SchelkunoffRodSurfaceImpedance

class pmrf.materials.SchelkunoffRodSurfaceImpedance

Bases: AbstractSurfaceImpedance

Solid round conductor, exact.

Mathematical Formulation

Schelkunoff’s internal impedance per unit length for a cylinder of radius \(a\) is

\[Z = \frac{\gamma}{2\pi a\sigma}\,\frac{I_0(\gamma a)}{I_1(\gamma a)}, \qquad \gamma=\sqrt{j\omega\mu\sigma}.\]
Since \(\zeta_c=\gamma/\sigma\) for a smooth bulk metal, the corresponding surface impedance is
\[Z_s = \zeta_c\,\frac{I_0(\gamma a)}{I_1(\gamma a)}.\]

As \(\gamma a\to0\), \(I_0/I_1\to2/\gamma a\) and \(Z_s\to2/a\sigma\), which is the exact dc limit. As \(\gamma a\to\infty\), \(I_0/I_1\to1+1/2\gamma a\), so the shape factor approaches HalfSpaceSurfaceImpedance with the leading curvature correction.

Validity

Exact for a homogeneous, isotropic solid rod carrying axially symmetric current. Numerical accuracy is limited by i0_over_i1().

References

Schelkunoff, S. A. (1934). The Electromagnetic Theory of Coaxial Transmission Lines and Cylindrical Shields. Bell System Technical Journal, 13(4), 532-579. Eq. (65).

impedance(omega, conductor: ConductorProperties, *, a, **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