Mathematical Analysis
Angle-preserving geometric transformations offer a powerful analytical path to solving the complex electromagnetic field equations of coplanar and microstrip transmission lines. In planar circuit design, conformal mapping converts the complicated boundaries of conductors and dielectric interfaces into a simpler, rectangular geometry where capacitance and impedance can be calculated directly. This transformation reduces the need for computationally heavy numerical solvers during the initial stages of high-speed board design.
Analytical Method
The calculation divides the electromagnetic region into multiple regions of uniform dielectric properties. This allows engineers to determine the effective dielectric constant and characteristic impedance of a trace by summing the individual capacitances. While the method assumes infinite ground planes and zero-thickness traces, it provides highly accurate starting values for strip configurations.
Design Application
Fast calculation of transmission line properties enables rapid optimization of trace widths and spacing during the layout phase. The accuracy of the resulting model is particularly high for structures like coplanar waveguides with ground planes on the same layer. As a result, the technique remains a valuable tool in signal integrity suites to generate fast and reliable trace geometries.
By using closed-form analytical equations generated through these maps, design software can sweep through thousands of trace geometries in seconds, a speed that would be impossible with full-wave field solvers.
