Stefan's Tesla-Pages

Technological background, part#4
(or 'how to produce your own indoor lightning')

coupling (k-factor, coupling coefficient k)
quenching
quality factor (Q)

Definition:

k is defined by the formula:
k = M / sqrt(L1 * L2)
where L1 and L2 are the inductances of the primary and secondary coils and M the mutual inductance between both coils.

The coupling factor k determines how quickly the energy is transferred between the primary coil and the secondary coil (and back). It is almost entirely determined by the geometry and physical positioning of the two coils relative to each other. If the coils are spaced far apart, the coupling is "loose" and the k-factor is low. Energy transfer between the coils is slow and takes many RF cycles. A tighter coupling means higher k and faster energy transfer. Typical coupling coefficient for standard Tesla coils are in the range between 0.05 and 0.2. The looser the coupling, the shorter the sparks. But if the coupling is to tight, there will be racing sparks along the secondary and breakdown between the primary and the secondary. High voltage RF is VERY difficult to insulate, especially along insulating surfaces. An RF voltage that may jump only 1" in air, can easily flash over 3-4" along the surface of a dielectric. You should build you TC so that you can raise the secondary relative to the primary to bring coupling within the proper range. Start with the secondary elevated (low coupling), and tune for best spark before going to full power. Then you can slowly increase coupling to get best full-power operation without racing sparks.

Measurement 1:

A very easy way to measure the coupling was presented by Dan Kline in his mail on the TCML on 03.09.1998. For his method you only nead a 10kOhm resistor, a 1uF capacitor, a "household power resistor" like hair dryer or space heater and a multimeter for AC current (approx. 10A) and AC voltages (~1V). You'll get the mutual inductance and calculate k with the knowledge of the inductances L1 (primary coil) and L2 (secondary coil). But now with his own words:

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Subject: coupling coefficient - Best Method
Date: 03.09.98 03:13:39 CETDST
From: tesla-at-pupman.com (Tesla List)
To: tesla-at-pupman.com

Original Poster: Dan Kline <SNIPed>

All,
I have tried all the suggestion I have received (Thanks Malcolm, Fr. Tom, John C., Mark Rzeszotarski). The best method I have found that does not require expensive equipment or great theoretical challenges consists of the following.

Apply a heavy 60 Hz AC current to the primary coil. This is best done by placing a space heater, hair dryer, etc. in series with the primary to limit the current to about 10 amps. Measure this current with a multimeter. Note that the space heater gives a fairly stable resistance. Light bulbs have a non-linear resistance through the AC cycle and distort the measurement (they must cool down substantially at the nodes of the AC cycle). Of course, use great caution with the live AC on the primary so as not to kill yourself. Only the isolated primary need be connected to the AC. The capacitors, transformers, and other wiring should be disconnected from the primary for this test. Be cautious of the AC finding its way on to the secondary!

Place a 10k ohm resistor and a 1uF capacitor across the secondary and measure the AC voltage. It will be on the order of say 100 mV AC. The resistor and capacitor will eliminate stray noise picked up by the secondary and swamp any resonance which is significant at these low levels.
(Remark: resistor and capacitor in series or in parallel?)

The mutual inductance is found by:

    M = V / (w * I)
Where:

    M = Mutual inductance in Henries.
    w = the line frequency in radians per second (377 for 60Hz or 314 for 50 Hz).
    I = The measured current in the primary in amps AC.
   V = The measured secondary voltage in volts AC.

As an example:
If the current in the primary is 10 amps and the frequency is 60Hz and you measure 0.100 volts AC, you would get:

0.100 / (377 * 10 ) = 26.52 uH for the mutual inductance.
k can then be found by using the formula:

k = M / sqrt(L1 * L2)
Where L1 and L2 are the inductances of the primary and secondary coils.

This method is rock solid in theory and easy to do. The accuracy is excellent. There is little that can go wrong compared to other methods and you don't need anything special other than a multimeter to do the test. The accuracy is dependant on the accuracy of your multimeter. My tests could easily get within 1%.

Measurement 2:

This method requires a storage scope. You can determine the coupling simply from a scope trace showing the ringdown of the secondary circuit - if you have a storage scope available. The waveform (small signal, no heavy sparking from secondary!) is an amplitude modulated sine wave with some "beating" at the beginning (where energy is tranferred back and forth between the primary circuit and the secondary circuit) and an exponential decay at the end. Take a closer look at the section between two notches of the beating at the beginning of the waveform. Count the number (n) of full sine cycles inbetween those two notches. Coupling factor k simply can be derived by k=1/n.

