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What's the minimum (physics first) to get an oscillator?
bdunham7:
Take a Geiger counter and connect the output, with appropriate buffering, to a clock divider circuit like the ones that divide down a 32768Hz crystal oscillator, set up to put out a 1Hz 50% duty cycle square wave. Now put the Geiger counter in a box along with a gamma-emitter like Cobalt-60 and adjust the sample position until you get approximately a 1Hz output from the divider circuit.
Is that an oscillator?
IanB:
--- Quote from: bdunham7 on May 24, 2023, 04:16:25 am ---Take a Geiger counter and connect the output, with appropriate buffering, to a clock divider circuit like the ones that divide down a 32768Hz crystal oscillator, set up to put out a 1Hz 50% duty cycle square wave. Now put the Geiger counter in a box along with a gamma-emitter like Cobalt-60 and adjust the sample position until you get approximately a 1Hz output from the divider circuit.
Is that an oscillator?
--- End quote ---
Well, that would depend on how you wish to define oscillator.
But in a physics sense, there is no feedback, no momentum effect, no second order dynamic behavior of the process.
It would seem more like a random number generator.
RoGeorge:
--- Quote from: Nominal Animal on May 23, 2023, 07:23:56 pm ---
--- Quote from: RoGeorge on May 23, 2023, 06:43:26 pm ---It gets even more bamboozling when thinking about numerical oscillators.
--- End quote ---
Numerical oscillators? Do you mean periodic functions and periodic sequences?
--- End quote ---
Something like that. That type of question comes from an email "debate". At some point I was trying to look at the behaviour of a (physical) band-pass filter as a variable impedance, where the energy at the filter input will be reflected back (when the input impedance does not match), or transferred to the output. Probably a wrong way to look at a filter, it was about switched filters and in another context, anyway,
TL;DR, the other guy asked "if you say a filter is an impedance mismatch reflecting back some input energy, then what is reflected back in a numerical filter, like a DSP?".
I should have stay to physical oscillators only, and not mention the numerical implementations, that's another talk entirely, sorry.
RoGeorge:
--- Quote from: IanB on May 23, 2023, 08:12:22 pm ---
--- Quote from: RoGeorge on May 23, 2023, 06:43:26 pm ---That is why asking to identify the essential components of a physical oscillator, at first.
--- End quote ---
For a harmonic oscillator the essential component is that the mathematical description includes some kind of 2nd order differential equation in time. With a first order system harmonic oscillation cannot happen.
Another kind of oscillator results from switching, for example if you use logic elements to make a square wave. However, if you analyze at a small enough time scale even logic circuits are analog, and somewhere in the description will be 2nd order characteristics.
--- End quote ---
For mathematicians, derivatives can mean a lot of things and do many tricks. To me, less skilled with math, first derivative means the slope of a plot, or speed in mechanics. And the second derivative means to me the curvature of a plot, like acceleration in mechanics. I prefer thinking rather in physical terms, like speed and acceleration, because it's something that can be directly experienced by everybody.
So, if I try to translate 2nd derivative in words, would it be correct to say an oscillator needs some sort of acceleration in its behavior? If yes, then a rotating disk would also be an oscillator. This might still be OK (saying "might" because just like ejeffrey, I'm tempted to think circular motion is not the same as an oscillator), but the 2nd derivative condition doesn't make any good if, for example, it's a linear acceleration. Something has to repeat, to be periodic, to alternate. I don't see how the 2nd derivative implies alternation, or am I misunderstanding the 2nd derivative condition you were mentioning?
To me, the problem of explaining something in terms of math is that there is no causality in an equation. An equation is an equivalence, a relation between it's terms, and that can only be true or false. It's like saying an oscillator is something that satisfy the oscillating conditions. That would be a circular definition, same as saying an oscillator is an oscillator.
I'm not dismissing your explanation, only trying to say what I didn't understood, or what difficulties I have with it.
T3sl4co1l:
--- Quote from: westfw on May 23, 2023, 06:25:34 pm ---
--- Quote ---The prototype harmonic oscillator in physics discussions is a spring
--- End quote ---
How about a pendulum ? Or does the externality of gravity complicate things?
