IncrementalInductanceCurrentDistribution

class pmrf.models.components.lines.microstrip.IncrementalInductanceCurrentDistribution(formulation: AbstractMicrostripFormulation = <factory>, slab_impedance: AbstractSurfaceImpedance = <factory>)

Bases: AbstractCurrentDistribution[MicrostripCrossSection]

Wheeler’s incremental-inductance rule, evaluated by autodiff.

This evaluates the rule itself rather than Wheeler’s 1942 closed-form fit to it, which is what WheelerCurrentDistribution implements. It is also distinct from WheelerMicrostripFormulation, the 1977 quasi-static impedance approximation, which is a formulation and not a current distribution.

Mathematical Formulation

The geometry weight is the rate of change of the air-filled impedance as every conductor surface recedes into the metal by a distance \(n\):

\[k_c = \frac{1}{Z_0}\frac{\partial}{\partial n} Z_a(W-2n,\ H+2n,\ T-2n)\bigg|_{n=0}\]

The strip narrows by \(2n\) and thins by \(2n\), and the strip-to-ground gap grows by \(2n\) – \(n\) from the ground plane receding and \(n\) from the strip’s underside. \(Z_a\) is formulation evaluated at \(\varepsilon_r = 1\), so the derivative is taken on the same closed form that supplies the line’s impedance. It is taken with jax.grad(), which is exact for a closed-form \(Z_a\) and avoids the step-size problem a finite difference would have here (\(\partial Z_a/\partial n\) is of order \(10^5\ \Omega/\mathrm{m}\) against a \(Z_a\) of order \(10^2\ \Omega\)).

The weight is real, frequency-independent, and differentiable in \(W\), \(H\) and \(T\), exactly as Wheeler’s fit is.

Validity

The derivative is taken on formulation’s own \(Z_a\), so a formulation that ignores \(T\) cannot respond to the strip thinning and the top face and sidewalls drop out of the weight entirely – an under-prediction of about 30% at \(W/H = 2.5\). Both that pairing and an unspecified thickness raise rather than degrade silently; supply a thickness-aware formulation such as HammerstadJensenMicrostripFormulation and a cross-section with t set.

Departure from Wheeler’s 1942 fit

Against this derivative, the 1942 fit is high: 385.7 against 318.3 per metre at \(W/H = 2.5\) (\(W\) = 4 mm, \(H\) = 1.6 mm, \(T\) = 35 um, \(\varepsilon_r\) = 4.335), a ratio of 0.825, closing to 0.958 by \(W/H = 50\). An independent 2D quasi-static field solve, a power-loss surface-impedance integral, and a volumetric PEEC solve all land in 306-320 for that case.

This sign is disputed. TraceGroundCurrentDistribution, a trace/ground current split developed from the same Holloway and Kuester source, reports the 1942 fit as 12-18% low at strong skin effect. The two cannot both be right; the contradiction is unresolved and is recorded rather than papered over. Neither result changes the microstrip default, which remains WheelerCurrentDistribution.

Note also that the wide-line limit does not recover the \(2/W\) prefactor of the 1942 fit: at \(W/H = 50\) the weight is 22.9 against \(2/W\) = 25.0. The \(2/W\) argument assumes a zero-thickness strip and neglects fringing, and with fringing \(C_{air}\) is larger, so \(k_c = -\varepsilon_0 C^{-2}\,\partial C/\partial n\) falls below it.

Validating against an external 3D solver

slab_impedance defaults to RootSumSquareSlabSurfaceImpedance, which corrects for finite strip thickness, while a 3D solver run with its skin-effect approximation neglects thickness in the current distribution and applies a plain half-space \(Z_s\) instead. Most solvers allow that approximation to be turned off, so check what the tool was run with. Where it is in force the two differ by 2-5% over \(t/\delta\) = 3.7-6.0, so a cross-tool comparison that leaves the default in place attributes that 2-5% to one tool or the other when it is purely a difference in surface-impedance model. Pass slab_impedance=HalfSpaceSurfaceImpedance() explicitly for such a comparison. See WheelerCurrentDistribution for the full note; the default is unchanged here.

References

Wheeler, H. A. (1942). Formulas for the Skin Effect. Proceedings of the IRE, 30(9), 412-424.

Hammerstad, E., & Jensen, O. (1980). Accurate Models for Microstrip Computer-Aided Design. IEEE MTT-S International Microwave Symposium Digest, 407-409.

cross_section_type

alias of MicrostripCrossSection

formulation: AbstractMicrostripFormulation

The closed form supplying \(Z_a\). It must be thickness-aware; see Validity. This is deliberately a field of the distribution rather than an argument of _distribute, so that which \(Z_a\) the derivative is taken on is explicit.

slab_impedance: AbstractSurfaceImpedance

Finite-thickness surface impedance. Validating against an external half-space solver needs HalfSpaceSurfaceImpedance() passed here explicitly; see Validating against an external 3D solver. See AbstractSurfaceImpedance for normalisation details.