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A while ago I posted some calculations about the workings of a DRSSTC on this list. They were based on a simple electrical model and dealt with CW types of operation. The model is on the one hand dependent on the parameters of the coil like inductances and capacitances, which are easily measured, on the other hand on streamer properties, where little is known. Some more details of the model can be found here.
My previous calculations were based on a "steady state" like operation, i.e. constant current and voltage amplitudes, because they are much easier than dynnamical calculations. My coil really doesn't behave like that but I think a CW type analysis still is worthwhile.
The plots below shows primary (blue) and secondary (base) currents (red) of a 200us burst at 100 bps.
The upper diagram has the primary tuned to 224kHz, the lower one to 195kHz. Secondary frequency is 247kHz. The vertical units are 1A for the secondary and 60A for the primary (upper plot) and 50A (lower plot). I've chosen the vertical scales in such a way, that equal amplitudes of primary and secondary currents will correspond to equal amounts of energy.
So what happens? Initially the primary and the secondary are quite out of tune. for the lower plot even more than for the upper one. The rise of current in the secondary during ramp up is correspondingly slower in the lower diagram. Then streamers begin to form and add capacitance to the secondary top, pulling it closer to resonance with the primary. This causes the secondary current to shoot up and it drains the primaries energy. After that a plateau is reached. That can only be seen in the upper diagram, since my scope stops sampling after 200us. (It will do longer periods but only 250 samples per trigger. For longer periods the time resolution is too low to catch all the wiggles).
It looks like my coil works much as a SGTC: First the primary tank charges up and then it dumps its energy into the secondary quite fast, producing a power output that is larger than what my bus supply can deliver. When I shorten the burst in order to cut off the plateau region, the streamers don't get shorter, only dimmer.
The plateau region shows a much smaller primary current than at the time, when the coil was out of tune. This is due to the resistance coupled into the primary. This resistance is largest, when both tanks are in tune. Sadly this also reduces the power input from the bus voltage. But more about this later.
The diagram below shows the measured phase shift in degrees between primary and secondary currents. The red curve is for the measurement with the lower primary inductance. The blue one for the other one. Actually the currents should be in phase initially and not start off with the 80 degrees as seen in the plot. I've used a little self wound current transformer which was not properly loaded causing al phase shift offset. I subtracted that out in my calculations.
The phase shift indicates, where the secondary resonant frequency is with respect to the frequency the coil runs on. If the operating frequency is lower than the secondary frequency, the shift is lower than 90 degrees. At 90 degrees the frequencies coincide and at larger angles, the secondary resonance drops below the operating frequency. An increasing phase angle is thus an indicator of increasing capacitive streamer load. The plot shows, that if the primary is tuned closer to the secondary the phase shifts are higher and the point, when the tanks get to the same resonant frequency is achieved earlier. The 90 degree point is never reached for the second measurement (Remember to subtract the 80 degrees phase offset from the CT).
In the 2 diagrams below the secondary voltage (kV RMS, blue) and the power delivered to the streamer (kW, red) for the 2 measurements are shown. The voltage V was calculated from the secondary current V=Is/(w*Cs) where w is the circular frequency and Cs the secondary top capacitance. The idea is, that the current going into the base of the secondary comes out at the top and charges the top load. This is not exactly true, since the secondary winding has its own capacitance but it is a good approximation. Usually one would calculate the voltage by Is*w*L2 but that has the problem, that the secondary voltage is not only caused by its self inductance but also from the primaries magnetic field.
The power was calculated from the power transferred from the primary to the secondary P=w*M*Ip*Is*sin(phase), where M is the mutual inductance M=k*sqrt(Lp*Ls) and phase is the phase between the primary and secondary currents. To get the power output to the streamer I corrected this power for the energy stored in the secondary. This can be obtained from the secondary current. Since I'm not sure about the phase offset my current transformer generates, I started the plots from the time on, where there is significant phase shift, i.e. 20 degrees. At larger angles an error in the phase offset doesn't matter as much.
I've modeled the streamer load as a capacitance parallel to the top capacitance and a resistance parallel to it. That doesn't make much sense from a physicists point of view, where a resistor and a capacitor in series are more realistic. For a constant frequency both models are equivalent, though. I chose the parallel circuit, because it is easier to handle technically.