--- End quote ---
We could do some interesting categorization with respect to the decay mechanism (or conversely, gain when applicable). Note that simple rotation decays over time, resulting in a reduced frequency. That's a very nonlinear sort of behavior compared to SHM. Depending on the loss mechanism, the decay might be exponential, or double-exponential or something else (compare constant frictional resistance vs. friction proportional to angular velocity*). Or it might be increasing, with a hyperbolic decay curve -- as for an orbital system (frequency ~ 1/energy) decaying under gravitational radiation.
*Well... there was a kernel of something more interesting in my mind, but it doesn't sound very special when written out, does it? Simple resistance is torque proportional to angular velocity, which gives a simple pole (exponential decay), the usual case. But there are mechanisms which are strongly dependent on rotation rate, or frequency (in quantum terms, E ~ ν, but the emission / coupling rate matters too, and I don't know offhand any figures for that), or say temperature (radiation ~ T^4).
Say we stick a magnet or battery to the rotor, so it's emitting EM waves at the rotation frequency; at most frequencies, it will be a short dipole so loss varies strongly with frequency. (Ah, but does it vary proportionally, or square law, or more? I don't recall offhand**.)
But these are more easily collected by specifying an amplitude- or rate-dependent function, and observe the special cases for loss = 0 (perpetual) and loss ~ rate (constant resistance), then solve for whatever the resulting decay curve is (or infer the opposite way just as well).
**Well, induced voltage (relative to a non-rotating reference frame) is proportional to rate, and, what, a short dipole has radiation resistance as 1/λ^2? Which will be the series equivalent resistance of the lossy capacitor, so...
V ~ ω
I ~ V / Xc ~ V ω ~ ω^2
λ ~ 1/ω
R ~ 1 / λ^2 ~ ω^2
P ~ I^2 R
P ~ ω^6 ??
--- Quote from: Nominal Animal on May 23, 2023, 06:34:27 pm ---One class is that any convex shape is analogous to a sphere. (There is a better mathematical term for this, but I can't recall it right now.)
--- End quote ---
Gaussian curvature..?
--- Quote ---There is a third one, too: the smaller the amplitude compared to the energies and distances involved in an oscillator, the closer it seems to match a harmonic oscillator. However, I myself do not know why this is, or even if it is just a practical result of a noisy universe (i.e., any non-harmonic components drowned in noise in real life), so I consider it more or less just a rule of thumb.
--- End quote ---
In general that's just linearization around a point; in this case, a small cycle around phase space. If the system is continuous, it approximates a SHO orbit.
I suppose for certain dynamic systems, we can employ transformations to convert limit cycles into limit points, and map non-circular trajectories onto unit circles (thus accounting for harmonic distortion). The existence of such a mapping, on a given system, might suggest its generalized-SHO-ness.
--- Quote from: RoGeorge on May 23, 2023, 06:43:26 pm ---It gets even more bamboozling when thinking about numerical oscillators. What would mean energy conservation for a numerical oscillator, or momentum, or seeking minimum energy equilibrium, or inertia? All these are very important and inescapable in a physical oscillator, yet they don't make much sense, and they seem totally arbitrary for a numerical oscillator.
--- End quote ---
NCOs are abstracted in a different direction: an iterated transformation (discrete-time) on a discrete state vector (i.e. bits in memory). We could indeed apply some implementation (whether transistors or otherwise), assign an energy or power cost to the states, or transitions therebetween, and model it the same way -- just with a lot of extra steps.
More tantalizing, I suppose, might be applying such maps as above -- suppose we implement an NCO using "lossless" digital logic, at such a clock rate that the loss tangent of the capacitances is sufficiently negligible, and using an LC oscillator (and whatever applicable phase-shift networks) to afford the various clock phases required for operation. Then we could have a digital description of a continuous oscillating system, which exhibits rather complex behavior (various period-multiplication effects as the counters roll over from time to time) but overall is nonetheless, not just coherent with, but indeed energetically coupled to, our reference oscillator.
The standard implementation today is very much more mundane; since our information-theoretic efficiency is piss (something like ~10^6 times more energy loss per bit transformation), it's basically a class A amplifier: all power dissipation all the time. And with all that wasted power, it really doesn't matter what happens. You have the space for a lot of complicated -- but thermodynamically irrelevant -- functions. :)
Tim
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