[ §1 · resonator ]

A magnifier is not a longer coil

Three coils and two couplings, argued for on the coupling the driver can hold. The one head-to-head with figures made less spark per inch of winding than its builder's own two-coil machine.

CommonGnd19 August 202623 min read

A magnifier splits the resonator in two. The primary drives the lower half, the upper half stands on a base of its own, and the spark leaves the top of a coil the primary never sees. The argument for it is coupling: the driver is wound at a k far above anything a DRSSTC runs. The one head-to-head with figures on both sides that this page could find is a builder's own, his magnifier measured on the night against his two-coil machine as he remembered it, and the magnifier made less spark per inch of winding.

What it is and how it works

Three tuned circuits in a chain instead of two. The first holds the energy and a switch sets it ringing. It is coupled magnetically to the second. The second is wired straight through to the third, which is where the voltage is wanted. Energy walks along the chain, and the object of the design is to have all of it standing in the last circuit at one instant, with nothing left behind in the first two.

Two coupled tanks have two modes, the energy sloshes between them, and complete transfer arrives at a beat between the two. Three coupled tanks have three modes, and the same trick only works if the three frequencies stand in whole-number ratios to each other: at any other tuning the three oscillations never all come to rest in the same place at the same time. That is a solved problem rather than a matter of taste. Antonio Carlos M. de Queiroz's design equations take the ratio you pick, call it the mode, and return every reactance in the chain from it.

The third mode is made by one part: a capacitance across the middle resonator, at the joint between it and the last one. Take that capacitance to nothing and the third mode goes with it. What is left behaves as two coupled circuits, one of which happens to be wound in two pieces. So the joint capacitance is not a stray to be tolerated. It is the part that decides which of the two machines you have built.

What the chain does not buy is voltage gain. The step up is set by the square root of the ratio of the first capacitance to the last, the same as it is with two circuits. What it buys is geography: the high end of the chain is at the far end of it, physically distant from the part carrying the energy in, and the middle circuit shields the first from the worst of the voltage.

Two things follow from that shape, and both of them are mechanical rather than a matter of judgement. The joint capacitance carries the transfer, so it is a high-voltage component in the middle of the chain and has to be built as well as tuned. And the one magnetic joint now reaches only part of the chain, so the coupling at that joint has to be higher than the figure the whole chain behaves at, simply to match a two-circuit machine. That second one is where the argument about magnifiers actually lives.

What it does in a coil, and what you decide

The three circuits get names. L1 is the primary and its tank capacitor. L2 is the secondary, which in a magnifier is a driver rather than the thing the spark comes off. L3 is the tertiary, the resonator, standing on its own base with the topload on it. The joint capacitance, C2, sits where L2 meets L3.

Two coupling numbers, and only one of them performs

The coupling you wind is between L1 and L2. The coupling the machine behaves at is between L1 and the whole resonator, and they are not the same number:

k123 = k12 * sqrt( L2 / (L2 + L3) )

Two of the builders here write that relation down. Steve Ward puts it on his build page as K1:2:3=K1:2*√((L2)/(L2+L3)) before he winds anything. Queiroz derives it on the mailing list in answer to Gregory Peters, who had asked whether, "In the unoptimised version", the tighter primary to secondary coupling buys a magnifier anything over a two-coil system, and whose own simulations said it did not. Hold that qualifier: Peters asked about the untuned machine and nothing else, and the untuned machine is what most of this page is about. The answer is the whole subject in one sentence:

The tighter coupling is just a consequence of the small part of the secondary-tertiary system that is coupled to the primary.

The square root is always under one, so k123 is always below k12, and how far below is set by how the inductance was split. Queiroz gives the worked case: "if the original secondary coil inductance is split in a 8:1 ratio, k12=3k123"*, which is eight parts of the inductance in the resonator to one in the driver [derived: sqrt(1/9) = 1/3, and the formula returns his factor of three]. Put a headline coupling through that. A magnifier wound at k12 = 0.5 on that split behaves as a coil at 0.167 [derived: 0.5/3], which is an ordinary DRSSTC's number and not a remarkable one. The tight coupling is the entry fee, not the prize.

