Article

Popsicle-Stick Analysis

by Dr. Howard Johnson. First printed in EDN magazine, March 7, 2002

You can simulate the magnetic field surrounding a pc-board stripline using a rubber sheet and a Popsicle stick. First, stretch the rubber sheet over a rectangular frame. Then square off the end of the Popsicle stick, and push it into the rubber (Figure 1). Your simulation shows the magnetic-field potential surrounding the pc-board trace.

A close look at the precise forces bearing on the end of the stick teaches you a lot about stripline behavior. First, note that the force required to support the rubber sheet is not uniform at all points at the end of the stick. Because the stick is flat across its end face and because the rubber sheet is assumably lightweight, the forces acting on the stick are practically zero everywhere except around the edges of its end face. As a result, you don't really need any meat in the center of the stick; a hollow stick would do just as well.

Next, check the distribution of force around the edges of the end face. It's not uniform either. With a round, hollow stick, you might expect a uniform distribution of force where the edges of the stick meet the rubber, but when a rectangular stick presses against the rubber, the corners bear a disproportionate share of the weight. You know this if you have ever erected a big tarp for a picnic shelter. Once you put the corner posts in place and stretch the tarp so it is taut, your work is nearly complete. The corners do most of the work of holding up the tarp, and all you must do to complete the job is provide minor supports every so often to keep the sides and middle from sagging (or leave them to sag). In Figure 1, the wire frame lines near the tip of the protrusion show the same effect—the sheet is stretched hardest at the corners.

Now to the point of this article, which is the correspondence between the rubber-sheet world and the electrical world. The flat end of the stick represents the cross section of a pc trace, the rectangular frame represents reference planes above and below the trace, and the protrusion of the rubber sheet mimics the magnetic potential. The slope of the rubber sheet at any point indicates the magnetic-field intensity. Contour rings drawn on the rubber sheet at constant heights above the frame show familiar "magnetic lines of force" typically used to depict magnetic fields.

A direct correspondence exists between the forces acting on the end of the Popsicle stick, which induce the curvature in the rubber sheet, and the density of electrical currents flowing in a pc-trace of a similar cross-section profile. These currents likewise induce curvature in the magnetic potential. Where the rubber sheet is flat (no curvature), there is no force, and in the electrical world, there is no current. Where the rubber sheet is most tightly curved (at the corners), the force acting on the stick and also the density of current are greatest.

From this simple analogy you may conclude that high-frequency current flows only around the periphery of a trace, not in the middle. Such is the case at frequencies that are sufficiently high that the skin depth shrinks to much less than the conductor thickness, a condition required for the analogy to hold. Some other conclusions that flow easily from Popsicle-stick analysis include the following:

1. The density of current near the corners of a trace profile, such as the forces acting near the corners of a Popsicle stick, exceeds the density of current elsewhere. Because traces tend to be thin and wide, a concentration at the corners has the same general effect as a concentration on either side of the trace. This effect is called "edge-current concentration," the mathematics of which are predicted by rubber-sheet analysis.
2. If you make a differential pair by placing a signal conductor and a return conductor in close proximity (a configuration that is represented by a signal stick pressing up on the rubber sheet and a nearby return stick pressing down), the greatest concentrations of current will prevail on the inside-facing edges of the two conductors. The rubber sheet is stretched the hardest at these edges; the sheet traverses from peak to valley in the least distance. This effect is called the "proximity effect."
3. If a stripline trace is offset towards one reference plane, the slope of the rubber sheet on that side (and thus the force) must necessarily increase, indicating a larger concentration of current on the side of the trace facing the reference conductor than on the opposing side. This effect increases the apparent resistance of a signal trace above the value predicted, assuming a uniform distribution of current around the periphery of the conductor.

I know people who have spent years understanding the mathematics behind these three conclusions but still don't understand why they happen. You've just grasped the whole subject in a few minutes, and you got to enjoy a Popsicle!