Approximate Analytical Calculation of the
Skin Effect in Rectangular Wires
Dieter Gerling
University of Federal Defense Munich, 85579 Neubiberg, Germany
Dieter.Gerling@unibw.de
Abstract- It is well-known that thick wires energized with highfrequency currents show the effect, that the current mainly
flows near the surface of the wire. This effect is quite well
described in literature for round wires, for rectangular wires no
description of similar simplicity could be found.
This effect may be relevant for future electric drives, e.g. in the
automotive industry. Therefore, the skin effect of rectangular
wires is analyzed in this paper for the application of hybrid
drives for passenger cars.
Index Terms—Analytical Calculation, Rectangular Wire,
Skin Effect
I.
INTRODUCTION
In the automotive industry, currently much effort is
dedicated to the development of hybrid and pure electric
drives. As the voltage of the battery is limited (in many cases
in the region of 120V), quite thick wires have to be used for a
certain power level. In addition, high frequencies are used to
reach a high maximum speed. These boundary conditions can
lead to the occurrence of the skin effect in such applications
(it is well-known that thick wires energized with highfrequency currents show the effect, that the current mainly
flows near the surface of the wire). Moreover, sometimes
rectangular wires are used to better utilize the slot area of
such machines.
For round wires the skin effect is well documented, see e.g.
[1]. For rectangular wires a description of similar simplicity
could not be found in literature. Therefore, in this paper
especially the skin effect of rectangular wires is investigated.
Even if it can be assumed, that for rectangular wires the skin
effect is similar to round wires, a proof of this assumption and
a detailed calculation method are necessary.
The main task of this paper is the elaboration of a simple
calculation method for the skin effect in rectangular wires.
II. THE SKIN EFFECT IN RECTANGULAR WIRES
A. Solution of Maxwell’s Equations
The following figure 1 shows the wire under investigation.
We assume that in this wire the current has only a zcomponent (i.e. into the sheet of paper or out of the sheet of
paper), but this component depends on the x- and ycoordinate. This means in terms of the current density:
G
G
J = J z ( x, y ) ⋅ ez
(1)
Fig. 1. System under investigation.
With
G
G
G
(2)
rot ( H ) = J = γE
the following conclusions can be drawn:
G
G
E = E z ( x, y ) e z and
G
G
G
H ( x, y ) = H x ( x, y ) e x + H y ( x, y ) e y ,
G
G
G
where e x , e y and e z are the unit vectors in x-, y-, and zdirection, respectively. Evaluating (2) in Cartesian Coordinates gives:
∂
∂
Hy −
H x = γE z
(3)
∂x
∂y
With
G
∂ G
(4)
rot ( E ) = −μ H
∂t
we get in Cartesian Coordinates
∂
∂
E z = −μ H x
(5)
∂y
∂t
and
∂
∂
−
E z = −μ H y
(6)
∂x
∂t
Differentiating (3) with respect to time and inserting (5) and
(6) gives
γ
∂
∂t
⇒
Ez =
∂
2
1 ∂
2
μ ∂x
2
Ez +
∂
2
1 ∂
2
μ ∂y
2
Ez
∂
(7)
E z + 2 E z = μγ E z
2
∂x
∂y
∂t
This differential equation shall be solved with the
following product set-up:
E z ( x, y, t ) = X ( x ) ⋅ Y ( y ) ⋅ T ( t )
(8)
Inserting this set-up to (7) gives (a prime means the total
differential with respect to the relevant variable):
X ′′ ( x ) ⋅ Y ( y ) ⋅ T ( t ) + X ( x ) ⋅ Y ′′ ( y ) ⋅ T ( t )
= μγX ( x ) ⋅ Y ( y ) ⋅ T ′ ( t )
⇒
μγ
T′ ( t )
T (t)
X ′′ ( x )
=
X(x)
+
(9)
Y ′′ ( y )
Y ( y)
The left side of (9) depends only on the time, the right side
only on the locus. Therefore, both sides must be constant.
