2-D Field Solver

The 2-D field solver in RF PCB Toolbox™ allows you to model and analyze the cross-sections of multi-conductor transmission lines in a multi-layered dielectric above a ground plane like a microstrip line. For more detailed information, see [1]

The four primary transmission line parameters are the resistance R, the inductance L, the conductance G and the capacitance C. You can obtain these parameters by applying a total charge on the conductor-to-dielectric interfaces and a polarization charge on the dielectric-to-dielectric interfaces and solving the electrostatic field created. The 2D field solver uses pulse approximation for the total charge σ_T and discretizes the contours of the conductor-to-dielectric and dielectric-to-dielectric interfaces using straight line segments. The density of the segments are controlled by the distribution of nodes along these contours. The figure shows the discretization for a single conductor transmission line in a single layered dielectric above a ground plane.

At any point in the YZ plane and above the ground plane, the potential ϕ is due to the combination of σ_T

and the image of σ_T about the ground plane. J represents the layers in the PCB, where J = J₁+ J₂. J₁ is the number of conductor-to-dielectric interfaces and J₂ is the number of dielectric-to-dielectric interfaces.

$$\varphi (p)=\frac{1}{2\pi e_0}{\displaystyle \sum _{j=1}^J{\displaystyle \int {\sigma }_T(p^{\text{'}})\left(\frac{\left|p-{\widehat{p}}^{\text{'}}\right|}{\left|p-p^{\text{'}}\right|}\right)}}dl\text{'}$$

where l_j is the contour of jth interface in the YZ plane, dl' is the differential element of length at on l_j and is the image of about the ground plane.

The electric field is given by . Substituting ϕ into this equation yields

$$E(p)=\frac{1}{2\pi e_0}{\displaystyle \sum _{j=1}^J{\displaystyle \int {\sigma }_T(p^{\text{'}})\left(\frac{p-p^{\text{'}}}{{\left|p-p^{\text{'}}\right|}^2}-\frac{p-{\widehat{p}}^{\text{'}}}{{\left|p-{\widehat{p}}^{\text{'}}\right|}^2}\right)}}d{l}^{\text{'}}$$

The normal component of a displacement field is continuous across dielectric-to-dielectric interface. You can derive the second integral equation by substituting the formula for E_p into the interface conditions for displacement fields. You can solve the set of integral equations for a total charge σ_T using the methods of moments [2]. The analysis includes skin-effect, conductor losses, and dielectric losses.

References

[1] Djordjevis, Antonije R., Miodrag B. Bazdar, Tapan K. Sarkar and Roger F. Harrington, Linpar for Windows: Matrix Parameters for Multiconductor Transmission Lines. Artech House, 1999

[2] Harringhton, R. F. Field Computation by Moment Methods. New York, Macmillan, 1968