WheelerMicrostripFormulation

class pmrf.models.components.lines.microstrip.WheelerMicrostripFormulation

Bases: AbstractMicrostripFormulation

Microstrip line formulation using the standard Wheeler approximations.

This implements Wheeler’s 1977 quasi-static approximation. It is distinct from the 1942 conductor-loss rule in WheelerCurrentDistribution and does not calculate conductor loss.

Mathematical Formulation

With ratio \(u = W/H\), the effective relative permittivity (\(\varepsilon_e\)) and the impedance of the same geometry in air (\(Z_a\)) are:

\[\varepsilon_e = \frac{\varepsilon_r + 1}{2} + \frac{\varepsilon_r - 1}{2} \frac{1}{\sqrt{1 + 12/u}}\]
\[Z_a = \frac{120\pi}{u + 1.393 + 0.667 \ln(u + 1.444)} \quad (u > 1) \qquad Z_a = 60 \ln\left(\frac{8}{u} + \frac{u}{4}\right) \quad (u \leq 1)\]
\[Z_c = \frac{Z_a}{\sqrt{\varepsilon_e}}\]

Complex \(\varepsilon_r\) carries dielectric loss through the same filling factor. The effective width is \(W\).

Validity

Derived for a zero-thickness strip on an isotropic, non-magnetic substrate. Finite thickness is ignored by this formulation but remains available to the current distribution. Modal dispersion requires a separate AbstractMicrostripDispersion.

References

Wheeler, H. A. (1977). Transmission-Line Properties of a Strip on a Dielectric Sheet on a Plane. IEEE Transactions on Microwave Theory and Techniques.

quasi_static(*, w, h, t, ep_r) → PlanarQuasiStaticResult

Calculate the quasi-static solution.

Parameters:

  • w (ArrayLike) – Width of the microstrip trace in meters.

  • h (ArrayLike) – Height of the dielectric substrate in meters.

  • t (ArrayLike | None) – Trace thickness in meters, or None when unspecified.

  • ep_r (jnp.ndarray) – Complex relative permittivity of the substrate, shape (npoints,).

Returns:

The effective permittivity, impedance and effective width.

Return type:

PlanarQuasiStaticResult

thickness_aware: ClassVar[bool] = False

The 1977 result is derived for a zero-thickness strip.