Re: [Phys-l] transformer question

[John Denker <jsd@av8n.com>]

On 08/13/2009 12:36 PM, Mangala Joshua wrote:
> The transformer equation in undergraduate texts is 
>
> Vp/Vs=Np/Ns             [1]
>
> However if we set up an experiment we observe that 
>
> Vp/Vs depends on the frequency of the primary source( or Vs changes with frequency) 
>
> What is the cause of this frequency dependence? 


The answer involves some interesting and useful physics.

The physics behind equation [1] includes the assumption that every
flux line that goes through the primary coil also goes through the
secondary coil (and vice versa).  This is called "tight coupling".
If that assumption is valid, equation [1] absolutely must be valid.  
It's only half a step removed from one of the Maxwell equations.
 -- Sometimes that is a good assumption.  On various occasions I 
  have relied on equation [1] being accurate to 1 part per million,
  when applied to a Gertsch ratio transformer,
 -- Sometimes tight coupling is not a valid assumption.  If you 
  have two coils that are only imperfectly coupled, to describe
  the physics you need more than the simple scalar Np/Ns ... you
  need a matrix, i.e. the mutual inductance matrix Mij.

In general we have 

      Vi = Mij (d/dt) Ij                    [2]

In the case of only one coil, Mij is a 1x1 matrix, and the 
matrix element is called the inductance.  In the case of two
coils, M11 is the self inductance of coil 1, M22 is the self
inductance of coil 2, and M12=M21 is the (scalar) mutual 
inductance of the two coils.  If there are more than two coils,
we can still speak of the diagonal elements as being the self
inductances, but there is no longer any notion of "the" mutual
inductance;  the matrix formulation is the only game in town.

For coils that are only vaguely in the vicinity of one another,
the Mij matrix is nearly diagonal.  That is, the self-inductances
will be large compared to the mutual inductances.

The frequency dependence of this-or-that observed signal will 
depend on the
 -- the Mij values;
 -- the impedance of whatever is hooked to the primary; and 
 -- the impedance of whatever is hooked to the secondary.


===========================

Additional physics:  
 a) Sometimes designers can guarantee tight coupling just by 
  controlling the geometry of the windings.  
 b) In other cases, the geometry is far from ideal, but they try 
  to "persuade" the flux lines to couple tightly by means of a high
  permeability core.  The problem with this scheme is that the
  permeability is a function of frequency ... so that the mutual
  inductance matrix Mij is itself a function of frequency.  This
  gets very messy very fast.  Algebraic solutions are out of the
  question;  the only way to predict the behavior is via finite
  element modeling.  If you have the FEM software and know how to
  use it, you can probably earn a nice living as a consulting 
  "magnet architect".


===============

Constructive suggestion:  If you *want* equation [1] to hold, so
you don't need to address the complexities, I recommend a bifilar
winding on a high-permeability toroidal core.

I have in the past bought toroids from these guys:
http://www.fair-rite.com/cgibin/catalog.pgm?THEAPPL=Inductive+Components&THEWHERE=Closed+Magnetic+Circuit&THEPART=Toroids#select:freq5

===

If you don't want to wind your own, you can just buy a high-accuracy 
signal transformer or balun.