SchelkunoffTubeSurfaceImpedance

class pmrf.materials.SchelkunoffTubeSurfaceImpedance

Bases: AbstractSurfaceImpedance

Cylindrical tube of finite wall thickness, exact.

Mathematical Formulation

For inner radius \(a\), outer radius \(b=a+t\), and current returning on the inner surface, Schelkunoff’s internal impedance is

\[Z_s = \zeta_c\, \frac{I_0(\gamma a)K_1(\gamma b)+K_0(\gamma a)I_1(\gamma b)} {I_1(\gamma b)K_1(\gamma a)-I_1(\gamma a)K_1(\gamma b)}, \qquad\gamma=\sqrt{j\omega\mu\sigma}.\]

Dividing by \(I_1(\gamma b)K_1(\gamma a)\) gives the bounded form used for evaluation:

\[Z_s = \zeta_c\,\frac{K_0(\gamma a)/K_1(\gamma a) + \big[I_0(\gamma a)/I_1(\gamma a)\big]\,\rho}{1-\rho}, \qquad \rho = \frac{I_1(\gamma a)K_1(\gamma b)}{I_1(\gamma b)K_1(\gamma a)},\]
The implementation uses scaled Bessel functions for \(\rho\), leaving only the decaying exponential \(e^{-2\gamma t}\) and avoiding overflow for thick walls.

Under strong skin effect, \(\rho\to e^{-2\gamma t}\) and the shape factor becomes \(\zeta_c\coth(\gamma t)\) with the cylindrical curvature corrections retained. As \(\gamma\to0\), the exact dc limit is

\[Z_s \to \frac{2a}{\sigma(b^2-a^2)} = \frac{1}{\sigma t} \cdot\frac{2a}{a+b},\]
which reduces to \(1/(\sigma t)\) for a thin wall.

For t=inf, the result is the infinite-wall limit \(\zeta_c K_0(\gamma a)/K_1(\gamma a)\).

Validity

Exact for a homogeneous, isotropic tube carrying axially symmetric current that returns on its inner surface. Accuracy is limited by the numerical Bessel evaluations.

References

Schelkunoff, S. A. (1934). The Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields. Bell System Technical Journal, 13(4), 532-579. Eq. (74).

impedance(omega, conductor: ConductorProperties, *, a, t=inf, **geometry) → Array

Return the surface impedance of this cross-section, in ohm per square.

Parameters:

  • omega (ArrayLike) – Angular frequency in rad/s. w is reserved for strip width.

  • conductor (ConductorProperties) – Evaluated metal properties. conductor.zs is the surface prefactor, including any surface treatment, and conductor.gamma(omega) is the bulk diffusion constant. Treat them as independent: \(\gamma=\sigma\zeta_c\) holds only for a smooth bulk conductor.

  • **geometry – Cross-section-specific dimensions in metres. Implementations ignore dimensions they do not use.

  • weight (ArrayLike, optional) – Caller’s geometry weight in inverse metres. Only RootSumSquareSlabSurfaceImpedance uses it to express its dc limit in the caller’s normalisation.

Returns:

Surface impedance in ohm per square.

Return type:

jnp.ndarray