## Potholes

Driving home from the Spokane, WA, airport one clear night, a steaming cup of coffee cradled in my hand, I took a short cut across the Colville Indian Reservation. Almost immediately—bam!—my truck hit a giant pothole. Hot java flew in every direction. I stopped the truck to see what I'd hit.

The pothole was about a foot across. It was filled with water, so it was difficult to see. It looked like it would be a hazard to other motorists, so I scrounged around for a big rock and dropped it into the hole.

The rock wasn't a perfect fit. It bulged in the center, but it seemed to be the right overall size for the hole. I backed up and tried driving over the hole again (no coffee this time). It was much better. Satisfied with my good deed, I continued the drive homeward.

This incident reminded me of a similar treatment used in transmission-line design. You can improve a big imperfection in a transmission line (such as a capacitive load) by adding a compensating imperfection to the line. One imperfection partially cancels the other. Going back to the driving analogy, as long as the residual imperfection is smaller than your wheels, you won't feel it.

Figure 1 illustrates the scenario. Adjustments to the transmission- line width on either side of the load partially compensate for the capacitive load. The load adds extra capacitance to the line, but the extra-skinny trace takes away a compensating amount of capacitance. The skinny-line adjustment in the figure can substantially reduce the reflected wave height of any incoming edge whose rise or fall time is slower than the effective delay of the adjusted segment.

Figure 1--A short section of a skinny pc-board trace partially compensates for a lumped capacitive load.

In this example, we selected the length of the adjusted segment (x) so that its total inductance and total loaded capacitance remain in balance, producing an effective impedance of Z0. You mathematically represent this impedance condition as:

[1]

Where,

• Z0 is the impedance of the surrounding transmission medium with delay of t sec per in.
• Z1 = Z0k is the impedance of the adjusted segment (unloaded)
• xt(Z0 k) and xt/(Z 0k) are the total inductance and capacitance of the adjusted segment (unloaded), respectively, and
• CL represents the capacitive load.

Solving for the adjusted trace length x tells you how long (in inches) to make the adjusted segment for a best compensating match. You may notice in this next formula that the skinnier you make the adjusted trace (the higher the Z1 and the greater the k), the shorter you can make the adjusted segment:

[2]

The skinny-trace compensation technique works only when the rise or fall time of the incoming edge is significantly slower (three to six times slower) than the effective delay of the adjusted segment. The effective delay, tLOADED, of the adjusted structure (including load) is:

[3]

If you have implemented the trace length required in [2], then [3] tells you whether the pothole-filling technique will be effective. Namely, when the time-constant Z0CL is much less than the signal rise time, it's easy to find a reasonable value of k for which tLOADED remains acceptably small. For example, given Z0=50Ω, CL=3 pF, and k=2, x works out to 0.55 in., and tLOADED equals 200 psec. The resulting structure remains practically invisible to any signal with a rise or fall time slower than 600 psec.

On the other hand, if Z0CL looms comparable to or larger than the signal rise or fall time, you won't be able to adequately compensate for such a large CL. To fix this problem, you need a smaller CL, a smaller Z0, or a slower rise and fall time.

In the automotive world, a similar effect applies: Potholes bigger than your wheels are not easily filled with a single rock.