# 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 {cite}`pozar2012,jackson1999`.  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).

## 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 {cite}`hammerstadjensen1980`.
- **Huray "snowball"** — the physics-based sphere-cluster model of
  Huray et al. {cite}`huray2007`; the
  loss-factor form implemented is Bracken's eq. (5)
  {cite}`bracken2012`.
- **Cannonball-Huray parameterisation** — sphere radius and coverage
  from a single $R_z$ datasheet number via close packing,
  after Simonovich {cite}`simonovich2015`.

A real $K(f)$ is non-causal as a time-domain impedance (noted by
Bracken {cite}`bracken2012`); 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** $E_{\tan} = Z_s(\omega)\,(\hat n \times \mathbf H)$
{cite}`leontovich1948,senior1960` 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
  {cite}`foster1924`; NNLS is Lawson and
  Hanson {cite}`lawsonhanson1974`.  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 {cite}`kronig1926,kramers1927`.
- 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 {cite}`maloneysmith1992`, Beggs,
Luebbers, Yee and Kunz {cite}`beggs1992`, and the
first-order-section approach of Oh and Schutt-Ainé
{cite}`ohschuttaine1995` 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.