Evaluating firstly the left side of (9) gives:
T′ ( t )
T′ ( t ) − c ⋅ T ( t ) = 0
=c
⇒
(10)
T (t)
With the set-up for a harmonic wave T ( t ) = e
jωt
jωt
we get:
As the right side of (12) is constant, both terms of the left
side of (12) must be constant (because they depend on
different coordinates). It follows:
X ′′ ( x )
2
=k
X (x)
⇒
X ′′ ( x ) − k X ( x ) = 0
⇒
X ( x ) = C1 sinh ( kx ) + C 2 cosh ( kx )
2
(14)
In total we get from (8), (12), (13) and (14):
E z ( x, y, t ) = [ C1 sinh ( kx ) + C 2 cosh ( kx )] ⋅
[ C3 sinh ( A y ) + C4 cosh ( A y )] ⋅ e jωt ,
2
(15)
2
k + A = jωμγ
B. Calculation of the Variables
From (5) and (15) we get
[ C1 sinh ( kx ) + C2 cosh ( kx )] ⋅
[ A C3 cosh ( A y ) + A C4 sinh ( A y )] ⋅ e jωt
1 1
μ jω
1 1
μ jω
A C3 [ C1 sinh ( kx ) + C 2 cosh ( kx )] = 0 (17)
⇒ C3 = 0
Analogously, we get from (6) and (15)
C1 = 0
Summarizing (15), (17) and (18), we get:
E z ( x, y, t ) = C ⋅ cosh ( kx ) ⋅ cosh ( A y ) ⋅ e
2
(18)
jωt
,
2
k + A = jωμγ
The constant C can be calculated from
jωt
I ( t ) = ˆIe =
a
b
2
2
∫ ∫ J z ( x, y, t ) dy dx
−
a
2
−
(19)
(20)
b
2
Solving (19) and (20) for the constant C we get:
C=
Î
4γ
k⋅A
a⎞
⎛
⎛
sinh ⎜ k ⎟ sinh ⎜ A
2
⎝
⎠
⎝
(21)
b⎞
⎟
2⎠
Equations (19) and (21) describe the solution of the field
problem inside the rectangular wire, but there is one
additional equation missing for the constants k and A .
Possibilities for such an additional equation are given in the
following three sections C to E.
(13)
Analogously we get:
Y ( y ) = C3 sinh ( A y ) + C4 cosh ( A y )
Hx = −
−
jωt
jωe − c e = 0 ⇒ c = jω
(11)
Evaluating in a second step the right side of (9) leads to:
X ′′ ( x ) Y ′′ ( y )
+
= jωμγ
(12)
X (x)
Y ( y)
⇒
H x ( y = 0, t = 0 ) =
C. Evaluation of the Magnetic Field Strength
In the following we assume that the absolute value of the
magnetic field strength for x = a 2 , y = 0 and x = 0, y = b 2
are identical. This assumption is valid as long as the
frequency is not too high (for DC current the outside
circumference of the wire represents a field line) or the wire
is (at least approximately) a quadratic one. The evaluation of
(5), (6) and (19) gives:
G⎛
a
a
⎞
⎛
⎞
H ⎜ x = , y = 0 ⎟ = Hy ⎜ x = , y = 0 ⎟
2
2
⎝
⎠
⎝
⎠
=
= −μ
∂
∂t
1
jωμ
⎛ a ⎞ ⋅ e jωt
⎟
⎝ 2⎠
k ⋅ C sinh ⎜ k
G⎛
b⎞
b⎞
⎛
H ⎜ x = 0, y = ⎟ = H x ⎜ x = 0, y = ⎟
2⎠
2⎠
⎝
⎝
Hx
(16)
[ C1 sinh ( kx ) + C 2 cosh ( kx )] ⋅
[ A C3 cosh ( A y ) + A C4 sinh ( A y )] ⋅ e jωt
Because of the symmetry, for y = 0 the value of H x must
be zero for any time; therefore it must be valid even for
t = 0 . It follows:
=
1
jωμ
(22)
⎛ b ⎞ ⋅ e jωt
⎟
⎝ 2⎠
A ⋅ C sinh ⎜ A
Consequently we get
⎛ a⎞
⎛ b⎞
k sinh ⎜ k ⎟ = A sinh ⎜ A ⎟
⎝ 2⎠
⎝ 2⎠
(23)
D. Calculation of the Symmetric Case
The symmetric case is characterized by a = b (quadratic
wire). For this we get because of the symmetry:
k=A
2
⇒
2 k = jωμγ
⇒
k=A =
1
δ=
2
ωμγ
and
C=
2
Î
k
4γ
⎡sinh ⎛ k a ⎞⎤
⎜
⎟
⎢⎣
⎝ 2 ⎠⎦⎥
2
(25)
E. Calculation of the (near) DC Case
Taylor’s series expansion for the hyperbolic sine function
with truncation after the first summand gives sinh ( x ) ≈ x
(this approximation is valid for small arguments x, i.e. for
example for low frequencies). This leads to
⎛ a⎞
2 a
k sinh ⎜ k ⎟ ≈ k
2
⎝ 2⎠
(26)
b⎞
⎛
2 b
A sinh ⎜ A ⎟ ≈ A
2
⎝ 2⎠
For the case of a DC current, i.e. ω = 0 , (19) leads to:
⇒
ω=0
⇒
A =
(24)
j,
δ
k=
k =A =0
E z ( x, y, t ) = C
(27)
1
δ
1
δ
j
j
2b
a+b
2a
, δ=
2
(31)
ωμγ
a+b
F. Evaluation of k and l for Arbitrary Geometry and Frequency
Evaluating (31) for the symmetric case ( a = b ) we get the
exact result (please refer to (24)). This means: The
approximation of the hyperbolic sine function, that leads to
k and A according to (31) is valid for
arbitrary geometry and low frequencies and
the symmetric geometry ( a = b ) and arbitrary
frequencies.
