[sdiy] inductors, B fields, and confusion

[Magnus Danielson]

From: Don Tillman <don at till.com>
Subject: Re: [sdiy] inductors, B fields, and confusion
Date: 12 Jun 2004 01:09:26 -0700
Message-ID: <m2d645tcy1.fsf at till.com>

Richard & Don,

>    > Which will sound louder, an amp driving a speaker cabinet with two 16 
>    > ohm speakers wired parallel, or the same amp driving two 4 ohm speakers 
>    > wired in series?  In both cases the amp is running at an 8 ohm load.  
>    > The assumptions to take into account are that the voice coils of the 
>    > speakers have the same diameter for all of the speakers and the gauge 
>    > of the voice coil wire is all the same.  Also, the magnet weights are 
>    > the same.  Consequently, there should be four times as many turns of 
>    > wire in the voice coils of the 16 ohm speakers than in the 4 ohm ones.  
>    > Does this mean that the magnetic field strength for the 16ohm speakers 
>    > will be stronger for a given current than the B field generated in the 
>    > 4 ohm speakers?  
>    > 
>    > Does this mean that the 16 ohm speakers would be louder or am I 
>    > skipping something?  Is there some sort of physical saturation point 
>    > that I'm ignoring? 
> 
> I believe so.
> 
> The bible on this is the original Neville Thiele article
> "Loudspeakers in Vented Boxes", parts 1 and 2.  Also the Richard
> Small articles.  These are nicely reprinted in the Loudspeakers
> Anthologies available from the Journal of the Audio Engineering
> Society.  

I also recommend getting and reading those articles. The Loudspeaker
Anthologies is full of lovely articles and I do recommend them to people
interested to learn about it. Harry F. Olssons book "Acoustical Engineering"
remains another classic (whenever a "new concept" is announced I usually find
it already covered one way or another in this book so "nothing new under the
sun" is valid for so much crap that we see).

> (I haven't looked at these for a while, so it's entirely possible I'm
> missing something here.)
> 
> Thiele states that the efficiency of a speaker is:
>   eff = (rho/4 pi c) (B^2 l^2 Sd^2 / Re Mm^2)

       rho
          0    B²l²Sd²
eff = ------ * --------
      4*pi*c   Re * Mm²

Where rho0 is the density of the air.

True, but there is another thing... are you really using the same gauge of
wire?

If we assume we have about the same cross-section area of copper wire on the
coil (the coil thickness doesn't vary very much), then we would require a
smaller gauge of wire. The resistance per meter of wire varies reciprocal to
the wire radius in square, that is reciprocal to the per-wire-area.

The cross-sectional of the coil will be the number of turns N times the
effective area (there is some space inbetween the wires) so we have:

A    = N * A     * geometry-factor
 tot        Wire

(The geometry factor handles the geometry-depending area-scaling which is due
to the detailed packing geometry of the wires. It's a constant for our
purposes. I'll assume it to be 1 for simplicity.)

Now, there is a connection between N and l due to the geometry, a rought
equalent would be:

l = 2*pi*r*N

where r is the effective radius of the coil wires as they are wound around the
coil. r is assumed to be a constant for us as well.

The wires area is found as

            2
A     = pi*r
 wire       wire

Where rwire is the wires radius.

The resistance Re is formed by

     rho * l   rho*2*pi*r*N
Re = ------- = ------------
      A           A        
       wire        wire    

where rho is the resistivity constant for copper wire.

It's just that N depends now on the Awire so:

    A
     tot
N = -----
    A
     wire

So 

     rho*2*pi*r*A
                 tot
Re = ---------------
          2
         A
          wire

If we now stuff this into the efficient formula we have:

l = 2*pi*r*N


    A
     tot
N = -----
    A
     wire

l = 2*pi*r*Atot/Awire

       rho                 rho             4*pi²*r²*A²         A²
          0    B²l²Sd²        0    B²Sd²             tot        wire
eff = ------ * -------- = ------ * ----- * ------------- * ---------------
      4*pi*c   Re * Mm²   4*pi*c    Mm²        A²          rho*2*pi*r*A
                                                wire                   tot

this reduces down to

       rho             2*pi*r*A      rho            r*A
          0    B²Sd²           tot      0   B²Sd²      tot
eff = ------ * ----- * ----------- = ---- * ----- * ------
      4*pi*c    Mm²        rho       2*c     Mm²     rho

Now look at this formula. The things we can play with to change the resistance
comes in the total area cross-section of the coil Atot, the radius of the coil
r and the resistivity of the coil wire rho. The impedance we select to create
is for all practical purposes an independent variable for the efficiency.

Oh, and if anyone wonders, here is is the speed of sound in air and not the
speed of light in vacuum. ;O)

> (details in the article)
> 
> The interesting parts here are Re, the dc resistance of the voice
> coil, and l, the length of wire in the magnetic field.  The 16 ohm
> speakers will be louder because they have some many more windings in
> the magnetic field.

So, I think that this statement is a bit premature. If the same gauge of wire
could be assumed, then it would be true, but I think that in general we will
find that the gauge of the wires varies similar to what I have modelled
basically since adding more copper adds to coil mass (~r*Atot dependent!)
and when considering the capability of power-handling these coils would be
roughly equalent so adding unnecessary copper would not be productive.

However, adding excess copper would slightly increase the Mm term (which should
be Mms) but that is a sum of many weights.

So, assuming the same cross-section area Atot of the coil, combined with the
same coil radius, then these two setups would be equalent!

The trouble with these things is that you have only a few parameters which is
truly independent. In this example a number of engineering concerns went into
the formula and changed the result.

Cheers,
Magnus