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Electrostatic & Magnetostatic Field Analysis

0 bytes added, 22:45, 11 May 2017
<math> \mathbf{E(r)} = - \nabla \Phi(\mathbf{r}) = \frac{1}{4\pi\epsilon} \int\int\int_V \frac{\mathbf{r - r^{\prime}} }{ | \mathbf{r - r^{\prime}} |^3 } \rho(\mathbf{r^{\prime}}) dv^{\prime} </math>
 
== 2D Quasi-Static Solution of TEM Transmission Line Structures ==
 
At lower microwave frequencies (f < 10GHz), multi-conductor transmission line structures usually support either a dominant transverse electromagnetic (TEM) propagating mode or a dominant quasi-TEM propagating mode. These modes are almost non-dispersive, and their behavior can be regarded as frequency-independent. As a result, it is usually possible to perform a 2D electrostatic analysis of a transmission line structure and compute its characteristics impedance Z<sub>0</sub> and effective permittivity &epsilon;<sub>eff</sub>. The "quasi-static approach" to modeling of a TEM transmission line involves two steps:
 
<ol>
<li>First, you have remove all the dielectric materials from your structure and replace them with free space (or air). Obtain a 2D electrostatic solution of your "air-filled" transmission line structure and compute its capacitance per unit length C<sub>a</sub>.</li>
<li>Next, obtain a 2D electrostatic solution of your actual transmission line structure with all of its dielectric parts and compute its true capacitance per unit length C.</li>
</ol>
 
Then effective permittivity of the transmission line structure is then calculated from the equation:
<math> \epsilon_{eff} = \frac{C}{C_a} </math>
 
and its characteristic impedance is given by:
<math> Z_0 = \eta_0 \sqrt{ \frac{C_a}{C} } </math>
 
where &eta;<sub>0</sub> = 120&pi; &Omega; is the intrinsic impedance of the free space.
 
The guide wavelength of your transmission line at a given frequency f is then calculated from:
 
<math> \lambda_g = \frac{\lambda_0}{\sqrt{\epsilon_{eff}}} = \frac{c}{f\sqrt{\epsilon_{eff}}} </math>
 
and its propagation constant is given by:
 
<math> \beta = k_0\sqrt{\epsilon_{eff}} = \frac{2\pi f}{c}\sqrt{\epsilon_{eff}} </math>
 
where c is the speed of light in the free space.
== Magnetostatics Analysis==
<math> \mathbf{H(r)} = \frac{1}{\mu} \nabla \times \mathbf{A} (\mathbf{r}) = \frac{1}{4\pi} \int\int\int_V \mathbf{J(r^{\prime})} \times \frac{ \mathbf{r - r^{\prime}} }{ | \mathbf{r - r^{\prime}} |^3 } dv^{\prime} </math>
 
== 2D Quasi-Static Solution of TEM Transmission Line Structures ==
 
At lower microwave frequencies (f < 10GHz), multi-conductor transmission line structures usually support either a dominant transverse electromagnetic (TEM) propagating mode or a dominant quasi-TEM propagating mode. These modes are almost non-dispersive, and their behavior can be regarded as frequency-independent. As a result, it is usually possible to perform a 2D electrostatic analysis of a transmission line structure and compute its characteristics impedance Z<sub>0</sub> and effective permittivity &epsilon;<sub>eff</sub>. The "quasi-static approach" to modeling of a TEM transmission line involves two steps:
 
<ol>
<li>First, you have remove all the dielectric materials from your structure and replace them with free space (or air). Obtain a 2D electrostatic solution of your "air-filled" transmission line structure and compute its capacitance per unit length C<sub>a</sub>.</li>
<li>Next, obtain a 2D electrostatic solution of your actual transmission line structure with all of its dielectric parts and compute its true capacitance per unit length C.</li>
</ol>
 
Then effective permittivity of the transmission line structure is then calculated from the equation:
<math> \epsilon_{eff} = \frac{C}{C_a} </math>
 
and its characteristic impedance is given by:
<math> Z_0 = \eta_0 \sqrt{ \frac{C_a}{C} } </math>
 
where &eta;<sub>0</sub> = 120&pi; &Omega; is the intrinsic impedance of the free space.
 
The guide wavelength of your transmission line at a given frequency f is then calculated from:
 
<math> \lambda_g = \frac{\lambda_0}{\sqrt{\epsilon_{eff}}} = \frac{c}{f\sqrt{\epsilon_{eff}}} </math>
 
and its propagation constant is given by:
 
<math> \beta = k_0\sqrt{\epsilon_{eff}} = \frac{2\pi f}{c}\sqrt{\epsilon_{eff}} </math>
 
where c is the speed of light in the free space.
<br />
Kazem Sabet
28,333
edits