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XYLENE POWER LTD.

SPHEROMAK SHAPE PARAMETER

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

INTRODUCTION:
As shown on the web page titled: THEORETICAL SPHEROMAK the spheromak energy density distribution outside the spheromak wall is:
Up = Uo {Ro^2 / [Ro^2 + (A R)^2 + (B Z)^2]}^2
and the energy density distribution inside the spheromak wall is:
Ut = Uto [Ro / R]^2

The spheromak wall intercepts the plane Z = 0 at radius R = Rs and at radius R = Rc where:
Rs = spheromak maximum outside wall radius
and
Rc = spheromak minimum inside wall radius

As shown on the web page titled: THEORETICAL SPHEROMAK the spheromak shape parameters Ro, So and Zf are given by:
Ro^2 = A^2 Rs Rc
and
So^2 = (Rs / Rc)
and
Zf = [A / B][(Rs - Rf)(Rf - Rc)]^0.5
= [A / B][(Rs - Rc) / 2]

Plasma spheromaks can be photographed and the photos used to determine So^2 and (A / B).

In analysis of spheromaks there are a number of energy densities of interest:
Uo - the energy density at R = 0, Z = 0
Utc - the energy density at R = Rc, Z = 0
Uto - the energy density at R = Ro, Z = 0
Uts - the energy density at R = Rs, Z = 0
Utf - the energy density at R = Rf, Z = Zf
U|(R, Z >> Ro) - the energy density at R >> Ro and Z >> Ro

At the spheromak wall there is a boundary condition that the inside energy density Ut must equal the outside energy density Up. That requirement at R = Rc, Z = 0 gives:
Uo {Ro^2 / [Ro^2 + (A Rc)^2]}^2 = Uto [Ro / Rc]^2
or
[Uto / Uo] = Rc^2 Ro^2 / [Ro^2 + (A Rc)^2]^2
= Ro^2 / [(Ro^2 / Rc) + A^2 Rc]^2
= A^2 Rs Rc / [(A^2 Rs Rc / Rc) + A^2 Rc]^2
= Rs Rc / [A (Rs + Rc)]^2
= (Rs / Rc) / {A^2 [(Rs / Rc) + 1]^2}
= So^2 / {A^2 [So^2 + 1]^2}
 

Thus:
(Uto / Uo) = [So / A (So^2 + 1)]^2
(Uts / Uo) = [So / A (So^2 + 1)]^2 [Ro / Rs]^2
(Utc / Uo) = [So / A (So^2 + 1)]^2 [Ro / Rc]^2
Utf / Uo) = [So / A (So^2 + 1)]^2 [Ro / Rf]^2
 

ELECTROMAGNETIC SPHEROMAK IMPLICATIONS:
On the web page titled: ELECTROMAGNETIC SPHEROMAK it is assumed the the energy densities are the result of electric and magnetic fields arising from net charge cirulating at the speed of light around a closed toroidal path containing Np poloidal turns and Nt toroidal turns. As a result of the toroidal geometric assumptions the length Lh of the spheromak current path is:
[Lh / 2 Pi Ro] = {(A / 2 So)[(Np (So^2 + 1))^2 + (Nt (So^2 - 1) Kc)^2]^0.5}
where:
Kc = (ellipse perimeter length) / (contained circle perimeter length)

Magnetic theory gives:
Bto = Muo Nt Ih / 2 Pi Ro
or
Uto = Bto^2 / 2 Muo
= [1 / (2 Muo)] [Muo Nt Ih / 2 Pi Ro]^2
= [Muo / 2] [Nt Ih / 2 Pi Ro]^2

Magnetic theory gives:
Bpo = Integral from Xv = Xvc to Xv = Xvs of:
(1 / Ro){(Muo Ih Xv^2) / {Xv^2 + (A / B)^2 [Xvs - Xv) (Xv - Xvc)]}^1.5}
{2 Np / [Pi Kh (1 + (A / B))(Xvs - Xvc)]}
dXv {[(Xvs - Xv)(Xv - Xvc)] + (A / 2 B)^2 [Xvs + Xvc - 2 Xv]^2}^0.5 / [(Xvs - Xv)(Xv - Xvc )]^0.5
 
