L.._
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 consequence, 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 least feel positive about
its truth of their ownknowledge. As an eminent practician remarked, after prolonged seepticism, 'When Sir W. Thompson says so, who can doubtit? "
These were the wordsof Heaviside in a pleafor 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
be distributed uniformly over the cross-section. With
alternating current, however, the current distribution is
not homogeneous and, if the frequency, conductivity
and dimensions of the conductor satisfy certain conditions, to be dealt with later, the current flows mainly
in a thin layer at the surface of the conductor. This
phenomenon is called the skin effect. It is an electrodynamic effect, that is to say, it is a result of the way in
which time-varying electric and magnetic fields and
electriccurrents areinterrelated. The skin effect phenomenori is quite different from the action of a Faraday
cage, for instance, which acts as a barrier to a static
electric field purely and simply as a result of the fact
Prof. Dr. H. B. G. Casimir is a member of the Board of Management of N. V. Philips' Gloeilampenfabrieken; Dr. J. Ubbink is with
Philips Research Laboratories, Eindhoven.
that the charges in the wall of the cage are mobile.
The tendency of the current to flow at the surface is
closely connected with the stable character of the
electromagnetic phenomena. We see this in the following way. Let us assume that inside the metal there is a
filamentary current I which is increasing in strength
(fig. 1). This current is associated with a rotational
magnetic field H around it which is also increasing. A
changing magnetic field induces a rotational electric
field E which, in turn, induces a current in the metal.
According to Lenz's law, the direction of E is such that
it opposes the increase of I, thus keeping the situation
stable. The figure shows that simultaneously at some
distance from I a current is generated parallel to I. The
net result therefore is that the current is forced outwards. Furthermore,
we see that this effect increases
with frequency (E is larger the more rapid the change
~
~~__ ~_
PHILIPS TECHNICAL
t
Fig. 1. A filamentary current J is accompanied by a magnetic
field H. As J and H increase, an electric field E is produced
whose direction near J is such' astto oppose the increase in J,
whereas further away from J the induced field E produces a
current in the conducting medium parallel to J.
ofH) and with conductivity (the larger the conductivity, the larger the current caused by E).
In this :first article, we shall consider in some detail
the configuration of the current for one or two special
situations.
The skin effect can be a disadvantage in transporting
alternating current energy along a wire or cable. To
keep the resistance low, the cross-section of the conductor should be as large as is practicable but the effect
of increasing the diameter is far less than with direct
current. For alternating currents it is advantageous to
use hollow cables (in power engineering), or braided
cable (in radio engineering). At microwave frequencies
the skin effect can be put to good use: the effect makes
it possible to transport and store electromagnetic
energy without radiation losses by using closed waveguides and resonant cavities.
At high frequencies the skin layer may be regarded as
a layer screening electromagnetic
radiation incident
upon the metal: as a result ofthe conducting properties
of the metal the radiation penetrates into the metal no
further than the depth of the skin layer. Even a single
electron, whether bound or free, possesses screening
properties to a certain extent: incident radiation is
scattered by the electron so that the power travelling
straight on is less than the incident power.
As cu(the angular frequency) or (J (the conductivity)
increases, the penetration depth ~ decreases. In simple
electron theory, (J is proportional
to the mean free
path I of the conduction electrons. As (J or cu increases,
there will therefore be an instant at which ~ becomes
smaller than I. The current density at a given point
will then no longer be dètermined
simply by the
local field intensity and the static conductivity, and the
simple theory of the skin effect will no longer apply.
This situation is referred to as the "anomalous skin
effect".
. As the frequency increases, other effects may become
significant, namely, relaxation effects: the electron is
REVIEW
VOLUME28
subject to many cycles of the alternating field between
two collisions and within the mean time that it spends
in the skin layer. Broadly speaking, the field then
"sees" in effect a layer of free electrons.
Finally, at still higher frequencies, the "plasma frequency" 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
Metals in the superconducting
state form a special
class of conductors. These will be discussed in part "rIL
We consider the two-fluid model, in which the electrons
are divided into two types, normal and superconducting. The superconducting electrons, although they dissipate no energy, do have screening properties (even at
cu = 0). They-therefore cause a skin effect: fields can
penetrate
only to the London penetration
depth,
which is independent ofthe frequency, and the "supercurrent" in this layer does not cause energy losses. The
normal electrons within this layer do, however, absorb
electromagnetic
enemy
for cu =!= 0, givmg some
(small) high frequency losses. There is, however, a
frequency limit above which the superconducting
electrons also absorb energy. This absorption is due
to the transfer of electrons from the superconducting
to the normal state by the radiation, via a quantum
process. A superconductor
differs very little from an
ordinary metal at frequencies above this frequency
limit (which lies in the microwave range).
