HollowayKuesterSlabSurfaceImpedance

class pmrf.materials.HollowayKuesterSlabSurfaceImpedance

Bases: AbstractSurfaceImpedance

Exact finite-thickness impedance for the total current in a planar strip.

Mathematical Formulation

Holloway and Kuester’s eq. (45) gives the coupled impedances of the two strip faces, \(Z_s=\zeta_c\coth(\gamma_c t)\) and \(Z_m=\zeta_c\operatorname{csch}(\gamma_c t)\). Their eq. (100) shows that the total current therefore sees

\[Z_s+Z_m=\zeta_c\coth(\gamma_c t/2),\qquad \gamma_c=\sqrt{j\omega\mu\sigma}.\]

The half-thickness argument follows from referring the impedance to total strip current. The limits are \(2/(\sigma t)\) at dc and \(\zeta_c\) under strong skin effect.

Validity

Neglecting the difference-current impedance \(\zeta_c\tanh(\gamma_c t/2)\) is valid for quasi-TEM coplanar waveguide and for microstrip whose characteristic impedance exceeds 40 ohm.

Normalisation

The impedance is referred to total strip current and therefore requires a geometry weight of \(1/(2W)\). This differs from the weight supplied by WheelerCurrentDistribution because Wheeler’s model also includes the ground plane and edge-current crowding. Use this formulation only with a compatible weight; see AbstractSurfaceImpedance.

References

Holloway, C. L., & Kuester, E. F. (1994). Edge shape effects and quasi-closed form expressions for the conductor loss of microstrip lines. Radio Science, 29(3), 539-559. Eq. (45), (100).

Rautio, J. C., & Demir, V. (2003). Microstrip Conductor Loss Models for Electromagnetic Analysis. IEEE Transactions on Microwave Theory and Techniques, 51(3), 915-921. Eq. (4).

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