[time-nuts] sine to square wave circuits

[Attila Kinali]

On Wed, 4 Oct 2017 16:44:17 +0100
Dave B via time-nuts <time-nuts at febo.com> wrote:

> Silly question, and forgive my ignorance.

It's not silly at all. It took me some time to understand it...
well... understand more of it. I'm far from understanding noise
and how it propagates through a circuit.

Disclaimer: What follows is what I think is true. It might not be.
If I am wrong, please correct me.

> But, why is a simple two transistor discrete Schmitt trigger circuit no
> good for this purpose?
> 
> 10MHz is not a problem, so what is bad about it?

First of all, you need a bit of understanding about amplifiers
and noise. Rodolphe and Enrico provide a nice introduction about it
in [1]. Summarizing you have two forms of noise that matter: white
noise and flicker noise. White noise is what we call the noise floor
and is there from DC up to optical frequncies. The contribution of
white noise to jitter is dependent on its power relative to the signal
power. Ie the SNR defines your ultimate jitter performance. Every
amplifier adds a bit of white noise. What we call the noise figure
is the amount by which the SNR decreases due to the amplifier.
Today, most amplifier circuits are in the order of 0.5dB and 5dB.
Which means that the jitter increases by a factor of 1.06 (ie 6%
increase) to a factor of 2 (100% increase). This calculation is
made under the assumption that the amplifier is linear. Ie that
there are no higher order terms in the Taylor expansion. This
is not generally true, but a good approximation for a well designed
amplifier. In case the non-linearities (higher order terms) cannot
be neglected, these can contribute to a higher level of white noise
than the noise figure of the amplifier would suggest and it becomes
also signal level dependent (this is, at least partially, due to 
AM to PM noise conversion). What's nice about white noise is that
you can filter it. Because your signal is at a well known frequency
(assuming it's sine) and white noise is everyhwere, you can filter
out that bit where it's everywhere else.

Flicker noise is the other big contributor to jitter. It originates
from noise close to DC that is upconverted by the non-linearity to
the signal frequency. Even small non-linearities will lead to this
upconversion, which means it cannot be completely avoided, only
limited by keeping the non-linearities small.

The problem with a Schmitt Trigger is its high gain with bad noise
figure and a highly non-linear circuit. The positive feedback
loop that the Schmitt Trigger has will feed back noise, that came
from the input, was amplified through the amplifier stage, amplified
again through the feedback stage, into the input. Describing it this
way it is easy to see that the noise of a high gain amplifier with
a positive feedback is even higher than what you would expect from
a normal, feed-forward only amplifier. Additionally, the positive
feedback is so high, that the Schmitt Trigger exhibits strong
non-linear behaviour, as it snapps-in to the new value.
Ie 1/f noise close to DC will be upconverted to signal frequency in
addition to having increased noise figure for white noise.
Another way to look at this is by seeing the signal as it crosses
through the switchig point of the Schmitt Triger. The amplitude noise
on the signal will cause the Schmitt Trigger to switch earlier or later
depending on whether the noise increases or decreases the slope of the
signal. And once the Schmitt Trigger switches, there is no going back.
Ie AM noise becomes PM noise.

Very much simplified, this boils down to: Hysteresis is bad!
A little bit less simplified, the sine to square wave converter
should be a circuit that works as much in the linear region as
possible, especially close to the zero crossing point. 

Now, to answer the question itself: A simple inverter is a
class B amplifiers with a gain of 10dB to 30dB (for todays CMOS
processes, I don't know the values for HC/AC/...etc). The packaged
"single chip" inverters are usually a chain of those inverters.
Ie we have a very high gain, but at the same time, the amplifier
saturates quickly. These amplifiers are not optimized for noise,
but for high gain and high slew rate (ok, not really, but that's
what the design boils down to) and exhibit relatively high non-linear
behaviour even close to the switch point. But even though they are not
optimized for noise and are non-linear, they are still "more linear"
than a Schmitt Trigger as there is no hard-switching with no way
back, but a regular amplification.



				Attila Kinali

	
[1] "Phase Noise in RF and Microwave Amplifiers",
by Rodolphe Boudot and Enrico Rubiola, 2013
http://rubiola.org/pdf-articles/journal/2012-TUFFC--Noise-in-amplifiers.pdf
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