Since the skin effect is based entirely upon the dynamic properties of electromagnetic fields and currents
as given by Maxwell's equations, it will be useful to
set down the four equations here:
+ J,
curl H
= oD/of
curl E
= -oB/Ot,
(2)
div B
=
0,
(3)
div D
=
(J.
(4)
(1)
The following points should be noted:
a) In what follows we shall in general regard the material as a medium with a given relative dielectric constant and permeability er and pr (so that in the material
D = eE, B = pH, with e = ereO,p = prpo), in which
the free electrons carry the current. er and pr are generally of the order of unity for non-ferromagnetic
materials. Interesting complications which may arise
when pr becomes much greater than unity (ferromagnetism) will be discussed in the last section of part 1.
b) Over a very wide-frequency range, the term oD/Ot
(the ·"displacement. current") in the metal is negligible
with respect to the current density J and may therefore
be ignored. When J can be represented simply as (JE,
1967, No. 9
SKIN EFFECT,
this amounts to taking w as negligible with respect tb
a/e. For copper at room temperature for instance,
a F::::i 108 (Qm)-l and a/ e F::::i 1019 S-l, which is very much
higher than the frequencies with which we shall be
dealing in this article. 'öD/'öt can become comparable
to J only in the relaxation range, where J becomes
smaller than (jE; even then 'öD/'öt begins to become
really significant only at frequencies near the plasma
frequency.
I
273
the wires are thin and long compared with their separation: rw« a e; L (rw is the radius of the wires,
L their length and a is the spacing). It follows from (1),
using Stokes's theorem, that if H is thè.field
at-a dis-,
.
tance x from a wire carrying a current I:
= I,
2nxH
hence
= I/2nx,
H
and the total flux through a surface bounded by two
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 currents in. the wires affect one another by induction so that the current in the innermost wires is
less than in the outer wires.
Let us consider three equidistant wires, 1, 2 and 3,
connected together at their ends and connected to an
a.c. source of angular frequency w (fig; 2).
If the frequency is so low that induction effects can
be ignored, the currents h, 12and Is through 1, 2 and 3
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
does h since 12 and Is are on the same side of (1, 2).
An increase in la therefore tends to oppose 12 and to
reinforce h; a decrease in la tends to reinforce 12; this
has the result that 12 lags slightly in phase behind I:
and Is. Once there is a difference between hand 12,
h - 12 represents a circulating current in (1,2) producing a flux which, in turn, largely determines the
difference between hand h. The net result is that 12
lags in phase behind Is. and Is but has virtually the
same amplitude.
The situation is easier to grasp at very high frequencies because of the consideration that the net
flux through a circuit must be almost zero, because
any net flux would produce almost infinitely high
e.m.f.'s. Let us assume that the fluxes through (1, 2) as
a result of h, 12and la are CP1, CP2 and CPa respectively
(fig.2b). Owing to the geometry and as Is. and la are
equal in phase (and also in magnitude), due to symmetry, CP1 and CPa are in opposite phase. Because the net
flux is zero, CP2 must also be in phase or in antiphase:
because 1 and 2 are symmetrical with respect to the
zone (1, 2) and 3 is further away, CP1 = Ah, CP2 = Ah
and CPa = ala, with a < A. It then follows from
CP1 = ([>2 + CPa and Ia = Is, that Ah = Ah
ah,
hence 12 = h(A - a)/A. 12 therefore has a smaller
amplitude than Is, and there are no phase differences.
For a quantitative calculation let us assume that
+
The total flux between 1 and 2 is therefore:
poL
(ah In - - 12In -a - Ia In -2a) =
2n
rw'
rw
a
1
[(h - 12) In -a - Ia In 2
= poL
2n
rw
.
Let us assume that the fields and currents vary with
time as exp (jwt). It then follows from (2) and Stokes's
--i
I
I
!
I
[
I
rv
L
_ ----
(--I
..... _
<P3
111,
I
I
---
----
I
t 13
12
I
1
1
1
I
_..-
I
(.....
!
I
I
----
I
I
I
i
_
'-"
__
-------/ 1---- ---_
---......-
a
a
__ ')
-
j----
I
Iv2rw
,i
-{
I
•
3
Fig. 2. a) Three parallel wires, connected electrically in parallel,
of length L, diameter 2rw and spacing a. Wais the flux between the
first two wires caused by the current in the third.
b) Qualitative indication of the fluxes through the circuit (1, 2)
at very high frequencies.
VOLUME 2S
PHILlPS TECHNICAL REVIEW
274
theorem, with B = /hoH, El = Rd!, E2 = Riiz (RI resistance per ~~~t_lengthof the wires) that:
RI(1t - h)L
= -
or, substituting 13
the frequency. On going to higher frequencies we must
bear in mind that RI is the effective resistance per unit
length, and that at higher frequencies it no longer
corresponds to the resistance at zero frequency - the
cause of this being, of course, the skin effect in each
wire.
jW/hoL
[(ft - h) In -a.]- Ia In 2 ,
-2:n:
rw'
= It:
In 2
(5)
1--,
a
with
a
a
.2:n:RI
(6)
In --J--.
rw
/hOW
=
The combined effect of the geometrical and electrical
factors is contained in the parameter a.
