XYLENE POWER LTD.

SPHEROMAK STRUCTURE:

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

SPHEROMAK CONCEPT:
The existence of stable quantum charged atomic and nuclear particles is enabled by an electro-magnetic structure known as a spheromak, in which electric and magnetic forces cancel each other. Net charge circulating around a complex closed quasi-toroidal filament path can form the electric and magnetic field energy density distributions required for spheromak existence. The electric and magnetic fields of a spheromak maintain its stable geometry and contain a stable amount of energy.
 

SPHEROMAK GEOMETRY:
Conceptually a spheromak wall is a quasi-toroidal shaped surface formed by the closed filament winding of a spheromak. This filament path conforms to the spheromak wall surface curvature. In plan view, looking along the Z axis, a spheromak wall is round. In cross section, looking along the spheromak minor axis, a spheromak wall is pseudo elliptical.

A spheromak is cylindrically symmetric about the spheromak Z axis (major axis of symmetry) and is mirror symmetric about the spheromak's equatorial plane where Z = 0.

The linear size of a spheromak is characterized by its nominal radius about the Z axis Ro. The spheromak minimum radius about the Z axis is Rc. The spheromak maximum radius about the Z axis is Rs. The height of the spheromak, parallel to the Z axis, is 2 Hm. Hence the relative geometry of a spheromak is specified by the parameters:
(Rc / Ro), (Rs / Ro) and (Hm / Ro). These parameters are the same for all isolated spheromaks. These spheromak relative geometry parameters are the same for all isolated spheromaks and are independent of the spheromak linear size and contained energy. The total energy content Ett of a spheromak is inversely proportional to its nominal radius Ro.
 

SPHEROMAK FIELDS:
For an isolated spheromak in a vacuum, at the center of the spheromak the net electric field is zero. In the region inside the spheromak wall the field is purely toroidal magnetic and the electric field is zero. In the region outside the spheromak wall the magnetic field is purely poloidal and the electric field is locally normal to the spheromak wall. In the far field the electric field is spherically radial. The current circulates within the filament winding that forms the spheromak wall that separates the inside region from the outside region.

The spheromak's mathematical field structure allows semi-stable plasma spheromaks and discrete stable atomic charged particles to exist and act as stores of energy.
 

SPHEROMAK WALL:
Outside the spheromak wall the field energy density U has electric and poloidal magnetic field components.
Thus outside the spheromak wall:
U = Up + Ue
= ([Bp(R, Z)]^2 / 2 Muo) + ([Epsilono / 2] [E(R, Z)]^2)

Inside the spheromak wall the field is purely toroidal magnetic:
U = Ut(R)
= (Muo / 2) [Nt I / 2 Pi Ro]^2 [Ro / R]^2

At the spheromak wall:
Ut(R) = Up(R, Z) + Ue(R, Z)
or
([Bp(R, Z)]^2 / 2 Muo) + ([Epsilono / 2] [E(R, Z)]^2) = (Muo / 2) [Nt I / 2 Pi Ro]^2 [Ro / R]^2

This equation defines the spheromak wall. The functions [Bp(R, Z)]^2 and [Ep(R, Z)]^2 are developed in terms of Ro on the web page titled Theoretical spheromak.

At R = Rc, Z = 0:
Utc = Btc^2 / 2 Muo
= (Bpc^2 / 2 Muo) + (Epsilono / 2) Eec^2]
and at R = Rs, Z = 0
Uts = Bts^2 / 2 Muo
= (Bps^2 / 2 Muo) + (Epsilono / 2) Ees^2]

Bto = Muo Nt I / 2 Pi Ro
or
Uto = Muo Nt^2 I^2 / 8 Pi^2 Ro^2
and
Utc = [Muo Nt^2 I^2 / 8 Pi^2 Ro^2] [Ro^2 / Rc^2]
and
Uts = [Muo Nt^2 I^2 / 8 Pi^2 Ro^2] [Ro^2 / Rs^2]
 

SPHEROMAK WINDING CONCEPT:
An approximate plan view of the current path (filament) of a theoretical elementary spheromak with Np = 3 and Nt = 4 is shown below. The blue lines show the current path on the upper face of the spheromak. The red lines show current path on the lower face of the spheromak. Note that the current path never intersects itself except at the point where the current starts to retrace its previous path.

