## Rainy-day Fun

I live in Seattle, where the rain drizzles steadily all winter. Heck, it rains all year. Because I work indoors, the weather usually causes me little concern. However, yesterday I encountered a big puddle in the parking lot outside my office. The puddle was large, brown, and uniformly 1/2 in. deep, with a sheet resistivity, ρ, of about 1200Ω per square. (Yes, I measured it.)

As I looked at the puddle, it occurred to me that you can use puddles to solve certain difficult problems in the design of high-speed transmission lines.

Specifically, you can use puddles to solve the determination of characteristic impedance and near-end crosstalk for arbitrary single-ended- and differential-transmission-line cross sections. To solve these problems, you use a basic equivalence between the pattern of electric-field lines inside a transmission line and the pattern of electric-field lines in a puddle (See Figure 1 below).

Figure 1—Use a flat and uniform conductive puddle to determine the characteristic impedance of a transmission line.

For you propellerheads, Table 1 shows the equivalence relations for quasistatic electric-field problems. Current in the conductive sheet corresponds to quasistatic charge in the 2-D transmission-line cross section.

Table 1—Equivalence relations

On the left, the table says that for a 2-D transmission line problem, the divergence of the electric field equals the charge density, q, divided by the electric permittivity of the dielectric medium surrounding the transmission line.

On the right, for a conductive-sheet problem, the divergence of the electric field equals the injected current density, J, divided by the electric conductivity of the resistive medium surrounding the transmission line.

In both cases we have a quasi-static problem with no changing magnetic fields, so the curl of the electric field is zero.

Simply put, the equations say that you can directly measure the resistance between two electrodes in a puddle, and that measurement will correspond proportionally to the characteristic impedance of an equivalent transmission-line cross section. To measure the resistance, I inject a signal of known voltage amplitude and measure the current across the sense resistor in figure 1. Remember to place some long, straight electrodes in the puddle representing the reference planes in your board stackup.

You can measure crosstalk, too. Just impress a voltage across any two electrodes in the puddle and directly measure the near-end crosstalk induced between any other pair of electrodes. For any high-frequency transmission-line problem in a uniform dielectric medium (such as a stripline), the correspondence will be exact.

2-D transmission-line cross section Conductive sheet
DIV E = q / ε DIV E = J / σ
CURL E = 0 CURL E = 0

And the experiments are fun. Just move the electrodes around in the water, and you can see the impedance change. What a terrific way to learn about coupled differential lines. You can directly observe the differential impedance shrinking as you bring two differential lines closer together.

If you decide to try this method, here are a few helpful hints:

• Use a plastic tank with a very flat bottom and carefully level the tank to within 1% of its depth.
• Form the electrodes from galvanized flashing. This material has a low surface resistivity when immersed in water, and it doesn't oxidize or corrode. Clean it with acetone. Use any model scale factor you want—scaling a 2-D cross section doesn't affect its impedance or crosstalk. Use flat strips to represent the power and ground planes.
• Use an ac excitation for your resistance measurements. This practice prevents problems with pesky battery-like voltages that appear when you immerse the electrodes in a conducting solution.
• Transformer-couple the excitation from a 100-Hz sinusoidal source and use a regular handheld digital meter in rms mode to measure the voltages. The 100-Hz excitation minimizes errors due to 60-Hz noise pickup.
• Build a calibration cell. I use two circular coaxial electrodes for calibration. The outer diameter is 5.63 times the inner diameter. If you embedded this coaxial cross section in a dielectric material with a relative permittivity of 4.3, it makes a characteristic impedance of 50Ω. Embedded in my tank, whatever resistance it makes I use as a calbration factor representing "50-ohms" in my tank.

My water comes from a well and contains plenty of electrolytes. If your water is too pure, add a little salt. A sheet resistivity of a few kilohms per square is convenient.

When properly calibrated, a conductive tank can be a valuable instrument. Considering the price of new electromagnetic-field-simulation software, I figure my puddle is worth about \$30,000. Even better, I didn't have to read a 600-page manual to learn how to use it.