As the frequency varies from 0 to infinity, the locus
of als parallel to the imaginary axis and hilt describes
a semicircle in the complex plane as sketched infig. 3.
Although, under the stated condition a» rw, the
effect can never become particularly large, it still
shows a few essential traits of the skin effect, as may
be seen from (5) and (6) and fig. 3; the "inner current"
is smaller than the "outer current" and lags behind it;
at low frequencies there are phase differences only,
which increase as the frequency increases and the resistance decreases. At very high frequencies the distribution of current between the wires, and hence the
field distribution outside the wires, is independent of
Fig. 3. The current ratio h/ft describes a semicircle in the complex plane as the frequency passes from zero to high values.
It is still practicable to calculate the current distribution under the set conditions for four wires. The
result becomes more and more complex as the number
of wires increases: if there are n wires, n - 1 equations
with n - 1 unknown quantities have to be solved.
. The current in the wires for three, four and five wires,
with a = 3, is shown in fig. 4. This real value of a will
9
Q
9
-
-
1.0
o
I
Q
Q
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 frequencies and the value 3 corresponds to a wire spacing/wire diameter ratio of about 10.
1967, No. 9
SKIN EFFECT, I
be realized at very high frequencies where the imaginary part becomes insignificant (there are therefore no
phase differences between the currents). The value 3 for
the real term corresponds to a value of about 20 for
a/rw. In each case the wires are distributed over the
same width and the total current is chosen such that
the average current per wire (a kind of current density)
is unity.
275
c5k is a quantity with the dimension of a length, and is
called the (classical) "skin depth". Equation (9) cannot
be solved by elementary means. However, if a dimensionless complex variable
=
X
is introduced
Although, as we have seen, it is possible to calculate
the "skin effect" for a small number of parallel wires,
there is little sense in performing this type of calculation
for an increasing number of wires in order to arrive, in
the limit, at the effect in a solid conductor. The analysis of this case is much better dealt with by considering
the conductor from the start as a continuum with continuously distributed fields and currents, as below.
•
c5k
the equation
x2
d2J
-
dx
2
+X
f
(11)
dJ
-
dx
+ x2J
. . .
(12)
= 0,
a Bessel differential equation of order zero. The only
solution which remains finite at x = 0 is the' Bessel
function of the first kind. The absolute value and arguJ_
!
I
The distribution of an alternating current over the
cross-section of a cylindrical wire is the standard
example in discussing the skin effect. As the problem
has circular symmetry, we look for a circularly-symmetric solution in which the electric field E and the current
density J are parallel to the axis of the wire and the
magnetic field H is perpendicular
to a plane passing
through the axis of the wire (fig. 5). E, Hand Jare
simply functions of the distance r from the axis of the
cylinder, with time dependence exp (jWI). Applying
Stokes's theorem to an area with a contour a of radius
1', it follows from (1) that for H in the wire:
=
. . . .
reduces to:
Wire of circular cross-section
2nr H_
l'
(I-J)-
I
I
I
r--
E,
E2
-
1
dr_
I
I
b
Ï"
r-
~
I
j_
r
2nr.J dr .
o
Differentiating
with respect to
1':
dH
H+r- =Jr.
dr
(7)
Ifwe now apply equation (2) and Stokes's theorem to a
surface with a contour b, we find:
dE
El - E2 = - -dr
dr
= -jwf1Bdr
.
I
With J
=
aE, it follows that:
.
H=-j-
1
dJ
(8)
«uur dr
Differentiating (8) and substituting for Hand
(7) gives a differential equation for J:
d2J
1 dJ
2j
- 2 +---J=O
dr
r dr
c5k2
where
Fig. 5. Diagram relating to the calculation of the current distribution in a cylindrical wire. Above: longitudinal cross-section.
Below: transverse cross-section.
'
..•..
ment of this Bessel function are tabulated, e.g. in the
Jahnke-Emde
tables, for a variable of the complex
dH/dr in
form (11). The amplitude of J (the modulus of the
Bessel function) is shown in fig. 6 as a function of
r/c5k for 0 < r/c5k < 10; the amplitude multiplied by
(9) the cosine of the phase (the argument of the Bessel
function) is shown by the dashed curves. This last curve
give's an idea of the distribution of the current at a
,(10)
given time. It can be seen from the figure that no
PHILlPS TECHNICAL
276
J
J
i
i
100
I
I
I
I
I
I
I
I
<,
'\
\
\
\
\
\
\
00
50
/
/
/
0
2
I
Fig. 6. The current distribution in a cylindrical wire. The amplitude of the current density is plotted vertically and r/(jk horizontally. The dashed line shows the current distribution at an arbitrary instant. Both curves are also given for small r/(jJ, with the
vertical scale enlarged 50 times.
