On poles and detuning in QCW coils

[Anders Mikkelsen]

In my recent work with ramped double resonant coils, I’ve looked deeper into the topics of poles, detuning, tank impedance and coupling. It was not clear to me at first how all these factors relate, but I’m getting more insight after a good amount of testing and simulation. I thought I’d document my findings here before I lose interest, if only to serve as a reference for myself while it’s still fresh. This is going to get a bit heavy and long, so let's start by laying some groundwork and defining some terms.

We assume that the primary tank is being driven by a voltage source, while using current feedback to control the switching. When I talk about the impedance of the tank at a given pole, this has two practical meanings. Firstly it determines how much current the system draws from the inverter, and secondly it determines how much gain the self oscillating feedback has. Since the absolute impedance depends on arc loading, I will mainly be looking at the relative impedance between the poles. This allows analysis of which pole a system will oscillate at, and also look at how each pole responds to arc loading.

Then what is the significance of the impedance the inverter sees? This is key to optimal Tesla coil design. If the impedance is too high, the coil will simply not draw enough power to give good output. And if it's too low then the OCP will trip, also limiting performance. Or if there is no OCP, then the bridge will not last for very long.

So my goal here is to try to define how all of these parameters fit together, in order to facilitate a systematic design process

With QCW coils, there are two main goals.

1: To ramp up the primary current slow enough that an arc has time to form, in order to clamp the secondary voltage to a low value. This allows long sparks from a short resonator without the risk of flashover across the secondary, and also allow high coupling to be used.

2: to control the ramping in such a way that sparks don't branch, allowing all the primary power to go into a single streamer and maximizing the arc length for a given primary peak power.

I've seen speculation that linear arc growth happens with a constant streamer tip velocity, and this matches experimental evidence, so we'll work from this assumption.

QCW coils seem to follow the Freu relation, with arc length proportional to the square root of power, so if we want linear tip velocity the power has to ramp quadratically with time.

When operating on the upper pole, capacitance from arc loading lowers the reflected arc impedance, independent of the relative tuning between the primary and secondary frequencies. On the lower pole the opposite happens, arc loading decreases the reflected arc impedance. When driven from a voltage source, the upper pole draws more current with increasing voltage, while the lower pole draws less.

I used to think that a coil with a primary tuned below the secondary would draw more power with arc detuning, as the secondary would be pulled closer to the primary resonant frequency, but this appears to not be the case from my simulations and measurements. It only depends on which pole we are operating at. To illustrate, I have my test coil implemented in LTSpice, allowing me to play with some values to demonstrate the effect.



If we start with the primary and secondary tuned at the same frequency, this is the primary impedance with and without 8 pF of arc loading. I have not modelled the resistance of the arc, in order to see the relative effect of capacitance on the poles.



Blue is without arc loading and green is with. Here we can see that the gain at the upper pole increases with capacitance, while the gain at the lower pole decreases. Note that the upper pole has more gain even when the coils are perfectly in tune. Lower impedance corresponds with higher gain here since we're driving it from a voltage source and the current is what we care about.



Now we have the primary tuned lower than the secondary. Now the lower pole has more gain, and gain at the upper pole increases with loading, compared with lower pole gain reducing.



This last plot is with the primary tuned higher than the secondary. This increases relative gain at the upper pole, and gain still increases at the upper pole compared to the lower pole when we add arc loading. So does this also happen in practice when we consider the resistive aspects of the arc? I've done some scope captures in operation that suggest that it does. This is a measurement of coil operating parameters when the system was on the border of switching poles, modulated by mains ramping. The first burst starts off at the upper pole but quicly jumps to the lower pole due to excess gain there (20 dB+ relative to the upper). The current envelope, channel 2, shows current decreasing with rising primary voltage (bus voltage on channel 3, bridge output voltage on channel 1). For the next burst, it managed to stick on the upper pole, and primary current increases with rising bus voltage as seen by the envelope shape.