For instance, if in the first ringdown you can count 5 "sines" (one local max (high) plus one local minmum (low) each), then k=1/5=0.2:

The approach of estimating "k" by taking the reciprocal of the number of peaks during the first primary ringdown is accurate to within 5-10% for the typical range of "k's" used in 2-coil systems.

Measurement 3:

Measure f_I and f_II of the secondary, with f_I = primary shorted and f_II = primary open circuit. As defined above, k is ...
(still have to find again where I've read about this method...)

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'Magic' k-factors:

Sometimes you might hear about 'magic' k-factors. Forget them! The theory is based on very simplyfied circuits and therefore those 'magic' k-factors simply are wrong...

Definition:

"Quenching" is the ability of the spark gap to stop conducting. This can only happen when the energy (both current and voltage) in the primary circuit is low for a significant amount of time. So this will only happen at the primary notches described above. If there are enough ions in the air between the gap, it will re-ignite if the voltage rises again and will quench at a later notch.

What is "good" about an early quenching?

The goal is to achieve a "good" quenching, that is an early quenching (preferable the first or second notch) so that no more power is dissipated in the spark gap as heat. The reasons for that are:
1) The heat errodes the electrodes.
2) The heat reduces the firing voltage of a static gap so that less energy could be transferred to the secondary.
3) An early quench reduces the average current in the primary circuit. Therefore the gap and capacitor are stressed less. Compared to a 4th notch quench (exponential decay neglected), a 2nd notch quench reduces the average current down to only approx. 3/7 (43%), a 1st notch quench even down to 1/7 (14%)!

Measurement:

You can observe the quenching on a storage scope. Simply use an antenna far enough from the secondary (so that it doesn't get hit).

How to achieve a good quenching:

1. Run a cool ( not necessarily fancy ;-) gap
The gap should have a big thermal mass and good cooling by a sufficient forced airflow which also helps to blow away the ions.
2. Use a high primary voltage
A high voltage (10-20kVeff) allows a smaller primary capacitance and (this the important one) a higher primary inductance. This high inductance increases the surge impedance and reduces the current in the primary circuit (=> less heat).
3. Be sure your tuning is exact
If you run the TC for the first time, you should not apply power for long times, just enough to do the fine tuning (frequency will change with the length of the sparks!). If your tuning is bad, not all the energy will be transferred to the secondary and the notch will not be a real notch since the envelope of the current waveform does NOT fall to zero! It would be very difficult to get a quench in this case.
4. Allow spark breakout
Of course, a toroid never can be to big. But a big toroid can prevent spark breakout. Therefore, a breakout point should be used (at least at the beginning unless the system is set up perfectly, then it can be removed later) at the toroid. This is to remove the power from the system as quick as possible. If you use a variac to slowly bring up the voltage without the use of a breakout point, you might not get breakout though the spark gap is firing until you cranked it up to a certain level. In this situation, all the power is dissipated in the gap and primary capacitor, a very destructive situation!  

Definition:

Q = X_L / R_AC = (2*Pi*Fres*L) / R_AC

with
Fres = resonance frequency of the coil
R_AC = resistance of the coil at resonance frequency (>R_DC because of skin effect and proximity effect)

another definition:

Q = (Fres/BW )
with
BW: Bandwidth of the coil

Goals:

Make the primary Q as high as possible. This means high inductance (and low R_AC). The impedance of the primary coil usually is at least one order of magnitude higher (approx. 20-40Ohms) than the resistance of the primary coil and of the spark gap (up to 2Ohms). Therefore the primary current will be lower for higher Q coils and the losses in the sparkgap will decrease. Spark gap losses reach up to a sifgnificant amount of the input energy, so this is really the place to look at! If the secondary is sparking well, the primary Q of the live TC is in the order of 10.    

The secondary Q will decrease dramatically if the sparks break out and it will go flatline if you have an power arc to ground. This means that for spark gap (!) driven coils it is useless trying to achieve a very high secondary Q (please read this sentence again!). Instead of this, its better to wind more turns of thinner wire for higher inductance because this will result in a higher primary inductance with the advantages mentioned in the paragraph above. So there is a compromise between inductance and resistance. High Q is required when the voltage in the secondary swings up before breakout. When the coil is sparking and when an arc is formed to ground (resulting in a closed circuit), the AC resistance should also be small. I would say go for about 1500-2000 turns in the secondary of spark gap driven coils.
OK, now lets have a look at CW-TCs [continious wave TCs like vacuum tube driven Tesla coils (TTTCs) or solid state driven Tesla coils (SSTCs)]. Here the secondary voltage is not defined by the energy dumped into the secondary at one shot but on the resonant rise due to continious energy transfer. If you shoot for a very small wire to achieve a high inductance of the secondary, the DC-resistance will increase dramatically. Also the AC-resistance (skin effect, proximity effect) will do. Q should be high here, go for thicker wire! As a rule of thumb, you should shoot for about 500-800 turns in the secondary coil of a CW Tesla coil. Else, you'll get a serious heating problem (once in a while, you can read about molten coil formers).
If you place a topload onto your secondary, the Q will drop a bit.

