## Whang That Ruler

A quarter taped to the end of a ruler lowers its resonant pitch.

Clamp a wooden ruler to your desktop so that it overhangs the edge of the desk by about 8 inches. Now, flick the end of the ruler (Figure 1). It resonates, doesn't it? You can easily change the resonant frequency. Tape a few quarters to the end of the ruler and observe that the resonant frequency decreases. Shorten the length of overhang and hear it increase. If you push the resonant frequency high enough, it becomes difficult to stimulate the resonance with the soft end of your fingertip. Overcome that difficulty by depressing the end of the ruler with your finger in a way that lets the ruler slip off the hard edge of your fingernail.

Figure 1—When you release the end of the ruler, it oscillates for many cycles before settling down.

At the risk of annoying your coworkers, whang the ruler over and over while you adjust the amount of overhang until you get a nice, musical resonance at about 100 Hz. Now, disconnect the clamp and hold the same length of ruler in your hand. Flick the end. Try to create the same resonant effect. I bet you can't do it.

When you clamp the ruler to the desktop, the clamp creates a solid, rigid mechanical impedance at one end of the ruler. The other end of the ruler remains free to move—the only limit being air resistance. Your stimulation creates a mechanical wave that bounces back and forth between these two endpoints, neither of which absorbs much of the mechanical energy. It therefore takes many back-and-forth cycles for the ruler to settle down. You have created a highly resonant system.

When you hold the ruler in your hand, the mechanical impedance of your hand lies close to the natural characteristic impedance of the ruler. Even if it is not a perfect match, your hand absorbs a significant portion of the energy in each cycle. The result is that the ruler cannot resonate.

The ruler supports transverse mechanical waves in just two directions: from one end of the ruler to the other and back again. Mechanical engineers call that phenomenon 1-D wave propagation. For any system like this one, an absorbing device at either end can totally damp the oscillations. Mechanical engineers use hydraulic shock absorbers, friction, air resistance, and rubber to absorb energy and create damping. Electrical circuits use resistive terminations.

A more complex system, such as a child's Indian drum, supports wave motion in two dimensions. Waves on the surface of the drumhead spread and reflect in many highly varied and complex patterns. An absorbing device at just one location fails to damp the drumhead. To completely silence the drum, you must either apply absorbing material around a large fraction of the circumference or ask the child to please stop whanging. Good luck with that tactic.

PCB (printed-circuit-board) traces in a simple, linear topology behave as a 1-D-wave-propagation medium. If the trace is long enough and if it lacks any good energy-absorbing devices, it will resonate, distorting your signals.

When I say a trace has a simple, linear topology, I mean that it is a point-to-point connection or, at most, a linear-bus structure with multiple transceivers arrayed along a single trace. More complex structures, such as H distributions, star clusters, or random hairball nets, support multiple modes of oscillation and may, like a drumhead, require terminations in multiple locations.

When I say "long enough," I mean that the end-to-end trace delay is a significant fraction of the signal rise or fall time. A delay as long as one-sixth the rise or fall time is significant, especially if the trace has a particularly low-impedance driver or a large capacitive load. I simulate all such traces. Keep in mind that even short traces resonate; the resonant frequency is just so high that you may not observe its effect using logic with a particular rise and fall time. Applying logic with faster edges to the same trace might make it ring, just as flicking the ruler with your fingernail stimulates resonance at a small length.

Applying a capacitive load to a PCB trace has much the same effect as a load of quarters on the end of the ruler: It lowers the resonant frequency of the structure, making it more likely that you will notice its effects.