The solutions (23) and (31) for the constants k and A are
based on the same physical constraints (both are valid for low
frequency or quadratic wire). Nevertheless, there is a
difference when evaluating these equations.
To estimate the deviation F between the solutions (23) and
(31) depending on geometry and frequency, the solution (31)
will be evaluated and the deviation analyzed like follows:
⎛ a⎞
⎛ b⎞
(32)
F = k sinh ⎜ k ⎟ − A sinh ⎜ A ⎟
⎝ 2⎠
⎝ 2⎠
This results in the following figure for the deviation F
(figure 2 shows the real part of the complex value of the
deviation F , the imaginary part is qualitatively similar to this
with just a slightly lower amplitude).
Consequently the constant C for the DC case becomes
C=
Î
γ ⋅a ⋅b
(28)
For the case of a near DC current, i.e. low frequency, (19),
(23) and (26) lead to:
2
2 a
k +k
= jωμγ
b
(29)
jωμγ 2b
⇒ k=
2 a+b
Analogously we get:
A =
jωμγ 2a
(30)
2 a+b
For the constant C (21) holds true.
With the skin depth δ (please refer e.g. to (24) or reference
[1]) we get from (29) and (30):
Fig. 2. Re { F } as a function of frequency f and edge length a.
In figure 2 the following conditions were considered:
a frequency of 0Hz to 1000Hz;
-
2
-
a copper wire of cross section A wire = 3.75mm ;
-
a temperature of 20°C (the influence of temperature
variation is similar to the influence of frequency
variation, please refer to (31));
an edge length variation between 0.1mm and 3.0mm
(the rectangular copper wire of 3.0mm times 1.25mm
is the application described in chapter III).
It can be deduced from figure 2 that the deviation F is
small, if low frequency or symmetric geometry is regarded
(for these cases both solutions deliver the exact result). If one
of these conditions is not fulfilled, the deviation because of
the used different approximations is increased.
-
G. Numerical Calculation of k and l for Arbitrary Geometry
and Frequency
At this time it can not be decided, if the deviation F
coming from the different approximate calculations of k and
A for arbitrary frequency and geometry has an important
influence on the current density distribution inside the wire.
Therefore, both solutions to calculate these parameters will be
compared. This can be done only numerically, because the set
of equations (19) and (23) is of transcendent nature. This
numerical solution is performed using the software package
MathCad. For the same conditions like in the previous
section, for varying the frequency between 0Hz and 20kHz,
and for varying the edge length a of the rectangular wire
between 0.1mm and 10.0mm an absolute value of the failure
−10
below 10 for each data point could be reached.
The following figure 3 illustrates the difference between
the analytical calculation (according to (31)) and the
numerical calculation according to (23). As an example the
parameter Re { k } is given. The results for the imaginary part
and for the parameter A are similar.
