ConductorProperties

class pmrf.materials.ConductorProperties(zs: Array, sigma: Array, mu_r: Array)

Bases: Module

Evaluated conductor properties.

zs and gamma() are independent: zs is the surface prefactor, which a surface treatment such as roughness may scale, while gamma() describes diffusion into the unmodified bulk.

Parameters:

• zs (jnp.ndarray) – Surface impedance in ohm per square.

• sigma (jnp.ndarray) – Bulk conductivity in S/m.

• mu_r (jnp.ndarray) – Complex relative permeability.

gamma(omega) → Array

Return the propagation constant inside the metal, in 1/m.

Mathematical Formulation

\[\gamma = \sqrt{j\omega\mu\sigma} = \frac{1+j}{\delta}, \qquad \delta = \sqrt{\frac{2}{\omega\mu\sigma}},\]

with \(\mu=\mu_0\mu_r\). This is the inverse complex skin depth: it governs how fast the field diffuses into the bulk, and it is what makes \(\gamma a\) and \(\gamma t\) the dimensionless “how many skin depths across is this cross-section” arguments of the Bessel and \(\coth\) expressions of surface_impedance. Despite the name it is not the propagation constant of the line.

It is deliberately computed from \(\sigma\) and \(\mu_r\) rather than recovered from zs as \(\sigma\zeta_c\). That identity holds only for a smooth bulk metal: any surface treatment which scales zs – roughness today, cladding or plating tomorrow – would otherwise inflate the diffusion constant, which surface texture does not change.

The value is zero at dc, where the \(\sqrt{\cdot}\) branch point is guarded so the gradient stays finite, and infinite for a perfect conductor. Callers are responsible for their own dc and perfect-conductor branches.

Parameters:

omega (ArrayLike) – Angular frequency in rad/s.

Returns:

Propagation constant inside the metal, in 1/m.

Return type:

jnp.ndarray

References

Pozar, D. M. (2011). Microwave Engineering (4th ed.), Section 1.7. Wiley.

mu_r: Array

sigma: Array

zs: Array