Making Resistors with Math

Brief, high-power pulses of electrical energy throbbing through intricate circuitry are the heartbeats of modern radar. But they are the bane of many an electronics engineer. Sometimes the high-frequency currents that are crammed into a pulse swirl through a simple resistance as if it were also a small coil (inductance); sometimes the pulses treat the resistance as if it were a capacitor. Either way, coil or capacitor, those unwanted effects introduce annoying problems.

In an effort to reduce such side effects, electronics experts have resorted to all sorts of tricks. But in most cases the best they could do was follow advice as old as Scottish Physicist James Clerk Maxwell, the father of electrical theory, who died in 1879. It was Maxwell who pointed out that resistors could be bent into hairpin turns so that their current flowed in two directions, canceling out capacitance or inductance. Later, Physicist Georges Chaperon wound resistances into intertwined coils with the same result.

Wandering Mind. Those solutions work well, but not quite well enough for today's high-power equipment. At Sandia Corp. in Albuquerque, Physicist Richard L. Davis was busy trying to devise improvements. One day he let his mind wander and remembered an old mathematical parlor trick, the Moebius loop. * Math suddenly merged with electronics, and Davis had what he was searching for: the design of a noninductive Moebius resistor.

A Moebius loop can be made by cutting a harrow strip of paper and gluing its ends together after giving the strip a half-turn. The loop that results has peculiar qualities. Most important, though the paper it is made of has two sides, the loop itself has only one surface. This can be proved by drawing a pencil line down the middle of the strip. The pencil line covers both sides of the paper and returns to the starting point without the strip's being turned over. When cut along the pencil line, the paper forms not two loops but one long, narrow loop. Cut once more in the same manner, the narrow loop becomes two interlocked loops.

Double Passage. Davis made a Moebius loop out of a strip of nonconducting plastic that had metal foil bonded to both sides to serve as an electrical resistance. He attached wires to the foil on opposite sides of the strip. When he sent electrical pulses through those wires, the current divided, flowed in both directions through the foil, and passed itself twice. Because of the double passage, the inductance was as low as Davis had hoped. He is delighted but still puzzled. The pulses apparently pass right through themselves, and he cannot be sure how or why his device works. "Maybe Maxwell could tell us," he says, "but he's dead."

*Named for German Mathematician August Moebius, 1790-1868.

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