= [Muo Ih Np / 2 Pi Ro] Integral from Xv = Xvc to Xv = Xvs of:
{(Xv^2) / {Xv^2 + (A / B)^2 [Xvs - Xv) (Xv - Xvc)]}^1.5}
{4 /[ Kh (1 + (A / B))(Xvs - Xvc)]}
dXv {[(Xvs - Xv) (Xv - Xvc)] + (A / 2 B)^2 [Xvs + Xvc - 2 Xv]^2}^0.5
/ [(Xvs - Xv)(Xv- Xvc)]^0.5
= [Muo Ih Np / 2 Pi Ro][INTEGRAL]
where:
Uo = (Bpo^2 / 2 Muo)
Xvc = (A Rc / Ro) = (1 / So)
Xvs = (A Rs / Ro) = So
Ih = circulating current.

and
[INTEGRAL] = Integral from Xv = Xvc to Xv = Xvs of:
{(Xv^2) / {Xv^2 + (A / B)^2 [Xvs - Xv) (Xv - Xvc)]}^1.5}
{4 /[ Kh (1 + (A / B))(Xvs - Xvc)]}
dXv {[(Xvs - Xv) (Xv - Xvc)] + (A / 2 B)^2 [Xvs + Xvc - 2 Xv]^2}^0.5
/ [(Xvs - Xv)(Xv - Xvc )]^0.5

Note that Kh is a function of parameter A and that the limits of the integral are functions of So. Thus [INTEGRAL] is a function of the parameters A and So.

Thus:

Uo = Bpo^2 / 2 Muo
= [1 / 2 Muo] {[Muo Ih Np / 2 Pi Ro] [INTEGRAL]}^2

Recall that:
Uto = [Muo / 2] [Nt Ih / 2 Pi Ro]^2

Hence:
Uto / Uo = ([Muo / 2] [Nt Ih / 2 Pi Ro]^2) / ([1 / 2 Muo] {[Muo Ih Np / 2 Pi Ro] [INTEGRAL]}^2)
= [Nt]^2 / [Np INTEGRAL]^2

The web page titled: THEORETICAL SPHEROMAK shows that:
[Uto / Uo] = [So / A (So^2 + 1)]^2

Equate the two expressions for [Uto / Uo] to get:
[So / A (So^2 + 1)]^2 = [Nt]^2 / [Np INTEGRAL]^2
or
[So / A (So^2 + 1)] = [Nt] / [Np INTEGRAL]
or
[Np / Nt] = {A (So^2 + 1) / (So [INTEGRAL])}
which is the (Np / Nt) value calculated from the center boundary condition.

This equation provides a three dimensional relationship between the three parameters {Np / Nt], A and So.
 

SPHEROMAK WALL BOUNDARY CONDITION:
(Np / Nt)^2 + Kc^2 [(So^2 - 1) / (So^2 + 1)]^2 = [4 A^4 / Pi^2]

Since Kc is a function of the parameter A this equation provides another 3 dimensional relationship between the parameters [Np / Nt], A and So.
 

COMBINE THE TWO BOUNDARY CONDITIONS:
Both boundary condition equations share the same So and (Np / Nt) values. Hence A must take a value which makes these two equations both give the same (Np / Nt) value for a common So value.
 

Substitute the center boundary condition equation:
[Np / Nt] = {A (So^2 + 1 ) / So [INTEGRAL]}
into the wall boundary condition equation to get:
{A (So^2 + 1 ) / So [INTEGRAL]}^2 + Kc^2 [(So^2 - 1) / (So^2 + 1)]^2 = [4 A^4 / Pi^2]
or
{A (So^2 + 1 ) / So [INTEGRAL]}^2 = [4 A^4 / Pi^2] - [Kc (So^2 - 1) / (So^2 + 1)]^2
or
[Kc (So^2 - 1) / (So^2 + 1)]^2 = [4 A^4 / Pi^2] - {A (So^2 + 1 ) / So [INTEGRAL]}^2
which formula will be used later on this web page.
 

