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Based on a recent measurement and some theory I've created an LTSpice model to predict arc current from arc voltage. Here are some of the results: Below are diagrams of a measurement of the difference (blue) of the secondary top current and the arc current. The current in blue shows therefore only that part going into the toroid capacitance. It is thus proportional to the top voltage. The measured arc current is shown in red. The measured voltage served as an input to the simulations.
High power measurement, a vertical unit is 1A for the toroid current (blue) and 0.5 A for the arc (red). Horizontal unit is 0.33us.
High power simulation
Medium power measurement, a vertical unit is 1A for the toroid current (blue) and 0.5 A for the arc (red).
Medium power simulation
Low power measurement, a vertical unit is 1A for the toroid current (blue) and 0.25 A for the arc (red).
Low power simulation
Waveform amplitudes and shapes agree quite well, although there is a tendency to underestimate the current at high power. Possibly this is due to the way I made my measurements: At high power about half of the current going from the secondary into the toroid is due to the arc. The other half comes from the capacitance of the toroid itself. I single-triggered the DSO on a secondary current threshold, so a single very conductive and branching arc will make the DSO trigger first, as I ramp up the bus voltage. At lower power, i.e. trigger thresholds set lower, the arc current is smaller compared to the current due to the toroids capacitance and the effect of particularly conductive arcs on the triggering is less.
The circuit for simulation is this:
On the left is a DRSSTC circuit. The right is the arc model. The basic idea is, that an arc can be modelled by a resistive wire. The capacitance is kept constant per unit length and the resistance depends on the power being dissipated in the wire. The power will heat up the wire (aka arc channel) and also create free electrons and ions, which makes it more conductive. When you run the model, you can watch the arc grow by looking at the voltages down the arc chain. Every resistor is about 8cm of the arc. Actually you will observe small voltages arbitrarily far down the chain. Steve Ward has made the observation, that arcs require at least 40kV to grow, so that the arc probably stops growing, when the voltage at its end drops below that value. You can also simulate ground arcs by shorting the chain to ground after the voltage level has reached a certain point along the arc.
The heart of the model is the sub circuit shown here:
The arc resistor is the current source G1. The current is given by
I = (V(N1)-V(N2)) * (4e-7 + 6e-11*V(N4)*V(N5))
The first factor, V(N1)-V(N2) is the voltage across it, so the current source acts as a resistor with a resistance of
R = 1/(4e-7 + 6e-11*V(N4)*V(N5))
The voltage source B2 calculates the power, which is averaged by 2 RC circuits. R1 and C1 have a long time constant of 300us. R2 and C2 have a short time constant of 20us. They were inspired by the fact, that the arc conductivity seems to have a quickly responding part as seen in the wiggles of current in the high power measurement and a slowly responding part at the start of the arc. Probably the long time constant accounts for the arc channel heat up and the short one reflects ion lifetimes. V(N4) and V(N5) are the time averaged powers. The constant 4e-7 allows the arc to start growing. Initially the power averages are 0 and with an infinite initial resistance, there would be no power dissipated and the arc model would not start. The bigger this value is, the earlier the arc will start growing. The constants 4e-7 and 6e-11 were chosen to get a close as possible match to the measurement. The choice of the arc capacitances depends on the calculated length of the arc. Each RC element in the arc stands for a certain piece of arc length. A bigger C will make the arc shorter. So basically the choice of C will change the scale of the simulation, i.e. the amount of arc length one RC element stands for.
Below is the result of a complete simulation, i.e. arc model and DRSSTC circuit. Note, that the diagrams above have not been made with a DRSSTC circuit, but with a voltage input to the arc derived from the measurements.
The initial rise and the later dip in voltages and currents are similar to measurement. The later oscillations in the amplitude are missing, though, and also the max current is too low, 1.8A instead of the measured 2.4A. I believe the discrepancies are partially due to the incorrectly simulated phase shifts between arc voltage and current. In reality, the phase changes between 60 and 45 degrees, while in the simulation, they stay in the vicinity of 60 degrees. In the simulation, the phase shift is a bit higher during peak currents, just as in the measurement but the predicted variations are much smaller. Also the arc seems to acquire strength too early. This is probably caused by too much conductivity at low power
The model here fails drastically when applied to the data of a measurement made on the Electrum by Greg Leyh. The current in the simulation is much too high and the simulated arc length too long.
Generally resistive wire type arc models show a voltage to current ratio coming from the total arc capacitance, i.e. length multiplied by capacitance per unit length. This is only weakly dependent on how arc resistance varies with power dissipation. I believe, that the capacitance per unit length of the Electrum arcs must be considerably lower than those of my measurements. I've tried using smaller capacitances in my model, but the arcs became even longer. That can be traced to too much arc conductivity in the model. The model is based on the conductivity being proportional to the square of the power dissipated. Probably the conductivity rises much slower with power for large arc currents. Measurements at higher powers may show a way in which to extend the model. I've included the schematics here in case you want to experiment with this. ]arc3circuit.zip[/file] For higher power levels additional arc RC elements should be added in order to accomodate longer arcs. Also note, that the measurements, the model is based on, used a breakout point. Breakout depends heavily on this.
