## 2-D Field Solver

The 2-D field solver in RF PCB Toolbox™ allows you to model and analyze the cross-sections of multiconductor transmission lines in a multilayered 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 *p* 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.

$$\varphi (p)=\frac{1}{2\pi {e}_{0}}{\displaystyle \sum _{j=1}^{J}{\displaystyle \int {\sigma}_{T}({p}^{\text{'}})\left(\frac{\left|p-{\widehat{p}}^{\text{'}}\right|}{\left|p-{p}^{\text{'}}\right|}\right)}}dl\text{'}$$

where *l _{j}* is the contour of

*j*th interface in the YZ plane,

*dl'*is the differential element of length at

*p'*on

*l*and $$\widehat{p}\text{'}$$ is the image of

_{j}*p'*about the ground plane.

The electric field is then given by φ

$$E(p)=\frac{1}{2\pi {e}_{0}}{\displaystyle \sum _{j=1}^{J}{\displaystyle \int {\sigma}_{T}({p}^{\text{'}})\left(\frac{p-{p}^{\text{'}}}{{\left|p-{p}^{\text{'}}\right|}^{2}}-\frac{p-{\widehat{p}}^{\text{'}}}{{\left|p-{\widehat{p}}^{\text{'}}\right|}^{2}}\right)}}d{l}^{\text{'}}$$

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*using the methods of moments [2]. The analysis includes skin-effect, conductor losses, and dielectric losses.

_{T}## 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