# Mutual Capacitance and Mutual Inductance Calculation

Parasitics are some of the most difficult quantities to calculate analytically from Maxwell’s equations or phenomenological models. For coupled transmission lines, whether part of an impedance-controlled bus or adjacent to each other in a PCB, the parasitics between the lines determine their crosstalk and coupled impedances during operation. Normally, these would be extracted from a complex PCB layout, or even between transmission lines, using a field solver or a set of complex equations.

What if you didn’t need a field solver to extract parasitics, but instead you could determine these from coupled-mode impedance models. To illustrate how to extract capacitances between coupled transmission lines, this article will present an analytical determination of mutual inductance and mutual capacitance. The method simply involves usage of standard equations to determine coupled-mode transmission line impedances, which are then used to calculate mutual inductance/capacitance directly.

## Parasitics Related to Coupled Line Impedances

The relation between parasitics and coupled line impedance comes directly from the odd-mode and even-mode impedance values for transmission lines. If we suppose for a moment that we want to know the mutual capacitance and mutual inductance between two lines with the same geometry, we can determine this as long as we know the even and odd-mode impedances for these two lines.

These traces will have different impedance if they are driven in odd-mode or even-mode. We can use these impedances to calculate the mutual capacitance and mutual inductance between these traces.

Note that the parasitics you would calculate between these lines apply regardless of how the lines are driven. This is because the mutual inductance and mutual capacitance are only functions of the line geometry, spacing, and materials; they do not depend on how the lines are driven. However, these parasitics do determine how signals propagate on transmission lines, and thus they determine the impedance in the odd-mode and even-mode.

### Mutual Capacitance

To calculate the mutual capacitance between two transmission lines, we have to introduce two new concepts: even-mode capacitance and odd-mode capacitance. These terms are introduced because they illustrate how the electric emitted from a trace and its reference plane superimposes on the electric field from the other trace. In this way, the two traces form different fringing fields depending on how the traces are driven.

For this reason, the mutual capacitance will depend on odd-mode or even-mode impedances, which are in turn related to even-mode and odd-mode capacitances:

Here, the even-mode and odd-mode capacitances are defined in terms of even-mode and odd-mode impedances, respectively:

With these formulas, we have a simple way to compute mutual capacitance. The dielectric constant will be a single value for striplines, while it will depend on the trace geometry with respect to the ground plane for microstrips.

One important point to note here is that losses are not being considered in the above method. It is implicitly assumed that the dielectric constant is real and the loss tangent is being ignored. Once the even-mode and odd-mode capacitance values are determined, they can be used in another formula to determine the mutual inductance.

### Mutual Inductance

To determine mutual inductance, we again start with the odd-mode and even-mode impedances. If we assume these values are known from some model or field solver solution, then the even and odd mode capacitances can be used to determine the mutual inductance values. The mutual inductance equation is:

The terms in the denominator state that the odd-mode and even-mode capacitances must be evaluated at epsilon = 1. The equation is universal to microstrips, striplines, or any other pair of coupled lines with matching geometry.

## Summary

As we can see from the above method, the results depend on an accurate calculation of odd-mode and even-mode impedances for the PCB traces being investigated. In summary, the determination of parasitics through this method follows a simple process:

1. Complete your routing floorplan, stackup design, and material selection
2. Determine even-mode and odd-mode impedances for the lines
3. Determine fringing fields as part of the even-mode and odd-mode capacitances
4. Calculate the mutual capacitance and mutual inductance using results from #3

How do we get the even-mode and odd-mode impedances to start the process? This is sometimes difficult to do analytically, especially considering the fact that analytical models are not available for every possible transmission line configuration. The best set of equations are Wadell’s equations, which cover a very diverse range of differential and common-mode interconnects. With these sets of equations, it is possible to analytically determine even-mode and odd-mode impedances, and then use these values to determine parasitics.

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