In the diagram yellow shows the region of toroidal magnetic field. Outside the yellow region is a poloidal magnetic field and a spherically radial electric field.

This elementary spheromak winding pattern was generated using a polar graph and formulae of the form:
R = Rc + K [t - t(2N)] where t(2N) = (2 N) (3 Pi / 4) and t(2N) < t < t(2N+1)
and
R = Rs - K [t - t(2N + 1)]
where:
t(2N + 1) = (2N + 1)(3 Pi / 4)
and
t(2N + 1) < t < t(2N + 2)
where:
N = 0, 1, 2, 3. Use:
Rc = 1000,
Rs = 4105,
K = (4140 / Pi)
Top to bottom connection points were depicted by adjusting the torus Rs to 4045 and Rc to 1060.
 

ATOMIC PARTICLE SPHEROMAKS:
Atomic particle spheromaks have a quantized charge that superficially appears to be at rest with respect to an inertial observer. Isolated stable atomic particles such as electrons and protons hold specific amounts of energy (rest mass). When these particles aggregate with opposite charged particles the assembly emits photons. This photon emission decreases the total amount of energy in the assembly, creating a mutual potential energy well.

In an atomic particle spheromak net charge moves at the speed of light C around a closed spiral filament path of length Lh. The spheromak net charge Qs is uniformly distributed along this current path. The uniform charge distribution along the current path and the uniform current cause constant electric and magnetic fields. The time until an element of net charge retraces its previous path is (1 / F) where:
F = C / Lh
is the characteristic frequency of the spheromak. Note that frequency F increases as filament length Lh decreases.

An isolated spheromak in free space has a distorted ellipse cross section. However, the fields of an atomic particle spheromak may be further distorted by externally imposed electric and magnetic fields.
 

LOCATION IN A SPHEROMAK:
A spheromak has both cylindrical symmetry about its main axis of symmetry and has mirror symmetry about its equatorial plane. Any position in a spheromak can be defined by:
(R, Z)
where:
R = radius from the main axis (Z axis) of cylindrical symmetry;
and
Z = height above (or below) the spheromak equatorial plane.
Phi = angle about th Z axis.
 

SPHEROMAK CROSS SECTIONAL DIAGRAM:
The following diagram shows the approximate cross sectional shape of a real spheromak.

Note that the cross section of a real spheromak is pseudo elliptical, not round.

In this diagram on the axis of symmetry is R = 0

At R = 0, Z = 0 the field energy density is maximum and is entirely due to the poloidal magnetic field.  

GEOMETRICAL FEATURES OF A SPHEROMAK:
Important geometrical features of a spheromak include:
Rc = the spheromak wall inside radius on the equatorial plane;
Rs = the spheromak wall outside radius on the equatorial plane;
Ro = the radius of an imaginary ring radius on the equatorial plane which indicates the nominal linear size of a spheromak;
Ho = the spheromak wall height above the equatorial plane at R = Ro;
Hm = the maximum spheromak wall height above the equatorial plane;
Np = number of poloidal filament turns about the major axis of symmetry;
Nt = number of toroidal filament turns about the minor axis in the spheromak;

The subscript c refers to spheromak wall inside radius (core) on the equatorial plane;
s refers to the spheromak wall outside radius on the equatorial plane.

When the spheromak filament has passed through the spheromak central hole Nt times it has also circled around the main axis of spheromak symmetry Np times, after which it reaches the point in its closed path where it originally started.

Define:
Lt = 2 Pi K Ro = one purely toroidal filament turn length;
Lp = 2 Pi Ro = one purely poloidal filament turn length;
 

SPHEROMAK FILAMENT PATH LENGTH Lh:
Electromagnetic spheromaks arise from the electric current formed by distributed net charge Qs circulating at the speed of light C around the closed spiral path of length Lh which defines the spheromak wall. On the equatorial plane the spheromak inner wall minimum radius is Rc and the spheromak's outer wall maximum radius is Rs.

