Electronics > RF, Microwave, Ham Radio

What to use for an array of very high Q resonators?

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I need an array of resonators (10...100 of them) with quality factor (Q) as big as possible, for a physics experiment.

Roughly, the experiment is about poking some energy at some of those high Q resonators, then observing what's happening and how the resonators exchange energy.  The math alone or a numerical simulation probably won't cut it, because I expect to see something new.  The resonant circuits have to be analogue by their nature, not digital.

By high Q I am thinking very low losses of energy, so I can study the vibrations and how the energy travels.  Assuming a single isolated resonator, once poked with some energy it should vibrate for as long as possible.  Preferably the resonator has to be a passive circuit.  A helping amplifier to increase Q might be allowed, but only as a last resort, and only if it doesn't turn the resonator into an oscillator.

The resonant frequency is not important, but I would prefer to keep it less than 100 MHz so I can easily measure or generate that with the instruments I already have.

To sum up the requirements:
- any technology of resonators goes, mechanical/electrical/ultrasound/etc.
- Q as high as possible
- 10 to 100 resonators, all the same (preferably possible to fine tune them individually)
- the exact frequency doesn't matter as long as all the resonators are tuned the same
- the resonance is preferably to be at a single frequency, if it happens at more than a single frequency, then the many frequencies must be far away enough, so to be possible to distinguish between multiple resonant frequencies
- helping circuits to increase Q are allowed, but only if they are analogue by their nature and do not self-oscillate
- it's a hobby project made at home, budget in the range of $100

So far I could only think of the 32768 Hz resonators used in real time clocks and wrist watches.

Q1. Any recommendation for a particular model to use?
Q2. Any other technology or idea to buy or to DIY very high Q resonators?
Q3. Any schematics or control techniques or tricks I should use to achieve very high Q?

Any other ideas welcomed, regarding how to study or how to experiment with an array of resonators.

Feel free to ignore my ignorance, but what is it that you expect not to be able to simulate about a general network of resonators? :-// I hope you're not extracting vacuum energy, because if you do then don't. :D

Well if you want to have a room full of expensive cavities each a good fraction of a meter in some dimension or other you could use
resonant cavities made from pipe or something like cylindrical cavities resembling metal waste bins, that's possible.
If you want to get more creative you can mechanically engineer them to be re-entrant, dielectrically loaded, or otherwise made more compact
for their Q factor and low VHF resonance frequency by some mechanical means.  The benefit is they're tunable mechanically and can be made / tuned to arbitrary frequencies.  The drawback, size, cost, and that they're mechanical so you will have to worry about vibration and temperature shifts causing mechanical changes in resonance etc.

You could use quartz crystals but they're not particularly tunable beyond a very small range.  Some people do build crystal filters out of ladder networks of up to dozens of quartz crystals each selected manually to be precisely only a few Hz to kHz apart in their resonant frequency so you can get a desired L / C network characteristic with a Q of NN * 1000's.
If you want them to be all the same frequency you certainly could order highly precisely tuned units or adjust the tuning of slightly differently tuned units until they overlap sufficiently.  You can to some extent spoil the Q by adding parallel resistance so that a network connection of artificially broadened resonances acts well-enough as if they had approximately the same center frequency relative to the bandwidth but you're not really tuning each one beyond some NNN Hz  away from its intrinsic mechanical resonance by parasitic external reactance.

0-100 MHz is awkward as a frequency range for high Q because the things that are naturally high Q are physically large and therefore expensive in that range besides the few commercially made quartz crystals that exist or other mechanically engineered resonators that have "high enough" Q.

If you want much higher frequency then you've got things like masers / lasers / spectroscopic molecules & atoms as well as more compact electromechanical things like dielectric resonators, EM resonant cavities, et. al.

You could use coaxial lines or other kinds of waveguides though the mechanical stability and Q and compactness are still not ideal in the HF /  VHF range.  They generally don't compare to true cavities for Q even in the UHF frequency ranges where both are practical.

On the other hand although you must be careful of numerical precision and stability there are plenty of cases where simulating high Q systems on modern computers is completely practical and fast / easy compared to building such a complex and difficult to stabilize / tune device.
I suggest simulation where practicable.

If you want to use quartz crystals, you should be aware of the characteristics of them and the electronic oscillators which use them.
There are various crystal cuts which are used and which produce oscillators with different frequency ranges, Q values, temperature sensitivities, and sensitivities to other parameters like aging / vibration / etc.

For instance the 32kHz watch oscillator crystals typically use a tuning fork resonator as opposed to bulk wave resonances in other common quartz crystals at much higher frequency ranges which might be AT cut, SC cut, et. al.

Some starting (tip of iceberg) references for crystals / oscillators / crystal filters:

Rev. Quartz Crystal Resonators and Oscillators

Accurate Clocks and Their Applications

Introduction to Quartz Frequency Standards DTIC

Characterization of Clocks and Oscillators - NIST

Crystal characterization and crystal filter design

Crystal Ladder Filters for All - ARRL


I will note that on the subject of crystal characterization / crystal filters, there's some discussion relating to using the "NANOVNA" multi-generational series of low cost VNA instruments for measuring crystals' parameters for instance so that they can be individually selected for use in multi-crystal filters.  Apparently particular instrument versions and non-default firmware / software may be required for best use.  It isn't clear to me how good the process is since there's also the variable of how good the frequency standard used by the VNA itself is and how people calibrate that instrument's frequency so that the result has absolute accuracy though I guess relative accuracy may be ok if the VNA uses a TCXO.
et. al.

Designing Narrow-Bandwidth Ladder Filters

Synthesis and Realization of Crystal Filters

9 MHz Filters Built With Inexpensive Crystals

Can you explain what you expect to find, that linear network theory doesn't establish?

