Linearity

What is the meaning of linearity, and why should I care?

Howard Johnson, PhD -- EDN, September 9, 2010

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I took a long puff on my pipe and answered slowly, “Well, linearity is one of two properties essential for good signal fidelity—audio or otherwise. The other property is time invariance. A linear, time-invariant system responds equally well to loud and to soft inputs, whether composed of one sound or many.”

“You are just waving your hands,” Breathe said. “I don’t buy it. There must be some more-concrete definition.”



“Yes,” I answered. “And your speaker probably distorts at the high setting, too, so that won’t be the same either, but what I’m saying is that, if you keep the volume in a reasonable range, then scaling works.”

“Does the term ‘reasonable range’ include a setting of zero?” he asked.

“Of course, zero is a perfectly valid input signal for any linear system,” I said. “The output would be zero.”

“But it’s not,” said Breathe. “Even when I set my guitar to zero, a little hiss always comes out of the speakers. So the amp is not, according to your definition, linear for either large-scale or small-scale inputs.”

At this point, I realized that, through earlier such conversations, I had already taught Breathe far too much about electrical engineering. His questions were becoming dangerous. My next columns will lay out for him, in a methodical but simple way, the whole concept of linear-time-invariant behavior so he can understand its importance, not only as a tool for modeling but as an ideal standard of behavior against which you can measure circuit performance.

On another topic, a recent article (Reference 1) generated a lot of reader responses. Here are some of the more oft-repeated ideas. First, the precise values for the 10% resistor scale (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, and 82) nearly fit an exponential scale. The steps are adjusted so that the tolerance bands in most cases overlap. For example, the nominal value of 68Ω−10% gives 61.2, slightly smaller than 56+10%, which equals 61.6. Only the gaps from 12 to 15Ω and 18 to 22Ω violate this rule. Because most of the bands overlap, almost any resistor you manufacture fits into some tolerance band somewhere on the scale. Few parts go to waste, and manufacturers love this fact. The other tolerance scales—20, 5, and 2% and so on—have similar overlapping properties.

Next, if you file the side of a carbon-composition resistor, notching through its outer coating into the bulk carbon layer, you can raise its resistance. This approach makes every resistor a “variable resistor.” A drop of lacquer reseals the outer coating. Don’t file too far!

Finally, the value of a carbon-composition resistor drifts with temperature and with age. If you want long-term stability, you must prebake your resistors.

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Reference

Johnson, Howard, PhD, “7% solution,” EDN, June 10, 2010, pg 22.

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