Let Np be the integer number of poloidal turns in filament length Lh and let Nt be the integer number of toroidal turns in filament length Lh.

The spheromak wall contains Nt toroidal turns equally spaced around 2 Pi radians about the main spheromak axis of symmetry.
Each purely toroidal winding turn has length:
Lt
so the purely toroidal spheromak winding length is:
(Nt Lt)

The spheromak wall contains Np poloidal turns which are spaced around the spheromak wall perimeter. The spheromak field calculations are based on the poloidal turns being concentrated at R = Ro. The purely poloidal turn length is:
Lp = 2 Pi Ro
and the purely poloidal winding length is:
(Np Lp)

If the spheromak wall was a straight round solenoid the total spheromak winding length Lh would be given by:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2

However, in reality, due to toroid curvature the formula for Lh is more complicated.

In one spheromak cycle period T the poloidal angle advances Np (2 Pi) radians. In the same spheromak cycle period T the toroidal angle advances Nt (2 Pi) radians.
In each case the frequency remains constant. Hence:
(poloidal angle advance) / (toroidal angle advance) = d(Phi) / d(Theta) = Np / Nt

Note that to prevent spheromak collapse, Np and Nt cannot be equal and cannot have common integer factors.
 

SPHEROMAK SHAPE PARAMETERS:
The relative shape of a spheromak is defined by the ratios:
(Rc / Ro), (Rs / Ro) and (Hm / Ro)

The winding of a spheromak is defined by the number of poloidal filament turns Np and the number of toroidal filament turns Nt.

The energy and frequency of a spheromak are a function of Lh. The ratio (Lh / Ro) is the same for all isolated spheromaks.
 

SPHEROMAK PARAMETER DEFINITIONS:
Define:
R = radial distance of a point from the major axis of symmetry of spheromak;
Z = axial distance of a point above the spheromak equatorial plane (Z is negative for points below the equatorial plane);
H = distance of a point on the spheromak wall above the spheromak equatorial plane;
Ro = characteristic spheromak radius
Ho = H|(R = Ro)
Hm = H|(R = Rm)
Ue = electric field energy density as a function of position outside the spheromak wall;
Up = magnetic field energy density as a function of position outside the spheromak wall;
U = Ue + Up + Ut = total field energy density at any position;
Upor = (Bpo^2 / 2 Muo) = magnetic field energy density at R = 0, Z = 0;
Lp = 2 Pi Ro
Lt = 2 Pi K Ro = single turn toroidal winding length
Utc = toroidal field energy density at R = Rc, Z = 0
Ut = Uto (Ro / R)^2
= toroidal magnetic energy density function inside the spheromak wall
Uts = Uto (Ro / Rs)^2 = toroidal magnetic energy density at R = Rs
Utc = Uto (Ro / Rc)^2 = toroidal magnetic energy density at R = Rc
Upor = _________
= poloidal magnetic field energy density at the origin on the spheromak axis.
 

SPHEROMAK FILAMENT PARAMETERS
Define:
Ih = filament current;
Lh = overall length of closed filament loop;
Dh = center to center distance between adjacent filament paths
As = outside surface area of spheromak wall
Q = proton net charge
Qs = net charge on spheromak
Nnh = integer number of negative charge quanta
Nph = integer number of positive charge quanta
Vn = velocity of negative charge quanta along charge hose
Vp = velocity of positive charge quanta along charge hose
C = speed of light
Nr = Np / Nt
= ratio of two integers which have no common factors. This ratio must be inherently stable.
 

SPHEROMAK CHARGE DISTRIBUTION ASSUMPTION:
Assume that the spheromak charge is uniformly distributed over the filament length.
 