The relation carries a condition, and the condition is the whole of section one: Queiroz ends the sentence with "Of course this assumes negligible capacitance across L2." The equivalence between a magnifier and a two-coil machine holds exactly when C2 is not doing anything. Anybody designing with this equation is designing the untuned case on purpose. Hold that, because it comes back.

The claim it is usually set against comes from Dex, quoted by Paul Nicholson on the mailing list in 2010 and reproduced here as it stands in the archive, run-together full stop and all:

1.Overall coupling in many magnifier systems is in 0.2+ range.That means magnifier is more efficient than a typical coil which usually has problems when coupling is that high.

Nicholson, who had just posted a distributed model of the one documented tuned magnifier, agrees with the mechanism in two sentences: "That's surely the simplest explanation for magnifier performance. Simply a fast energy transfer being more efficient." It is the first of two points he answers, and he offers it as the simplest explanation rather than the proven one.

Note which number Dex is quoting. He says overall, which is the comparable one, so the claim is on the right axis and can be tested. It says a magnifier gets to live at an overall coupling that would give a two-coil machine racing sparks. Everything below is one machine being asked whether it did.

What Ward built, and what he gave up to run it

Steve Ward's DRSSTC Magnifier II was designed backwards from that argument, and he says so:

Firstly, I wanted the overall coupling (between L1, L2, and L3) to be about .25 as it would be for a 2-coil system. I then assumed the coupling between L1:L2 to be about .5.

So the divisor was fixed at 0.5 before the wire went on, which is a split with three quarters of the inductance in the tertiary [derived: 0.25/0.5 = 0.500, squared gives L2/(L2+L3) = 0.25, so L3 = 3*L2]. The winding lengths do not show it. His secondary is 19 inches of winding and his tertiary 21, nearly equal, while the inductance behind them splits one to three.

He did not get to keep 0.5. Flashover between the coils is what a driver coupling of that size buys, and it arrived:

After raising my secondary up 6" from its initial height, I was finally able to avoid flashover between the coils.

Writing to the list that night:

i did some calculations and it seems i had to reduce my K (L1:L2) to about .34, which yields an overall K (L1:L2:L3) of about .17. My systems seem to need a K of .22 or so to really perform

Both of those figures came through the same divisor. The 0.17 is the 0.34 put through the 0.5 he assumed at design time, so dividing it back out returns the 0.5 that was put in and discovers nothing about the built machine. What it does say, inside his own arithmetic, is that the whole fall from the designed 0.25 to the built 0.17 is driver coupling handed back to the insulation and nothing else. Neither figure is tight enough to carry more than that: both are his "about" numbers at two digits, and at 0.335 and 0.174 the divisor would come out at 0.52 [derived]. To reach the 0.22 he says his machines need, at that split, the joint would have to run at k12 = 0.44 [derived: 0.22/0.5], which is most of the way back to the 0.5 he could not keep.

He had already been dropping it before that night. In his own first-light post he writes "The coupling is also much too high" and plans to remove an inch or two of winding from the bottom of the secondary to cure it. Read that against the relation above: the bottom turns of L2 are the ones nearest the primary, so taking them off lowers k12, and it lowers L2, which is inside the divisor. The overall coupling falls twice for one cut, and it was already the number he said was too low. The two coupling figures on a magnifier do not have independent adjustments, and the one you can reach with a saw is the one that is not doing the work. That reading is ours; both of his sentences are his.

Which denominator

Ward's stated goal was "10-12 foot sparks from the smallest resonator possible". He did not get the sparks. The resonator was 21 inches and the spark came in short of the bottom of the range he was aiming at, and he says why on the build page: "the limiter was cutting in a lot at full power, limiting me to about 9' max". The 108 inches he did get divides two ways, on his own two numbers:

against the resonator alone   108 / 21  =  5.14   [derived]
against the total winding     108 / 40  =  2.70   [derived]

His own two-coil machine returns 3.02, and it returns 3.02 on both measures at once, because a two-coil machine's resonator is all of its winding. That is why the choice of denominator is a question you only have to answer about a magnifier. Answered one way Ward's machine wins by 1.70 times; answered the other it makes 10.7 per cent less spark for each inch of wire; nothing moved but the line under the 108 [derived, both from the block below].

