Robert Powell from Capstone Visual needs to carry some high-speed network traces across his two-layer pc board. The board consists of 62-mil-thick FR-4 material. No solid plane layer exists anywhere on this board.
To manage skew between the two traces, Robert wants to route them exactly on top of one another—one on the top layer and one on the bottom. Robert wrote to me, asking for a way to calculate the trace width necessary to make a good 100Ω differential configuration using this geometry.
Before I address this issue, I should congratulate Robert for recognizing that any trace configuration with a uniform cross section makes a perfectly good transmission line. You can use one trace with a solid reference plane, two traces side by side, two traces above and below one another, or four traces in a quad configuration; the possibilities are endless. Just keep the geometry consistent, and then all that matters are the impedance, delay, attenuation, and crosstalk.
Robert's traces are short compared with the extent of his network connection, so the trace delay and trace loss are immaterial.
Regarding the differential impedance, you can use an "image-plane" method to calculate the impedance of that configuration.
Figure 1 shows a cross-sectional view of two round conductors, assuming an air dielectric. (The conductors need not be round; any shape gives the same type of picture.) The electric-field patterns in the region surrounding the traces emanate from each trace perpendicular to its surface. A lot of lines flow directly from the positive trace to the negative trace. The lines bulge on either side. Voltages at the top of the drawing are positive, and voltages at the bottom are negative. Everywhere along a line drawn through the middle the voltage equals exactly 0 V.
The dotted, green horizontal line bisecting the diagram represents an imaginary plane, or image plane, separating the two conductors. "Symmetry" in this case means that, if the differential pair is properly balanced, the voltage everywhere on the image plane remains zero at all times. Therefore, concerning the differential mode of propagation, no circuit can distinguish between, first, the original unreferenced configuration and, second, the same configuration with a real, physical, 0V reference plane added in the middle.
To calculate the impedance of an above-and-below differential-pc-board-trace setup with no reference plane, just lie to your 2-D field solver. Tell it that there is a solid reference plane dead center in the middle of the stackup. Separately compute the impedance from the top trace to the plane (a normal microstrip arrangement) and double that number to get the complete differential impedance.
The image-plane method makes quick work of the impedance calculation, but what about crosstalk? Lacking a solid reference plane, Robert's traces will be particularly sensitive to crosstalk from nearby sources (that is why I do not recommend a non-referenced approach for general use), but at least it won't suffer reflections!