On Saturday, December 11, 1999 3:00:00 AM UTC-5, BillyFish wrote:   
   > Essentially, the following problem was posed on this newsgroup:   
   > ***************   
   > Consider a transformer wound on a large toroidal core using a high    
   > permeability material so that very little magnetic field is outside the   
   core.    
   > That is, there is little leakage reactance.  Put a primary winding along a    
   > small length around the circumference of the core.  Put a similar secondary    
   > winding on the diametrically opposite of the primary.  Connect the primary   
   to    
   > a low impedance ac power source and the secondary to a variable resistance    
   > load.  As the load resistance changes, current in the primary and secondary    
   > changes in such a way as to keep the flux in the core relatively constant.   
   >    
   > Using the Poynting theorem, for example, how does power get transferred from    
   > the primary to the secondary?  The flux in the core is not greatly affected    
   > by the power.  That flux is also longitudinal.  There is no change in the E    
   > field.  The same voltage is across each winding at low and high loads.   
   >    
   > Suppose you set up a plane symmetrically between the two winding cutting the    
   > core into two halves.  If you integrate the Poynting vector over this plane,    
   > I do not see that the E x H to be very different for high and low resistive    
   > loads.  There is no physical current flow across the plane other than    
   > displacement current.   
   > **********   
   > This problem vexed me.  After I got up to go to the bathroom last night, I   
   > could not go back to sleep.  I pondered the problem, and I believe I have the   
   > answer.  It was partially formed in a conversation with someone who had some   
   > glimmerings but not the full insight.  The description above is, not   
   > surprisingly, a *red herring*.   
   >    
   > One key to the problem is to realize that the leakage reactance of a   
   > transformer is *independent *of the core!  The core increases the magnetizing   
   > inductance and coupling coefficient but has NO effect on the leakage   
   reactance.   
   >  This is well known to designers of pulse transformers, for example.  In   
   > equivalent circuit diagrams, current from the primary to the secondary   
   > transfers *through* the leakage reactance.  Most transformer engineers do not   
   > think in terms of Poynting's theorem.   
   >    
   > In a transformer as described above, the main portions of the core, that are   
   > not covered by windings, act as two pole pieces.  A magnetic field component   
   > fringes between them.  It is driven by the bucking currents flowing in the   
   two   
   > windings producing an H field proportional to the ampere turns in each   
   winding.   
   >  This H cannot be reduced by using a high permeability core material.  The   
   core   
   > enables this leakage field to be distributed over a larger volume.  Without   
   > this core, the leakage would be local to the individual windings.  This H   
   field   
   > produced by opposing currents in the primary and secondary  windings.  It   
   > provides an H that can be crossed with an E field to give a power transfer   
   from   
   > primary to secondary.   
   >    
   > Where does the E field to do this come from?  The magnetic field B through   
   the   
   > core is proportional to the voltage across the primary and secondary and 90   
   > degrees out of phase with this voltage.  According to Faraday's law, this   
   flux   
   > produces an E field through the core hole proportional to the rate of change   
   of   
   > flux inside the core.  Thus, this E field is proportional to the voltage in   
   > each winding and 90 degrees out of phase with the flux.  The result is that   
   the   
   > transverse components of the E and H fields, for resistive loads, are in   
   phase   
   > and contribute to a real transfer of power from primary to secondary.   
   >    
   > I do not know if this description for energy transfer has ever been presented   
   > before.   
   >    
   > William Buchman   
      
   Tomtech,   
      
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