This site takes the total winding. The driver is copper that had to be wound, tuned, insulated and stood upright; a ratio that leaves it out compares a machine against a part of a machine. The other reading is not dishonest, though, and it is worth knowing what it is for: if what you are buying is standoff, the distance from the spark to the electronics, then the resonator is the number you care about and the driver is allowed to be as large as it likes. Ward bought exactly that and says so: "I am happy to say that the sparks were like 10' away from the electronics". Queiroz names the same benefit and immediately qualifies it, which is the next section.

The part that decides whether you built one at all

Queiroz, on tuning the secondary system:

If no tuning of the secondary system is done, the system is just a regular Tesla coil loaded with a parasitic capacitance at the union of the secondary and tertiary coils.

That is the test, and it is a definition rather than an opinion: C2 tuned, and you have three modes and the transfer the design equations promise; C2 left to whatever the geometry gives, and you have a two-coil machine carrying a stray. He lists what people use to set it deliberately: a terminal on the secondary coil, a transmission line down to the base of the tertiary, another terminal at the base of the tertiary, or a high-voltage variable capacitor. And he puts the difficulty where the work is: "The main difficulty in making a real magnifier is finding a practical way to tune the secondary system, by adjusting the capacitance across the secondary coil."

Now put that beside the design equation two sections up, the one Ward states on his own page. It holds when the capacitance across L2 is negligible. A magnifier designed through that relation is designed as the untuned case, and by Queiroz's definition the untuned case is a regular Tesla coil with its resonator wound in two pieces. That is not a criticism of the build. It is a statement about what the head-to-head below is evidence for, and putting the two together is ours: Ward's equation is on his page and Queiroz's condition is in his message, thirteen years apart.

Ward's machine had a C2 whether he designed one or not. There is a corona ring on top of the secondary, and when corona ran the length of the tertiary he "stuck a spare toroid under it, and that seemed to tame everything". He reached for that toroid to stop corona, and says so twice; the flashover was a different fault on the same machine and he cured it by raising the coil. But a terminal at the base of the tertiary is item three on Queiroz's list of ways to build the tuning capacitance, so whatever else the toroid did it moved C2. That last step is ours. The toroid, the corona and the reason for both are his.

Somebody reaching for the same lever on purpose has the same trouble telling what it did. Acid Byte's magnifier, a spark gap machine rather than a solid state one, runs a resonator at 375 kHz off a primary at around 358 kHz, and the entry that matters here is the one where he "added terminal capacitance to the secundairy coil" and reports the spark "more fluid and bigger now". Take that as an illustration and not as evidence. The same post opens "Changed the topload for a step bigger on the resonator", which is the other thing that grows a spark, so two variables moved in one edit; no measurement went with either, so it is a builder's eye and not a figure; and he nowhere says he was tuning the secondary system. What is on the record is that he added capacitance at exactly the place Queiroz says decides the machine.

Queiroz built the tuned case himself and it did not go well either. His 3:4:5 magnifier tuned close: "The three resonance frequencies were measured as 232, 307, and 385 kHz. The ideal values would be 228, 304, and 380 kHz." Then, in his words, "The performance of the system in a full power test, however, was not as good as the performance of a conventional Tesla coil with the same elements." The lumped C2, a pair of Leyden jars made from plastic cups, was "evidently consuming significant energy, with intense corona around the edges of the plates, and becoming warm quickly". Take that capacitor out, leaving only the distributed capacitance, and the system "falls close to a higher mode" and the performance came up "close to the performance of my Tesla coil". So the designed mode lost to his two-coil machine and the accidental one drew level with it. No spark lengths were published, so this is a second comparison without a table, going the same way. He offers three causes and rules none of them out, closing with "Further tests will reveal what is happening." They are the capacitor's own losses, a spark gap that may not quench when the notches are narrow, and a structural one worth carrying away. For the same peak output voltage a three-resonator system delivers less rms voltage than a two-resonator one, "0.4341 for mode 3:4:5" against a flat 0.5 while the gap conducts, which is 13 per cent less [derived: 0.4341/0.5]. A streamer is grown by what arrives, not by the peak, so a magnifier that hits the same volts can still make the shorter spark. That is his mechanism and his arithmetic; whether it is the one that decided his bench is not established.

