Etesla6 math and physics, take-2

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Peter Lawrence by way of Terry Fritz <teslalist-at-qwest-dot-net>" <Peter.Lawrence-at-Sun.COM>

Terry (and Paul),
                  another stab at "Etesla6 in a nutshell", comments and
corrections will be much appreciated.


1. The potential (aka voltage) field is computed:

    a. using a "relaxation" method - which is a fancy term for grid neighbor
       averaging - of which there are several numerical techniques (e.g. SOR,
       Jacobi, Gauss-Seidel, etc). These methods require known conditions
       along all boundaries and compute unknown values within the region
       defined by the boundaries.

    b. the boundaries are defined by the TC and an external surface, which
       could be the rectangular box defined by the floor, ceiling, and walls
       around the TC but in Etesla6 is a cylinder of those approx dimensions
       to take advantage of the axial symmetry of the TC to reduce the
       computational complexity (computations are actually only done in a
       meridional plane)

    c. the voltage of the toroid is assigned arbitrarily at 1-volt, the
       voltage along the secondary coil is assigned by a voltage profile
       that came from TSSP with 1 at the top and 0 at the bottom, and the
       voltage of the surrounding floor/ceiling/ walls cylinder is assigned
       zero.

    d. the relaxation-averaging method is applicable because an electric
       potential field obeys "poisson's equation"

       <http://titan.colorado.edu/courses.d/AFEM.d/AFEM.Ch02.d/AFEM.Ch02.pdf>

       appears to be a good on-line starting point for finite element
       analysis of fields, examples in this chapter include heat and fluid
       flow, electrostatics (what Etesla6 does), and magnetostatics. Its a
       book I'd consider buying if/when it gets published.


2. The E-field is computed as the gradient of the potential field,
    this is by definition in the math and physics of E&M.

    a. this can be done for the entire region around the TC to generate
       pretty pictures

    b. in order to compute the charge on the TC this only needs to be done
       along a "gaussian surface" that completely encloses the TC and is
       completely enclosed by the 0-volt outer boundary. Etesla6 chooses
       a sphere as its gaussian surface.



3. The charge inside the gaussian surface is the sum of the E-field normal
    to the surface at each point on the surface
    (ie, abs(E-field) * cos(angle between E-field and surface normal),
     or, dotprod( E-field, Normal), in either case a scalar value)

    One of the beautiful theorems in math is that the value computed for
    the charge inside the surface is independent of the size/shape of the
    surface, given that the field is <Paul can you supply the correct
    mathematical definition here>, which is true of electric potential
    and electric flux fields.


4. The capacitance of the TC is the charge (from 3) divided by the volts
    (from 1.c 1-volt was assumed)


5. The resonant frequency of the TC can now be better computed with the
    standard F = 1/(2*PI*SQRT(L*C)) equation, but using C from step 4
    rather than the "Medhurst + 0.8*Toroid" heuristic formula.

    (as I recall Etesla6 asks the user for a measured L of the secondary,
     does it use any corrections for L based on TSSP results?)


-Peter Lawrence.