Re: TC Inductance Formulas...

Hi Malcom,

At 09:46 AM 2/21/96 +1200, Malcom wrote:

>> L:(Average)    (SUM)  0.166435839  166.43583937  166435.83937
>>                (AVG)  0.027739307  27.739306562  27739.306562
>> L[m]:(Average) (SUM)  0.167945794  167.94579413  167945.79413
>>                (AVG)  0.027990966  27.990965688  27990.965688
><and again>
>> L[p]:(Average) (SUM)  0.166568649  166.56864907  166568.64907
>>                (AVG)  0.027761442  27.761441511  27761.441511
>I don't think you said what the measured inductance was (I may
>have missed it.)

No, you didn't miss it, I forgot to include it.  I have 3 different 
indutance meters in the lab that I used, unfortunetly one of the meters
must be broken because it gave me and inductance reading of approximately
15 mH, which I knew was way off, so I was limited to having only two
measured results, here they are.

Measured Inductance of 6" Secondary Coil (22AWG):

Inductance                     H             mH           uH
Measurement      :(IM#1)  0.027759164  27.759164000  27759.164000
                  (IM#2)  0.027759241  27.759241000  27759.241000

(Averages)       :(SUM)   0.055518405  55.518405000  55518.405000
                  (AVG)   0.027759203  27.759202500  27759.202500

Note the average of the two measured inductances comes very close to the
calculation of average "prime" inductance L[p] = 27.761441511mH.  Now I
can accept this, the only problem is that I don't believe the only way to
calculate the closest possible inductance of a coil, without direct
measurement, is using 6 different formulas and then taking the average
of those 6 formulas to arrive at a calculated inductance with an uncertainty
of +/- 0.01 (as compared to the actual measured inductance of the coil).

There has to be a better formula for the induction of a closewound coil...

I have pretty much established that the "prime" calculation (as I called
it) is the best/closest approximation.  I would imagine this is because it
takes into account the actual insulation around the conductor in the coil 
whereas the, "nominal" does not, and the "mean" actually does, but it over
compensates by measureing all the way to the center of the conductor rather
than stop at the outside of the conductor.

Now of the 6 different formulas (i1-i7) that I have for calculating inductance
in a closewound coil:  3 of the formulas give an inductance of approximately
26mH and 3 other formulas give an inductance of approximately 29mH and the
one is just plain wrong altogether.  None of the formulas gets even close (in 
scientific terms) to the 27mH of the actual measured inductance.

>      Comment - I too like exactitude but in the real world this is
>physically not possible. All the science I know quantifies the
>degree of uncertainty or experimental error. For example, the "end 
>effect" of a dipole aerial means that best tune is obtained with the 
>less-than-ideally calculated element length. I suspect there are end 
>effects associated with coils as well and any coil will couple 
>however loosely into its surroundings. You will probably find that 
>slightly different formulae are needed to get the closest values for 
>different aspect ratios.

I was just hopeing there was someone who knew of a more exacting formula for 
calculating the inductance of the secondary.  I know it is not really that
essential to have an exact inductance calculation for practical tesla coil
uses, but if a scientist were to use this tesla coil apparatus in their 
research (which I am) they would require the closest, most exacting
calculation they could find, especially if they had to do a write up on
the inner workings of the device (in this case the tesla coil).  I can't just
say, "Well the exact inductance dosen't really matter, this inductance is
2mH off from the actual measured value of the coil, but that's alright", it
really wouldn't go over to well in the scientific community.  Besides I am
a perfectionist...it really bugs me :)

Since there does not appear to be a more exacting calculation for inductance
I figured I would make/derive one.  The only problem with that is, as you 
mentioned, there would probably be different formulas for different diameters
and aspect ratios.  Another problem I have (I have alot of those...) is I 
only have a 6" coil left out of the 5 secondaries I had (the others were 
burnt up, either bad form material, human error, or bad wire).  If I had other
specifications and actual measured inductances on other size coils, say a 4",
 8", 10", 12" and 15", I could work everything out and see just what effect
the diameter and aspect ratio has on the inductance (or which formula best
suits the coil).  This information is only useful of course if I know exactly
what type of wire was used in the winding (in order to find the thickness of 
the insulation). So if anyone has measured the actual inductance of there coil
, knows the type and manufactor of the wire used, and would not mind taking the 
time to send me the specifications, please do (hint...:).

