DRSSTC inverter vs. secondary impedance inequality discussion

[Uspring]

Mads wrote:

As from https://highvoltageforum.net/index.php?topic=90.msg462#msg462 we have the equation: Vmax/Imax < k  / (2*pi*f*C)

That's a generous upper limit on the primary capacitance. It's assuming perfect tuning and a choice of secondary L best matched to arc loading. My coils run best at about half the limit. If you choose a too big primary C, i.e. a low impedance primary, then you will run into overcurrent terminating the burst. A small C will not allow to push as much current into the primary as you want to. Detuning by choosing different L taps can compensate for this. That's easier than "tapping" the MMC.

[Mads Barnkob]

Mads wrote:

As from https://highvoltageforum.net/index.php?topic=90.msg462#msg462 we have the equation: Vmax/Imax < k  / (2*pi*f*C)

That's a generous upper limit on the primary capacitance. It's assuming perfect tuning and a choice of secondary L best matched to arc loading. My coils run best at about half the limit. If you choose a too big primary C, i.e. a low impedance primary, then you will run into overcurrent terminating the burst. A small C will not allow to push as much current into the primary as you want to. Detuning by choosing different L taps can compensate for this. That's easier than "tapping" the MMC.

That is also what I tried to shed a bit of light on in the last reply, that there is no real answer, but a bundle of entwined and interdependent equations which will still point in each their direction

Even what I suggested in the last thread, might be a too low inductance primary as it would hit peak current at ~60us?

[Uspring]

The inequality

Vmax/Imax < k  / (2*pi*f*C)

is derived from a steady state consideration, i.e., where does primary current end up after a long burst, when everything has settled down to constant values.

but due to the low primary inductance, we might hit 600A real fast,

I agree. Primary rampup is a transient situation. Before a steady state is reached, primary current has to reach a significant level, then energy has to be transferred to the secondary, then the arc has to grow to its final length. Due to the slowness of the latter 2 processes, there might be a considerable overshoot of primary current. I've seen that in my coils. Slowing down rampup by increasing primary impedance avoids that to some extent. Steve Wards older coils as far as specs are available as well as mine seem to roughly run at

Vmax/Imax ~ 0.5 * k  / (2*pi*f*C)

Here are some values for your coils. I couldn't find any data on k, though.

DRSSTC  Vmax/Imax                   1 / (2*pi*f*C) in ohms
#1              0.7                                    5.2
#2        0.28, 0.5  (Rev1 and 2)           6.5
#3              0.5                                    5.2
#4              3.6                                    33

mostly there is a factor of about 0.1 between these values. That would correspond to 0.5*k for a k of 0.2 in the above equation.

[Mads Barnkob]

I think it makes sense that my 4 coils are very even and leans up against Wards numbers, as that is how I learned to build Tesla coils, from Wards website

I do not believe that I have designed for as high a coupling as 0.2, but more like in the range of 1.3 to 1.6. So corrected to 0.15

DRSSTC  Vmax/Imax                   0.5 * k / (2*pi*f*C) in ohms
#1              0.7                                    0.39 (was before I had proper measurement equipment)
#2        0.28, 0.5  (Rev1 and 2)           0.49 (this has a long steady state operation)
#3              0.5                                    0.39 (never had time to measure it properly, due to size its never practical, dummy load measurements are not useful)
#4              3.6                                    0.3 (impossicoil, got no idea about this bastard  

We agree that low impedance coils really never operate in a steady state? That the fast ramp up of current and then turn off, it is not getting there.

But in the small #2 http://kaizerpowerelectronics.dk/tesla-coils/kaizer-drsstc-ii/ it has a very steady ending from the long on-time, resembling a... spear head.

[Uspring]

We agree that low impedance coils really never operate in a steady state? That the fast ramp up of current and then turn off, it is not getting there.

Yes. And an OCD turn off will cause much of the energy stored in the primary tank to return into the bus caps and not ending up in the arc. So it is not desirable.
The scope shot shows, that in the steady state at the end the power input to the coil is much reduced compared to the bridges capabilities. The OCD is set much higher than the current at the end, since the burst wouldn't have continued over the initial overshoot of current. That can be improved by either upper pole operation or possibly, by pulse skipping, which both avoid the overshoot.

