in the following way. The current now flows through a for case Ca),R« Ök, may easily be found. Consider an thin broad surface band. This band has a length of Q Fig. 11. Lines of magnetic induction inside and outside a ferromagnetic sphere in an initially homogeneous alternating magnetic field in three frequency regions. a) "Low-frequency": R « 15k,b) "medium-frequency": 15k« R « f.trc5k, c) "high-frequency" ftrc5k « R. elementary ring in the sphere as shown in fig. 12. We about 2nR and a cross-section of about ÖkR. The heat once more assume that the field and current are pro- developed per second is roughly: portional to exp (jwt). With Stokes's theorem, equa- 2nR nJ 2R2Ök tion (2), J = aE and B = 3 floH, we find for the m P R:; (Re JÖkR)2 . -- = . current density in the ring J = -~ jwaflor H. The actual aRök a instantaneous current is ReJ drdz and the resistance If (jj is the total flux through the sphere (amplitude is 2nr/adrdz, whence the energy dissipation per (jjm), then from (2) and Stokes's theorem: second is 2nRJ = -jwa(jj, 2nr 2nr--- dP = (ReJ drdz)" --- = - (Re J)2 drdz . so that: adrdz a W (jjm2 PR:; -- 2nflÖk The flux through the sphere differs in the two cases (b) and (c). In case (b) the external field pattern and hence I I the flux through the sphere are the same as in the static Idz: Ddr case, so that cp = nR2 X 3floH. In case (c) the flux lines are tangential to the surface, so that no demagnetiza- tion effects occur. Therefore, in the skin layer, B = flH and (jj = 2nRökB = 2nRHflök. Substituting for the flux in each case gives: b) P I'::::i 9n WfloHm2R3 (.!!_), 2 flrÖk c) P I'::::i 2nwflOHm2R3 (fl~k) '. Fig. 12. Annular volume element (radius r at distance z from the centre) of a sphere, used in calculating the power dissipated [3] in case (a), R « 15k•The axis ofthe ring is parallel to the magnetic An exact calculation gives the same results for field. these limiting cases except that the numerical factor 282 PHILIPS TECHNICAL REVIEW VOLUME28 9n/2 in (b) becomes 3n and the factor 2n in (c) becomes 3n/2. If we put ,_ --. ---- 2 1_---li =l-- 6nfloHm R3 = W and ~tr(Jk/R= a, p _- r:r and once more introduce COc according to (21), and using (10), the final result is: i For case (a), i.e. R« 15kor co/wc« 1/flr: 2 P = gW W~tra-2 = gcocW (:c) (22a) For case (b), i.e. (Jk« R« flr(Jk or l/flr« co/Wc« flr: Wc/f1.rl _W (22b) Fig. 13. The power P dissipated in a ferromagnetic sphere in an alternating magnetic field as a function of the angular frequency For case (c), i.e. flr(Jk« R or flr« «[os«: 00, both plotted on a logarithmic scale, according to equations (22a, b, c) (the factors 1/5, 1/2 and 1/4 being omitted). The solid 1 2 line applies for f1r = f1rl » 1 (T < Tc), and the dashed line for P = !coWa = !cocWfli/2 (:J / (22c) f1r = 1 (T > Tc). The double arrow indicates the range in which "temperature hysteresis" may be expected. In jig. 13 P is plotted against w/wc for each of these cases, on logarithmic scales, for a given value of flr, the Curie point. If the amplitude of the high-frequency namely flr1. The factors g, -} and t have been omitted fieldis increased,Pshifts upwards, or, effectively,Pc shifts in plotting this diagram. downwards with respect to P. The temperature that A peculiar phenomenon resulting from this inter- the body assumes is given by the intersection of the play of skin effect and ferromagnetism is "temperature two curves. As the amplitude increases the cycle passes hysteresis" [4]. If a conducting body is brought into a high-frequency field, the temperature assumes a value such that the heatdevelopedis equal to the heat radiated. The heat radiated is highly temperature-dependent (cc T4 or T5). In the ferromagnetic case, the following A phenomenon can now occur. The body is gradually 8 brought near to a high-frequency coil. Initially the P.Pe temperature increases gradually, and the body begins to glow a pale red. At a certain point the body will i suddenly become