[Lh / 2 Pi Ro] STABILITY CONDITION:
Let:
M = (1 / 3), (1 / 2), 1, 2 , 3.

On the web page titled PLANCK CONSTANT it is shown that for:
(Lh / 2 Pi Ro) stability it is necessary that at constant So:
- M dNt = dNp
and that:
[Np / ( M Nt)] = [(So^2 - 1) Kc]^2 / [(So^2 + 1)^2]
= constant
or
[Np / ( M Nt)]^0.5 = [(So^2 - 1) Kc] / [(So^2 + 1)]
or
[(So^2 + 1)] [Np / ( M Nt)]^0.5 = [(So^2 - 1) Kc]
or
So^2 (Kc - [Np / ( M Nt)]^0.5) = (Kc + [Np / (M Nt)]^0.5)
or
So^2 = (Kc + [Np / (M Nt)]^0.5) / (Kc - [Np / ( M Nt)]^0.5)

However, M must be chosen so that the spectrum of (Np / Nt) values is such that Np and Nt have no common factors. This spectrum of (Np / Nt) values is given by:
(Np / Nt) = {[P - 2 N] / N}
where N is a positive integer
which gives:
dNp = - 2 dNt

Recall that:
- M dNt = dNp
Hence:
M = 2

Recall that:
[Np / ( M Nt)] = [(So^2 - 1) Kc]^2 / [(So^2 + 1)^2]

The spheromak wall boundary condition gives:
(Np / Nt)^2 + Kc^2 [(So^2 - 1) / (So^2 + 1)]^2 = [4 A^4 / Pi^2]

Combining these two expressions with M = 2 gives:
[Np / Nt]^2 + (1 / M)[Np / Nt] = [4 A^4 / Pi^2]
or
[Np / Nt]^2 + (1 / 2)[Np / Nt] - [4 A^4 / Pi^2] = 0
which has the quadratic solution:
[Np / Nt] = {- (1 /2) + [(1 / 2)^2 + 4(1) 4 A^4 / Pi^2]^0.5} / 2
= - (1 / 4) + (1 / 4) [1 + 64 A^4 / Pi^2]^0.5
= (1 / 4){- 1 + [1 + 64 A^4 / Pi^2]^0.5}

Thus:
[Np / Nt]^2 = (1 / 4)^2{- 1 + [1 + 64 A^4 / Pi^2]^0.5}^2

Recall that the wall boundary condition gives:
(Np / Nt)^2 + Kc^2 [(So^2 - 1) / (So^2 + 1)]^2 = [4 A^4 / Pi^2]

Thus: Kc^2 [(So^2 - 1) / (So^2 + 1)]^2 = {[4 A^4 / Pi^2] - (Np / Nt)^2}
= {[4 A^4 / Pi^2] - (1 / 4)^2 {- 1 + [1 + 64 A^4 / Pi^2]^0.5}^2}
= {[4 A^4 / Pi^2] - (1 / 4)^2 {1 + [1 + 64 A^4 / Pi^2] - 2 [1 + 64 A^4 / Pi^2]^0.5}}
= {- (1 / 4)^2 {2 - 2 [1 + 64 A^4 / Pi^2]^0.5}}
= {(1 / 8) {-1 + [1 + 64 A^4 / Pi^2]^0.5}}
which equation can be readily rearranged to give So as a function of parameter A.

To find So and A we must simultaneously solve the equation:
Kc^2 [(So^2 - 1) / (So^2 + 1)]^2 = X^2 = {(1 / 8) {-1 + [1 + 64 A^4 / Pi^2]^0.5}}
and the equation:
[Kc (So^2 - 1) / (So^2 + 1)]^2 = X^2 = [4 A^4 / Pi^2] - {A (So^2 + 1 ) / So [INTEGRAL]}^2

Solving these two equations together requires itteration because A and So are both components of [INTEGRAL].