I'm wondering about the 0.5 * k / (2*pi*f*C) value for DRSSTC4. That would imply a k of 0.02 (?)

Another remark to Procrastinators coil: It looks like he tuned primary and secondary res to be quite close (initially). That defeats the purpose of upper pole operation, since once secondary fres drops below primary fres, the upper pole won't follow the secondary fres downwards as the arc grows. It will stay up high, blocked by the primary fres.

[Mads Barnkob]

I do not even know what the coupling of DRSSTC4 is, its folded primary does not really make it fit into any calculators I know of.

DRSSTC 4      
Vbus   320   
Max current   50   
Fresonance   100000   
Coupling   0,3   
C    0,047   uF
      
Vmax/Imax   <   k*sqrt(L/C)
8,148733086      10,15882615

That is properly a better estimate

[Uspring]

The relation you proposed

Vmax/Imax   <   k*sqrt(L/C)

is similar to the one I derived here: https://highvoltageforum.net/index.php?topic=1705.msg13007#msg13007 except from a factor of 2, i.e:

Vmax/Imax = k/2 * sqrt(Lpri/Cpri)

Considering the assumptions and approximations I made in the derivation, both versions are acceptable. My version suggests a larger preferred impedance. If the primary impedance is too large to get enough current into the tank, there is always the possibility of detuning, which will increase primary current. If the primary impedance is too low and current too large, improving tuning might not always be possible.

[davekni]

The relation you proposed

Vmax/Imax   <   k*sqrt(L/C)

is similar to the one I derived here: https://highvoltageforum.net/index.php?topic=1705.msg13007#msg13007 except from a factor of 2, i.e:

Vmax/Imax = k/2 * sqrt(Lpri/Cpri)

Based on some LTSpice simulation, I'm guessing that the actual answer is the geometric mean between the above two:
Vmax/Imax = k/sqrt(2) * sqrt(Lpri/Cpri)

I simulated a case roughly matching Costas' DRSSTC parameters, sweeping a simple R+C arc load model searching for the highest resonant primary impedance.  For at least this one example, the result matched k/sqrt(2) almost perfectly.  This result may be an artifact of the particular primary detuning simulated.  I haven't tried other cases.

Simulation schematic with sweep parameter ranges narrowed down around maximum impedance (lowest peak primary current):


Coupling factor:                       0.14
Primary resonant frequency:     229kHz
Secondary resonant frequency: 257kHz    (w/o arc load)
Primary impedance at 229kHz:  15.8 ohms
Primary resonant impedance with optimum arc loading:  1.56 ohms

Sweep of primary current over 9 arc load value sets:


Primary current w/o arc load showing both poles:


At least in this case, the lower and upper poles almost merge with optimum arc loading.  Don't know if that is universally true.  If so, that might explain the difference between k/2 presuming lower pole and k/sqrt(2) result of this one simulation.

[Uspring]

Thank you for the calculation. I've made a simulation with a somewhat lower arc load:
Green curve Rarc=700k, Carc=1pF
Red curve Rarc=350k, Carc=2pF

The green curve shows 3 ZCS frequencies at 222, 253 and 260 kHz, which are the frequencies at which the dotted green (phase) curve is at 180 degrees. The 222 and 260 kHZ are the lower and upper pole frequencies. The corresponding effective resistances are about 0.18 and 2 Ohms indicating a very effective power transfer at the upper pole.

In the case of stronger loading (red curve) there is only one ZCS frequency left at 221 kHz with about 0.4 Ohms resistance. If arc load increases while operating at the upper pole, the operating frequency probably jumps down to the lower pole at some point. That happens at larger loads for better matching primary and secondary resonance frequencies and for larger coupling. I've made some very simplifying assumption about the operating frequency in the derivation of the k*sqrt(L/C) relation, which don't always hold for particular ways of tuning, i.e. upper pole operation with primary tuned low or for large arc loads, where ZCS frequencies vanish. That makes a range of resistance values around k*sqrt(L/C) possible.

I don't know if the poles coalesce in the case of strong damping. But surely only one ZCS frequency remains for sufficiently large loading.