The diagrams below show the evolution of streamer capacitance (red in pF) and resistance (blue in 50k units). The resistance can be calculated from the streamer power and the voltage R=V^2/P. The calculation of the capacitance is more involved but once the Q of the secondary is known (from R) and the phase shift it is not so difficult. The plots are somewhat jagged, probably due to the low sampling rate of my scope. Also my whole evaluation hinges on the assumption of slow changes in phases and amplitudes, so this has to be taken somewhat sceptically. I am only really confident about the values in the flat region at the end of my first measurement.
I'm aware of 2 streamer models, one implemented in Terry Fritzs scantesla and one by Steve Conner, the hungry streamer model. AFAIK scantesla assumes a 220k resistance in series with a 1.5pF capacitance per foot streamer length. Conners model assumes a streamer capacitance of 25pF per m streamer length and a resistance, which is equal to the capacitive reactance. The results I got seem to match the hungry streamer model much better. I'd be interested in the logic behind that model.
The streamer length I obtained in my experiments were about 2.5 (first plot) and 3 feet (second). I got the best results if I tuned my primary as low as possible. This is limited by my OCD, which kicks in before my primary current gets lowered by the start of streamer loading. A breakout point sticking out long allows me to tune the primary lower. But even at lowest primary tuning my primary current will drop way low after the streamer has started. So then the power input from the bus is reduced much below my IGBTs capabilities. I'd be happy if I'd find a way to keep up the primary current for a longer time. One idea is to radically increase secondary inductance in order to make the secondary resonance curve flatter (low Q). That would make the the primary current initially lower preventing my OCD to trigger and later keep the current higher, when the secondary has been pulled into resonance. In my measurement only about 1.6J are stored at most in my primary, while my bus caps have a storage capability of about 100J and it seems that only the 1.6J contribute to the streamer length.
I'm intrigued by the coil from MRacerxdl. The coil shows 1.2m sparks (more than mine) but seems to have a maximum energy in the primary of 0.6J (less than mine). It looks to me like he is successfully growing sparks beyond the time of discharge of the primary into the secondary i.e. in the plateau region.
Registered Member #30
Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706
Excellent work! I'll have to think about it some more before I'm in a position to say anything helpful.
My hungry streamer model is just the maximum power transfer theorem. I argue that the streamer will grow until it's as close to being a conjugate match of the coil's output impedance as the laws of plasma physics will allow. Then it won't grow any more because further growth would decrease the power transfer into it.
wrote ... It looks to me like he is successfully growing sparks beyond the time of discharge of the primary into the secondary i.e. in the plateau region.
Any well designed and well-tuned DRSSTC should do this. It's the secret of the high performance. For instance, my DRSSTC (which wasn't a particularly outstanding example ) would produce a 5 joule bang in 150us, when the primary circuit was only capable of storing about 1J. Most of the energy goes straight from the DC bus capacitors to the streamers, with the coil system acting as an impedance matching transformer.
Edit: Here's another example where it made bangs of about 16J in 300us, with only about 2-3J of primary energy storage.
My hungry streamer model is just the maximum power transfer theorem. I argue that the streamer will grow until it's as close to being a conjugate match of the coil's output impedance as the laws of plasma physics will allow. Then it won't grow any more because further growth would decrease the power transfer into it.
Do you have an equilibrium kind of thing in mind? Assume that the streamer has stopped growing and assume that its resistance gets lower when it gets hotter. Also assume that the temperature is just so that its resistance maximizes power consumption. Then the streamer won't have the tendency to get even hotter since that requires more power and power input decreases. It might be colder, though. That would be the case if an increase in temperature could not be sustained by the corresponding increase in power. That being said, I got from the long constant tail in my measurement a capacitive reactance of 130k and a resistance of 110k, which is a deviation well inside my measurement errors.
Any well designed and well-tuned DRSSTC should do this. It's the secret of the high performance. For instance, my DRSSTC (which wasn't a particularly outstanding example ) would produce a 5 joule bang in 150us, when the primary circuit was only capable of storing about 1J. Most of the energy goes straight from the DC bus capacitors to the streamers, with the coil system acting as an impedance matching transformer.
How did you do that? My coil reacts rather violently to the streamer capacitance.
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A streamer load measurement with a DRSSTC
I'll have to think about it some more before I'm in a position to say anything helpful.