# Linearity of power amplifiers

How are power amplifiers linear? Although they have some distortion they are still considered linear.

I tried referencing many materials, but I couldn't find an answer. The characteristics of transistors are non-linear for large signals and hence the output waveform for say a sine wave should be some mixture of sine and exp - but we don't see that.

• The important concept here is "negative feedback."
– jonk
Jun 14, 2022 at 7:10
• Not a answer yo rour Q but rather a punctualization, PAs are only linear up to a certain point. Techniques such as predistortion are a hot research topic nowadays as PAs used in RF applications are very non-linear for those applications, and the non-linearity only gets worse as the power is increased. Jun 14, 2022 at 7:44
• Are you speaking of audio frequency amps or RF amps? Jun 14, 2022 at 7:59
• For audio amps see this for driving power stages ... cieri.net/Documenti/Altri%20marchi/… Jun 14, 2022 at 8:05
• Antonio51 - 17 likes. Jun 15, 2022 at 14:36

Expanding on my earlier comment... When there is excess gain available, NFB (negative feedback) is used to greatly improve linearity with non-linear devices and structures. (The following is from some old notes I'd saved. And yes, it's late to the game, as you have accepted an answer, but I don't care.)

Let's take this non-linear equation:

$$y = e^x$$

And now take 10% of the output and feed it back as NFB. The resulting equation is:

$$y = e^{x-0.1\, y}$$

That solves as:

$$y = 10\:\operatorname{Lambert-W}\left(\,0.1\: e^x\,\right)$$

Now compare the two curves:

(The above chart came from this Wolfram link.)

And that's the effect of just 10% NFB applied to an exponential curve.

NFB will do that to ANY equation. Let's grab a different one:

$$y = 4\, x^2 - 6\, x + 3$$

Apply 10% feedback to it to get:

$$y = 4\left(x-0.1\, y\right)^2 - 6\left(x-0.1\, y\right) + 3$$

Find now two solutions:

\begin{align*} y &= 10\, x + 5\: \pm 5\,\sqrt{10\, x - 2}\end{align*}

Plotting those:

(The above chart came from this Wolfram link.)

Now try:

$$y = e^{\sin(x)+1}$$

That's got to be a mess, right? An exponential of a constant plus a sine wave?

Apply 30% NFB:

(The above chart came from this Wolfram link.)

Note that the sinusoid is nearly recovered from the non-linear mess! (This fact alone can be quite important for audio.)

Take ANY equation you want and feed back some fraction of the output as negative feedback, back into the input variable, and the resulting equation is moved towards linearity.

NFB is very, very powerful medicine.

• Very nice mathematical demonstration. Will try it with my simulator. :-) Jun 14, 2022 at 10:10
• @Antonio51 Thanks. Sometimes, I think people need a mathematical demonstration to "see" with their eyes. NFB is incredibly powerful. Many people don't realize just how powerful it is. (Assuming sufficient excess gain is available to make it work right, of course.)
– jonk
Jun 14, 2022 at 10:14
• Great demonstration. I have understood that NFB is powerful and can linearize non-linear waveforms, however, could elaborate further how NFB has the tendency to linearize non-linear waveforms. Is it because of the inherent nature of feedback trying to make reference equal to output? Jun 15, 2022 at 10:18
• @VasuSuresh I gather you are referring to the idea that a sine or cosine isn't "linear" in some usual mathematical sense. But what NFB does do is help remove self-similarity and expose underlying functions. This gets into fractals, which I'm not prepared to elucidate here.
– jonk
Jun 15, 2022 at 11:11

For audio amplifiers, from this link, this is a "try" for one technique used in the pre-driver stage.
NB: here without feedback.

EDIT: Added simulation (simplified sytem) for NFB.

Made with microcap v12. Link to file test.

And for low level, no "cross" distortion.

For information. Simplified system showing characteristics (NFB applied).

The "input" is in black, the others are for a change in gain. Higher the gain, "smaller" the non-linearity.

