The mathematical formulation of the lateral
skin-effect problem for a thin conductor will now be derived
simultaneously for the elliptic section (fig. l*a*) and for
the rectangular section (fig. l*b*) with the common
notation 2*h* for the thickness, *h* being given by (9) in
the first case and equal to the constant *b* in the second. The
linear current density is the discontinuity of the tangential
component *H _{x}* of the magnetic field. By symmetry we
have for a right-handed coordinate system:

(40) |

On the other hand, the true current density (A/cm^{2})
is

(41) |

Finally, Lenz's law yields:

(42) |

It is convenient to introduce the vector potential *A*
which has only a *z*-component. We then have:

; |
(43) |

whereas the voltage drop *ZI* along the
conductor is

(44) |

In (44), *E _{z}* is expressed in terms
of

(45) |

On the other hand, by elimination from (40)-(4l)-(42), we obtain the Biot boundary condition

(46) |

When the magnetic fields are eliminated from (46) by
(43), the resulting relation shows that the derivative of (45) with
respect to *x* is zero. Biot's condition is thus equivalent to
saying that the impedance computed by (45) is independent of *x*.
The form (45) of the boundary condition is to be preferred to (46),
since it yields the impedance without additional effort. Finally, the
problem amounts to solving the Laplace equation for *A _{z}*,
with the condition (45) on the conductor (i.e. for

(47) |

at large distance, corresponding to the vector
potential of a filament of current *I* at the origin, with a
coaxial return of large radius *D*.

We introduce the conformal transformation

(48) |

yielding

(49) |

The transformation is one-to-one with the restrictions

In the (*x*,*y*)-plane, the curves of
constant *v* are homofocal ellipses (*fig. 7*); the
segment *y* = 0,
is the infinitely flat ellipse *v* = 0, and *v*
increases outwards to infinity. The curves of constant *u* are
the hyperbolae of *fig*. 7 but there is a cut along the
segment *v* = 0, so that *u* is positive in the
upper half-plane Re *y* > 0 and negative in
the lower half-plane. The semi-infinite segment y = 0,
corresponds to *u* = 0, whereas the segment *y* = 0,
corresponds to
.

**Fig. 7.**
The conformal representation
.

The expression of *A _{z}* is of the
form

(50) |

where the first two terms give the principal value (47), because (48) yields

whereas the sum (extending from *n* = 1
to
)
with undetermined coefficients *A _{n}* is the general
harmonic function, vanishing at infinity, and having the appropriate
quadrantal symmetry.

For *y* = + 0, and hence *v* = 0,
*u* > 0, then by (49):

so that (45) becomes:

(51) |

The coefficients *A _{n}* must now be
determined so that (51) takes a constant value for all

, |
(52) |

so that the denominator in (51) simplifies to

(53) |

With the notations (15)-(16) and (12), and with the substitution

k A_{n} |
(54) |

(51) multiplied by (53) becomes

(55) |

Replacing the product of cosines occurring in the left-hand side of (55) by cosines of sums and differences, one obtains a Fourier series, and its identification with 1 yields the infinite system

(56) |

of linear equations in *z* and the unknown
coefficients *B _{n}*.

Disregarding temporarily the first two equations contained in (56), one obtains the three-term recurrence relation

_{-1} + B+1 = 2(1+_{n}n/k)B(_{n}n = 2,
3, ...) |
(57) |

which is very similar to the recurrence relation

(58) |

for Bessel functions. For *v* = *n* + *k*,
*s* = *k*, (58) shows that

C J(_{n+k}k) |
(59) |

satisfies (57), with *C* an arbitrary constant.
In fact (59) is the solution of the second-order difference equation
(57) in our case, because the other linearly independent (51)
solution (involving a Bessel function of the second kind) is excluded
on physical grounds since it makes all coefficients *B _{n}*
infinite at d.c. Since (57) holds down to

(60) |

(used for *v* = *s* = *k*),
we obtain

(61) |

and (17).

From the known expression (50) for *A _{z}*,
where

The expansion (22) of (17) can be deduced from the
three-term recurrence relations (57). In the equivalent network of
fig. 4, they correspond to the Kirchhoff relations between the
currents in branches incident to a common node. and the following
electrical proof is therefore equivalent to a mathematical discussion
of (57). Consider the ladder network of *fig. 8*, where the
series admittances are denoted *Y*_{1}, *Y*_{2},
... and the shunt admittances *Y _{a}*,

(62) |

for the node voltages, and the solution *V*_{0}
is the input impedance. From a comparison of (56) and (62), it is
found that the solution *z* of (56) is the input impedance of
the ladder network of fig. 4 where the elements are normalized
(*k* is taken as the complex frequency). Since the input
impedance of fig. 4 is (22), we have indirectly obtained the
continued-fraction expansion of the function (17).

**Fig. 8.**
A ladder network.

For the rectangular section, *h* is constant in
the denominator of (51). After multiplication by the known Fourier
series

(63) |

and substitution of (54), (51) becomes

(64) |

The linear system resulting from (64) is analogous to (56) except that the matrix is now (28). This establishes (27).

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