This explains why the upper pole seems to work so much better for linear spark growth; the coil naturally draws more current as the bus voltage is increased, giving something resembling a quadratic power ramp with a linear voltage ramp. It should still be possible to get sword sparks on the lower pole, but it probably requires sharper ramping and more careful tuning of the ramp shape. There is also another issue with lower pole operation as we will see later.

So what's the significance of primary vs. secondary tuning? The main effect is changing the relative impedance of the upper and lower poles. Tuning the primary lower will increase the gain at the lower pole, while tuning it higher will increase the gain at the upper pole. When the primary and secondary are tuned to the same frequency, both poles have similar impedance, with the upper pole having somewhat more gain, around 6 dB with the coupling I’ve been using (0.35).

So how do we choose how to tune the primary in relation to the secondary? Independently of the tuning, the inverter sees a resistive load at either pole, so we are free to choose the amount of detuning without any penalty on how much power is drawn from the inverter and delivered to the arc, but this only applies if the system has no losses. If we assume operation at the upper pole, there are three practical consequences of detuning the primary:

1: It changes the relative impedance of the pole, requiring the primary impedance (sqrt(Lpri/Cpri)) to be adjusted in order to get the target power draw from the inverter

2: It changes the apparent Q factor for a given spark load, which is the ratio of reactive to real power in the tank. This has direct consequences for the tank circuit sizing, as a higher Q means higher voltage across the primary capacitor and coil for the same power processed. Therefore it also has a direct impact on both losses and dimensioning of the primary circuit. This might be less consequential for smaller coils where the resonant capacitor is moderately sized, but it quickly becomes a major cost if we want to push QCW coils beyond a few hundred kilowatts peak.

3: It determines which pole a self-oscillating driver will operate at, due to the circuit wanting to oscillate at the pole with highest gain.

So from a perspective of making the coil most efficient, we want the primary tuned close to the secondary in frequency. The optimum seems to be when the primary frequency is close to the secondary frequency considering arc loading. This is not possible for a plain self-oscillating (UD2.7 and similar) driver, as the lower pole would have more gain without spark detuning.

This is where the advantages of PLL drivers show up, as we can limit the operating range to never lock to the lower pole, allowing tuning for minimum primary VARs and minimum primary and secondary losses. When compared with a PLL driver, an UD2.7 would need more gain at the upper pole, which means that the primary needs to be tuned above the loaded secondary frequency. This lowers the impedance of the upper pole, so we need to increase the tank impedance to keep the inverter current below the limit of the bridge. And higher primary impedance means more voltage across the primary capacitor and coil for a given primary current.

The self-oscillating driver as described in [] falls somewhere in between, as it allows the primary to be tuned lower without losing upper pole lock. For the UD style driver, the upper pole needs to have some excess gain without spark loading (maybe 6 dB? I did not quantify this as I haven’t experimented with it), which in practice means the primary has to be tuned at or above the secondary unloaded frequency. For the self-oscillating driver, it still manages to stick to the upper pole with 14 dB of excess gain at the lower pole, but not with 20. This allows the primary to be tuned lower, reducing primary impedance for a given load current and therefore also losses.

With a PLL style driver, the primary can be tuned even lower, in order to match the loaded secondary frequency or even below. Tuning the primary even lower increases primary VARs again, but it can be a useful tool to raise the primary impedance at the upper pole without rewinding or tapping the primary coil. This trick has been used by several people I talked to. The extreme case of this is tuning the primary so low that the coil behaves more like a single resonant coil, which can work very well if the coupling is high enough.

On the topic of coupling, how do we choose this? I see no reason to not make it as high as practical, as higher coupling reduces the loaded Q of the system, decreasing losses as well. The upper limit comes from trying to avoid flashovers, so it’s a tradeoff between practical geometries and the performance advantage. Ferrite can be used inside the secondary (or outside the primary) to increase it, with a clear cost and benefit.

