cycles, cycling and cyclical structures

[Magnus Danielson]

From: Martin Czech <czech at Micronas.Com>
Subject: cycles, cycling and cyclical structures
Date: Wed, 26 Apr 2000 09:47:06 +0200 (MET DST)

> Time to throw a new idea into the dungeon...
> 
> 
> >BTW. All that cycling isn't good for you, you need to get into the lab, inhale
> >some sane soldering fumes like the rest of us ;)
> 
> That's what my mom always told me... 
> No serious, as long as the pulse is not above 165 bpm I'm still able to
> think, and some great ideas were born riding through the black forrest.  

You could not escape that one easilly, could you? You react just like any
adict ;)

> Recently I thought about test structures for delay measurements.
> You may know that ring oscillators are used for this purpose. This
> means a ring of inverters, of course an odd number. It is not so easy
> to determine the speed of CMOS integrated logic from static parameters,
> so an inverter chain or ring is still a good idea.

You can easilly show that an even number inverter ring can get stuck in two
diffrent patterns, both involve every other being stuck at 0 and those between
them get stuck at 1. This lock pattern can not exist in an odd number of
inverter ring where you have at least one inverter which is about to change
state. However, both types can also get stuck in the metastable linear mode
where noise, temperature gradients etc may force the ring into one or another
change. You can naturally do some tricks to force oscillations but this means
somehow changing the curcuit.

> OTOH there are still hundreds of attempts to patent such ring oscillators,
> it is very hard not to jump out of the window if you look into the actual
> patent surveys...

I can beleive you on that one...

> All right, what is it all about? As long as you have an odd number of
> inverters in the ring, it should start oscillating as soon as power is
> switched on.  I say it should, but we all know that amplifiers oscillate
> , whereas oscillators always get stuck ;->. Old saying. Well, they do
> run most of the time, but if you take a lot of inverters, say 21, the
> pattern which is running arround in circles is not exactly determined.

Yes, you can have multiple switch events in such a ring. Also, the delay of
each inverter is not exactly the same, which adds somewhat to complexity. The
frequency readout is an average over those inverters 0->1 and 1->0 switch
delays.

> Now, this is a pain in the @ss for parameter measurement people,
> why bother with that, we are set for music. Exactly we are, and this
> undetermined behaviour is something interesting. Inverters were used
> before, just think of J.H.'s WASP clone. E.N. has a great deal of digital
> pseudorandom cycling stuff, but AFAIK it is all clocked logic.
> 
> This proposal is different, because it is analog. It remembers me of the
> modes that a resonator like a church organ pipe can have. You need a ring
> of inverters, the slower they are, the better. (I can't hear 1MHz). We
> can regulate the speed via Vdd and or by controling the current for each
> individual inverter.  You know the "clocked inverter", totem pole which
> has two PMOST in series and two NMOST.  The outer MOST can be used for
> switching or current throtteling.  We can use other elements as well,
> there are single CMOS inverters in SMD, or we can use OTAs, op-amps...
> 
> 
> There are a lot of possibilities to create such a cyclic resonator in the
> audio range, it is like a organ pipe that is bend into a torus.  We can
> tap the ring, mix the taps. We can apply a forced initial condition
> via set transistors. We can apply an external signal. Finally we can
> go completely linear, using integrators and avoiding input overdrive.
> This last option may have noise problems, however. Also the frequency band
> is limited due to the inverters. Better use phase shifters instead.

Allpass filters, not phase shifters...

> The most interesting things will of course happen, if the ring is
> long, i.e.  if the individual section delay time is much shorter then
> the total ring time.  Because only then a complex pattern other then a
> sine wave can exist in the ring.

I agree here... you need a long ring in order to make sense.

> There are interesting questions:
> 
> -Once a pattern has settled in the ring, will it survive a voltage
> controlled ring mode tuning?

You can make it survive, but you may have to increase the ring gain if you
do too quick changes. You will also need to balance the ring gain somehow.
I think that a varying the ring gain changes many diffrent properties.
Further, changing the allowed signal level in the ring will naturally affect
the balance between linear/non-linear properties. Some non-linearity can be
good in order to feed some overtone energy into the ring so that you ensure
the existence of these frequencies in the ring.

> -Will the pattern survive at all, or will it alter infinitely due to
> amplification of noise? Or will it lock into a natural mode of the device
> due to parasitics?

If we are talking about a fully linear responce, then you will see that the
ring will take the responces of the allpass filters and bring them together
and the number of poles you are able to put at or near the jw-axis will come
into play. Also, you can play dirty tricks with zeros. One thing you can do
is to make subrings by doing loopback chains ot feedforward chains (in a ring
these are equivalent!!!) to mess things up further.

> -Will it be possible to force an initial state, and will this state
> circulate for a longer time?

You can create some initial state by insertion of a signal somehow. This is
a highly complex resonator and tuning it to remain the state should be
possible althought possibly hard.

> -How will an external signal interact with the resonance modes, and how
> will the nonlinearity act upon that?

If you have it perfectly balanced at resonance mode, then will frequencies
near the resonance peaks die off very very slowly where as other parts fall of
quicker. The nonlinearity (given "weak" nonlinearity) may create overtones
which lies near some resonance not existing in the original signal. Note that
you may loose energy on nonlinearities so gain or signal level will greatly
affect the properties.

> -This is a nonlinear resonance system. Is it possible to find a route
> to chaos?  

You are on a path leading into chaos here.

> -Anyone been there?

I have seen people building Chaotic oscillators using a combination of linear
and nonlinear devices. You should be able to find some of it over the web.
A quick search gave me these:

http://www.physics.utah.edu/~p373/hamlet/lessons/chaos/chaos/chaos.html
http://www.hypertextbook.com/chaos/
http://website.lineone.net/~edandalex/chaosynt.htm
http://www.industrialstreet.net/chaosmetalink/LiveSites.asp?CategoryID=4

... and much more

Also, I have seen someone do various saxophone sounds using analog curcuits.
It was one the web somewhere, that migth be interesting material aswell.

> -How will this sound like? Can we get away from the usuall
> sine-tri-sqa-saw stew?
> 
> Lots of questions. Ihave no answers for. That makes life interesting!
> Now, who is biting the worm? I guess a software simulation will not do,
> because of the very nonlinear nature of the device.
> 
> 
> As you may see, cycling is not totally useless for diy. (I sometimes see
> Jan Ulrich and other Telekom staff rushing on the other side of the road
> which is kind of funny... makes me feel infinitely small)

Cheers,
Magnus