RootSumSquareSlabSurfaceImpedance

class pmrf.materials.RootSumSquareSlabSurfaceImpedance

Bases: AbstractSurfaceImpedance

Resistance-only blend for a finite planar conductor.

Mathematical Formulation

\[Z_s=\sqrt{R_{dc,sq}^2+\Re(\zeta_c)^2}+j\Im(\zeta_c),\qquad R_{dc,sq}=\frac{1}{\sigma W t\,k},\]

where \(W\) is the strip width, \(t\) its thickness, and \(k\) the caller’s geometry weight in inverse metres. Multiplication by \(k\) recovers the exact dc resistance \(1/(\sigma Wt)\) and the half-space high-frequency limit. This is a ParamRF convention.

Why it is the planar default

Unlike HollowayKuesterSlabSurfaceImpedance, this formulation uses the caller’s weight and therefore matches both asymptotes under a frequency-independent planar weight. See AbstractSurfaceImpedance.

Validity

This is not an exact transition model. It retains the half-space reactance \(\Im(\zeta_c)\propto\sqrt{\omega}\), whereas a finite slab’s low-frequency reactance approaches \(\omega\mu t/6\). Internal inductance is therefore inaccurate when the thickness is comparable to or less than the skin depth.

References

ParamRF convention; no source paper.

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.

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