EQUATORIAL PLANE:
On the spheromak's equatorial plane:
Z = 0
For points on the spheromak's equatorial plane the following statements can be made:

For R = 0 the net electric field is zero;
For R < Rc the toroidal magnetic field is zero;
For R < Rc the magnetic field Bp is purely poloidal;
For R = 0 the magnetic field is parallel to the axis of cylindrical symmetry;

For Rc < R < Rs the electric field is zero;
For Rc < R < Rs the poloidal magnetic field is zero;
For Rc < R < Rs the toroidal magnetic field Bt is proportional to (1 / R).

For Rs < R in free space the electric field Ero is spherically radial;
For Rs < R in the far field the electric field Ero is proportional to (1 / R^2);
For Rs < R in free space the toroidal magnetic field is zero;
For Rs < R in the far field the poloidal magnetic field Bp is proportional to (1 / R^3);
 

FILAMENT CURRENT:
Ih = [Qp Np Vp + (- Q Ne Ve)] / Lh
= Q C / Lh
 

SPHEROMAK FILAMENT WINDING GEOMETRY:
A very important issue in understanding natural spheromaks is grasping that:
Np cannot equal Nt and that Np and Nt can have no common factors other than one. Otherwise the windings would fall on top of one another or the spheromak would collapse.
 

SPHEROMAK FILAMENT PATH:
Consider a quasi-toroidal winding with a poloidal axial length at radius Ro of 2 Pi Ro.

This winding has Np round poloidal turns effectively located at radius Ro at Z = 0 and has Nt toroidal turns centered at radius:
Rx = [(Rs + Rc) / 2].
These toroidal turns have a horizontal radius of:
[(Rs - Rc) / 2].

Hence the wall to wall width of the spheromak at its equator is:
(Rs - Rc)
and its height is:
2 Hm.

The minor axis of the spheromak toroidal winding lies along R = Rx, Z = 0 .

Rx = [(Rs + Rc)/ 2]

Any point on this winding can be identified by its Theta and Phi values. Theta is the angle about the spheromak minor axis Rx measured with respect to a radial line from Rx to the orgin. Phi is the angle about the major axis of symmetry measured with respect to the same radial line. Hence at the filament closure point:
Phi = 0, Theta = 0, or Phi = 2 Pi Np, Theta = 2 Pi Nt,
where Np and Nt are positive integers with no common integer factors.

The range of Phi is:
0 < Phi < 2 Pi Npo radians;
The range of Theta is:
0 < Theta < 2 Pi Nto radians

The value of dTheta / dPhi is given by:
dTheta / dPhi = Nto / Npo

In a spheromak dTheta / dPhi is constant at its operating point.

A very important issue is that dL / d(Theta) is constant. Hence if Theta advances linearly with time so also does L. Hence the net charge moves along the filament at the constant speed of light C.

The filament winding path is partly described by:
R = Rx + [(Rs - Rc) / 2] cos(Theta)
where
Rx = (Rs + Rc) / 2
and
Phi / Theta = [d(Phi) / d(Theta)] = Np / Nt
where:
Phi is an angle about the Z axis.

dY = R d(Phi)

dL^2 = (dH)^2 + (dR)^2 + (dY)^2

R = Rx + [(Rs - Rc) / 2] cos(Theta)
or
R^2 = Rx^2 + 2 Rx [(Rs - Rc) / 2] cos(Theta) + [(Rs - Rc) / 2]^2 cos^2(Theta)

Hence:
R^2 Np^2 / Nt^2
= Rx^2 Np^2 / Nt^2 + Rx [(Rs - Rc)] cos(Theta)(Np / Nt)^2 + [Np^2 / Nt^2][(Rs - Rc) / 2]^2 cos^2(Theta)

Differentiation of R gives:
dR = - [(Rs - Rc) / 2] sin(Theta) d(Theta)
 

GENERAL SOLUTION:
dL / dt = C
needs to be constant to be consistent with a constant speed of light and a constant frequency:
(d(Phi) / dt = constant.

dL^2 = (dH)^2 + (dR)^2 + [R d(Phi)]^2

(dL / dt)^2 = (dH / dt)^2 + (dR / dt)^2 + (dY /dt)^2 = C^2

R = Rx + [(Rs - Rc) / 2] cos (Theta)
where Rs and Rc are set by the spheromak wall position at H = 0.

