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Re: TC Inductance Formulas...

>From: Tim Chandler <tchand-at-slip-dot-net> originally, with a reply by Malcolm Watts
>Subject: Re: TC Inductance Formulas...
><snip>
>>
>>     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 
>>figure.
>
>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???
>
Hello Tim, Malcolm, and other interested folks,
        I have a keen interest in understanding R.F. resistance and have
been investigating it a little. (Much more yet to be done.)  I have examined
Malcolm's data to gain some insight on the problem.  First, a little intro:
        The D.C. resistance of an inductor can be easily measured, and
agrees pretty well with formulas for the resistance of wire based on bulk
resistivity and geometry factors.  When an inductor is used at A.C.
frequencies, skin effects crop up.  Basically, the changing magnetic and
electric fields within the wire cause the current to tend to travel on the
outer surface of the wire.  As a result, the A.C. resistance for a piece of
wire is substantially higher than the D.C. resistance.  It is proportional
to the square root of the operating frequency.  Typically, the A.C.
resistance is about 1.25 to 1.75 times the D.C. resistance of the wire, at
Tesla coil frequencies.  
        However, there is another effect which is much more dominant (at
least in closewound coils).  It is called proximity effect.  Basically, the
magnetic and electric field of one wire affects the current distribution in
an adjacent wire, causing the current distribution to be confined even more
than that due to skin effects alone.  As a result, the R.F. resistance of a
coil (which includes D.C. resistivity, skin effects and proximity effects
combined) is substantially higher than the D.C. resistance.  Proximity
effects have been studied in several simple models, such as the case of a
pair of wires in a "go and return" circuit.  I have seen no decent
literature on proximity values for solenoidal coils, only a few anecdotal
observations.
        Another confounding factor in Tesla coils is the observation that
the addition of a toroid on top of a helical resonator significantly alters
the R.F. resistance of the coil.  Apparently, electric field effects from
the toroid alter the current distribution within the wires near the toroid,
changing the proximity effect substantially.  Both the size and the position
of the toroid affects the total R.F. resistance.  In Malcolm's data, the
R.F. resistance is about 4 times the D.C. resistance, for a 3:1 H:D
closewound coil with a toroid on top.  Proximity effects are reduced
substantially in spacewound coils, (but so is inductance, which is important
to maximize).
        The R.F. resistance as I interpret it is the total effective
resistive losses in the coil.  If you use total R.F. resistance instead of
D.C. resistance in the formula Q=wL/R, you get numbers that make sense.  I
do not know if the same value applies to the formulas listed at the
beginning of this post, but I suspect it does.  If anyone can shed
additional light on this one, I'd appreciate it.
Regards,
Mark S. Rzeszotarski, Ph.D.