Copyright (C) 2019 S.V. Matsievskiy
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
Program source files may be found at https://gitlab.com/matsievskiysv/beadpull
General equation:
\[ \frac{\Delta f}{f} = - \frac{V_1}{4 W}\left[ \frac{ \varepsilon_0 E_{0i}^2 }{ p_i + (\varepsilon-1)^{-1} } + \frac{ \mu_0 H_{0i}^2 }{ m_i + (\mu-1)^{-1} } \right]. \]
Simplified equation:
\[ \frac{\Delta f}{f} = \frac{ - \displaystyle\sum_{i=1}^{N}{k^E_i E_{0i}^2} + \displaystyle\sum_{i=1}^{N}{k^H_i H_{0i}^2} }{W}, \]
where \[ k^E_i = \frac{V_1}{4}\frac{\varepsilon_0} {(\varepsilon - 1 + p_i)} \] is an electric field form-factor, \[ k^H_i = \frac{V_1}{4}\frac{\mu_0}{(\mu - 1 + m_i)} \] is a magnetic field form-factor.
Normalized electric field:
\[ \xi = \frac{E}{\sqrt{P Q}} = \sqrt{\frac{\Delta f}{2 \pi k^E f_0^2}} \left[\frac{\text{Ohm}^{1/2}}{\text{m}}\right]. \]
Normalized magnetic field:
\[ \varsigma = \frac{H}{\sqrt{P Q}} = \sqrt{\frac{\Delta f}{2 \pi k^H f_0^2}} \left[\frac{\text{???}}{\text{???}}\right]. \]
\[ \Delta \dot S_{11} = \dot S_{11} - \dot S_{11}^0 = \dot C \dot E^2, \]
where \(\dot S_{11}^0\) is a complex reflection in absence of bead.
Formula for normalized electric field:
\[ \hat E_n = \sqrt{\frac{|\dot S_{11}^n - \dot S_{11}^0|} {2 \pi k^S f_0^2}}. \]
Formula for electric field phase:
\[ \varphi_{n} = \frac{\varphi_{n} - \varphi_{0}}{2}. \]
Using the following equation: \[\Delta f = \frac{f_0}{2 Q_{load}} \tan{\Delta \varphi} \approx \frac{f_0 \Delta \varphi}{2 Q_{load}}, |\Delta \varphi| \le 5^\circ\]
Normalized electric field:
\[ \xi = \frac{E}{\sqrt{P Q}} = \sqrt{\frac{\Delta \varphi}{4 \pi k^E f_0 Q_{load}}} \left[\frac{\text{Ohm}^{1/2}}{\text{m}}\right]. \]
Normalized magnetic field:
\[ \varsigma = \frac{H}{\sqrt{P Q}} = \sqrt{\frac{\Delta \varphi}{4 \pi k^H f_0 Q_{load}}} \left[\frac{\text{???}}{\text{???}}\right]. \]
| Bead material | E-field | H-filed |
|---|---|---|
| Metallic bead | ↓ | ↑ |
| Dielectric bead | ↓ | ↓ |
Dielectric bead with \(\varepsilon > 1, \mu = 1\) only measures electric field; with \(\varepsilon = 1, \mu > 1\) only measures magnetic field.
For \(\varepsilon \gg 1\) or \(\mu \gg 1\) some equation approximations do not hold!