Conductor losses#

Magnelio offers two routes for wall losses on good conductors: a perturbative post-processing route (default) and a broadband time-domain surface-impedance boundary (opt-in).

Perturbative wall losses (post-processing)#

With lossless PEC walls in the field solve, the dissipated power is evaluated perturbatively from the tangential magnetic field,

\[ P_{\text{loss}}(f) = \tfrac12\, R_s(f) \sum_{\text{wall}} w\,|H_{\tan}|^2, \qquad R_s = \sqrt{\pi f \mu / \sigma}, \]

(monitors/wall_loss.py, postprocessing/wall_loss.py, DD-082). This is the classical power-loss perturbation method of microwave engineering [16, 39]. Two accuracy refinements are in-house (DD-087): exact conformal wall areas on curved conductors (removing the \(4/\pi\) staircase over-count) and a conformal tangential-H sampling rule (uncut-face booking with a normal-direction walk).

The surface current#

On a conductor the tangential magnetic field at the wall is the surface current density, \(\mathbf J_s = \mathbf n \times \mathbf H\) [A/m], with \(\mathbf n\) the outward normal. The normal component of H vanishes there, so the cross product with the full vector is the same thing as with its tangential part, and \(|\mathbf J_s| = |H_\text{tan}|\).

recording.surface_current(mesh) — on a field monitor’s recording or spectrum, or on a single FieldState — returns one value per wall patch of the mesh: its position, its outward normal, its conducting area and the current on it. The field must carry all three magnetic components over the whole grid (fields=["H"], no corners), because a patch the monitor does not cover has no current to state.

The patches are the same enumeration the wall loss is booked from, which is what makes the two consistent. The magnitude is that booking split by patch — \(|\mathbf J_s|^2 A = \sum w |H|^2\) over the samples the patch booked — so power_loss(R_s) reproduces MonitorWallLoss exactly, not approximately. The direction is \(\mathbf n \times \mathbf H\) with the weight-averaged H of those samples and the patch’s own normal, which on a curved conductor is the direction of the sub-cell wall vector rather than a staircase axis.

Two things to know before reading numbers off it.

The absolute current is as accurate as the wall sampling, which is calibrated on the loss. The wall samples sit a small step off the surface and their weights (with the curvature pullback) compensate that for \(|H|^2\); the current is linear in H, and the residual displacement has opposite sign on a convex and a concave wall. Measured on an air coax: the inner conductor’s total current lands within 1.3 % of \(\sqrt{P/Z_0}\) and the shield within 5.0 %, and neither refines away. The distribution — where the current runs, and how it crowds at an edge — is far better resolved than that integral, and it is what a current picture is read for.

A port plane is not a wall. Where a conductor leaves the model through a boundary, that face holds the feed’s cross-section: the structure continues through it, and counting it as surface overstates both current and loss (9.5 % on the coax above). The mesh cannot tell that face from one where the domain simply ends inside metal — a port sits on a PEC face and is substituted at run time — so the call asks once, and exclude_faces=("zmin", "zmax") names the port planes while exclude_faces=() says there are none.

js.show(geometry) draws it: one arrow per patch over the model, coloured by \(|\mathbf J_s|\), with density= thinning a fine mesh out so the arrows stay readable.

Surface roughness#

Roughness enters the perturbative chain as one real, frequency-dependent multiplier \(K(f)\) on the surface resistance, \(R_{s,\text{rough}} = K(f)\,R_{s,\text{smooth}}\) (materials/roughness.py, DD-088). Implemented models:

  • Hammerstad — the classical RMS-height curve fit of Hammerstad and Jensen [40].

  • Huray “snowball” — the physics-based sphere-cluster model of Huray et al. [41]; the loss-factor form implemented is Bracken’s eq. (5) [42].

  • Cannonball-Huray parameterisation — sphere radius and coverage from a single \(R_z\) datasheet number via close packing, after Simonovich [43].

A real \(K(f)\) is non-causal as a time-domain impedance (noted by Bracken [42]); this is admissible here because the perturbative chain evaluates power per frequency bin and never forms a time-domain impedance.

Broadband time-domain SIBC#

SIBC is currently an opt-in: wall_model="sibc". The surface-impedance boundary condition realises the Leontovich condition \(E_{\tan} = Z_s(\omega)\,(\hat n \times \mathbf H)\) [44, 45] directly in the leapfrog update (solver/sibc.py, DD-091):

  • \(Z_s(\omega)\) — smooth-metal \(\sqrt{j\omega\mu/\sigma}\) or the causally completed rough impedance — is fitted as a Foster/Stieltjes ladder \(c_0 + \sum_p c_p s/(s+b_p)\) with non-negative coefficients by NNLS on fixed log-spaced poles (materials/surface_impedance.py). Foster’s canonical positive- real ladder form is classical network synthesis [46]; NNLS is Lawson and Hanson [47]. Because every branch is elementarily passive, the time-domain recursion is dissipative by construction — stability is unconditional at the unchanged lossless CFL, independent of fit accuracy — an in-house result (internal derivation dossier investigations/sibc/DERIVATION.md, kept outside the public repository).

  • The causal reactance of a rough surface is completed from the real roughness excess \((K-1)R_s\) by a subtracted Kramers–Kronig quadrature [48, 49].

  • The per-branch states are advanced with the trapezoidal rule and folded into the H update like a magnetic surface conductivity.

Approximating a surface impedance by a low-order rational function and convolving it recursively in FDTD is an established technique: Maloney and Smith [50], Beggs, Luebbers, Yee and Kunz [51], and the first-order-section approach of Oh and Schutt-Ainé [52] are the closest published antecedents; Magnelio’s specific NNLS-Foster construction with its unconditional dissipation identity is in-house.

The conformal booking of SIBC faces (which faces carry the damping term and with which geometric weight \(G_f = A_f/l^2_{\text{dual}}\)) reuses the DD-087 conformal wall-area machinery.