It doesn't sound like you're going to find anything new in terms of known physics or analysis; but it's definitely not obvious to one not experienced in the subject, and in that respect, you stand to find quite a lot of new-to-you behavior!

The critical factor is the coupling.  What kinds of signals/fields are you coupling from/to, and how do you propose to adjust that coupling?

Very high Q resonators aren't very important, to the extent that Q factor is greater than system Q.  There are multiple ways to measure Q:

If we consider a component (substantially L or C) by itself, we can consider its terminal/port Q factor as the ratio of reactance to resistance, or vice versa depending on whether we're using series or parallel equivalent impedance.

If we measure the Q of a resonator, we get high values at most frequencies, but (port) Q drops to 0 at resonance -- that is to say, at resonance, reactance goes to zero, and resistance is finite.  We should use a different measure of Q, for example coupling very lightly (k < 1/Q) to the resonator and measuring its spectral width.  The maximum bandwidth is obtained with minimal coupling (which still needs to be nonzero, lest we have nothing to measure at all).  So when we say "resonator Q", we typically mean the combined Q factor of the L and C equivalent making up that resonator, rather than the Q of its measured terminal impedance.

If we place a component into a network, now its resistance and reactance add to the impedance it's embedded in.  For example, consider an inductor of 15.9uH with Q = 10 at 1MHz (X_L = 100R, ESR = 10R).  If we embed it in series between a 50R source and load (terminator), total loop resistance is 110R and reactance is 100R, for a system Q of slightly less than 1.  We get a first order low-pass filter with this being the cutoff frequency.  The minimum insertion loss is 10% (assuming ESR = DCR).

So there are different ways to express Q, and we must be careful which one we're talking about.  The sharpness of a filter is determined by system Q, while system Q and insertion loss are determined (and limited) by component or resonator Q.

Some resonators can't be coupled tighter than a certain amount, due to quirks of their design.  It's not possible to pull a quartz crystal more than a few kHz, or equivalently, get more than about the same bandwidth, because of the ratio of effective resonator impedance to holder reactance; it's a series-parallel transformation which limits the coupling factor.

When you're constructing resonators yourself, you have pretty broad control over Q.  A cavity, coaxial or helical resonator, you simply connect the coupling link higher up on the structure, or enclose more area in the cavity.  In this way, it's reasonable to get fairly low system Q's, maybe 3-5.  At higher Q, mere proximity between resonators suffices (a small (magnetic) coupling factor), or a window in the shield between them, etc.

At very low system Q, coupled-resonator approximations break down, and you're largely better off with a conventional ladder network.  Which, on a related note: if you've ever had much of a play around with bandpass filter calculators, you've probably found some pretty grotesque -- obviously intractable -- values pop up, particularly at high Zo and low bandwidth (high Q).  The reason is, each shunt and series branch is coupled to the next, by virtue of its impedance* as a ratio to Zo: the series tanks being Q times above, the parallel tanks being Q times below.  You can very easily get fF capacitors this way, which obviously won't work out; the reciprocally massive inductors they're paired with, obviously will have much more self-capacitance, or to ground or other things nearby, and so the network will be fundamentally different than what is shown (namely, there's a capacitor divider in the middle of it, effecting an impedance match, which the calculator doesn't know about so the response goes all wrong).

*Resonator impedance Zr = sqrt(L/C).

For an array of resonators, the most general description is to have a matrix of coupling factors between them.  This will be a symmetrical matrix (k_nm = k_mn) because the system is reciprocal.  For practical filters, it will also be a very diagonal matrix: coupling along the main diagonal gives poles, while coupling on the off-diagonals give zeroes.

A flat matrix (~equal coupling between all nodes) gives something of an EBG (electromagnetic band-gap) structure, for the same reason a crystal of atoms (closely coupled wave functions) gives an energy bandgap (or if you prefer, it's still frequency response, but what's being filtered is electron waves, because matter is waves, after all).

With a small network, we can observe line splitting directly: two resonators over-coupled, effectively resonate against each other in parallel and series modes, thus exhibiting two resonant frequencies with some amount of notch between them, the spread being k12 / kp1 (p = port) or something like that.  That is, the resonators are coupled more closely to each other than to the system.  If we add more and more resonators to the network, each one contributes a new resonance, and the splitting continues; eventually we find there's a seemingly prohibited span of frequencies in the middle (Fo +/- bandgap/2), with forests of resonances (or overlapping as whole passbands) above and below the gap.  (In the case of condensed matter, there are ~10^23 possible modes, so dense that we really can't tell, and don't care, about them individually; instead we integrate, assuming they form a continuum, and from there, derive various electronic and statistical-mechanical properties of the material.)

And if, rather than coupling resonators to zero-dimensional ports, they are distributed through space; then we need to take account of 3D fields and polarization, and we can construct all sorts of cool things like metamaterials, but calculating/simulating/constructing them is another matter...  If you intend to do experiments like arrays of antennas, understand that coupling drops off quickly (1/r^3 or faster in the near field, to 1/r^2 far) so you aren't likely to get much interesting behavior (i.e. pole splitting) at more than a modest distance.  So you for example, get modest bandwidth on a Yagi (stack of ~closely coupled dipoles), but a bunch of antennas loose in a field is just that, a bunch of antennas that can be treated as independent absorbers/reflectors (as the case may be).

Heh so... not that a statistical mechanical analog is likely all that helpful.  Stat mech is NOTORIOUSLY hard, much harder than network theory (which doesn't even require integrals)...  So, if that doesn't mean much to you, I understand.  In that case, now at least you know that a complex and deep property of matter, arises in the same way that behaviors of your immediate-future experiments do.



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