## Linearity

Breathe already knew too much. His questions were becoming dangerous.

My good friend Breathe asked me recently, "What is the meaning of linearity, and why should I care?"

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."

"There is," I replied. "It’s tricky to state the whole thing, so I’ll begin with a necessary condition, meaning that every linear system must at least do this task. The condition is called scaling. Scaling means that, if you turn up the volume on the system input, the system response scales proportionately. Your guitar amplifier, for example, has the property of scaling." (Breathe plays a fine old arch-top jazz guitar. He uses a Mackie mixer driving a linear studio-quality monitor to produce a clean sound. He doesn’t need distortion because his technique is impeccable.)

"I don’t believe that," Breathe said. "Look, if I tweak the volume knob on my guitar to 5, it sounds one way. If I turn it up to 10, the club manager comes over and tells me to turn it down. So the response is totally different in those two cases."

"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 the volume on my guitar amp 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.

My good friend Chris "Breathe" Frue is a talented musician, a trained audio engineer, and an excellent conversationalist. He wants to learn more about equalizers, a subject pertaining to both audio and high-speed digital systems.

On another topic, my recent article about standard resistor values (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 slightly. For example, the nominal value of 68Ω minus 10% gives 61.2, slightly smaller than 56 plus 10%, which equals 61.6. Only the gaps between 12 to 15Ω and 18 to 22Ω have no overlap. 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; manufacturers love that. 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 can prebake your resistors to pre-age them so they won't drift as much later.

### Reference

[1] Johnson, Howard, PhD, 7% solution. EDN, June 10, 2010, pg 22.

### Other articles in the Basic EE series:

• Linearity -- Linearity is one of two properties essential for good signal fidelity, audio or otherwise. The other property is time-invariance. EDN 9/9/2010
• Superposition -- Linear superposition opens the door to many advanced methods of circuit analysis. EDN 10/7/2010
• Time Invariance -- Hard clipping obeys time-invariance, but not superposition. A tremolo circuit obeys superposition, but varies its gain with time. EDN 11/4/2010
• Impulsive Behavior -- Stimulate any linear system with one short, intense pulse, and you see a response characteristic of that particular system. EDN 12/2/2010
• Undo Machine -- The signal distortion caused by some linear time- invariant processes can be completely un-done. EDN 1/6/2011