white-hot. If it is now gradually removed again, it will continue to glow white until far beyond the point where it first became white hot and then suddenly reverts to the pale red colour. This behaviour may be explained in the following way. As a result of its dependence upon fl1', the heat developed, P, is highly temperature-dependent in the region of the Curie temperature Tc. Let us suppose that flr has a high value flr1 when T < Tc and is unity when T> Tc. P is also plotted for flr = 1 in fig. 13 (then region (b) is non-existent). We see from this -T ~-78-Tc-1D Tó-T£-") that- there is a frequency range in which P increases ~-Ta 78·-Té-Td ....---~ abruptly when T is raised above Tc. We consider a Fig. 14. The heat developed, P, and the heat radiated, Pc, as a frequency in this range. function of the temperature T (logarithmic scales) for a ferro- Fig. 14 gives a diagram of the heat developed, P, and magnetic sphere in an alternating magnetic field with a frequency at which P rises abruptly when f1r jumps from f1rl to 1 at the the heat radiated, P«, plotted against the temperature Curie temperature (in the range indicated by the double arrow on logarithmic scales. The curve for P« is a straight in fig. 13). A point of intersection of P and Pc defines a steady state. When the intensity of the alternating field is increased, P line (with a slope of 4 or 5). P exhibits an abrupt rise at shifts upwards, and thus Pc shifts downwards with respect to P, so that situations A, B, C, D and E arise successively. On tra- versing the full cycle A ... E ... A, the temperatures correspond- (4] See: J. L. Snoek, Newdevelopmentsinferromagneticmateri- ing to situations between Band D are different on the forward als, Elsevier, Amsterdam 1947. journey from those on the return journey. 1967, No. 9 SKIN EFFECT, I 283 through situations A, B, C, D and E. Between Band D Pea, fhr = 1) ~ PCb, fhr = fhrl) there are two stable equilibrium temperatures (see, for and example, situation C). As the amplitude increases, the P(c, fhr = 1) ~ PCb, fhr = fhrl), temperature rises abruptly from To to Tn' at the mo- where the a, band c refer to our three frequency re- ment that the line D leaves the lower bend in curve P. gions. Using (22a, b, c) we find: As the amplitude decreases the temperature remains high until, as B leaves the upper bend in curve P, it 25 W y- -4 «-« l fhrl. drops back from TB' to TB. The frequency range in fhrl Wc which temperature hysteresis may be expected (indi- For our sphere (diameter 1 mm, a = 1.2 X 107 [2-1 m-I, cated in fig. 13 by a double arrow) is given by the fhrl = 5000) this means that the frequency f must be conditions: in the range 100 Hz «f « 3 MHz. Summary. This first part of three articles on the skin effect con- cross-section, the surface current distribution is the same as the tains a general Introduetion followed by a discussion of the charge distr.ibution over the surface of a charged conductor of current distribution for several configurations. The essence ofthe the same shape. This consideration is used in the problem of the skin effect is illustrated by the case of a number of parallel flat strip. The problem becomes purely one-dimensional for two wires in one plane, connected in parallel. The standard problem strips which together form a flat coil. Finally the case of a ferro- of the cylindrical wire is used to introduce the concept "classical magnetic sphere in a magnetic alternating field is discussed. The skin depth". The problem ofthe distribution ofthe current over a combination of skin effect and ferromagnetism can lead to conductor of arbitrary shape is stated in general terms. If the "ternperature hysteresis", an effect in which the temperature skin layer is thin, the problem of the distribution in the surface variation of the sphere is hysteretic with respect to an increasing is reduced to a potential problem. For a cylindrical wire of any and a decreasing field amplitude. '0