One further point to make is that, since the relative upper pole gain rises with arc loading, the driver tends to stick on this pole as arc capacitance increases its gain relative to the lower pole. On the lower pole, the gain is reduced relative to the upper pole with arc loading, making pole jumping a bigger risk with feedback drivers. Lower pole operation also modifies the secondary voltage distribution in such a way as to make primary-secondary flashovers more likely, so this further emphasizes upper pole operation as the right choice for QCW operation.

This all agrees with the common recommendation of “upper pole operation, primary tuned below the secondary”, but it also explains by how much the primary should be detuned and why it’s needed and advantageous.

So where does that all lead us? With these observations, we can develop a systematic design procedure for optimal QCW type coils, which I will get into in a later post.

This is mostly a collection of statements, but I'm of course open to being incorrect here, so any further input, corrections or clarifications are appreciated.

[Continuum]

Hello!
First of all, hats off to your research, I follow it closely with great interest!

Very insightful post backed up by some great simulation results and measurments, but I have some trouble understanding what you say about changing the primary tank impedance to limit the primary current:

This lowers the impedance of the upper pole, so we need to increase the tank impedance to keep the inverter current below the limit of the bridge.

This fragment i dont fully understand - isn't it the case that at resonance only the loss resistance of primary coil + transformed resistance of the secondary coil (R₁ + R₁') remain?
Resistance transformed into the primary from the secondary is R₁'=R₂*omega²*M²/Z₂², where notation 2 denotes a secondary coil parameter and 1 a parameter of the primary coil.
Equation taken from: https://books.google.com/books/about/Poradnik_radioamatora.html?id=yXRDNAEACAAJ&redir_esc=y (book sadly not availble in english and rather hard to find in general, im lucky to have a copy)

In such case changing the reactances of the primary tank while maintaining the same tuning wont affect the resonant resistance but the reactive power in the primary tank (resonant voltage rise on the L and the C), thus the Q, and with that the time constant of the resonant circuit - the time it takes for the current to ring up. With higher Q it takes longer for the current to ring up, so it might somewhat slow down the onset of OCP, but i dont see how it would limit the current overall..

Anyway, thank you very much for sharing your findings, they inspire me (and many others im sure) to keep coiling and tinkering! Kind regards

[Anders Mikkelsen]

Hello!
First of all, hats off to your research, I follow it closely with great interest!

Very insightful post backed up by some great simulation results and measurments, but I have some trouble understanding what you say about changing the primary tank impedance to limit the primary current:

This lowers the impedance of the upper pole, so we need to increase the tank impedance to keep the inverter current below the limit of the bridge.

This fragment i dont fully understand - isn't it the case that at resonance only the loss resistance of primary coil + transformed resistance of the secondary coil (R₁ + R₁') remain?
Resistance transformed into the primary from the secondary is R₁'=R₂*omega²*M²/Z₂², where notation 2 denotes a secondary coil parameter and 1 a parameter of the primary coil.
Equation taken from: https://books.google.com/books/about/Poradnik_radioamatora.html?id=yXRDNAEACAAJ&redir_esc=y (book sadly not availble in english and rather hard to find in general, im lucky to have a copy)

In such case changing the reactances of the primary tank while maintaining the same tuning wont affect the resonant resistance but the reactive power in the primary tank (resonant voltage rise on the L and the C), thus the Q, and with that the time constant of the resonant circuit - the time it takes for the current to ring up. With higher Q it takes longer for the current to ring up, so it might somewhat slow down the onset of OCP, but i dont see how it would limit the current overall..

Anyway, thank you very much for sharing your findings, they inspire me (and many others im sure) to keep coiling and tinkering! Kind regards

Thanks for the feedback!

For the equation, what is M here? Is this mutual inductance?