Re { k }
Re { k}
a in 0.1mm
a in 0.1mm
frequency in 100Hz
a)
Fig. 3.
frequency in 100Hz
b)
Re { k }
as a function of frequency f and edge length a
a) analytical calculation according to (31)
b) numerical calculation according to (23)
H. Evaluation of the Edge Condition
Because of the special behavior of the electromagnetic field
at the edges of the regarded geometry, a certain edge
condition must be fulfilled in addition to the solution of
Maxwell’s equations. This edge condition means that (please
refer to [2]) “the electromagnetic energy density must be
integrable over any finite domain even if this domain contains
singularities of the electromagnetic field. In other words, the
electromagnetic energy in any finite domain must be finite”.
The Poynting’s vector
G G G
S = E×H
(33)
describes the energy, that flows per time unit through the unit
G
area perpendicular to S . As the edge condition must be
fulfilled for any finite time interval, this means that
Poynting’s vector must be finite over any finite domain.
The Poynting’s vector in the case of the rectangular wire is:
G G G
G
G
S = E × H = E z H x e y − E z H y ex
(34)
Inserting (5), (6) and (19) into (34) gives:
G
G
G
S = Ez H x ey − Ez H y ex
(
= C cosh ( kx ) cosh ( A y ) e
jωt
) ⎛⎜ − ∫ E z dt ⎞⎟ eG y −
⎝ μ ∂y
⎠
∂
1
( C cosh ( kx ) cosh ( A y ) e jωt ) ⎛⎜ ∫ E dt ⎞⎟ eG x
⎝ μ ∂x
⎠
∂
1
z
Further we get:
G
jωt
S = − C cosh ( kx ) cosh ( A y ) e
(
1
) jωμ ⋅
G
⎡⎣ C A cosh ( kx ) sinh ( A y ) e jωt e y +
jωt G
C k sinh ( kx ) cosh ( A y ) e e x ⎦⎤
=
−1
jωμ
(C
2
cosh ( kx ) cosh ( A y ) e
j2 ωt
(35)
)⋅
G
⎡⎣ A cosh ( kx ) sinh ( A y ) e y +
G
k sinh ( kx ) cosh ( A y ) e x ]
For a certain operating point the parameters k , A , and C
(in addition to ω and μ ) are fixed values. As the wire has
finite dimensions, even the variables x and y are finite.
G
Therefore, Poynting’s vector S is finite over any finite
domain.
This means that the edge condition is fulfilled for the
solution given above; i.e. this solution is valid for rectangular
wires (as long as the approximations are valid).
III. CURRENT DENSITY DISTRIBUTION
From (19) and (21) the frequency dependent current
density distribution in a rectangular wire can be calculated.
For the calculation in this section the time t = 0 , the
temperature ϑ = 20°C , a total current of Î = 10A and a
rectangular copper wire with a = 3.0mm times b = 1.25mm
( 0 ≤ x ≤ a , 0 ≤ y ≤ b ) cross section is assumed.
In a first step, the analytical calculation according to (31)
for the parameters k and A is used to get the result for the
current density distribution. Figure 4 shows this current
density distribution for 4 different frequencies.
The following figure 5 shows the frequency dependent
current density distribution for the same boundary conditions
like in figure 4. The difference to this former calculation is
that during this evaluation the numerical calculation for the
parameters k and A according to (23) is used to obtain the
result for the current density distribution. Some slight
differences to figure 4 can be noticed at high frequency.
The time dependent current density distribution is
illustrated in the following figure 6. As there are some slight
deviations (for high frequency) between both calculation
methods, only the results of the numerical calculation
according to (23) are given. The calculation was performed
for a frequency of f = 20kHz .
Fig. 4. Current density distribution in a rectangular copper wire (cross
section 3.0mm times 1.25mm, current Î = 10A , time t = 0 ,
temperature ϑ = 20° C ): analytical calculation according to (31).
Fig. 6. Current density distribution in a rectangular copper wire (cross
section 3.0mm times 1.25mm, frequency f = 20kHz , current
Î = 10A , temperature ϑ = 20° C ): numerical calculation
according to (23).
IV. ENTIRE SET OF MAXWELL’S EQUATIONS
Fulfilling the entire set of Maxwell’s equations, the
following equation (36) has to be checked:
G
∂
∂
∂
div ( E ) =
Ex +
Ey +
Ez = 0
(36)
∂x
∂y
∂z
G
As E = E z ( x, y ) is true, the condition (36) is fulfilled.
Fig. 5. Current density distribution in a rectangular copper wire (cross
section 3.0mm times 1.25mm, current Î = 10A , time t = 0 ,
temperature ϑ = 20° C ): numerical calculation according to (23).
