# Multi-Stage Common-Source Amplifier - When is it no longer a small-signal?

I have a question about the actual definition of a small-signal signal. We use it all the times in amplifiers. But how small is a small-signal? When does it stop becoming a small-signal?

Take for example, a common-source amplifier cascaded with another common-source amplifier.

[Excuse the poorly drawn sinusoids]

Let's say that both transistors are biased with a 200mV VDS headroom over Vgs-Vth in the saturation region. Let's assume stage 1 has a gain of 5 and stage 2 has a gain of 10.

If we apply a 10mV peak to peak signal at the input, which is our small-signal input. It becomes a 50mV signal at the input of the second stage and a 500mV signal at the output stage.

A 50mV signal at the input of the second stage - how is that a small-signal? Wouldn't we now see distortion because our gm2 is behaving less and less like a constant slope but changing due to this "small-signal" 50mV affecting the bias point? [Q1]

What's the point in calculating output voltage swings if we go anywhere near them, we no longer see a small-signal behaviour. Or are they only important for the output stage, because it doesn't go to any amplifier afterwards [Q2]

• "small signal" is more about analysis that is susceptible to "differential" mathematics than about the signal being below some particular amplitude. Commented Feb 10, 2021 at 1:43
• Small signal AC analysis is virtually no signal at all; it's an analysis of the circuit with virtually no AC magnitude level at all. Commented Feb 10, 2021 at 9:03

It is no longer small signal when you decide it is no longer small signal.

Strictly, "small signal" analysis means you're finding the linearized behavior of the system at an operating point -- i.e., $$\dx_{out}/dx_{in}\$$, and it's only valid for $$\\lim_{\Delta x_{in} \to 0}\$$.

In a more general sense -- how much distortion can you tolerate or ignore? It's "small signal" up to that level of distortion.

1. Wouldn't we now see distortion because our gm2 is behaving less and less like a constant slope ?

yes. There are many ways to measure this distortion.

2.It depends on your specs. for distortion

## Methods:

1. Sweep DC slowly with very small output AC swing, measure gain with a DSO a/b=gain calculation for AC coupled signal
2. measure %THD using FFT or spectral density for a sine and correlate gain change from peak to peak with the generation of harmonics
3. AC couple output and measure the difference ratio in +/-peaks. For example a 1V swing might result in difference in gain for each Vdc+ac output peak and this will change with DC sweep over the so-called linear range.
4. In RF, one measures output power at the 3rd harmonic intercept of the fundamental for a distortion benchmark. (IP3 or TOI)

My method (3) compares the 0 to peak of each polarity using an AC coupled output after settling to 0Vdc average.
ΔVp/Vpp/2 = gain distortion ratio = THD

I have not empirically proven this, but experimentally is close enough when you don't have a distortion analyzer.

What you will find is for increasing input +ve peak $$\g_m\$$ may increase decrease for -ve peak but the average may be constant over the "linear range".

Yet changing the AC input level at the same DC bias may result in almost the same gain, yet the asymmetry of the gain increases with output swing becomes noticeable. So the Av measurement is valid, but as you suspected, the THD or peak-to-peak gain variation may become a concern, leading you to consider what levels are acceptable and may consider negative feedback after several stages to reduce the distortion effects.

• Your method (3) would severely underestimate odd-order distortion. It'll work great for an amplifier that's primarily even-order distortion, like a single-ended amplifier, but a really well-matched push-pull amplifier (old-style tube & transformer is an extreme example) would show very little in your test, and yet could have severe 3rd-harmonic distortion. Commented Feb 10, 2021 at 16:41
• Thanks Tim. Yes I should have included that assumption for these single-ended designs as well as BW for phase shift on 2nd harmonic which affects this. But still useful in this case Commented Feb 10, 2021 at 17:45