.
REVIEW
VOLUME28
checked with (10» may be used to give an idea of the
skin depth at various frequencies. For copper at room
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
a) 15k= 1 cm (or, more accurately, 0.9 cm) at 50 Hz;
b) 15k= 1 micron at a wavelength of7 cm (microwave
region).
The skin depth in the microwave region is thus very
small. Experimental verification of equation (10) in the
microwave region therefore requires very careful attention to the surface quality. Equation (10) èan, for
example, be tested by measuring the Q (quality factor) of
a resonant cavity. Ij Q is a direct measure of th~ power
dissipation in the wall and the dissipation is directly
related to the skin depth. The measured Q of a resonant cavity is generally considerably lower than tl.lat
predicted by (10), partly because the grooves produced
during machining aré deeper than the skin depth, so
that the surface becomes effectively much larger.
Little improvement is achieved by polishing such a
surface. It is however possible to approach the theoretical value of Q very closely by taking the greatest
possible care in machining the surface. Gevers [1]
obtained 98 % of the theoretical Q for a resonant cavity machined with a feed much smaller than the radius
of curvature of the point of the tool: the tool used had
a radius of about 100 fLm, and the feed was about
1 fLm.
Conductor of arbitrary shape
significant amplitude or phase variations occur while
rjch < 1, or, in other words, the skin effect is not encountered in wires thinner than the skin depth.
For r » 15kwe obtain a relatively large increase in the
current density. If we try J = exp (ex) as a solution of
(12), we see that the relative value of the second term
decreases as x increases. We can therefore consider the
extreme case in which the second term is neglected
and in this case the simple exponential function is, in
fact, a solution. The current is now substantially at the
surface of the wire, i.e. R » 15k,where R is the radius
of the wire. It is therefore convenient to introduce the
variable z = R - r. Returning to (9) and omitting the
second term, we find immediately as a solution:
J
= Jo exp [-(1 + j)Zjt5k] ,
(13)
Jo being the current density at the surface. Bearing in
mind that this still has to be multiplied by exp (jwt), it
will be seen that the variation of the current density is
given by an attenuated wave travelling inwards. The
attenuation is very large: at the first point where the
phase is opposite to that at the surface, zjt5k = n, the
amplitude is exp (-n) = 0.05 times that at the surface.
The following rules of thumb (which can easily be
If we consider a conductor of any shape in which an
alternating current is flowing, then in order to calculate the distribution ofthe current in the conductor, we
have to find solutions of Maxwell's equations in the
conductor and in the space around it that are compatible at the bounding surface.
Once more we shall restrict ourselves to a single frequency low enough to permit the displacement current
to be neglected and again assume that the currents and
fields have the time dependence exp (jwt). If we now
take the curl of (1) and substitute J = 0 outside the
conductor and J = O'Eand curl E = -jwp,H (from
(2» inside it, we find (since curl curl = grad div-tl
and div H = 0):
= 0,
inside the conductor: tlH = jWf-loH.
outside the conductor: tlH
(14)
(15)
It is not practicable to search for solutions to these
equations that are compatible at an arbitrary boundary
surface. The case of the cylindrical wire was so simple
because the solutions for the regions inside and outside
tn M. Gevers, Measuring the dielectric constant and loss angle
ofsolids at 3000 Mc/s, Philips tech. Rev. 13, 61-70,1951.
1967, No. 9
SKIN EFFECT, 1
the wire can in fact be obtained
independently
other,
ourselves
provided
with circular
that we restrict
symmetry.
tions
is no problem,
both
solutions
and only
ternal
The matching
for the phase
are constants
these constants
solution
distribution
conductor
have circular
symmetry.
The problem
very
from
which
it follows,
with
of the
we
does not
Let L be a length
considerably
curvature
down
much
into two parts:
vary
along
than
per-
than
L is then
the skin depth,
much
itself or the apparatus
the conductor
the component
very
slight.
surface
(length
pletely
containing
Gauss's
the
theorem,
"box"
the
is
at the
Ok)
re
COIll-
(see fig. 7). The com-
the
H«, varies
inward
box.
flux
From
very
(3) and
passing
through
the surface,
boundary
r-(1 + j)Z/Ok]
Ho exp
with
the
current
condition
that
,
distribution
already
in (13).
found
Part (b) of the problem
with
can be formulated
a simple
boundary
as a poten-
condition,
since
It
H can be derived from a magnetic potential tp, i.e. H = grad ip. Since div H = 0
we have at once 6.cp = div grad cp = O. The boundary
condition
for this potential
is Hn = àcp/àz = O. In
follows
other
from
region
outside
the
conductor.
this that
words,
the magnetic
minate
at
right
ductor.