Additionally,
G
div ( B ) = 0
(37)
G
G
G
must be fulfilled. With H = H x ( x, y ) e x + H y ( x, y ) e y and
G
G
B = μH we get
G
div ( B ) = 0
∂
⇔
∂x
With (5) and (6) this leads to
∂
∂
∂ ⎛ 1
Hx +
Hy =
⎜−
∂x
∂y
∂x ⎝ μ
Hx +
∂
∂y
Hy = 0
⎞ ∂ ⎛1 ∂
⎞
∫ ∂y E z dt ⎟⎠ + ∂y ⎜⎝ μ ∫ ∂x E z dt ⎟⎠
∂
=
∂
∂
⎡∂ ∂
⎤
E z dt −
E z dt
∫
∫
⎢
⎥⎦
μ ⎣ ∂y ∂x
∂x ∂y
=
∂ ∂
⎡ ∂ ∂
⎤
E z dt − ∫
E z dt
∫
⎢
⎥⎦
μ ⎣ ∂x ∂y
∂x ∂y
1
1
=0
Therefore, even condition (37) is fulfilled.
V. COMPARISON TO LITERATURE DATA
There are only very few papers dealing with the skin effect in
rectangular wires. In addition, not the time dependent current
density distribution is given in [3, 4], but the absolute value
of current density or electrical field strength versus the wire
cross section. Therefore, a qualitative (but no quantitative)
comparison is possible and illustrated in the following figures
7 and 8. It can be deduced that the results obtained with the
proposed calculation method correspond quite well with the
FEM results published in literature. Nevertheless, there is
some deviation along the edges of the wires. Most probably
this comes from the approximative assumption mentioned in
section II.
Fig. 7. Comparison to literature data (left: data published in [3], right:
solution according to this paper for the same boundary conditions.
VI. CONCLUSION
In this paper the skin effect of rectangular wires is
investigated. In addition to the well-known effect of
frequency and beside the influence of the wire geometry
(edge lengths) even the influence of temperature and material
(e.g. copper or aluminum) can be deduced very easily from
the formulae given.
For the rectangular wire an analytical solution could be
found that is exact for the symmetric case (i.e. quadratic wire)
and nearly true for low frequency and arbitrary geometry. For
arbitrary geometry and frequency, the solution given is just an
approximation. Two different mathematical descriptions for
the approximation are given, but physically both are based on
the same assumption. The solutions deviate from each other
depending on frequency and geometry, but there is no
criterion to decide which one fits better to reality.
Evaluating Poynting’s vector, it could be proven that the
necessary edge condition according to [2] is fulfilled for the
solution given.
The analytical calculation of the current density
distribution deduced in this paper shows (in accordance to the
FEM results known from literature) that with increasing
frequency the current is displaced more and more to the
corners of the wire. This is different to round wires, where the
entire circumference is conducting the current.
For the example of winding characteristics of nowadays
hybrid electric drives for the automotive industry (typical
rectangular copper wire with a = 3.0mm times b = 1.25mm
cross section), it could be shown that even for the critical case
of high frequency up to f = 1000Hz , the approximate
analytical solution is fairly good enough to describe the
current density distribution. Therefore, even the AC loss
calculation will be fairly good using the approximate
analytical solution. Consequently, for practical AC loss
calculation this simple and time efficient approximate
analytical calculation method may be used to evaluate the
application envisaged.
REFERENCES
[1]
[2]
[3]
[4]
Fig. 8. Comparison to literature data (left: data published in [4], right:
solution according to this paper for the same boundary conditions.
K. Simonyi, Theoretische Elektrotechnik, VEB Deutscher Verlag der
Wissenschaften, Berlin, 1980 (in German).
J. Meixner, “The behaviour of electromagnetic fields at edges,” IEEE
Transactions on Antennas and Propagation, 20(1972)4, pp. 442-446.
Fichte, L. O.: “Berechnung der Stromverteilung in einem System
rechteckiger Massivleiter bei Wechselstrom durch Kombination der
Separations- mit der Randintegralgleichungsmethode“, Ph.D.
dissertation, Helmut-Schmidt-University, Hamburg, 2007 (in German)
Faraji-Dana, R; Chow, Y.: “Edge condition of the field and a.c.
resistance of a rectangular strip conductor“, IEE Proceedings
127(1990)2