# Resonant charging design

```Original poster: "R.E.Burnett by way of Terry Fritz <twftesla-at-qwest-dot-net>" <R.E.Burnett-at-newcastle.ac.uk>

Hi guys,

About a week ago,  I said that I would go through my method for
designing the AC resonant charging circuit for a tesla coil.

What I am going to explain is a way in which I propose that the
ballast and tank capacitor values for a particular system can be
determined at the design stage.  i.e.  You choose the rotary speed,
and how many kW of power you want,  and use the following steps to
obtain the required ballast inductance and tank capacitance.  As a
bonus this method also achieves good power factor.

I should say that this is very much "simulation based" at this
stage,  as I haven't built lots of different systems to prove every
combination.  However,  I have used Microsim extensively to verify
my work,  and have got promising results.  What little practical
work I have been able to do has also closely matched the predicted
performance.

Ok,  here goes......

Let's say I want to design a Tesla Coil to process 10 kW of real
power,  using a 12 kV power transformer,  and a 371 BPS asynchronous
rotary gap.  Power feed is from 250V -at- 50Hz,  and at this stage I
have no idea what ballast inductance or tank capacitor value would
give good results.

One possible approach would be to try many combinations of ballasts
and capacitors in simulations until the combination that results in
best performance is found.  This is what I did and it was _VERY_ time
consuming !  I did this so hopefully you don't have to take this
hit and miss approach.

STEP 1: Determine what the resonant frequency of the charging
circuit should be for your chosen rotary firing rate (BPS.)

We know that the ballast inductor at the primary side of the power
transformer resonates with the tank capacitor at the secondary.
There have been many references to LTR caps.  We are not concerned
with whether the cap is larger or smaller than a particular value.
We are only concerned with what the actual resonant frequency of the
charging circuit is.  In this design approach there is nothing
special about resonance at 50 or 60 Hz.

For a whole range of break rates,  I tried simulating charging
circuits with different resonant frequencies.  Particular attention
was paid to the charging waveforms and power factor.  As I explained
before,  it is the resonant frequency of the charging circuit which
determines all of the timing related stuff like how quickly the cap
charges between firings,  and the power factor.  So the resonant

The graph here...

http://hot-streamer-dot-com/temp/reschrg1.gif

...displays what I think the resonant frequency of the charging
circuit should be for different rotary break rates to get a power
factor of 0.85

In our example the rotary break rate is 371BPS so the charging
circuit should be designed to resonate at 85.2 Hz.

That ensures a PF of 0.85 at 371BPS,  and ties down the product of
the ballast and the tank capacitor,  but there are still many
combinations that would give the same resonant frequency.   Indeed
all combinations which give the correct resonant frequency do give
the same charging waveforms and good power factor !  As long as
the product of L and C is right for the chosen BPS,  the resonant
frequency is right and the charging circuit works correctly.

Choosing various combinations of L and C only changes one thing.
The _POWER THROUGHPUT_.

STEP 2: Determine what the ballast inductance should be.

If changing the ballast and tank cap values together only alters the
power throughput,  then we can adjust these values to get whatever
power level we want.  However if we don't know where to start,  how
can we pick a value for one of them without guessing and trying
something first ?

Well,  a good place to start is by ballasting the transformer to
whatever power throughput we desire.  In our example,  the desired
power throughput is 10kW,  so we need to work out what ballast
inductance will limit the transformer to 10kVA when its secondary
winding is shorted.

10kW represents a current of 40 Amps at 250 Volts.  Therefore the
ballast choke should present an impedance of 6.25 Ohms.  At our
supply frequency of 50Hz,  that requires 19.9 mH of inductance.

This figure of 19.9mH for the ballast is a "first-approximation".
Although it draws 10kVA when the transformer is short-circuited,
it is unlikely that this will give exactly 10kW of real power
throughput at our chosen break rate.  Life is just not that
easy ;-)

STEP 3:	Adjust the ballast inductance depending on chosen break rate.

If we used the ballast value calculated above,  the real power
throughput would be somewhat less than our desired 10kW.  The actual
shortfall depends on the rotary break rate.

The graph here...

