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INTRODUCTION:
This web page describes the basic structural elements of a spheromak.
SPHEROMAK COMPONENTS:
The essential components of a spheromak are: a single layer quasi-toroidal closed filament path defining a spheromak wall through which a constant charge per unit length circulates at the speed of light, internal toroidal magnetic field energy, external poloidal magnetic field energy and external radial electric field energy. For the spheromakigeometry to be stable the combined field energy density at the inner surface of the spheromak wall must everywhere exactly equal the combined field energy density at the outer surface of the spheromak wall.
SPHERMAK WINDING CONCEPT:
An approximate plan view of the filament path of a theoretical elementary spheromak with three poloidal turns (Np = 3) and four toroidal turns (Nt = 4) is shown below to illustrate a winding with a both poloidal and toroidal turns. 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 winding closes so that the current can 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 radial electric field. In the straight portion of the spheromak core both the axial and radial electric fields are zero.
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, and has reached the point in its closed path where it originally started.
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.
SPHEROMAK PHOTOGRAPH:
A photograph of a real plasma spheromak taken by General Fusion circa 2012 is shown below:
More plasma spheromak photographs are shown in the paper:
Spheromak formation and sustainment studies at the sustained physics experiment using high-speed imaging and magnetic diagnostics by Romero-
Talamas et al , Physics of Plasmas, vol.13 (2006).
LOCATION IN A SPHEROMAK:
A spheromak wall has cylindrical symmetry about its main axis of symmetry and has mirror symmetry about its equatorial plane. Any position in a spheromak cross section 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.
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 on the spheromak wall where (dZw / dRw) = 0.
Note that Ro indicates the nominal linear size of a spheromak;
Ho = the spheromak wall height Zw above the equatorial plane at R = Ro;
Np = number of poloidal filament turns about the major axis of symmetry;
Nt = number of toroidal filament turns about the ring R= Rp, Z = 0.
The subscript c refers to spheromak wall inside radius (core) on the equatorial plane;
s refers to the spheromak wall outside radius (surface) on the equatorial plane.
SPHEROMAK CROSS SECTIONAL DIAGRAM:
The following diagram shows the approximate cross sectional shape of a spheromak.
Note that on this diagram:
y = Z and x = R
For this diagram:
Ho = 2.156
and
Rs = 2.156
and
Rc = 0.4802
Note that the cross section of a real spheromak is not round.
In this diagram on the main axis of symmetry is R = 0
for the purpose of drawing this diagram an elliptical spheromak cross section was assumed. In reality the spheromak cross section is more distorted.
It is shown at Spheromak Wall, that in cross section the spheromak wall has a distorted shape. In cylindrical coordinates the spheromak wall's Zw value peaks at R = Ro, Z = +/- Ho and intersects the Z plane at R = +/- Rc and at R = +/-Rs. The spheromak minor axis is at R = Ro, Z = 0.
Conceptually a spheromak wall is a quasi-toroidal shaped closed surface, with a distorted cross section,formed by the closed filament current path of a spheromak. This filament path defines the spheromak wall surface curvature. In plan view, looking along the Z axis, a spheromak wall is round at both its inner radius Rc and its outer radius Rs. In cross section, looking along the spheromak minor axis, the spheromak cross section is distorted with maximum width (Rs - Rc) and maximum height 2 Ho.
SPHEROMAK GEOMETRY:
The spheromak wall separates the region inside the spheromak wall from the region outside the spheromak wall.
The spheromak wall 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 spheromak wall is defined by a distorted toroidal shaped single layer filament winding carrying current I with net charge Q continuously circulating at speed of light C around a closed path of length Lh.
The filament's charge (4 Q / 3) is uniformly distributed along the filament length Lh.
This filament winding contains a blend of Np poloidal turns around the spheromak major axis of symmetry (the Z axis) and Nt toroidal turns around the spheromak minor axis.