It should
find a potential
L, parti-
to the surface,
a flat
layer
wallof
the mag-
a distance
to the surface,
upper
of
than
in the magnetic
"'" L, thickness
the skin
perpendicular
over
over
consider
and width
radius
smaller
exciting
perpendicular
Now
the surface?
the smallest
netic field, such as a coil. The variation
outside
In conforrnity
=
curl H = 0 in the
vary in a direction
greater
smaller
of the surface.
the conductor
H
tial problem
then breaks
the
the fields are zero in the interior,
the
of the surface,
along
little
in the metal:
=jw,uaH,
vZ~
that
in which
even if the conductor
do field and current
ponent
vary
so that
o2H
current
to the surface?
b) How
little
fields
~
The ex-
to the dimensions
do field and current
pendicular
then
o2H/oz2,
plane
radial
case
and the radii of curvature
the problem
cularly
the
6.H "'"
for
boundary
by the
the limiting
can simplify
field
cause
in the wire.
skin layer is thin compared
but
amplitude
to be matched.
affected
If we now consider
a) How
to sol utions
of the two soluand
at the
have
is not
of each
277
equipotential
angles
to
be noted
which
the
surfaces
surface
that we must
of
For every revolution
ing wire,
there
around
is a certain
increase
tercon-
be prepared
to
function
of
is not a single-valued
position.
the
a current-carryin the magnetic
potential.
If we now restrict
necessarily
(b) can
ourselves
to a cylindrical,
circularly-symmetric,
be reduced
but not
conductor,
to a potential
question
problem
in another
current distribution
along the sur/ace is the same as the distribution of the
charge over the surface of an insulated, charged conductor of the same shape. In other words, the boundary
way.
Under
condition
these
conditions
the
is that the potential
is constant
at the surface
of the cond uctor.
That
this is so may
magnetic
be seen from
field may be described
A
the following.
by a vector
potential
A such that
Fig. 7. The magnetic fields at the surface
enclosing the skin layer.
of a flat volume
for A we choose
the walls
and,
of the box is the same
since
no
fl ux passes
HnL2 is, at most,
the fluxes through
pensate
one
In the limiting
therefore
disregard
the magnetic
We choose
the surface
problem
a local
is
case
thus
at
most
Ok«
where
wall,
L, we can
to
at the surface.
co-ordinate
and z-axis perpendicular
system
very simple.
with
origin
to the surface.
div A = O.
(17)
the gauge:
It follows
at once from (I) (with
curl curl = grad div -6.)
corn-
components:
has now become
(16)
6.A
in
Be-
The solution
current
that interests
is parallel
oD/Ot
=
us is the one in which the
to the longitudinal
(the x direction),
pendicular
to it and all the quantities
of x (the
longitudinal
problem
such
of the
field is per-
are independent
two-dimensional).
a vector
Ax
direction
the magnetic
is ,evidently
situations,
direction:
J = 0 and
o.
conductor
describe
= 0,
that
about
perpendicular
to the tangenrial
field is tangential
Part (a) ofthe
lower
H;
flux,
of Ht.OkL (or even less,
the component
with respect
the
the side walls virtually
Hn
another).
(ol<fL)Ht.
the surface
through
of the order
because
as the outward
=
curl A
element
potential
= A, Ay = Az
To
A in the
= 0, is
a
VOLUME28
PHILlPS TECHNICAL REVIEW
278
convenient choice. Equation (16) is then equivalent
to Hy = bA/~z,.l{z.= -bA/by.
Let us again take at some point on the surface a local
system ofaxes, with the z axis perpendicular to the
surface. In our limiting case <5k « L the magnetic field
has, as we saw, no vertical component at the surface:
Hz = 0, so that M/by = 0, which means that A is
constant at the surface. The two-dimensional problem
thus reduces to finding a scalar function A(y, z) which
satisfies !lA = 0 outside the conductor and is constant at the surface.
Flat strip
We shall use the above to calculate the current
distribution over the width of a flat strip for the case of
a very small skin depth. If we consider a flat strip (width
. B, thickness D) in which an alternating current flows
in the longitudinal direction, we can distinguish between various frequency ranges. At very low frequencies (direct current) the current is uniformly distributed
over the cross-section. At higher frequencies there is a
range in which the current is still uniformly distributed over the thickness (<5k » D) but no longer over
the width. The calculation of the distribution over the
width in this situation has been carried out by Belevitch,
Gueret and Liénard [2]. At very high frequencies the
current flows in a skin layer that is thin compared to
the strip thickness (<5k« D), and here, again the
distribution is not uniform over the width. It is this
distribution that we shall now calculate.