http://hot-streamer-dot-com/temp/reschrg2.gif

...shows what the real power output is as a fraction of the
"ballasted VA"  for different rotary speeds.  Notice that low speeds
give almost the full "ballasted VA" as real power, but higher speeds
give progressively less real power for the same ballast setting.

For our chosen rotary speed of 371 BPS,  it can be seen that the
running Tesla Coil would process only 64% of the 10kVA that it was
ballasted to draw with a short-circuit.   Therefore we must multiply
the ballast inductance by 0.64 to correct for this shortfall and meet
the desired power throughput criteria.

0.64 x 19.9mH = 12.74 mH

The "corrected" ballast inductance is 12.74mH,  and this is the value
that we must use to get our full 10kW of power throughput when running.

STEP 4:	Calculate the required tank capacitor value.

Now that we know the ballast inductance,  and the resonant frequency
of the charging circuit,  we can finally calculate what the tank
capacitance should be.

First we need to imagine that the ballast inductor is at the high-
voltage side of the power transformer, instead of at the low voltage
primary side.    This enables us to deal with the ballast inductor and
tank capacitor like a series resonant circuit.  The equation:

F = 1 / [ 2 pi sqrt (L x C) ]  can then be used as normal.

In our example the power transformer steps 250 Volts up to 12000 Volts,
so it has a turns ratio of 48.  Impedances are transformed by the
square of the turns ratio,  so our 12.74 mH ballast inductor becomes
48 x 48 x 0.01274  =  29.35 H  when referred to the secondary side.

Re-arranging our resonant frequency equation and plugging-in the values
for F (85.2) and L (29.35) gives us a tank capacitance of 119 nF.

And that is it.

To summarise the ballast is 12.74 mH and the tank capacitor is 119nF.
This gives us a system that processes 10kW of real power at 0.85 PF
with our 12kV transformer and 371 BPS rotary spark gap.  (Running a
Microsim simulation with these parameters gave me 10080 W of power,
11999 VA,  and therefore a PF is 0.84)

There are two useful advantages of knowing that the approximate
power factor is 0.85:

1. We can estimate the VA or current draw from the 250 Volt supply.
In this case the real power is 10kW,  so the VA = 10000 / 0.85
The VA = 11.76 kVA,  or 47 Amps from our 250 Volt supply.

2. We can calculate the optimum PFC capacitance to improve PF further.
With a PF of 0.85,  the VARs are roughly equal to _half_ the
watts.  Therefore we need 5kVAr of PFC correction capacitance
across the supply to achieve the absolute maximum power factor.
That is a capacitive current of 20 Amps at 250 volts requiring
255uF of capacitance at 50Hz.  This will cancel any remaining
reactive current and maximise power factor.

Increasing or decreasing the power:

Remember that all the timing type behaviour is determined by the
resonant frequency.  If you want to double the power throughput of your
design,  all you need to do is double the tank capacitance and halve
the ballast inductance.  (Assuming the power transformer can cope ;-)

This will process twice the power due to the reduction in impedance.
However,  the charging waveforms,  peak voltages,  and power factor
are not changed because the resonant frequency of the charging circuit
has not changed !

There is nothing particularly special about the design method described
here.  All I have done is run a large number of simulations to find out
what works best at various break rates.  Then I have condensed this data
into two graphs showing:

1. What the resonant charging frequency should be for various BPS.
2. How much the ballast needs to be adjusted to get the desired power.

Now you can work backwards from Watts and BPS to get ballast inductance
and tank capacitance reasonably easily.  Think of this method as being
like a lookup table.

Congratulation if you have read this far,  I appreciate that this stuff
is heavy going !!!  This kind of approximations and calculations are
ideally suited to a computer program,  so I intend to produce a simple
program to do all this stuff automatically.  It will ask for your
intended power level in kW and your rotary speed in BPS.  It will then
calculate the suggested ballast and capacitor values as explained above,
but much faster than using graphs and a calculator.

I hope that the above information is error free,  and has provided some
insight into the work that I have been doing for some time.  If you
have any questions,  criticisms or suggestions I will try to deal with
them as quickly as possible.

Regardless of whether you understand the method or you just use the
computer program,  I hope this work will lead to bigger sparks for all ;-)

Cheers,

-Richie Burnett,
(Newcastle, UK)

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