Secondary Coil Turns

From:  Mark S. Rzeszotarski, Ph.D. [SMTP:msr7-at-po.cwru.edu]
Sent:  Wednesday, June 10, 1998 6:46 AM
To:  Tesla List
Subject:  Re: Secondary Coil Turns

Hello All:
Steve Young stated in part:
>> In many postings I have read, the general advise is to not exceed about
>> 1,000 turns on the secondary of disruptive TCs.  For example, Bert Pool's
>> excellent "Building Conventional Tesla Coils" states secondary coils should
>> be at least 400 turns, but no more than 1,000 turns.
        I have done considerable modelling of this as well as
experimentation, and several comments are in order: 
        First, consider the secondary as a quarter wave helical resonator
with a capacitive tophat.  In this scenario, the maximum voltage on top
depends on resonant rise, and this flattens out for H:D ratios above 2.5:1
or so.  In other words, there is diminishing return on Vtop as you increase
this ratio.  3:1 is a good compromise for a max value unless the coil is
really low in inductance (small diameter).  
        Consider a coil with a fixed H:D of 3:1.  Skin effects are frequency
dependent such that as the operating frequency increases, the current tends
to flow mainly along the outer surface of the wire.  This increases the
effective resistance (losses) of the coil.  Generally, one should use a wire
diameter that is at least N skin depths thick, where N might be chosen to be
6-10 or so, depending on your preferences.  In addition proximity effects
cause the current in the coil to be further restricted to mostly along the
inner diameter of the coil, due to electromagnetic fields.  As a result, the
effective resistance of the coil is often 3-5 times the DC resistance of the
wire.  As finer wire is used on a secondary, the operating frequency drops
since Ls is increasing, and the skin depth gets thicker.  At some point the
coil losses become excessive, causing Vtop to max out.  This happens to
occur at around H:D of 3:1.
        Now consider the pure lumped element model.  It says that the
voltage on top depends on the ratio Ls/Lp or equivalently Cp/(Ct+Cd), where
Cp = primary capacitance, Ct = toroidal capacitance, and Cd is the
distributed capacitance of the secondary.  A taller coil will have a higher
Cd, reducing Vtop.  In addition, you need a substantial Ct to store energy
for those nice long streamers.  If Ct is small, the energy in the spark must
be extracted from Cd, and the effective resistance of the coil comes into
play, reducing the amount of energy you can extract from the system, causing
for those wimpy purple sparks.
        From the Ls/Lp standpoint, one would think that you should max out
Ls and minimize Lp for maximum voltage rise.  However, the current induced
into the secondary due to the primary depends directly on M, the mutual
inductance between the primary and secondary.  M in turn is the product of
Np times Ns times G, where Np is the number of primary turns, Ns is the
number of secondary turns, and G is a geometric factor which takes into
account the position and geometry of the two coils.  This implies a high Np
and high Ns might be preferred, which reduces Vtop of course, since Lp and
Ls are proportional to Np^2 and Ns^2 respectively.  Hence, again we have a
tradeoff which limits Ls to some extent.  We can adjust our coupling
(increase the G in the equation for M), and couple more energy into the
system, but then we might not be able to quench the spark to keep the energy
locked in the secondary.

Much to ponder.

Mark S. Rzeszotarski, Ph.D.