We take <5k« D, so that the above considerations
apply, and D « B. With this latter condition we can
idealize the strip as infinitely thin so that its cross
section becomes a line and our boundary condition
becomes: A is constant over the line. The two-dimensional potential problem .can now be solved by conformal mapping. We take co-ordinates X, Y in the
plane of our problem. Consider the complex variable
Z=X
jY and write down an analytical function w
+
ofZ:
IV
=
U
+ j V = IV(Z).
.This relation maps the Z-plane on to the w-plane.
The relationship tlA = 0 remains valid in the w-plane
for it follows from the Cauchy relations for analytical
functions,
bu
bv
bv
bX
bY'
bX
bu
=-
bY'
that:
tlzA
in which
=
bX
+ (bV)2J
bY:
[(bU)2
!lwA ,
so that the requirement !lzA = 0 becomes !lwA = O.
Let us assume that the line on which A is constant is
Y = 0, -1 ~ X ~ +1. Now, by choosing a relation
between wand Z which maps this line on to a circle,
we reduce our problem to a simple circular-symmetric
problem-Such a relation is
Z'=
tew + l/w).
The circle Iwl = 1 described by IV = exp (j-&) as -&varies
is converted into Z = cos -&,i.e. X = cos -0-, Y = 0,
which describes our line. Taking IV = r exp (j-&),then
for a circular-symmetrical A outside the circle, !lA = 0
is equivalent to:
1 bA
-=0.
ör
b2A
-+br
r
2
The solution is A = e In r, in which e is an arbitrary
constant. From (1) and Stokes's theorem the surface
current density Js, i.e. the current per unit width of
surface Js = J Jdz, is equal to the tangential magnetic
field, and hence:
bA
J« = lim -.
bY
Y-+O
For points close to the surface, r ~ 1: let us put
r =1
e (s « 1). Then:
+
Y
= Im Z = ter - l/r)sin -&~ e sin-&
and
bA
- =
bY
hence
öln r
e
e·--=bY
r
br
-=-
bY
e ös
-=-----.
r bY
e
e
Js=--=.
sin -&
(1
e
+ e)sin ij. ,
.
Vl-X2
The current along one face of the strip is:
.
+1
10
=
edX
.J VI
X2
=
en
-1
(the total current is 210, since both faces carry current).
To facilitate comparison with the parallel-wire problem we normalize so as to obtain an average surface
current density ofunity. We therefore put e = 2/n (the
strip Y = 0, -1 ~ X ~ + 1, has a width of 2). Js is
plotted in fig. 8 for e = 2/n.
We note that, under the conditions <5k« D« B,
the current distribution over the width is independent of the frequency and the conductivity. Further,
the solution given here forms an asymptotic approximation in the range of (lower) frequencies where, in
contrast with the above conditions, the skin depth is
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,
1.0
&.
x
e
Fig. 8. The distribution
of
the surface current over the
width of a flat strip (width B,
thickness D) with Jk « D «
B (the solid curve). The curve
is normalized to give a "surface current density"
of unity. The points indicated
by
triangles, crosses and circles,
taken from fig. 4, represent
the current distribution
over
three, four or five wires respectively ford = 3 (cf. fig. 4).
I
279
H is constant. As the field must vanish for z -+ ± 00,
H = 0 everywhere outside the strips; we assume that
H = Hobetween the strips. Inside each strip:
d2H
-_
=jwf.loH.
dz2
+
With the boundary conditions H = 0 for z = a
d
and H = Hofor z = a, we find for the field in the
"upper" strip (z > 0):
sinh a (a
+d-
z)
H=Ho-------sinh ad
05
(18)
'
+
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):
J
OL_----------------~
=
-aHo
cosh a (a
+d-
sinh ad
z)
, . .
(19)
VZZZZZZ?ZZZ7IZZ7ZZZZZZZ77?J
One might be inclined to think that the problem of
the flat strip could be reduced to a one-dimensional
problem by considering the limiting case of an infinitely wide strip (where one need consider only the coordinate perpendicular to the plane of the strip). This
is however not possible because the contribution
towards the magnetic field from distant current strips
cannot be neglected. The field due to a current strip of
width dX at a distance X (measured along the width)
is proportional
to dX/X, and the integral of such a
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 infinitely wide strips placed opposite to each other and
carrying current in opposite directions. We shall deal
with this briefly.
A one-dimensional
wide strips
problem: two opposite and infinitely
Suppose that two strips of thickness d at a distance
2a extend to an infinite distance in both length and
width (see jig. 9). In this system, the current can flow
lengthwise (equal but opposite currents in the two
strips) with the magnetic field along the width, the
current and field depending only on the co-ordinate z
perpendicular to the plane of the strips. Even if the condition Ok« d is not fulfilled this one-dimensional
problem is very simple.
