Re: Mutual Inductance, Self Inductance, Wire Length And The Neumann Equation (fwd)

["Tesla list" <tesla@xxxxxxxxxx>]

Original poster: List moderator <mod1@xxxxxxxxxx>



---------- Forwarded message ----------
Date: Sun, 10 Dec 2006 17:48:16 -0600
From: Shaun Epp <scepp@xxxxxxx>
To: Tesla list <tesla@xxxxxxxxxx>
Subject: Re: Mutual Inductance, Self Inductance,
     Wire Length And The  Neumann Equation

Jered,

I don't know why wire length is so important to you.  Four times the wire 
length for a 1/4 wave seleniod does not equal the coils operating 
wavelength.

Using wirelength in the equations also removes some important parameters for 
calculating inductance of a selenoid.    The Area of the coil, the length of 
the seleniod, the number of turns (squared) are what makes the inductor. 
Using "wirelength" takes away the fact that it has to be formed into a coil, 
not just a straight piece of wire.

L = u N^2 A
    ------------
          l

I realize that the about equation and the wirelength version that you quote 
are only off by 4pi, but why muddy up the picture of what a coil is made up 
of.



I used Wintesla and a calculator to work out my tesla coil as an example.

Wire length for the secondary coil is 1415.1 feet, which works out to 
frequency of 173.9 khz (assuming 1/4 wavelength resonance).  Wintesla says 
that my coil resonate at 306.9 khz without a topload.  This takes into 
acount the geometery of my coil (which gives an inductance), the self 
capacitance to ground and interwinding capacitance.  I confirmed this using 
standard equations for inductance, self capacitance given by wintesla (since 
I don't have the self capacitance equations handy), and the resonant 
frequency equation.  If I add my topload (which it too small), the frequency 
drops to 191.7khz.  I have measured my coil to resonate at about 200khz.

This should prove that 1/4 wave (wirelength) resonance is only a rough 
aproximation for the operating wavelength of a selenoid!


Shaun Epp

See below for other comments:

----- Original Message ----- 
From: "Tesla list" <tesla@xxxxxxxxxx>
To: <tesla@xxxxxxxxxx>
Sent: Friday, December 08, 2006 10:16 AM
Subject: Mutual Inductance, Self Inductance, Wire Length And The Neumann 
Equation


> Original poster: "Jared Dwarshuis" <jdwarshuis@xxxxxxxxx>
>
>
> Mutual Inductance, Self Inductance, Wire Length And The  Neumann Equation
>
>
>
>
>
> The Neumann equation reads:
>
>  M = u / 4pi    closed integral path (a) closed integral path (b)   [ 
> dl(a) dot dl(b) /   r ]
>
>
>
> We will demonstrate that closed integral paths (a) and (b) are the lengths 
> of wire used in the inductor or transformer.
>
>
>
> Let us examine an air cored solenoid:
>
>
>
> Suppose we wind a short coil directly on top of a long coil such that all 
> of the magnetic flux  going through the long coil also must go through the 
> short coil.

This is never the case with a standard tesla coil.  There is less flux 
reaching the top turns of the coil that at the bottem where the primary is 
wound.


>
>
>
> Thus:  flux B = pi  Rsqrd  u  N  I
>
>
>
> Then:  M(ab) = N(b) flux (ab) / I(a)
>
>
>
> So:  M(ab)  = u pi Rsqrd N(a) N(b) / a
>
>
>
> Now we will multiply the numerator and denominator by 4pi and regroup.
>
> So: M(ab) = u/ 4pi   (2pi R N(a) ) (2pi R N(b) ) / a
>
> Or: M(ab) = u/4pi  (wire length (a) ) ( wire length (b) ) / a
>
>
>
> We can now clearly identify the components of the Neumann equation.
>
>
>
> Closed integral path (a) is simply the wire length of solenoid (a)
>
> Closed integral path (b) is simply the wire length of solenoid (b)
>
>  1/r is simply 1/a
>
>
>
> Now we can write the self inductance of a solenoid as:
>
>
>
> L = u (wire length)sqrd / 4pi height
>
>
>
> Equivalently:
>
>
>
> L = u /4pi height      (wire length) (wire length)
>
>
>
> (L ) is also in the form of the Neumann equation. This shows that self 
> inductance is actually a form of mutual inductance.
>
>
>
> Now we can write (u ) as:      u = 1/ e Csqrd
>
>
>
> Then: M =  (1 / 4pi e height)   (wire length (a) / C ) (wire length (b) / 
> C )
>
>
>
> Then: L =  (1 / 4pi e height)   (wire length / C ) (wire length / C )
>
>
>
>  (1/ 4pi e height) represents an inverse capacitance based of the distance 
> our fields have traversed through the permittivity of free space
>
> (wire length / C) gives units of time
>
>
>
> With (M) we can have two different wire length, with (L) we have a single 
> wire length that must be squared.
>
>
>
> End derivation.
>
>
>
> A very brief overview of considerations regarding M and L
>
>
>
> When we have a primary around the secondary of a Tesla coil, we can no 
> longer simply call the primary or secondary a self inductance. Because we 
> are coupled to a second set of windings it is now properly called M ( a 
> mutual inductance)
>

They are usually modelled separately as part of the same circuit.  (  Xl and 
Xm )

>
>
> Perhaps you have noticed that the primary tap points calculated using f = 
> 1/ 2pi sqrt(LC) can be off significantly.   This is especially true for 
> tightly coupled coils.
>
That's because of Xm
>
>
> As we couple a coil more tightly, the primary/secondary interaction starts 
> to become more M like, and less like two separate( L).   When a coil is 
> tightly coupled we will need more primary turns than calculated using: f = 
> 1/ 2pi sqrt(LC).
>
I'd say they are both there

>
>
> A tightly coupled coil depends on the extra turns that we added to 
> accumulate 'leakage inductance'. It is the sum of the uncoupled inductance 
> that we need for resonance. Uncoupled inductance means self coupled (the 
> magnetic field is self contained within the inductor).
>
Where do you get this from?

>
>
> A very lightly coupled coil  has so little M to consider that we can 
> ignore the term entirely and pretend that both primary and secondary are 
> entirely L.   The magnetic field for a lightly coupled coil has a weak 
> interaction between primary and secondary so it is essentially self 
> coupled and behaves as a self inductance.
>
Maybe for calculating inductance and resonant frequency but for a tesla coil 
there is always coupling, otherwise it wouldn't be a tesla coil.

>
> Sincerely:
>
> Jared Dwarshuis
>
> Dec. 06

I've never taken the level of physics or mathematics that you have taken but 
you messages always comes down to the basic equations and the wirelenth. 
The stuff that you come up with makes you look silly.