The numbers

One builder, two machines, and only one of them ran on the night. Ward measured the magnifier on 30 April 2005 and quotes his two-coil machine from earlier sessions in the same message: the 136 inches carries no date, and the 120 inch run is explicitly "the last run". Everything not marked derived is quoted from his message to the list or from his build page.

Ward's magnifier, measured that night
  spark                    108 in    2.74 m   [derived, 25.4 mm/in]
  secondary winding         19 in
  tertiary winding          21 in
  total winding             40 in
  spark / total winding    2.70               [derived]
  spark / resonator        5.14               [derived]
  driver coupling k12      0.34
  overall coupling k123    0.17
  primary current limit    850 A

Ward's two-coil machine, recalled from earlier runs
  spark                    136 in    3.45 m   [derived, 25.4 mm/in]
  winding                   45 in
  spark / winding          3.02               [derived]
  an earlier run           120 in    [derived, 12 in/ft, from his 10']
                                     under 800 A, k = 0.24
  spark / winding          2.67               [derived, assuming 45 in again]

what he says he needs
  overall coupling         0.22

His own verdict on those figures, in the same message:

I think 9' is pretty good for the total length of windings involved. 19" for the secondary and 21" for the tertiary coil give 40" winding for 108" of spark... not shabby. The 2-coil system did 136" spark vs 45" winding, so its still winning! But, i had my current limiter set a bit "tight" tonight at 850A, so the maggy couldnt really let loose.

And what falls out of the arithmetic:

The magnifier makes 10.7 per cent less spark per inch of winding [derived: 1 - (108/40)/(136/45)].
His own two two-coil results are 11.8 per cent apart from each other [derived: 1 - 120/136]. That is wider than the gap the head-to-head found, but only just, and it is not a noise figure: the two runs are not repeats. The 120 inch run is stated at under 800 A with k = 0.24 and the 136 inch run carries neither, so the spread measures whatever he changed between them and cannot be used to size the significance of anything.
Against that earlier run the magnifier is level, 2.70 against 2.67 [derived, and it assumes the two-coil winding was the same 45 inches both times, which he does not say].
The overall coupling is 23 per cent below what he says he needs [derived: 0.05/0.22], and below the 0.2 or better that Dex says many magnifier systems run at. On both readings the machine under test was not at the coupling the argument for magnifiers rests on.

Three things about that block before it is quoted anywhere. The two spark figures were not gathered the same way, and may not have been gathered on the same evening: the 108 was a horizontal strike to metal caging measured exactly, and the 136 has no stated method and no stated date. The 0.17 is a calculation, not a reading: it is k12 times a divisor he assumed at design time, which is the same defect the coupling page documents across eight published figures. And the limiter was cutting in, so the magnifier's row is not the machine at full stretch.

How much was left in it is not computable from what he published. He puts the input voltage at under three quarters and the burst at about 12 cycles at 38 kHz, which is 0.32 ms of run time [derived: 12/38 kHz], but the current limit is what was setting the energy and no primary inductance is published to turn amps into joules. The one comparison that survives is between the two currents, and it runs the wrong way for the magnifier: at equal primary inductance the tank energy goes as the square of the current, so 850 A against under 800 is at least 13 per cent more energy in the tank for a spark 10 per cent shorter than that run's [derived: (850/800)^2 and 108/120, and at n = 0.31 to 0.33, the arc-length exponent this site uses, the extra energy is worth about 4 per cent of arc length]. Whether the two machines shared a primary is not stated, so read that as indicative and not as a result. His own summary of the efficiency question is the fairest line in the thread: "Im not 'sold' on the magnifier for its efficiency yet. In fact, i find the efficiency to be somewhat less i think than normal. But it might come down to using a non-optimized setup."