Equation (1) becomes J = dH/dz. In the region
between and outside the strips, J = 0 and therefore
and that in the lower strip follows by replacing z by
-z and multiplying the entire expression by -1. The
current per unit width in the upper strip is:
('
a+d
Jdz
l'ddzH
a+d
=,
ü-r
dz
=
[Ht
d
=
(20)
-Ho.
a
No complications arise in taking the d.c. limit of (19),
00, i.e. a -+ O. Expanding
(19) and neglecting all
but first order terms in a, J = =Hejd, as is consistent
with (20). The system may be considered as a deformed
single-turn coil.
Ok -+
Fig. 9. Two strips of thickness
infinity in length and width.
Ferromagnetic
d at a spacing
2a extending
to
sphere
In ferromagnetic metals the skin effect occurs in a
frequency range completely different from that in the
non-ferromagnetic
metals. For a metal in which
flr = 5000 and
0 = gOcojJper,
the value of uo is
1000 times greater than it is for copper, so that the skin
depth is only 1 mm at frequencies as low as 1 Hz.
[2]
V. Belevitch, P. Gueret and J. C. Liénard,
un ruban, Rev. HF 5, 109-115, 1962.
Le skin-effe!
dans
PHILIPS
280
TECHNICAL
VOLUME 28
REVIEW
In the previous sections we always thought of the magnetic flux lines (which are continuous through the
surface) is given by:
alternating current in the conductor as being produced
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 betanç = BnI/ÉtI, tan e = BnefBte.
ing produced by induction. We shall now consider the
. In a static magnetic field, tan cp has a value of the
case of a ferromagnetic sphere in an alternating magorder
of unity (the field within the sphere is homogenenetic field. An interesting complication is that the tenous),
so
that for a ferromagnetic material tan e » 1:
dency of the alternating field and therefore of the flux
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.)
Let us now consider alternating fields of frequencies
may be seen as a conflict between these opposing
such that ~k« R: the fields are confined to a layer of
tendencies [3].
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:
!lr~k
the frequency. In reality, ftr is in general a function of
tan 'IjJ
the frequency. Employing the usual notation for comR
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 characterLet us now consider a sphere of radius R in an ini- ized by flux lines perpendicular to the surface (ferrotially 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),
R::j
--
•
2
Wc=--
Fig. 10. Refraction of the lines of magnetic induction at the surface of a ferromagnetic sphere.
1
W
-«-,
(a)
= Bni,
Hte = He or Bs« =
H. B. G. Casimir, Philips Res. Repts. 2, 42-54, 1947.
of the
1
ftr
W
-« -«
flr
Wc
ftr ,
W
(c)
!lr« -
.
Wc
In the case of a sphere 1 mm in diameter (i .e.
= 5 X 10-4 m) and a = !acopper = 1.2 X 107 (Qm)-l,
Wc
5 X 105 radfs. For such a sphere, with {lr = 5000,
the frequency boundaries W = wcf ftr and W = flrWc of
the medium-frequency region become: W = 100 and
W .- 2.5 X 109 radfs, corresponding roughly to 20 Hz
and 400 MHz.
R::j
Bi: ff.lr,
from which it follows that the "refraction"
[3]
(b)
R
Bne
(21)
the boundaries between regions (a), (b) and (c) are the
frequencies given by:
Wc
and Gauss's theorems and equations (1) and (3), one
arrives at the well-known boundary conditions that
must apply at the surface (the suffixes nand t refer
to the normal and tangential component, i and e refer
to internal and external):
.
poaR2 '
]967, No. 9
SKIN EFFECT, I
The difference between the three cases shows up
particularly in the power dissipated by the sphere.
This depends on u, a and w in different ways in the
three cases. If we borrow from magnetostatics the result that (because of demagnetization effects) the
magnetic flux density in a sphere with a high flr in an
originally homogeneous static magnetic field His B =
3floH (i.e. independent of flr), the energy dissipation
for case Ca),R« Ök, may easily be found. Consider an
281
Let Jm be the amplitude of J; then (Re J)2 ~ tJIit2.
Integration over the sphere gives:
P
=
6n wflHm2R3
5
(~)2,
flrÖk
where H,« is the amplitude of H.
The dissipation in cases (b) and (c) may be estimated
in the following way. The current now flows through a
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
once more assume that the field and current are proportional to exp (jwt). With Stokes's theorem, equation (2), J = aE and B = 3 floH, we find for the
current density in the ring J = -~ jwaflor H. The actual
instantaneous current is ReJ drdz and the resistance
is 2nr/adrdz, whence the energy dissipation per
second is
2nr
dP = (ReJ drdz)" ---
adrdz
2nr---
=-
(Re J)2 drdz .
a
about 2nR and a cross-section of about ÖkR. The heat
developed per second is roughly:
P
R:;
2nR
(Re JÖkR)2 . -aRök
=
nJm2R2Ök
a
.
