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Gradient Model
==============
*Calculate the surface impedance of a rough metal surface using the Gradient Model (GM)*
[](https://doi.org/10.5281/zenodo.4694878)
Example: Surface Impedance of Gold
----------------------------------
Surface impedance of gold (conductivity: 4.1e7 S/m) between 280 GHz and 360 GHz:
<p align="center">
<img src="https://raw.githubusercontent.com/garrettj403/GradientModel/main/examples/results/wr3p0-surface-impdance-300k.png" width="800">
</p>
To use these results in HFSS, select `Assign boundary > Impedance...` and then copy/paste these values into the dialog box:
```
50 nm surface roughness:
Real: -1.6644e-25 * Freq^2 + 5.1199e-13 * Freq^1 + 6.1107e-02
Imaginary: -3.4813e-25 * Freq^2 + 2.2603e-12 * Freq^1 + 7.2291e-02
ur: -1.3448e-23 * Freq^2 + 5.0514e-11 * Freq^1 + 3.9464e+00
75 nm surface roughness:
Real: -1.5186e-25 * Freq^2 + 6.0379e-13 * Freq^1 + 5.8012e-02
Imaginary: -4.5447e-25 * Freq^2 + 3.0960e-12 * Freq^1 + 7.9819e-02
ur: -2.5105e-23 * Freq^2 + 9.6127e-11 * Freq^1 + 5.8256e+00
100 nm surface roughness:
Real: -1.4427e-25 * Freq^2 + 7.0489e-13 * Freq^1 + 5.5407e-02
Imaginary: -5.6445e-25 * Freq^2 + 3.8871e-12 * Freq^1 + 8.8834e-02
ur: -3.9840e-23 * Freq^2 + 1.5236e-10 * Freq^1 + 8.0508e+00
```
References
----------
Gradient model:
- G. Gold and K. Helmreich, “A Physical Surface Roughness Model and Its Applications,” IEEE Trans. Microw. Theory Tech., vol. 65, no. 10, pp. 3720–3732, Oct. 2017, doi: [10.1109/TMTT.2017.2695192](https://doi.org/10.1109/TMTT.2017.2695192).
- K. Lomakin, G. Gold, and K. Helmreich, “Analytical Waveguide Model Precisely Predicting Loss and Delay Including Surface Roughness,” IEEE Trans. Microw. Theory Tech., vol. 66, no. 6, pp. 2649–2662, Jun. 2018, doi: [10.1109/TMTT.2018.2827383](https://doi.org/10.1109/TMTT.2018.2827383).
Closed-form solution:
- D. N. Grujic, “Closed-Form Solution of Rough Conductor Surface Impedance,” IEEE Trans. Microw. Theory Tech., vol. 66, no. 11, pp. 4677–4683, 2018, doi: [10.1109/TMTT.2018.2864586](https://doi.org/10.1109/TMTT.2018.2864586).
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