I did a quick simulation, and modifying the primary characteristic impedance with everything else being the same, only modifies the reflected resistance of the secondary loading but otherwise the behavior of the tank is the same. Doubling the primary impedance also doubles the reflected load resistance at the pole frequencies. To have the same power transfer, this would require 1/sqrt(2) times the current and sqrt(2) times the voltage. Since primary VARs scale with I^2, the reactive power as seen by the tank components is therefore the same. As Q is the ratio of reactive to active power, it will also stay the same.

All of this assumes a fixed resistive load on the secondary, while the arc load is more complex and dynamic, with both resistance and capacitance that depends on the arc size and shape. The conclusions should still hold though.

[Continuum]

For the equation, what is M here? Is this mutual inductance?

Yes, sorry, forgot to include that in my original post.

I did a quick simulation, and modifying the primary characteristic impedance with everything else being the same, only modifies the reflected resistance of the secondary loading but otherwise the behavior of the tank is the same

Im entirely guilty of not simulating this - I stuck to writing down equations on paper, hence my question. I missed an important detail while analysing the equation i mentioned - with the increase of the primary impedance, the mutual inductance grows as well, making R₁' increase, which is in line with your observations.
I now opened LTSpice as well, copied your setup, and im surprised how accurate that doubling of the reflected secondary impedance is when the primary characteristic impedance is doubled. its bang on!

As Q is the ratio of reactive to active power, it will also stay the same.

I simulated this as its seriously unintuitive to me - you are right!
Pic shows the calculated Q value for your coupled coils setup along with the standard impedance plot.

Head scratcher, with the coils coupled the Q remains the same, once decoupled the Q rises (as id expect) with higer characteristic impedance. This does make a lot of sense, but one has to abandon their intuitions which would hold for standard single resonant circuits. Coupled coils keep proving to be difficult to intuitively understand Thank you for some valuable insight!

[davekni]

Great thread!  I've always used LTSpice to optimize coil designs.  Hardest part is guessing/modeling arc impedance accurately enough.  Helpful wider view of optimization here.

The self-oscillating driver as described in [] falls somewhere in between, as it allows the primary to be tuned lower without losing upper pole lock.

With sufficiently high coupling, self-oscillating driver works fine at upper pole with optimum detuning.  I've tested both PLL and self-oscillating drivers for my QCW coil.  After giving up on phase-shift ramp generation (since using IGBTs at 480 to 440kHz), I now use simple self-oscillation driver with external buck converter ramp generator.

I've seen speculation that linear arc growth happens with a constant streamer tip velocity, and this matches experimental evidence, so we'll work from this assumption.

QCW coils seem to follow the Freu relation, with arc length proportional to the square root of power, so if we want linear tip velocity the power has to ramp quadratically with time.

Makes sense.  However, my bit of experimenting gave better non-branched arc performance with a linear Vbus ramp than with adding any significant quadratic term to Vbus ramp.  Optimum power seems to be somewhat less than quadratic since bus current doesn't increase as steeply as does voltage.

[Uspring]

Your diagrams show nicely, that gain depends mostly on the proximity between operating frequency and secondary resonance frequency. The secondary resonance is located at the peak between the poles.
The performance of the coil is (when it is run at one of the poles) determined by the location of the poles relative to the secondary resonant frequency. Both poles lie outside of the frequency range between primary and secondary resonance. If e.g.the primary is tuned lower than the secondary, the upper pole will be closer to secondary fres than the lower pole. So the upper pole has an advantage.
Also, when arc loading decreases secondary fres, the upper pole will follow it downwards, so detuning is less of a problem.

It is, though, possible to run at the lower pole. It will generally require higher primary currents than an upper pole coil, but that can be resolved by increasing primary tank impedance. This is at some cost in efficiency, since more turns increase losses.

[Anders Mikkelsen]

Your diagrams show nicely, that gain depends mostly on the proximity between operating frequency and secondary resonance frequency. The secondary resonance is located at the peak between the poles.
The performance of the coil is (when it is run at one of the poles) determined by the location of the poles relative to the secondary resonant frequency. Both poles lie outside of the frequency range between primary and secondary resonance. If e.g.the primary is tuned lower than the secondary, the upper pole will be closer to secondary fres than the lower pole. So the upper pole has an advantage.
Also, when arc loading decreases secondary fres, the upper pole will follow it downwards, so detuning is less of a problem.

