_ground_return_effective_width

pmrf.models.components.lines.microstrip._ground_return_effective_width(w, h)

Effective width of the ground-plane return current, in metres.

Mathematical Formulation

Holloway and Kuester’s quasi-static ground-plane current density for a strip of width \(W\) at height \(H\) is the strip’s own current convolved with the image-filament kernel \(\,[\pi H(1+(x/H)^2)]^{-1}\). Referring the ground loss to the total line current defines an effective width \(W_g = I^2\big/\!\int J_g^2\,\mathrm{d}x\), which Parseval’s theorem reduces in closed form. With \(u = W/H\),

\[\frac{W_g}{W} = \frac{\pi u} {2\left[u\arctan(u/2) - \ln\!\left(1 + u^2/4\right)\right]}.\]

The limits are \(W_g \to 2\pi H\) for a narrow strip – the filament result, independent of \(W\) – and \(W_g \to W\) for a wide one, where the return current is confined under the trace. Representative values are \(W_g/W\) = 3.667, 3.896 and 14.394 at \(u\) = 1.94, 1.80 and 0.44.

Parameters:

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

• h (ArrayLike) – Substrate height in meters.

Returns:

Effective ground-plane width in meters.

Return type:

jnp.ndarray

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.