Skin effect

From:  Antonio C. M. de Queiroz [SMTP:acmq-at-compuland-dot-com.br]
Sent:  Saturday, April 04, 1998 12:17 AM
To:  Tesla List
Subject:  Re: Skin effect

Bill the arcstarter wrote:

> Who wrote this? I missed the start of this thread...  Sounds like fun!

I wrote.

> I do know that, for a series lumped RLC - where you are viewing the
> impedence across the terminals of the lumped cap - it can be shown that,
> at resonance (note 1), the net system impedence is equal to Z0 = L/RC,
> where R,L,C are the lumped values, and Z0 is the system impedence across
> the cap.
> I'm curious if perhaps the measurement of Rac is being confused with the
> measurement of Z0 = L/RC.  This resonant circuit will transform the
> actual "lumped" ac resistance by the L/((Rac)C) operation.

One of the methods that I used was to search for the maximum attenuation
of the circuit:

Vin o----R-----+--oVout
               L (the coil, with Rac resistance and self capacitance)
               C (lumped variable capacitor, or open circuit)

With C chosen to make the LC tank resonate, Rac can be obtained from the 
relation Vout=Vin*Rac/(R+Rac) at the frequency when Vout/Vin is minimum. 
If the frequency gets closer to the self-resonance frequency of the coil,
the self-capacitance of the coil (or transmission line effect)
appears as a capacitance in parallel with the LC series combination and 
another capacitance in parallel with C (approximately). If C=0, only the 
self-capacitance of the coil remains. It appears then as the full self-
capacitance at the position of C, and as a fraction of it (1/3) across the 
LC combination (to simulate an impedance pole at 2 times the frequency of 
the impedance minimum, due to the 1/2 wave resonant mode of the coil). 
The analysis of the exact effect is rather complex, but simple to simulate.
The effect is that the impedance reduces to ~Rac, with great precision, at the
minimum of Vout/Vin.
(In my case, using the computed Rac=142 Ohms, Z0=32 mH/(142 Ohms*5.34 pF)=
42.2 MOhm, too high to be confused with 1 KOhm.)
> (NOTE 1) Notice that, for the above discussion, "resonance" is defined
> as the frequency at which the net impedence is purely resistive.  This
> is NOT the same as the frequency which yields the peak voltage (although
> it is close).  This frequency, expressed in rad/sec is W0 = sqrt(
> (L-R^2*C)/(L^2*C) ).  Note that if R ==0 then this is equivalent to the
> classic W=sqrt(1/LC), etc.

> I'm working on a derivation for W1 and Z1 such that output voltage is
> peaked, which is more to the needs of us coilers...  Anyone want to
> maximize a ratio of a 6th and a 4th order polynomials?!  Where's a
> freshmen when you need one??? :)

What are W1 and Z1?

Measurement of resonator coil parameters is really rather fun. Did
someone try to characterize a coil as a transmission line? If you try
you may find some surprises (more about this later).

Antonio Carlos M. de Queiroz