Wcalc Technical Notes



The models used by Wcalc are either closed form or psuedo-closed form models. In cases where an exact solution of the electromagnetic fields exists, for example in the coaxial transmission line or infinitely thin conductor stripline case both with ideal conductors, then the exact solution is used. In all other cases, various approximate solutions have been applied.

Many of the approximate solutions (found in the literature, not developed by the Wcalc author) have been developed by optimizing an approximating function to match a reference value obtained through exhaustive numerical solutions to Maxwell's equations. The approximating functions are typically chosen so they approach known analytic solutions in extreme cases. These models are typically much much to complex for manually applying with a hand calculator but are trivially simple for a modern or even not so modern computer to solve. This makes Wcalc be very fast. In many cases the error in the approximation is less than the error due to physical tolerances. However, one should heed any warnings issued to Wcalc when physical dimensions are outside the usual range for accuracy.

Incremental Circuit Model

The transmission line structures analyzed by Wcalc currently are all either transverse electromagnetic (TEM) or quasi-TEM lines and are modeled by the incremental circuit model shown below.

[Incremental Circuit Model]

Before applying this model, it is important to understand some of the limitations. In general, the equations for characteristic impedance and velocity are derived assuming perfect conductors (resistivity of the material is zero). Then conductor loss is added with the assumption that the fields are not significantly changed by the finite conductance of the metal.

This approach works well for the capacitance per length of line calculation. This is because when a static potential is applied to the line, there is no current flow in the x or y directions. We are assuming that the line is uniform in the z-direction so that the x-y geometry fully specifies the line cross section. The implication of this is that the microstrip model or coupled microstrip model can accurately predict capacitance of circuit board traces at low frequencies.

Inductance calculations, in contrast to capacitance calculations, are a bit more complex. Consider the case of the coaxial transmission line shown below.

[Coaxial Transmission Line]
The standard analysis found in text books assumes that the current flows in an infinitely thin sheet at the surface of the conductors. Ampere's law tells us that the magnetic fields are confined to the dielectric layer in between and the calculation of inductance yields the usual inductance per length for coax. This inductance per length may be calculated directly using the quasi-static fields or by calculating capacitance and finding the inductance from the capacitance and velocity of a TEM line.

At low frequencies, the inductance found based on TEM transmission line models becomes incorrect. This is because at low frequencies, the current is not confined to a thin sheet at the conductor surfaces but rather is distributed across the conductor. This modifies the magnetic fields and changes the inductance. Currently Wcalc does not do anything to account for this effect. If you are designing circuits where you plan on doing things like using a short (compared to 1/4 wavelength) line as an inductor, you must be concerned with this effect. The good news is that the quality factor of the inductance formed by a line at low frequencies is usually low enough to discourage its use anyway.

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