dY = d(R Phi)
or
dY / dt = R d(Phi) / dt

dR = - [(Rs - Rc) / 2] sin(Theta) d(Theta)

(dR / dt) = - [(Rs - Rc) / 2] sin(Theta) [d(Theta) / dt]

(dR / dt)^2 = [(Rs - Rc) / 2]^2 sin^2(Theta) [d(Theta) dt]^2

d(Theta) / dt = constant

d(Phi) / dt = constant

At time t = 0 :
Theta = 0
and
Phi= 0

At time t = T = 1 / F:
Theta = 2 Pi Nt
and
Phi = 2 Pi Np

Thus:
[d(Theta) / dt] T = 2 Pi Nt
and
[d(Phi) /dt] T = 2 Pi Np
which give:
(1 / T) = {[d(Theta) / dt] / 2 Pi Nt)
= {[d(Phi) / dt] / 2 Pi Np)

Hence:
[d(Theta) / Nt] = [dPhi / Np]
or
d(Phi) / d(Theta) = [Np / Nt] d(Phi) / dt = [Np / Nt] d(Theta) / dt dY / dt = R d(Phi) / dt
= R [d(Phi) / d(Theta)] [dTheta / dt]
= R [Np / Nt] [d(Theta) / dt]

Thus:
[dY / dt]^2 = R^2 [Np / Nt]^2 [d(Theta) / dt]^2
= {Rx + [(Rs - Rc) / 2] cos(Theta)}^2 [Np / Nt]^2 [d(Theta) / dt]^2
= [Rx^2 + Rx (Rs - Rc) cos(Theta) + [(Rs - Rc) / 2]^2 cos^2(Theta)} [Np / Nt]^2 [d(Theta) / dt]^2

Recall that:
(dL / dt)^2 = (dH / dt)^2 + (dR / dt)^2 + (dY /dt)^2 = C^2
= (dH/ dt)^2 + [(Rs - Rc) / 2]^2 sin^2(Theta) [d(Theta) dt]^2 +
+ [Rx^2 + Rx (Rs - Rc) cos(Theta) + [(Rs - Rc) / 2]^2 cos^2(Theta)} [Np / Nt]^2 [d(Theta) / dt]^2
= C^2

In order for the speed of light C to be constant:
(dH / dt)^2 = [(Rs - Rc) / 2]^2 cos^2(Theta) [d(Theta) dt]^2 +
+ [Rx^2 + [(Rs - Rc) / 2]^2 sin^2(Theta)} [Np / Nt]^2 [d(Theta) / dt]^2
- Rx (Rs - Rc) cos(Theta)cos^2(Theta) [Np / Nt]^2 [d(Theta) / dt]^2

Thus:
(dL / dt)^2 = [(Rs - Rc) / 2]^2 + {Rx^2 + [(Rs - Rc) / 2]^2} [Np / Nt]^2

Recall that:
d(Theta) / dt = [2 Pi Nt / T]

Hence:
dL / d(Theta) = constant d(Phi) / d(Theta) = (Np / Nt)
dL^2 = (dH)^2 + (dR)^2 + [R d(Phi)]^2
R = Rx + [(Rs - Rc) / 2] cos(Theta)
dR / d(Theta) = - [(Rs - Rc) / 2] sin(Theta)
Which equals zero at Theta= 0 and Theta = Pi corresponding to R = Rs and R = Rc.

[dR / d(Theta)]^2 = [(Rs - Rc) / 2]^2 [sin(Theta)]^2

At R = Rc:
(dR / d(Theta)) = 0
[(dH / d(Theta)] = large negative

At R = Rs:
[dR / d(Theta)] = 0
[(dH / d(Theta)] = large positive

(dH / d(Theta)) = (dH / dR)(dR / d(Theta))
= - (dH / dR) [(Rs - Rc) / 2] sin(Theta) [dR / d(Theta)]^2 = [(Rs - Rc) / 2]^2 [ sin(Theta)]^2