Registered Member #30
Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706
Very interesting! Thanks for sharing I'll try running it against the simulations I've already done for my upcoming DRSSTC.
I thought the ion lifetimes would be tens of milliseconds, though. By changing the break rate of a Tesla coil, you can see that above about 100bps, the streamer is mostly reusing the channel from the previous bang, instead of starting with a fresh one. This shows that it must still be at least somewhat conductive after tens of milliseconds.
I would be interested to see how your modelled streamer length varies with break rate for constant bang energy. The experiments I've done show a "knee" around 100Hz and traditionally the most efficient coils in terms of spark length per watt input seemed to run at 100 or 120Hz. Higher break rates than this tend to make the arcs brighter rather than longer. You gain a lot more spark length going from 25Hz to 100, than you do going from 100 to 400.
I'll try running it against the simulations I've already done for my upcoming DRSSTC.
I'm very interested in how your simulations compare to mine. Please follow up on that.
I thought the ion lifetimes would be tens of milliseconds, though. By changing the break rate of a Tesla coil, you can see that above about 100bps, the streamer is mostly reusing the channel from the previous bang, instead of starting with a fresh one. This shows that it must still be at least somewhat conductive after tens of milliseconds.
The wiggles in current amplitude in the middle and at the end of the burst follow the voltage changes rather rapidly. In my measurement post, I had also plotted the conductivity of the arc and it seems to change closely following the voltage wiggles. Whether this is can be due to ion lifetimes I don't know, but it was the only explanation I could think of. I should check the literature on that. The BPS dependency you observed is probably coming from the heat remaining in the arc channel between bursts. Heat thins air and this causes lower breakdown voltages. I tried to make this part of the model with the longer (300us) time constant. Actually 300us is too short to account for the knee at 100 BPS. I'm not too sure about that 300us value. I've used 600us bursts in my measurement and I would need much longer bursts to pin that down. Thinking about this, I'm noticing an error I've made. My simulations were single shots, while the mesurement was made at 100 BPS. Probably the simulation will change somewhat, if I simulate 100 BPS, but as having said, the 300us time constant will create a knee likely only at much higher break rates.
@Dr. Dark Current:
I wonder if the arc model would work also for "QCW"-type sparks.
In principle it should work. Most QCWs run at higher frequencies, though, and I'm in doubt how well the capacitance values I've chosen fare there. The Electrum e.g., running at 35kHz, seems to be different. My capacitance values seem to be due to heavily branching arcs, since they are much higher than that of a straight wire. The branches are just lumped into one arc in the model. QCW's can make nice straight arcs.
Registered Member #2292
Joined: Fri Aug 14 2009, 05:33PM
Location: The Wild West AKA Arizona
Posts: 795
This is very cool, I have been working to get a decent spark model for some time now, you seem to have done this in spades!
Just a thought, I noticed that with most QCWs that the voltage and primary current grow up to a large amount just before they break out due to the toroid at which point they fall drastically and then start growing again, but at a slower rate. The toroid seems to stand off the voltage until it reaches a point in which it's high enough to ionize the air. In my simulation I modeled this as a TVS type setup with a breakdown voltage similar to what would be seen as the breakdown voltage at the toroid. In a SGTC or DRSSTC this effect is not really noticeable due to the fact that the first cycle has enough energy to break out from the toroid immediately, where a QCW needs 10’s to 100’s of cycles to get to the same level.
I'm also interested in what Steve is talking about. I have noticed this in my DRSSTCs for years. Low break rates tend not to generate as large of sparks. But there is a sweet spot, Steve's is at 100Hz, but I have noticed it from 80Hz up to 250Hz depending on the size of the coil. My larger coils tend to have a lower sweet spot than my smaller higher frequency coils. Another example is DR. Spark’s FATBOY, that coil performs better at lower break rates than any coil I have ever seen...
4hv really needs more post like this, thanks for posting!
EDIT:
One last thought I wanted to add in regards to your phase shift. Is it possible that maybe the arc in real life has some amount of inductance that is contributing to create a larger phase shift? This may explain why you see a shift in real lift but not in simulation. Just a thought.
In a SGTC or DRSSTC this effect is not really noticeable due to the fact that the first cycle has enough energy to break out from the toroid immediately, where a QCW needs 10’s to 100’s of cycles to get to the same level.
With SGTCs and with your monster DRSSTCs you are certainly right. If you look at the measurement with my little coil, arc current begin is delayed several cycles after begin of secondary rampup. This is probably due to my rather low bus voltages and the high impedance of my primary tank.
The model probably won't describe BPS dependence the way it is currently parametrised. A 300us decay time amounts to about 30 decay times between the bursts at 100 bps. Since I've used an exponential decay of energy (RC circuit), there will be not much energy left at the beginning of the next burst. I believe, that the assumption of an exponential decay of heat is inaccurate. Temperature probably drops slower at the end of the cooling period.