>     Last comment - if you are after accuracy in resonance measurement
>use NO resistance, series or shunting with the coil. The following is
>the "real" resonance formula :
>    fr = 1/(2xPI) x SQRT( 1/(LxC) - R^2/(4xL^2) )
>I think I got that mouthful correct (I stand to be corrected). Anyhow,
>by plugging the figures in you can see why the usual no-resistance
>formula is generally good enough for a quick and dirty ballpark 

Yeah, I have a formula that looks something like that:

     f[r] = (1 - (R[L]^2 * C) / L) / (2 * pi * (LC)^0.5)

          where, f[r] = resonant frequency (Hz)
                 R[L] = resistance of inductor (ohms)
                 C    = capacitance (F)
                 L    = inductance (H)

If you were going to calculate the resonant frequency with your above formula
what would you use as the resistance (R) value?  You could directly measure
it with a ohm-meter (maybe) if the coil was already constructed, but if it
wasn't how would you calculate it?  Use the resistance of conductor itself?
I recently e-mailed Mark Graalman to find out what formula TESLAC II uses to
calculate the RF Resistance in the secondary, this is what he sent me:

                                  /----------           /
                            .02 \/    F Mhz.           /
          RF Resistance = ( --------------------  )   /     2
           (per foot)             Dia. inches        /
   It is for copper only

I do not even know/understand the difference, if any, between RF Resistance
(above) and resistance used in the resonant frequency formula above.  I haven't
really played around with the above formula yet, haven't had time.  Can anyone
clarify for me the difference, if there is any, between the 2 resistance's???

Also, as I stated before, I myself actually did not measure the resonant 
frequency of the coil, but rather a friend did.  He said he used a signal 
generator, a resistive load, a capacitance, and the coil, all in series, and
a scope across either the capacitance or the resistive load.  His little
schematic he drew for me looked like chicken-scratching, but as I can make of 
it, it looks something like this:

       |            R                |
      /-\                             O
     (SGU)                            O
      \-/                           L O
       |                              O
       |                              O
       |             C               |
       +-----------o( |o-------------+
                   |   |
                   |   |

     SGU = Signal Generating Unit
     R   = Resistive Load
     C   = Capacitance
     L   = Inductor (tesla secondary coil)

Or something to that effect, I am not exactly sure wheather the capacitive and
resistive components are supposed to be switched around, I dunno...

Comments on this anyone?  Does this configuration work?  Or is the whole idea
of using the above circuit to measure the resonant frequency totally wrong?

Amazingly the measured values he gave me for the coil were incredibly close to
my calculated f[r,p] using Ed Harris's formula for ballpark resonance:

(NOTE:  the 6 measurements were taken at different locations within the lab,
        and then averaged together for the below result)
Measured F[r]:       (AVG)  289.3691  

...and now my calculated results:

Self-Resonant Frequency Calculations (non-inductance dependent)

                                Hz              kHz             MHz
F[r]              (rf5)    289748.97438    289.74897438     0.289748974
                  (rf6)    289466.57582    289.46657582     0.289466576

F[r]:(Average)    (SUM)    579215.55019    579.21555019      0.57921555
                  (AVG)     289607.7751     289.6077751     0.289607775

F[r,m]            (rf5)    288146.14045    288.14614045      0.28814614
                  (rf6)    287865.30406    287.86530406     0.287865304

F[r,m]:(Average)  (SUM)    576011.44451    576.01144451     0.576011445
                  (AVG)    288005.72225    288.00572225     0.288005722

F[r,p]            (rf5)    289607.06112    289.60706112     0.289607061
                  (rf6)    289324.80088    289.32480088     0.289324801

F[r,p]:(Average)  (SUM)      578931.862      578.931862     0.578931862
                  (AVG)      289465.931      289.465931     0.289465931

Note, again, the closeness of the actual measured value to the "prime" value.
The calculated ballpark resonance (Ed's formula) which is non-inductance
dependant, meaning it is strictly based on the physical parameters of the coil
itself, has an uncertainty based on the actual measured resonance of only
+/-0.0968 kHz.  Not to shabby, I hesitate to call this a "ballpark"
resonance.  Of course all this is for not, if the above measured resonant
frequency was 
measured wrong. If it was, then the above is a extreme coincidence.

Sorry for the long feed everyone...

>    Further comments welcome.



P.S.  For those interested in the formulas I used to calculate the inductance,
      resonance, and such, let me know, and will either post them or mail
      them to you personally, whichever...

| Timothy A. Chandler                ||   M.S.Physics/B.S.Chemistry     |
| NASA-Langley Research Center       ||   George Mason University       |
| Department of Energy               ||   Department of Physics         |
| FRT/Alpha - NASALaRC/DOE JRD/OPM   ||   Department of Chemistry       |
| CHOCT FR Designation #82749156/MG09||   OPC-EFC                       |
|                 Private Email Address:  tchand-at-slip-dot-net               |