It is, though, possible to run at the lower pole. It will generally require higher primary currents than an upper pole coil, but that can be resolved by increasing primary tank impedance. This is at some cost in efficiency, since more turns increase losses.

That's definitely an interesting way to analyze it, by considering the relative proximity of the uncoupled frequencies. What was not obvious to me before is that the secondary frequency moving away from the upper pole actually increases the upper pole gain when fed from a voltage source, facilitating quadratic power ramping and linear spark growth, while the opposite happens on the lower pole. Do you have any insight on whether quadratic ramping is optimal for linear spark growth? It does seem that way from my experiments, as I found it very hard to grow linear sparks on the lower pole when ramping the bus voltage.

Great thread!  I've always used LTSpice to optimize coil designs.  Hardest part is guessing/modeling arc impedance accurately enough.  Helpful wider view of optimization here.

Thanks! Hopefully this can serve as a starting point for further analysis. The behavior of the arc load is still a big uncertain, and it would be great to investigate how this affects the behavior in greater detail.

The self-oscillating driver as described in [] falls somewhere in between, as it allows the primary to be tuned lower without losing upper pole lock.

With sufficiently high coupling, self-oscillating driver works fine at upper pole with optimum detuning.  I've tested both PLL and self-oscillating drivers for my QCW coil.  After giving up on phase-shift ramp generation (since using IGBTs at 480 to 440kHz), I now use simple self-oscillation driver with external buck converter ramp generator.

I think my coupling is a bit on the low side for this, but it's good to hear that it can work better with higher coupling. There's also some freedom in tailoring the phase lead network to further suppress gain at the low pole, which becomes easier when higher coupling moves it father from the upper pole.

I've seen speculation that linear arc growth happens with a constant streamer tip velocity, and this matches experimental evidence, so we'll work from this assumption.

QCW coils seem to follow the Freu relation, with arc length proportional to the square root of power, so if we want linear tip velocity the power has to ramp quadratically with time.

Makes sense.  However, my bit of experimenting gave better non-branched arc performance with a linear Vbus ramp than with adding any significant quadratic term to Vbus ramp.  Optimum power seems to be somewhat less than quadratic since bus current doesn't increase as steeply as does voltage.

Interesting observation.



Looking at my scope shots, it seems like the current is growing faster than linear with the bus voltage. I suspect this is caused by arc branching, due to the mains ramp being too fast for an arc of this length to grow linearly. I'm working on a buck modulator now, to see if the current stays lower if I suppress branching while keeping everything else the same.

For the equation, what is M here? Is this mutual inductance?

Yes, sorry, forgot to include that in my original post.

I did a quick simulation, and modifying the primary characteristic impedance with everything else being the same, only modifies the reflected resistance of the secondary loading but otherwise the behavior of the tank is the same

Im entirely guilty of not simulating this - I stuck to writing down equations on paper, hence my question. I missed an important detail while analysing the equation i mentioned - with the increase of the primary impedance, the mutual inductance grows as well, making R₁' increase, which is in line with your observations.
I now opened LTSpice as well, copied your setup, and im surprised how accurate that doubling of the reflected secondary impedance is when the primary characteristic impedance is doubled. its bang on!

As Q is the ratio of reactive to active power, it will also stay the same.

I simulated this as its seriously unintuitive to me - you are right!
Pic shows the calculated Q value for your coupled coils setup along with the standard impedance plot.
Head scratcher, with the coils coupled the Q remains the same, once decoupled the Q rises (as id expect) with higer characteristic impedance. This does make a lot of sense, but one has to abandon their intuitions which would hold for standard single resonant circuits. Coupled coils keep proving to be difficult to intuitively understand Thank you for some valuable insight!