(R - Rx) = [(Rs - Rc) / 2] cos(Theta)
dR / d(Theta) = [(Rs - Rc) / 2] [- sin(Theta)]
[dR / d(Theta)]^2 = [(Rs - Rc) / 2]^2 [sin(Theta)]^2

cos(Theta) = [2 (R - Rx) / (Rs - Rc)]
sin(Theta) = {1 - [2 (R - Rx) / (Rs - Rc)]^2}^0.5

Rx = (Rs + Rc) / 2

sin(Theta) = {1 - [2 (R - Rx) / (Rs - Rc)]^2}^0.5
= {1 - [4 (R^2 - 2 R Rx + Rx^2) / (Rs - Rc)^2]}^0.5
= {[(Rs - Rc)^2 - 4 (R^2 - 2 R Rx + Rx^2)] / (Rs - Rc)^2}^0.5
= {[Rs^2 - 2 Rs Rc + Rc^2 - 4 R^2 + 8 R Rx - 4 Rx^2]^0.5 / (Rs - Rc)}
= {[Rs^2 - 2 Rs Rc + Rc^2 - 4 R^2 + 4 R (Rs + Rc) - (Rs + Rc)^2]^0.5 / (Rs - Rc)}
= {[Rs^2 - 2 Rs Rc + Rc^2 - 4 R^2 + 4 R Rs + 4 R Rc - Rs^2 - 2 Rs Rc - Rc^2]^0.5 / (Rs - Rc)}
= {[ - 4 Rs Rc - 4 R^2 + 4 R Rs + 4 R Rc]^0.5 / (Rs - Rc)}
= {2 [ - Rs Rc - R^2 + R Rs + R Rc]^0.5 / (Rs - Rc)}
= {2 [- Rc (Rs - R) + R (Rs - R)]^0.5 / (Rs - Rc)}
= {2 [(R - Rc) (Rs - R)]^0.5 / (Rs - Rc)}

H(R) = H|(R = Rc) + Integral from R = Rc to R = R of [dH(R) / dR] dR
= Integral from R = Rc to R = R of [dH(R) / dR] dR

Now find Lh:
dLh^2 = (dH)^2 + (dR)^2 + [R d(Phi)/ d(Theta)]^2[d(Theta)]^2
= (dH)^2 + (dR)^2 + [R Np / Nt]^2[d(Theta)]^2
= {[dH / d(Theta)]^2 + [dR / d(Theta)]^2 + [R Np / Nt]^2} [d(Theta)]^2

dLh = {[dH /d(Theta)]^2 + [dR / d(Theta)]^2 + [R Np / Nt]^2}^0.5 d(Theta)

= {[dH / d(Theta)]^2
+ [dR / d(Theta)]^2
+ [R Np / Nt]^2}^0.5 d(Theta)


= {[dH / d(Theta)]^2
+ [(Rs - Rc) / 2]^2 [sin(Theta)]^2
+ [R Np / Nt]^2}^0.5 d(Theta)


= {[dH / d(Theta)]^2
+ [(Rs - Rc) / 2]^2 [sin(Theta)]^2
+ R^2 [Np / Nt]^2}^0.5 d(Theta)


Recall that:
R = Rx + [(Rs - Rc) / 2] cos(Theta)
R^2 = [Rx^2 + 2 Rx [(Rs - Rc) / 2] cos(Theta) + [(Rs - Rc) / 2]^2 cos^2(Theta)]
Thus: dL = {[dH / d(Theta)]^2
+ [(Rs - Rc) / 2]^2 [sin(Theta)]^2
+ [Rx^2 + 2 Rx [(Rs - Rc) / 2] cos(Theta) + [(Rs - Rc) / 2]^2 cos^2(Theta)] [Np / Nt]^2}^0.5 d(Theta)

Choose dH / d(Theta) to make dLh proportionl to d(Theta). Then:
[dH / d(Theta)]^2 = [(Rs - Rc) / 2]^2 cos^2(Theta) + [(Rs - Rc) /2]^2 sin^2(Theta)] [Np / Nt]^2
- 2 Rx [(Rs - Rc) / 2] cos(Theta)[Np / Nt]^2