If (jj is the total flux through the sphere (amplitude
(jjm), then from (2) and Stokes's theorem:
2nRJ
= -jwa(jj,
so that:
W
PR:; --
(jjm2
2nflÖk
I
I
Idz:
Ddr
The flux through the sphere differs in the two cases (b)
and (c). In case (b) the external field pattern and hence
the flux through the sphere are the same as in the static
case, so that cp = nR2 X 3floH. In case (c) the flux lines
are tangential to the surface, so that no demagnetization 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
c) P
Fig. 12. Annular volume element (radius r at distance z from
the centre) of a sphere, used in calculating the power dissipated
in case (a), R « 15k•The axis ofthe ring is parallel to the magnetic
field.
I'::::i
(.!!_),
flrÖk
2nwflOHm2R3 (fl~k)
'.
An exact calculation [3] gives the same results for
these limiting cases except that the numerical factor
PHILIPS TECHNICAL
282
VOLUME28
REVIEW
9n/2 in (b) becomes 3n and the factor 2n in (c) becomes
3n/2. If we put
,_
6nfloHm2R3 =
W
and
~tr(Jk/R = a,
2
P
= gW W~tra-2 =
gcocW
For case (b), i.e. (Jk« R«
.
i
(22a)
(:c)
flr(Jk or l/flr«
co/Wc« flr:
Wc/f1.rl
(22b)
For case (c), i.e. flr(Jk« R or flr«
P = !coWa = !cocWfli/2
_-
p
and once more introduce COc according to (21), and
using (10), the final result is:
For case (a), i.e. R« 15kor co/wc« 1/flr:
--------
1_---lir:r=l
«[os«:
(:J
12
/
(22c)
_W
Fig. 13. The power P dissipated in a ferromagnetic sphere in an
alternating magnetic field as a function of the angular frequency
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
line applies for f1r = f1rl » 1 (T < Tc), and the dashed line for
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
downwards with respect to P. The temperature that
in plotting this diagram.
A peculiar phenomenon resulting from this inter- the body assumes is given by the intersection of the
two curves. As the amplitude increases the cycle passes
play of skin effect and ferromagnetism is "temperature
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
8
phenomenon can now occur. The body is gradually
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
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
-T
(then region (b) is non-existent). We see from this
~-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 ferromagnetic sphere in an alternating magnetic field with a frequency
Fig. 14 gives a diagram of the heat developed, P, and
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
i
(4]
See: J. L. Snoek, Newdevelopmentsinferromagneticmaterials, Elsevier, Amsterdam 1947.
P, so that situations A, B, C, D and E arise successively. On traversing the full cycle A ... E ... A, the temperatures corresponding to situations between Band D are different on the forward
journey from those on the return journey.
1967, No. 9
SKIN EFFECT, I
through situations A, B, C, D and E. Between Band D
there are two stable equilibrium temperatures (see, for
example, situation C). As the amplitude increases, the
temperature rises abruptly from To to Tn' at the moment that the line D leaves the lower bend in curve P.
As the amplitude decreases the temperature remains
high until, as B leaves the upper bend in curve P, it
drops back from TB' to TB. The frequency range in
which temperature hysteresis may be expected (indicated in fig. 13 by a double arrow) is given by the
conditions:
Summary. This first part of three articles on the skin effect contains a general Introduetion followed by a discussion of the
current distribution for several configurations. The essence ofthe
skin effect is illustrated by the case of a number of parallel
wires in one plane, connected in parallel. The standard problem
of the cylindrical wire is used to introduce the concept "classical
skin depth". The problem ofthe distribution ofthe current over a
conductor of arbitrary shape is stated in general terms. If the
skin layer is thin, the problem of the distribution in the surface
is reduced to a potential problem. For a cylindrical wire of any
283
= 1) ~ PCb, fhr =
fhrl)
=
fhrl),
Pea,
fhr
P(c,
fhr = 1) ~
and
PCb, fhr
where the a, band c refer to our three frequency regions. Using (22a, b, c) we find:
25
-4
fhrl
W
«-«
l
Wc
y-
fhrl.
For our sphere (diameter 1 mm, a = 1.2 X 107 [2-1 m-I,
fhrl = 5000) this means that the frequency f must be
in the range 100 Hz «f « 3 MHz.
cross-section, the surface current distribution is the same as the
charge distr.ibution over the surface of a charged conductor of
the same shape. This consideration is used in the problem of the
flat strip. The problem becomes purely one-dimensional for two
strips which together form a flat coil. Finally the case of a ferromagnetic sphere in a magnetic alternating field is discussed. The
combination of skin effect and ferromagnetism can lead to
"ternperature hysteresis", an effect in which the temperature
variation of the sphere is hysteretic with respect to an increasing
and a decreasing field amplitude.
'0