ITTERATIVE SOLUTION:
a) Assume an initial value of A = 1.000

b) Use the equation:
X^2 = {(1 / 8) {-1 + [1 + 64 A^4 / Pi^2]^0.5}}
to find the corresponding value of X.

c) Use the ellipse equations:
h = [(A - 1)]^2 / [A + 1]^2
and
Kh = [1 + (h / 2^2) + (h^2 / 2^6) + (h^3 / 2^8)
+ (5^2 h^4 / 2^14) + (7^2 h^5 / 2^16) + (21^2 h^6 / 2^20) + ....]

and
Kc = [1 + A] [Kh / 2]
to find the corresponding value of Kc.

d) Use the equation:
Kc^2 [(So^2 - 1) / (So^2 + 1)]^2 = X^2
or
Kc [(So^2 - 1) / (So^2 + 1)] = X
or
(So^2 - 1) = (X / Kc) (So^2 + 1)
or
So^2 (1 - (X / Kc)) = (1 + (X / Kc))
or
So^2 = [(1 + (X / Kc)) / (1 - (X / Kc))
to find the corresponding value of So.

e) Insert these values of So, A and Kc into equation:
X^2 = [4 A^4 / Pi^2] - {A (So^2 + 1 ) / So [INTEGRAL]}^2
to find a new value of X^2, where [INTEGRAL] is given by:
[INTEGRAL] = Integral from Xv = Xvc = (1 / So) to Xv = Xvs = So of:
{(Xv^2) / {Xv^2 + A^2 [Xvs - Xv) (Xv - Xvc)]}^1.5}
{4 /[ Kh (1 + A)(Xvs - Xvc)]}
dXv {[(Xvs - Xv) (Xv - Xvc)] + (A / 2)^2 [Xvs + Xvc - 2 Xv]^2}^0.5
/ [(Xvs - Xv)(Xv - Xvc )]^0.5

f)Calculate a new value of A using the equation:
X^2 = {(1 / 8) {-1 + [1 + 64 A^4 / Pi^2]^0.5}}
or
(8 X^2 + 1)^2 = [1 + 64 A^4 / Pi^2]
or
A^4 = (Pi^2 / 64)[(8 X^2 + 1)^2 - 1]

g) Repeat as necessary to obtain convergent solutions for So and A.

h) Then find the corresponding value of [Np/ Nt] using the equation:
[Np / Nt] = (1 / 4){- 1 + [1 + 64 A^4 / Pi^2]^0.5}
 

SPECIAL CASES:
At A = 1.000:
[Np / Nt] = 0.4339479041
which gives:
[Np / M Nt]^0.5 = 0.4658046286

Recall that:
So^2 = (Kc + [Np / (M Nt)]^0.5) / (Kc - [Np / ( M Nt)]^0.5)

At Kc = 1:
So^2 = (1 + 0.4658046286) / (1 - 0.4658046286)
= 1.4658046286 / 0.5341953714
= 2.743948576

Note that the spheromak will collapse if:
[1 + 64 A^4 / Pi^2] = (X^2 / Y^2)
where x and Y are integers.

For Y = 1 then X =:
4, 9, 16, 25, 36 or 49 (squares of integers)

For Y = 1 the corresponding values of A and (Np / Nt) are given by:
A^4 = (3 / 64) Pi^2, (Np / Nt) = (1 / 4) or A = 0.8247270826
A^4 = (8 / 64) Pi^2, (Np / Nt) = (2 / 4) or A = 1.053907365
A^4 = (15 / 64) Pi^2, (Np / Nt) = (3 / 4) or A = 1.233254638
A^4 = (24 / 64) Pi^2, (Np / Nt) = (4 / 4) or A = 1.387020095
A^4 = (35 / 64) Pi^2, (Np / Nt) = (5 / 4) or A = 1.524217536
A^4 = (48 / 64) Pi^2, (Np / Nt) = (6 / 4) or A = 1.649869663