There are no filament winding cross overs.
The winding is stable in part because the integers Np and Nt are constant and share no integer factors except unity.
GEOMETRICAL PARAMETERS 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;
Np = number of poloidal filament turns about the major axis of symmetry;
Nt = number of toroidal filament turns about the minor axis of symmetry.
The subscript c refers to spheromak wall inside radius (core) on the Z = 0 plane;
s refers to the spheromak wall outside radius on the Z = 0 plane.
The subscript o refers to values measured at R = Ro.
SPHEROMAK SCALING:
An important property of spheromaks is that they maintain the same relative geometry over changes in linear size. Hence Ho, Rs and Rc are all proportional to Ro.
The height to width ratio of the spheromak cross section is:
Ka = [2 Ho / (Rs - Rc)]
Note that Ka is independent of Ro. For a circle Ka = 1.
Let subscript f designate a value corresponding to Ro = Rof. Then:
Rs = Rsf (Ro / Rof);
Rc = Rcf (Ro / Rof);
Ho = Hof (Ro / Rof);
Rs - Rc) = (Rsf - Rcf)(Ro / Rof)
= Kb Ro
where:
Kb = (Rsf - Rcf) / Rof
The scaling factor is Kb.
Hence:
(Rs - Rc) = Kb Ro
It is convenient to define constant Kr by:
Kr = Rs / Rc = Rsf / Rcf
Then:
(Rs - Rc) = Kb Ro;
gives:
(Kr + 1) Rc = Kb Ro.
Thus as Ro increases both [Rs- Rc) and Ho proportionately increase.
The spheromak wall is formed by charge circulating at the speed of light C around a complex quasi-toroidal shaped closed filament path of length Lh which has both Np poloidal turns Nt and toroidal turns. The spheromak parameters Np and Nt are both integer constants independent of Ro.
One purely poloidal turn has length Lp where:
Lp = 2 Pi Ro Ca.
One purely toroidal turn has length Lt where:
Lt = [Ct Ro]
where Ct is a constant.
The length Lh of the closed spheromak filament is given by:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2
Since Np and Nt are both constants independent of Ro and since both Lp and Lt are proportional to Ro, hence Lh is proportionl to Ro.
In summary, size of a spheromak's wall is defined by (Ro / Rof) and the shape of a spheromak is fully defined by Hof, Rsf and Rcf. The spheromak may be larger or smaller, as set by (Ro / Rof) but:
Kr = Rsf / Rcf
and
Hof / Rcf
remain constant.
Recall that:
Ka = [2 Ho / (Rs - Rc)]
= [2 Hof (Ro / Rof) / [(Rsf(Ro / Rof) - Rcf (Ro / Rof)]
= [2 Hof / (Rsf - Rcf)]
= [2 Hof / Rcf (Kr + 1)]
Note that the parameters:
Kr = Rsf / Rcf
and
Ka = 2 Hof / (Rsf - Rcf)
= 2 Hof / Rcf (Kr - 1)
are purely functions of fundamental physical and mathematical constants.
SPHEROMAK FIELDS:
The charge (4 Q / 3) circulates along a mixed toroidal and poloidal closed path defined by the filament winding that forms the spheromak wall.
For an isolated spheromak in a vacuum, at the exact center of the spheromak (at the origin) the net electric field is zero and the magnetic field is purely poloidal.
In the region inside the spheromak wall the magnetic field is purely toroidal.
On the Z = 0 plane inside the spheromak wall and inside the spheromak core the radial electric field is zero.
On the Z = 0 plane outside the spheromak at R = Rs the spherically radial electric field is:
(So / Epsilono)(Ro / Rs)^2
In the region outside the spheromak wall the magnetic field is purely poloidal and the electric field is sphericaly radial being locally normal to the spheromak wall.
In the spheromak core where R < Rc the radial electric field components cancel each other out leaving only axial electric field components which are close to zero in the straight portion of the spheromak core.
