Mutual Inductance, Self Inductance, Wire Length And The Neumann Equation

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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.



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)



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.



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).



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).



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.



Sincerely:

Jared Dwarshuis

Dec. 06