• Interesting, can you please share the microcap file? I'd like to play with it Jun 14, 2022 at 9:51
• dropbox.com/s/a90i2w9z8mwu63e/… Jun 14, 2022 at 9:56
• Thanks. if I increase I3 to 50u I get clipping at about 2V output and saturation in Q4/Q6... weird. Jun 14, 2022 at 12:58
• I did not check all ... And all values of devices were missing in the original schematic ... It is why I said it is a "try" :-) All characteristics not checked ... I3 should be the AC "input" of all the power part. Jun 14, 2022 at 15:23
• Looks similar to the Sansui circuit. diyaudio.com/community/threads/… Jun 14, 2022 at 16:20

In a circuit, the transistors can be under several sets of conditions, which influence the amplitude and nature of the distortion being generated. If you were to look at the transfer function of a circuit, plotting input on the X axis and output on the Y axis, the more it looks like a straight line from input to output, the less total distortion. But the sharpness of any bends or kinks in the transfer function (its second derivative, to be more accurate) determines the order of this distortion.

A smooth transfer function with low second derivative, looking somewhat like a portion of exponential or a parabola (left) will generate low-order harmonics whose amplitude decreases as signal amplitude decreases. This is typical of small signal class-A stages.

However a transfer function with sharp bends or kink (right) will have higher second order derivative, and will generate more high-order harmonics that are more difficult to get rid of with feedback due to falling open loop gain at high frequency. If the kink is near zero amplitude, then the distortion amplitude won't necessarily decrease with decreasing signal amplitude. This is typical of class-AB or class-B stages.

So, your transistors can be used in:

• Small signal: variations of both Vce and Ic are small compared to the bias point. This is the most linear condition, like the curve on the left.

• Large signal: large variations of Ic, to the point of turning off the transistor. This will generate the most distortion, and it produces high-order harmonics due to the sharp change in characteristics near turn-off. This is like the curve on the right.

But there are also:

• Large variations of Vce, which means Early and other effects can't be ignored, but small variations of Ic, so the exponential characteristic remains quite linear. This generates a higher amount of low-order distortion.

Here's a typical low-end opamp, JRC4558. Link to image source and circuit analysis.

In this circuit, unless the opamp is clipped or slewing, the high open loop gain means the input signal of the input stage is tiny. So Q2-3-4-5 work under small signal conditions, at least for current.

Q2-3 Vce can vary quite a lot if the opamp is in non-inverting configuration, which introduces a bit of low-order distortion. Some opamps fix this by adding a cascode to the input stage.

Since S2-3 have no emitter resistor, the transconductance of the input stage will be a tanh function, only linear near zero input signal. This type of input stage works best on small signals, which means it requires large open loop gain. If the signal has too sharp edges, using a significant proportion of available slew rate, or is at a high frequency relative to the available bandwidth, then the input stage will have to output more current to charge compensation capacitor C2. This requires a larger differential input signal, which causes an excursion out of the linear center part of the tanh characteristic, and thus more distortion.

Q6 works as a current gain stage, again small signal.

Q10 also works as current gain stage, with large voltage swings, so it will show the effect of varying hFe with Vce and self-heating. This

The output transistors Q11-Q12 work under large signal conditions because in normal use, they are expected to turn off as the output stage goes in class-AB. In addition, they see large current swings, so it is important the transistor has hFe constant enough with varying current.

The output stage is usually the largest source of distortion in the whole amp, simply because it is the only part that works under large signal conditions. All the other transistors work in class A. The only way to mitigate that is to remove the emitter resistor to get a better transfer function, but this is only possible if thermal runaway is avoided with a tight thermal tracking circuit. Otherwise, the only thing available to correct it is feedback.

• This nicely shows, why the kinks in the transfer function (Class B) are so problematic for wannabe-linear power stages. And IMO stabilizing a Class AB under all conditions to have no kink, is as hard or harder as using a highly oversampled Class D instead. Jun 14, 2022 at 8:52
• Yup. Although analog class D has much better potential than digital class D which is pretty much abandoned for hi-fi. (mostly because it can do PSRR, I guess) Jun 14, 2022 at 9:53
• The UA4558 dial opamp is 48 years old. The Japanese copy was called JRC4558 years ago but today it is NJM4558. Modern opamps have much less noise and distortion and better high frequency bandwidth. Jun 14, 2022 at 16:26
• EDIT: The original dual opamp was RC4558 (not UA4558) and is still made by Texas Instruments today. The Japanese copy produces more noise. Jun 14, 2022 at 16:36
• Yes these opamps are pretty bad. They're ubiquitous though. Jun 14, 2022 at 17:10