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XYLENE POWER LTD.

SPHEROMAK SHAPE PARAMETER

By Charles Rhodes, P.Eng., Ph.D.

INTRODUCTION:
The spheromak shape parameter:
So^2 = (Rs / Rc)
where:
Rs = outside radius
and
Rc = inside radius
is one of the most important spheromak characterization parameters. Experimental data shows that for plasma spheromaks:
So^2 ~ 4.2

Precise theoretical analysis elsewhere on this web site indicates that for quantum charged particles:
So^2 ~ 3.99596

On the web page titled: PLANCK CONSTANT AND FINE STRUCTURE CONSTANT it is stated that So ~ 2.0. The material on this web page supports that assertion.

The total static field energy contained in an electromagnetic spheromak can be expressed as:
Ett = [(Muo C^2 Qs^2) / 32] [1 / Ro] [4 So (So^2 - So + 1) / (So^2 + 1)^2]
where:
[1 / Ro]
= [(Pi Fh / C So)[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5

Hence:
Ett = [(Muo C^2 Qs^2) / 32] [4 So (So^2 - So + 1) / (So^2 + 1)^2]
[(Pi Fh / C So)[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
 
= [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)^2]
[Np^2 (So^2 + 1)^2 + Nt^2 (So^2 - 1)^2]^0.5
 
= [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)^2]
[Np^2 + [Nt^2 (So^2 - 1)^2 / (So^2 + 1)^2]^0.5 (So^2 + 1)
 

Define:
Nr = Np / Nt
and
R = (So^2 - 1) / (So^2 + 1)

Then:
Ett = [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)]
[Np^2 + Nt^2 R^2]^0.5

When a spheromak is at its stable state:
Np^2 (So^2 + 1)^2 ~ Nt^2 (So^2 - 1)^2

Hence:
Nr = (Np / Nt)
~ (So^2 - 1) / (So^2 + 1)

Hence:
Nr^2 ~ [(So^2 - 1) / (So^2 + 1)]^2
= R^2

The spheromak static field energy is given by:

Ett= [(Muo C Qs^2 Pi Fh) / 8] [(So^2 - So + 1) / (So^2 + 1)]
Nt [Nr^2 + R^2]^0.5
 
= [(Muo C Qs^2 Pi Fh Nt) / 8] [(So^2 - So + 1) / (So^2 + 1)][Nr^2 + R^2]^0.5

However, the web page titled ELECTROMAGNETIC SPHEROMAK shows that the spheromak boundary condition is:
[Nr^2 + R^2] = {1 / [(Pi / 2)^2 - (F / R)^2]}
where F indicates the electric field ratio at the spheromak wall at R = Rc, H = 0.

Generally 0.5 < F < 1.0. Note that F = 1.0 for a metal sphere and 0.5 for a flat plate.

Hence:
Ett = [(Muo C Qs^2 Pi Fh Nt) / 8] [(So^2 - So + 1) / (So^2 + 1)]
{1 / [(Pi / 2)^2 - (F / R)^2]^0.5}

In order for the Planck Constant h and Fine Structure Constant Alpha to be really constant the product:
[(So^2 - So + 1) / (So^2 + 1)]{1 / [(Pi / 2)^2 - (F / R)^2]^0.5}
must be constant at the spheromak operating point.

At So = 2.0 the equation for Ett simplifies to:
Ett = [(Muo C Qs^2 Pi Fh Nt) / 8] [(So^2 - 1) / (So^2 + 1)]
{1 / [(Pi / 2)^2 - (F / R)^2]^0.5}
 
= [(Muo C Qs^2 Fh Nt) / 8] [R]
{Pi^2 / [(Pi / 2)^2 - (F / R)^2]^0.5}
 
= [(Muo C Qs^2 Fh Nt) / 4]
{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5
 

Make substitution:
Muo C Qs^2 = 2 h Alpha
to get:
Ett = [(2 h Alpha Fh Nt) / 4]
{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5
 

or since:
Ett = h Fh
(1 / Alpha) = [ Nt / 2] {(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5

Note that Nt must be a positive integer. Comparison with experimental data shows that this equation is consistent with:
Nt = 274
and
{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]}^0.5 = 1
or
{(Pi R / 2)^2 / [(Pi / 2)^2 - (F / R)^2]} = 1

Rearranging this equation gives:
(Pi R / 2)^2 = [(Pi / 2)^2 - (F / R)^2]
or
(F / R)^2 = [(Pi / 2)^2 - (Pi R / 2)^2]
or
F^2 = R^2 [(Pi / 2)^2 - (Pi R / 2)^2]
= (R Pi / 2)^2 [ 1 - R^2]