Inductance probably has some effect on phase. I've omiited it, since arc resistances are on the order of a few 100 Ohms/cm. To make a significant effect on phase shift. the inductively caused impedance must be of similar magnitude. But that would imply several hundred uH/cm, which doesn't seem plausible.
Registered Member #15
Joined: Thu Feb 02 2006, 01:11PM
Location:
Posts: 3068
Goodchild wrote ...
I'm also interested in what Steve is talking about. I have noticed this in my DRSSTCs for years. Low break rates tend not to generate as large of sparks. But there is a sweet spot, Steve's is at 100Hz, but I have noticed it from 80Hz up to 250Hz depending on the size of the coil. My larger coils tend to have a lower sweet spot than my smaller higher frequency coils. Another example is DR. Spark’s FATBOY, that coil performs better at lower break rates than any coil I have ever seen...
I've noticed this as well. Around 100Hz is a sweet spot indeed. Interestingly enough, you would think that you would simply get longer arcs at the lower PRFs because the capacitor droop isn't as much as the bus voltage (average) tends to be a somewhat higher than at higher PRFs (dependent on how much recharge current you are getting from the line and size of your bus capacitors. But seems the opposite is true as you mentioned.
Registered Member #2292
Joined: Fri Aug 14 2009, 05:33PM
Location: The Wild West AKA Arizona
Posts: 795
In regards to what you are talking about with the heat keeping the channel ionized. Maybe it still is exponential but the arc is just a lot hotter than we may think it is. If it is indeed very very hot say 5000c or more maybe this time constant can be extended leading to the factor we see in tesla coils. It is possible to say it could get this hot, in a lot of cases we are putting a lot of energy into that sparks( 10's of Kw in a lot of cases) and only a fraction goes to producing light or EM field, so the rest has to be in heat.
Take what I'm saying with a grain of slat however, I am only speculating off the top of my head.
Regardless I think you have a great model here. I may just use it in the near future for Calculating a secondary MMC in out new supper large QCW. It's difficult to calculated frequency drift in the secondary circuit without an accurate model for sparks. Mainly what I was looking for was a capacitance per unit length of sparks or an equation perhaps modeling this capacitance. I'm wondering to how you came up with the value for your simulation? I'm assuming it was a measure and match sort of thing?
For QCW spark capacitance, my plan was to get an accurate read of secondary L and C with an LCR bridge. After this run the coil for some burst length and record the change in frequency at several sampled points along a single burst. Assuming L in the sparks is negligible, a value for added C per unit length may be found using the original L,C and the shift in frequency. It bet it turns out this value is none linear in nature.
From a theoretical point of view, there shouldn't be a sweet spot for some bps.Arc length should increase for higher values, although slowly beyond 100 bps if burst length and bus voltage stay the same. I think, though, that these conditions usually are not met. Arc length won't profit much from higher bps beyond 100Hz, while power consumption really goes up. The effect of remaining heat between bursts is quite strong. If the arc channel cools say to 600K between bursts, the air will be half as thick as at 300K, reducing breakdown voltage by a factor of 2 by Paschens law. The arc temperature still has to be brought up again to about 6000K (plasma temperature) but the initial heat will give it an advantage. If you run your coil on top of a big mountain, you will probably also get longer arcs, provided you find a wall plug.
Eric wrote:
If it is indeed very very hot say 5000c or more maybe this time constant can be extended leading to the factor we see in tesla coils.
That's a good idea. I have to be careful, though, since a longer time constant will also affect arc growth at shorter time scales. Likely I will need another time constant somewhere to account for the bps effect..
It is possible to say it could get this hot, in a lot of cases we are putting a lot of energy into that sparks( 10's of Kw in a lot of cases) and only a fraction goes to producing light or EM field, so the rest has to be in heat.
Yes it is definitely heat. In terms of a light source, a TC is a poor performer.
Mainly what I was looking for was a capacitance per unit length of sparks or an equation perhaps modeling this capacitance. I'm wondering to how you came up with the value for your simulation? I'm assuming it was a measure and match sort of thing?
I started off with some arbitrary arc capacitance. Then I adjusted the parameters (i.e. 4e-7 and 6e-11) so they would fit the measured currents. Then I followed the voltage amplitude down the chain to get the arc length. Comparing that to the measured arc length gives the value how much arc length a single RC element stands for. If in the simulation your arc gets too long, you can add some RC elements. On the other hand the length of arc per element should not get too big, i.e. less than 10cm. If that happens, you should decrease the capacitance in the element and start over adjusting the parameters.
Anyway, I'm hungry for data to improve the model on. If you have some, I will gladly try to adjust it and share the results.
Registered Member #2292
Joined: Fri Aug 14 2009, 05:33PM
Location: The Wild West AKA Arizona
Posts: 795
When I finally get to testing this 1/8th scale QCW in about a week or two I will forward on any data I collect. I plan to use the process I outlined above and we will see how that turns out.
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A dynamical arc model
I'll try running it against the simulations I've already done for my upcoming DRSSTC.
