[sdiy] inductors, B fields, and confusion

[Magnus Danielson]

From: "Scott E." <yahudinyhwh at sbcglobal.net>
Subject: Re: [sdiy] inductors, B fields, and confusion
Date: Sat, 12 Jun 2004 10:56:53 -0700
Message-ID: <40CB43E5.9040705 at sbcglobal.net>

Scott,

> Hmmm... I think I'll just stick to 8 ohm drivers. :)

Don't be supprised to have me delivering a full explanaition on why
specifically 8 ohm drivers is the ultimately bad choice! ;O)

Actually, I just got the hunch that something was terribly wrong so I embarked
on this exercise without for sure knowing the answer. This happends from time
to time. This time I was able to prove it fairly quickly. Sorry for the
ASCIImatics...

Cheers,
Magnus

> Scott E.
> ============================================================
> Magnus Danielson wrote:
> >        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)