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Nonlinear Distortion
First a little background. When we talk about feedback and nonlinear distortion, we are inevitably talking about gain devices – tubes, Bipolar transistors, Jfets and Mosfets. And when we talk about gain devices, we usually get around to nonlinear distortion. Here are the basic players shown in simple circuits:




All of these examples have current flowing from pin 1 to pin 2 as a function of the voltage between pin 3 and pin 2. In the example of the tube, the current from Plate to Cathode is largely determined by the voltage between the Grid and the Cathode. The other examples have similar relationships but vary in the details. We think of pin 3 as the control pin and signal presented here is amplified to a larger signal passing through the other two pins.
All gain devices have imperfections in their gain. Out of the box, you can measure their gain and for a given test, each will have a different value. If you take a single device and measure the gain, you will find that it varies with the current flowing through the device. It also varies with the voltage across the device as well as the temperature of the device. All three of these will create nonlinear distortion.
Distortion is what you get when the gain is not constant. If the devices had perfectly constant gain under all conditions, they would have no distortion. We are mostly concerned about what happens to an audio signal when it is amplified by a device whose gain figure is changing in response to the signal. The signal goes in and comes out with a different shape. The gain is not perfectly straight  it is bent or nonlinear.
An important thing about distortion: when you run a signal through a device which is even slightly nonlinear, you have changed the signal forever. You can use various techniques to reduce distortion after the fact but you can't go back.
The problem is greatly compounded when complex signals consisting of many frequencies all travel through the gain device at the same time; or when a simple signal is passed through a number of nonlinear gain stages in series. Of course you can do both and as we'll see later, these can add up to a perfect storm of distortion.



Harmonics
When you amplify a single tone (sine wave) and the gain device distorts it, the output contains the original tone plus a series of harmonic tones which are whole multiples of the original frequency. If the original tone is 1KHz, then the output will contain 1KHz plus maybe some 2KHz (second harmonic), 3KHz (third harmonic), 4KHz (fourth harmonic) and so on.
Audio signal alternates between positive and negative values. If the nonlinearity of the transfer curve is symmetric with respect to positive and negative, then the harmonics will be odd  third, fifth, seventh and so on. If the transfer curve is bent nonsymmetrically, the harmonics will be even  second, fourth, sixth etc. Figure 2 shows a sine wave with a high 2nd harmonic content. This variety of distortion is often seen in overdriven tubes operated in singleended Class A mode.





Figure 3 shows a sine wave with a high 3rd harmonic content, recognizable as a 'soft' clip, occasionally seen in overdriven pushpull Class A tube circuits.





A gain device's transfer curve can be expressed as a polynomial: V = a + bX + cX² + dX³... The 'a' term is a DC offset component and the 'b' is the linear coefficient, reflecting the distortionfree performance, which would be a straight line. The c, d and etc. are coefficients to a power series representing the 'bent' nonlinear portions of the curve.
Similarly, a voltage waveform can be expressed as a sum of harmonic frequencies, each with its own amplitude and phase coefficients. These two ways of looking at things nicely correspond to each other.