Integrating dLh from Theta = 0 to Theta = (Nt 2 Pi) gives:
Lh = {[(Rs - Rc) / 2]^2 + Rx^2 [Np / Nt]^2 + [(Rs - Rc) / 2]^2[Np / Nt]^2}^0.5 2 Pi Nt
Lh / 2 Pi = { Nt^2 [(Rs - Rc) / 2]^2 + Rx^2 [Np]^2 + [(Rs - Rc) / 2]^2[Np]^2}^0.5
[Lh / 2 Pi]^2 = Nt^2[(Rs - Rc) / 2]^2 + Np^2 {Rx^2 + [(Rs - Rc) / 2]^2 }
= Nt^2 [(Rs - Rc) / 2]^2 + Np^2 {[(Rs + Rc) / 2]^2 + [(Rs - Rc) / 2]^2 }
= Nt^2 [(Rs - Rc) / 2]^2 + [Np^2 / 4] {2 Rs^2 + 2 Rc^2}
= Nt^2 [(Rs - Rc) / 2]^2 + Np^2 {(Rs^2 + Rc^2) / 2}
= Nt^2 [(Rs - Rc) / 2]^2 + Np^2 {Ro^2}
where:
Ro^2 = [(Rs^2 + Rc^2) / 2]

This is the optimum expression for [Lh / 2 Pi]^2.

The [dH / d(Theta)] term is chosen to make dL / d(Theta) constant independent of (Theta). The spheromak wall is centered at:
Rx = [(Rs + Rc) / 2]
with horizontal radius:
[(Rs - Rc) / 2]

However:
[Lh / 2 Pi]^2 = Nt^2 [(Rs - Rc) / 2]^2 + Np^2 {(Rs^2 + Rc^2) / 2}

This expression is carried forward to the web page titled: Spheromak Winding Constraints and to the web page titled: Theoretical Spheromak.
 

SHAPE OF SPHEROMAK WALL:
Recall that:
[dH / d(Theta)]^2 = [(Rs - Rc) / 2]^2 cos^2(Theta) + [(Rs - Rc) /2]^2 sin^2(Theta)] [Np / Nt]^2
- 2 Rx [(Rs - Rc) / 2] cos(Theta)[Np / Nt]^2

Hence:
[dH / d(Theta)] = {[(Rs - Rc) / 2]^2 cos^2(Theta) + [(Rs - Rc) /2]^2 sin^2(Theta)] [Np / Nt]^2
- 2 Rx [(Rs - Rc) / 2] cos(Theta)[Np / Nt]^2}^0.5

(dH / d(Theta) = [dH / dR] [dR / d(Theta)] = [dH / dR] [(Rs - Rc) / 2][- sin(Theta)]

Recall that:
R - Rx = [(Rs - Rc) / 2] cos(Theta)
or
cos(theta) = 2 (R - Rx) / (Rs - Rc)

Hence:
dH / dR = {cos^2(Theta) + sin^2(Theta)] [Np / Nt]^2
- [2 (Rs + Rc) / (Rs - Rc)] cos(Theta)[Np / Nt]^2}^0.5 / {- sin(Theta)]}
= {cos^2(Theta) + sin^2(Theta)] [Np / Nt]^2
- [4 (Rs + Rc) (R - Rx) / (Rs - Rc)^2][Np / Nt]^2}^0.5 / {- sin(Theta)]}

Note that at R = Rx and Theta = Pi / 2:
[dH / dR] = [Np / Nt]
 

FINDING THE PEAK IN THE SPHEROMAK WALL:
Recall that:

The peaks in the spheromak wall occur at Theta values where:
dH / dR = 0.
Hence at the peak in the spheromak wall:
0 = { cos^2(Theta) + sin^2(Theta)] [Np / Nt]^2
- [4 Rx / (Rs - Rc)] cos(Theta)[Np / Nt]^2} / {- sin(Theta)]

To solve this equation we need to first find the value of (Np / Nt).

Note that the peak in the spheromak wall is displaced from Rx.

This web page last updated on August 23, 2024.

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