Thanks for looking into this! I'm guilty of relying too much on simulation due to having a less strong theoretical background.

[davekni]

Looking at my scope shots, it seems like the current is growing faster than linear with the bus voltage. I suspect this is caused by arc branching, due to the mains ramp being too fast for an arc of this length to grow linearly.

Yes, you are seeing quite different behavior than on my QCW coil.  Perhaps my QCW is too atypical with its relatively low Q (lossy) primary coil.  Will be interesting to see how current behavior changes with a slower ramp.
If it is of interest, below thread includes plots from my QCW coil including charge measurements of top load and breakout point along with primary I and V.  These are for ~20ms ramps.  My house is too small for longer ramps and I don't have any measurements from runs at other locations.  I didn't see any huge change with branching or not.  No captures of fast ramps, however.
   https://highvoltageforum.net/index.php?topic=1950.msg17747#msg17747

[Uspring]

Do you have any insight on whether quadratic ramping is optimal for linear spark growth?

I agree with you, that power ramping such that the arc grows with constant velocity is optimal. It boils down to the question how arc length relates to its power consumption. The empirics are not conclusive, though. There is e.g. the Freau formula, i.e. L ~ sqrt(P). Then there are Davids measurements, which suggest a different dependency:

They show a very small dependency of arc length on voltage, which implies a very high conductivity of the arc almost all the way from breakout to the tip. The arc resembles a piece of wire. The current into the arc is thus approximately proportional to its length, since the capacitance of a piece of wire is proportional to its length and the current is caused by the capacitance. The current voltage product of the arc is thus roughly also proportional to the length, leading to a linear relation between length and power.
There are a number of caveats to this argument, one being, that wire capacitance depends on wire thickness (albeit only logarithmically) and arcs are thicker near the breakout point. Also, the voltage measurement, derived from charge flowing into the top load, is a bit distorted by charges carried by the arc itself. Still I think, that Freau needs to be corrected for in the case of QCW arcs.

Then there are measuremnts from Steve Ward. He measured length versus power input to his coil for QCW arc and found a nearly linear relationship up to 2 m of arc length, i.e. about 18 kW/m. He later built a much more powerful QCW, but there this linear relationship failed. One of the problems was, that he needed to speed up the arc growth speed, since for very long bursts the arcs started to curl near the breakout, an effect, which was seen by others, including David. But there might be other reasons. I'd think a slightly above linear power ramp is your best bet at < 2m.

[Mads Barnkob]

Do you have any insight on whether quadratic ramping is optimal for linear spark growth?

They show a very small dependency of arc length on voltage, which implies a very high conductivity of the arc almost all the way from breakout to the tip. The arc resembles a piece of wire. The current into the arc is thus approximately proportional to its length, since the capacitance of a piece of wire is proportional to its length and the current is caused by the capacitance. The current voltage product of the arc is thus roughly also proportional to the length, leading to a linear relation between length and power.
There are a number of caveats to this argument, one being, that wire capacitance depends on wire thickness (albeit only logarithmically) and arcs are thicker near the breakout point. Also, the voltage measurement, derived from charge flowing into the top load, is a bit distorted by charges carried by the arc itself. Still I think, that Freau needs to be corrected for in the case of QCW arcs.

In my dangerous experiment of a "qcw", I did a wire simulation and it is a very practical and easy simulation with near-arc-like-results.

https://kaizerpowerelectronics.dk/tesla-coils/kaizer-drsstc-iv/

Secondary circuit test results
Setup with a 80 cm long wire with 3 bend wires hanging over and pointing down to be “branches”. Signal from signal generator connected to ground terminal on the secondary coil, ground left floating. Signal into oscilloscope captured from open loop probe hanging next to secondary coil.
Unloaded result: 101 kHz, 80 cm wire result: 91 kHz and 80 cm wire with branches result: 88 kHz.