Recall that:
[Np / ( M Nt)] = [(So^2 - 1) Kc]^2 / [(So^2 + 1)^2]

Hence for So^2 = 4:
(Np / Nt) = M [(So^2 - 1) Kc]^2 / [(So^2 + 1)^2]
= 2 [9 / 25] Kc^2
= 0.72 Kc^2
 

DETERMINATION OF Np and Nt:
Recall that:
(Np / Nt) = (P - 2 N) / N
= (P - 2 Nt) / Nt

Hence:
P = Nt (Np / Nt) + 2 Nt
where:
P, Nt and [Nt(Np / Nt)] are all integers. To find Nt increment Nt from 1 upwards until the quantity:
[Nt (Np / Nt)]
is a perfect integer.
At that state record:
P, Nt, Np
 

FIND THE VALUE OF [Lh / 2 Pi Ro]:
Once the Np, Nt and So values have been determined from the definition of Lh on the web page titled: ELECTROMAGNETIC SPHEROMAK we can calculate:
[Lh / 2 Pi Ro] = {(A / 2 So)[(Np (So^2 + 1))^2 + (Nt (So^2 - 1) Kc)^2]^0.5}
 

Thus if:
So = 1.656577506
So^2 = 2.744249035
A = 1.000
Kc = 1.000
Np = 125
Nt = 288
we find that:
[Lh / 2 Pi Ro]
= {(1 / 2 (1.656577506))[(125 (3.744249035))^2 + (288 (1.744249035))^2]^0.5}
= {(0.3018271093) [219,053.1381 + 252,349.2151]^0.5
= 207.2307112
 

FIND THE VALUE OF (1 / Alpha):
The web page titled: PLANCK CONSTANT provides the formula:
(1 / Alpha)
= [Pi / 2 A^2 B] [Lh / 2 Pi Ro]{So [So^2 - So + 1] / [(So^2 + 1)^2]}

which can now be calculated.
 

[Pi / 2 A^2 B] = 1.570796325

{So [So^2 - So + 1] / [(So^2 + 1)^2]}
= {(1.656577506) [2.744249035 - 1.656577506 + 1] / [(3.744249035)^2]}
= {(1.656577506) [2.087671529] / [14.01940084]
= 0.2466859843

Thus for A = 1 the theoretical value of (1 / Alpha) is:
(1 / Alpha) = [Pi / 2 A^2 B] [Lh / 2 Pi Ro]{So [So^2 - So + 1] / [(So^2 + 1)^2]}
= [1.570796325] [207.2307112] [0.2466859843]
= 80.30054065

Clearly A = 1.000 gives the wrong prime number P. A very small error in the ratio (Np / Nt) will lead to a wrong choice of P. Assuming that the error in (Np / Nt) is small in order to match experimental results the P value should be about:
(137.036 / 80.3005) X 701 = 1196.28 The likely P values are the primes 1193 or 1201. Other nearby primes are:
1187, 1213 and 1217.

Recall that:
Np = P - 2 Nt
Nt ~ P / 2.433947904
For P = 1193
Nt = 490
Np = 213
[Np / Nt] = 0.4346938776

For P = 1201:
Nt ~ P / 2.433947904
= 493.437

If Nt = 493 then Np = 215 and (Np / Nt) = 0.4361

If Nt = 494 then Np = 213 and (Np / Nt) = 0.43117

Thus the more likely value of P is 1193.

A good estimate of [Lh / 2 Pi Ro] is:
[Lh / 2 Pi Ro] ~ [207.2307112] [(137.036 / 80.3005)]
= 353.6474585

Note that even a small deviation of Kc from unity causes a small change in the calculated value of (Np / Nt) which in turn has a large effect on the choice of prime number P. The selection of prime number P is the dominant component of (1 / Alpha).

This web page last updated January 11, 2020.

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