At R = Rs, Z = 0 and extending out to R = infinity the external electric field is spherically radial.
The circulating current forms a purely toroidal magnetic field inside the spheromak wall and a purely poloidal magnetic field outside the spheromak wall.
The net electric charge Q on the filament and in the core causes an electric field outside the spheromak wall normal to the spheromak wall surface.
At the exact center of the spheromak, due to spheromak symmetry, the electric fields all cancel to zero and the magnetic field is purely poloidal and axial.
Outside the spheromak wall there are both a poloidal magnetic field and a radial electric field. Outside the spheromak wall the toroidal magnetic field is zero.
Hence a stable isolated spheromak has field energy components due to an external radial electric field, an external poloidal magnetic field, and an internal toroidal magnetic field.
The toroidal magnetic field in a spheromak may either clockwise (CW) or counter clockwise (CCW) with respect to the spheromak's poloidal magnetic field.
An isolated spheromak retains its size, shape and energy due to its own electric and magnetic fields. The spheromak wall is located on the locus of points where the combined field energy densities on both sides of the spheromak wall are equal. The spheromak wall position is stable because the second derivatives of field energy densities with respect to spheromak wall position are positive everywhere at the spheromak wall.
A spheromak has a characteristic radius Ro. On loss of an increment of spheromak energy the characteristic radius of a spheromak increases but the relative spheromak geometry remains unchanged. Also the number Np of poloidal turns and the number Nt of toroidal turns both are independent of Ro. The turn lengths Lp and Lt both change with changes in Ro. Hence the filament length:
Lh = [(Np Lp)^2 + (Nt Lt)^2}^0.5 is proportional to Ro.
The charge per unit area on the spheromak wall is proportional to the charge per unit length on the filament and is inversely proportional the filament spacing. Since the number of turns is fixed, the charge per unit length on the filament decreases with increasing Ro. Since the number of turns is fixed the distance between adjacent filament turns increases with inncresing Ro. These two effects combine to make the surface charge per unit area inversely proportional to R^2.
SPHEROMAK POTENTIAL ENERGY WELL:
The charge and charge circulation produce the electric and magnetic field energy density distributions required for spheromak existence. At the spheromak wall the inside and outside field energy densities are equal. The derivatives of the field energy densities create a total energy minimum at the spheromak wall position.
The net charge and the charge motion along the closed filament current path cause the electromagnetic forces on the filament of circulating charge to net to zero. Note that inertial forces and relativistic phenomena apply to plasma spheromaks but do not apply to isolated quantum charged particle spheromaks.
The average field energy density inside the spheromak wall is less than the average field energy density that would pertain if the spheromak wall did not exist. Hence a spheromak acts as a potential energy well, which gives the spheromak long term stability. The electric and magnetic fields of a spheromak contain a stable amounts of energy.
An important property of spheromaks is that the total energy contained in each of its fields is proportional to (1 / Ro) where Ro is the nominal spheromak linear radius. This property leads to the Planck Constant.
The spheromak's total field enegy Ett is the sum of the Toroidal magnetic field energy, the poloidal magnetic fiel energy and the electrostatic field energy. Each of these field energy components is inversely proportionalto Ro. Hence the total field energy is inversely proportional to Ro. The spheromak frequency F is inversely proportional to Ro via the filament length Lh.
Hence the change in energy is proportional to the change in frequency via the Planck constant.
A spheromak forms a potential energy well because the volume integrated toroidal magnetic field energy density within the spheromak wall is less than the sum of the volume integrated radial electric and poloidal magnetic field energy densities displaced.
FUNNEL FORMATION:
Inside the spheromak wall the internal electric field is zero. In order to suppress the internal electric field due to the filament winding concentration in the straight portion of the core there must be an off setting opposite static charge in the stright portion of the core. The filament has a uniform charge per unit length. Some of the filament turns are poloidal. To achieve increased charge cocentration near the center of the spheromak with poloidal turns the distance between adjacent poloidal turns must be less at R = Rc than at R = Rs. This issue leads to the formation of funnels at the entrance to and exit from the central core.