Measurement 1:

To measure the Q of a TC secondary, you'll need an o'scope & an f-gen. You'll get Q and fres from this measurement. This method was purposed by RWB on the GTL in 1999.

Setup:

Place the secondary coil (without primary coil!) in the middle of an open space in your lab (the most difficult part ;-) well away from anything which could influence the measurement. It should be elevated so that eddy currents in your conductive floor (concrete) and ceiling are minimized. Attach a wire (will act as an antenna) approx. 1m from the top end of your secondary (i.e. hang it froom the ceiling). Connect this antenna with a coax cable to your o'scope without using the o'scopes ground.  The "hot line" from the f-gen has to be connected to the bottom end of the secondary. You have to use the low impedance output from the f-gen here! The f-gen ground will be left unused, too. Connect the f-gen to the second channel of the o'scope.

Measurement:
Set the f-gen to the calculated resonance frequency of the secondary. You'll get a small signal from your antenna at channel one on the o'scope. Adjust frequency until you get maximum amplitude, then you've found the real resonance frequency (will differ some percent from the calculated value). It is important to stay well away from the coil so that you don't disturb the measurement. Set the Volts/div so that you'll see this maximum amplitude well. Write down the exact resonance frequency and amplitude. Also write down the amplitude of the signal you feed from the f-gen into the secondary coil. Now adjust the frequency so that you'll get exactly 70.7% of the peak amplitude. Do this for the higher and the lower frequency (both sides of the resonance frequency). Write down those two frequencies as well as the amplitude of the signal you feed from the f-gen into the secondary coil in each case. Repeat all those measuements at least three times. It is difficult to adjust the frequency and read all those values with the required precision, please take time and a good set of instruments. If your f-gen does not provide sufficient readout, a separate frequency counter is required.
Important: If the signal fed into the coil varies in amplitude (due to the changing load the secondary presents near resonance), the output impedance of your f-gen is to high and you can't do the measurement this way! You'll get a to low Q since the 70.7%-values present a smaller load and the high and low frequency will move to higher resp. lower values.
Calculation:
For the calculation, you first have to check if the measured values of the different measurements are confident. The calculate the mean values. The final calculation goes like this (example, values from RWB, bare secondary without topload):
Fres = 227.459kHz
Fhigh (0.707*max.Amplitude) = 228.072kHz

Flow (0.707*max.Amplitude) = 226.915kHz
Bandwidth BW = Fhigh-Flow = 1.157Khz
Q= (Fres/BW) = 227.459kHz / 1.157Khz ~197

The described procedure is not easy and requires a very good equipment, therefore I want to present another one.

Measurement 2:

For the second method to measure Q of a TC secondary, you'll need an o'scope , an f-gen and a 10 turn non inductive variable resistor (approx. 2kOhms). You need to know L before and you'll get Q and fres. This method was purposed by RWB on the GTL in 1999, too.

Setup:

Use the same setup as in the method described before. Connect the variable resistor between the f-gen and the secondary.
Measurement:
Adjust to resonance and set the output of the f-gen to an exact amplitude. Now adjust the resistor so that you'll have exactly 50% of that amplitude across the resistor. Re-adjust the frequency and resistance until you have exactly 50% of the amplitude at really the resonance frequency. Disconnect the variable resistor and measure to what value you've set it. This value is the same a the AC-resistance of the secondary at resonance frequency, because we've buildt a voltage devider (secondary and resistor) and adjusted for equal voltages. The AC resitance of the secondary will be multiple its DC resistance, so be sure you use a resistor of sufficient resistance.
Calculation (example again with values from RWB):
L = 35.5mH (measured before, e.g. with LCR meter)

Fres = 227.459kHz
R_AC at Fres (read out from variable resistor) = 264.24 ohm

(for comparison: R_DC = 17.3 Ohm, so R_AC~15*R_DC!)
With the formula
Q = X_L / R_AC = (2*Pi*Fres*L) / R_AC ,

this results in a Q of:
Q = (2*Pi*227.459*10^3*35.5*10^-3) / 264.24 = 50735.43 / 264.24 ~ 192,
that's nearly the same as before.
This second method is much simpler and faster but not as accurate as the first one (difficult to adjust the variable resistor).

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