What will get you

Plan the insulation before you wind, and size it by the voltage the driver will hold off rather than by the coupling figure. A k12 of 0.34, or of 0.5, is a real 0.34 or 0.5 between two windings a few inches apart. On Ward's machine that bit: he raised his secondary six inches to stop the coils flashing over, and he was on the list hunting 60 mil polyethylene sheet in his own words "for stuffing between the primary and secondary of my new magnifier". It does not bite everywhere, and the coupling number alone will not tell you whether it will. Queiroz built a driver at k12 = 0.3504, a flat primary under a short solenoid, and writes that the arrangement "produces the required quite high coupling, but doesn't present serious insulation problems"; his machine is a spark gap coil fed from a neon sign transformer, Ward's is a DRSSTC running 850 A in the primary. So the two machines on this page at essentially the same coupling went opposite ways: 0.34 bit Ward and 0.3504 did not bite Queiroz, and the one attempt at 0.5 flashed over outright. Three builds, and the coupling figure sorts none of them. What separates the two at 0.35 is everything else about them, the source, the current and the spacing, and none of that travels with a k.

The joint capacitor is a high-voltage part and it is easy to underrate. Queiroz's simulations of his own machine put 26 kV across C2 for a gap voltage of 5 kV during the transfer, "and somewhat more after the gap quenching", which is over five times the input [derived: 26/5]. His first attempt at building it, a pair of plastic-cup Leyden jars, coronaed at the plate edges and warmed up quickly. A capacitor that dissipates at the joint is sitting exactly where all the energy has to pass.

Corona along the tertiary is not cosmetic. Ward saw it end to end on L3 at first light and treated it as a coil about to short itself out, which is what it is. A toroid at the base of the resonator fixed it.

The thing that is genuinely safer is worth naming, because it is the reason to accept the rest. The spark leaves a coil that is not standing on the primary, so the metres between the arc and the electronics are metres you designed in rather than metres you hoped for. Queiroz allows it and attaches the caveat in the same message: a magnifier "may have better insulation between the high-voltage terminal and the primary system than a regular two-coils system, but is not optimized." Read that as a licence to put the driver where you want it, not as a licence to stand nearer.

What goes wrong

The spark is no longer than a two-coil machine of the same total winding. Then it is doing what the one measured comparison did, and the ratio to check is spark over all of the winding, not spark over the resonator.
The overall coupling ends up below what your two-coil machine wanted. You gave k12 back to the flashover margin, and the divisor took its share of what you gave. Compute k123 after every mechanical change, not before.
Removing turns from the bottom of the driver makes it worse, not better. It lowers k12 and L2 at once, and both of them lower k123.
The machine behaves exactly like a two-coil machine. Then it is one. If C2 was never tuned, Queiroz's definition says what you have is a regular coil with a parasitic capacitance at the union, and no amount of driver coupling changes that.
Performance drops when you add the tuning capacitor. Queiroz's case: the capacitor itself was eating the energy. His machine ran better with the lumped part removed and only the distributed capacitance left.
The three measured frequencies are not in the ratio you designed for. Expect this, and expect it to be hard to fix. Nicholson, who modelled the one well-documented tuned system, calls it "quite a problem to know which to adjust and by how much", with three variables available: topload height, C2, and the primary tap.
A simulation says the tighter coupling buys nothing. For the untuned machine it is right, and the untuned machine is the only thing it was asked about. Gregory Peters put the question as "In the unoptimised version, does the tighter primary:secondary coupling offer any performance increase over a regular two coil system? My simulations don't suggest it does", and Queiroz's answer is the divisor: the tighter coupling is a consequence of coupling into part of the resonator, not a gain over a coil that couples into all of it. Neither of them says the same about a tuned machine, and neither does this page.

Where next

Coupling: what it buys, and why it is not length, which is where the ordinary bands come from and why the longest arcs are not the tightest coupled. Everything on this page is that argument with one extra divisor in it.
Critical coupling, and why every coil is far above it, for the absolute mark under both numbers.
The two poles, and which one the driver lands on, which is what two coupled resonances do to a driver. A magnifier has three, and the mailing list's word for choosing among them is a mode rather than a pole.
How to wind and measure a Tesla coil secondary, because in a magnifier that page is about L3 and the driver is a different part with a different job.

Nothing on this page was measured here. The head-to-head is Steve Ward's, his magnifier measured on the night of 30 April 2005 against two-coil figures he recalls in the same message; the relation between the two coupling figures is written by Ward and derived by Antonio Carlos M. de Queiroz; the case for magnifiers is Dex's and Paul Nicholson's; the tuned build that lost to its own maker's two-coil machine is Queiroz's. The ratios and the percentages are arithmetic on the builders' own figures and are marked where they appear.

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