TescheTubeSurfaceImpedance

class pmrf.materials.TescheTubeSurfaceImpedance

Bases: AbstractSurfaceImpedance

Tubular conductor of finite wall thickness, via Tesche’s equivalent circuit.

Mathematical Formulation

With \(q=(a/(a+t))^2\), the tube’s per-unit-length dc resistance and internal inductance are

\[R_{dc} = \frac{1}{2\pi a t\sigma},\qquad L_{int} = \frac{\mu}{2\pi}\left[\frac{\ln(1+t/a)}{(1-q)^2} + \frac{q-3}{4(1-q)}\right],\]
combined as for the solid rod. The equivalent surface impedance is
\[Z_s = R_{dc,sq} + \frac{\zeta_c}{1+\zeta_c/(j\omega L_{int,sq})}, \qquad R_{dc,sq}=\frac{1}{t\sigma},\quad L_{int,sq}=\mu a\left[\frac{\ln(1+t/a)}{(1-q)^2}+\frac{q-3}{4(1-q)}\right].\]

Validity

This formulation has the same approximation as TescheRodSurfaceImpedance: its strong-skin limit is \(\zeta_c+R_{dc,sq}\) rather than \(\zeta_c\). For \(t\to\infty\), \(R_{dc,sq}\to0\) and \(L_{int,sq}\to\infty\), reducing the result to HalfSpaceSurfaceImpedance. This limit is used by TescheCoaxialFormulation when the outer-shield thickness is unspecified.

References

Tesche, F. M. (2007). A Simple Model for the Line Parameters of a Lossy Coaxial Cable Filled With a Nondispersive Dielectric. IEEE Transactions on Electromagnetic Compatibility, 49(1), 12-17.

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