Hence:
F = (R Pi / 2) [1 - R^2]^0.5

At So = 2:
R = (So^2 - 1) / (So^2 + 1)
= (3 / 5)

Thus:
F = (R Pi / 2) [1 - R^2]^0.5
= (3 Pi / 10) [1 - (9 / 25)]^0.5
= (3 Pi / 10)(4 / 5)
= 0.24 Pi
= 0.75398

Thus if an electric field analysis confirms that F ~ 0.75398 we can reasonably conclude that the spheromak shape factor So = 2.0

Hence, in spite of the complexity of the electric field integral calculation we are motivated to use it to solve for F in order to accurately determine So.

Note that an upper limit on F at So = 2 is imposed by:
(Pi / 2) > (F / R)
and
R = (3 / 5) or
F < (R Pi / 2) = 0.30 Pi = 0.942

Since the maximum value of F = 0.942 the inequality:

(Pi / 2) > (F / R)
together with:
F < 0.942 imposes a lower limit on R of
R = 0.942 / (Pi / 2)
= 0.5996

Recall that:
R = (So^2 - 1) / (So^2 + 1)

Thus the corresponding lower limit on So^2 is given by:
(So^2 + 1) R = So^2 - 1
or
So^2 (1 - R) = 1 + R
or
So^2 = (1 + R) / (1 - R)
= (1 + 0.59969) / (1 - 0.59969)
= 1.59969 / 0.40031
= 3.99614
which is close to where a spheromak actually operates.
 

DEPENDENCE OF So ON F:
The operation of spheromaks as contemplated on this web site relies on the spheromak shape parameter So being at or close to:
So = 2.0

This parameter value is set by the boundary condition: Nr^2 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]

The boundary condition makes Nr = (Np / Nt) a function of R which is in turn a function of So. Thus via the boundary condition F determines R and Nr which in turn determine So.

In a real spheromak typical parameter values are:
Np = 240, Nt = 274, So ~ 2.0, R ~ (3 / 5) and F ~ 0.24 Pi = .75398

When the electromagnetic field environment of a spheromak changes F will slightly change causing a corresponding small change in So.

To understand the stability of So we have to understand how the spheromak boundary condition sets So.

The boundary condition is:
Nr^2 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]
where:
R = (So^2 - 1) / (So^2 + 1) and
Nr = (Np / Nt)

Recall that from the web page titled: ELECTROMAGNETIC SPHEROMAK in a stable spheromak:
Nr^2 ~ 2 R^2

Substitution into the boundary condition gives: Nr^2 + R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]
or
3 R^2 = 1 / [(Pi / 2)^2 - (F / R)^2]

Rearranging the boundary condition formula gives:
[(Pi / 2)^2 - (F / R)^2] [(3 R^2] = 1
or
[3 (Pi R / 2)^2 - 3 (F)^2] = 1
or
R^2 = (1 + 3 F^2) (1 / 3) (2 / Pi)^2

The parameter F lies in the range:
0.5 < F < 1.0

At F = 0.5:
R^2 = (1 + 3 F^2)(1 / 3) (2 / Pi)^2
= (1 + (3 / 4))(1 / 3) (2 / Pi)^2
= (7 / 4)(1 / 3)(2 / Pi)^2
= 0.2364

At F = 1.0:
R^2 = (1 + 3 F^2)(1 / 3) (2 / Pi)^2 = (4)(1 / 3) (2 / Pi)^2 = 0.5403

Thus:
0.2364 < R^2 < 0.5403
or
0.48621 < R < 0.7351

Recall that:
R = (So^2 - 1) / (So^2 + 1)
or (So^2 - 1) = R (So^2 + 1)
or So^2 {1 - R} = {1 + R}
or
So^2 = (1 + R) / (1 - R)

At R = 0.48621:
So^2 = (1 + R) / (1 - R)
= (1 + .48621) / (1 - .48621)
= 1.48621 / 0.51379
= 2.8926

At R = 0.7351:
So^2 = (1 + R) / (1 - R)
= (1 + 0.7351) / (1 - 0.7351)
= (1.7351 / 0.2649)
= 6.5500

Thus So^2 lies in the range:
2.8926 < So^2 < 6.5500

Hence:
1.700 < So < 2.5592

This conclusion supports the approxmate and precise mathematical analyses elsewhere on this web site which indicate that a spheromak operates at So ~ 2 with:
R = (3 / 5)
= 0.60
and
R^2 = (9 / 25)
= 0.36

This web page last updated October 10, 2018.

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