SPHEROMAK FILAMENT PATH LENGTH Lh:
Let Lt = one purely toroidal filament turn length;
Let Lp = one purely poloidal filament turn length;
Let Np be the integer number of poloidal turns in filament length Lh,
Let Nt be the integer number of toroidal turns in filament length Lh.
The theoretical purely poloidal winding length is Np Lp.
The Theoretical purely toroidal winding length is Nt Lt.
However, in reality these two windings are orthogonal. Hence the total winding length Lh is given by:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2
TURN ADVANCE:
In one spheromak cycle period:
T = (1 / F)
the poloidal angle advances Np (2 Pi) radians.
In the same spheromak cycle period:
T = (1 / F)
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 spontaneous spheromak collapse, Np and Nt cannot be equal and cannot have common integer factors.
SPHEROMAK DIMENSION SCALING:
Assume that a spheromak has specified linear dimension ratios.
Then:
(Rs - Rc) = Kb Ro.
The relative shape of the spheromak is defined by:
Ka = [2 (Ho / (Rs - Rc)]
and
Kr = (Rs / Rc),
where Ka and Kr are independent of Ro.
[(Rs - Rc) / 2] = Kb Ro
Then:
Ho = Ka Kb Ro
The parameters [(Rs - Rc) / 2] and Ho both scale with Ro. In a spheromak changing the spheromak linear dimension Ro causes[(Rs - Rc) / 2]
to change in accordance with:
[(Rs - Rc) / 2 ]= Kb Ro
where (Ro) varies with the spheromak energy.
Hence, the relative shape of the spheromak is defined by Ka and the size is defind by Kb.
SPHEROMAK WINDING SCALING:
The winding of a spheromak is defined by the number of poloidal filament turns Np and the number of toroidal filament turns Nt.
The winding length Lh is given by:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2
where Np and Nt are both positive integer constants.
The purely poloidal turn length Lp and the purely toroidal turn length Lt both scale with Ro.
Hence Lh scales with Ro.
SPHEROMAK WINDING CURRENT:
The winding current I is given by:
I = (4 Q / 3 Lh) C,
where C = speed of light.
SPHEROMAK ENERGY AND FREQUENCY:
The frequency F of a spheromak is:
F = C / Lh
Since Lh scales with Ro, F is proportional to (1 / Ro).
The field energy of a spheromak has three components:
Et = toroidal magnetic field energy component;
Ep =poloidal magnetic field energy component;
Ee = electric field energy component
Since Ett and F are both proportional to (1 / Ro), the ratio [Ett / F] is a constant known as the Planck constant.
SURFACE CHARGE:
The spheromak wall is formed by a winding with charge (4 Q / 3) and length Lh. Lh is proportional to Ro while (4 Q / 3) is a constant. Hence the charge per unit length on the winding is:
(4 Q / 3 Lh),
which is proportional to (1 / Ro). Hence as Ro increases the charge per unit area S on the spheromak wall decreases due to this effect.
The linear dimension of a spheromak is proportional to Ro. However, the numbers of turns Np and Nt are constants independent of Ro. Hence the distance between adjacent turns increses in proportion to Ro. Hence, as Ro increases the charge per unit area S on the spheromak wall decreases due to this effect. The combination of these two effects makes So proportional to (1 / Ro^2).
Let S|(R = Ro) = So. Then if Ro changes to R, then:
S = So (Ro / R)^2
As shown at Spheromak Wall the surface charge density is of fundamental importance in determination of spheromak wall geometry.
FILAMENT LENGTH:
[Lh]^2 = Nt^2 [Lt]^2 + Np^2 {Lp}^2
This expression is carried forward to the web page titled: Spheromak Winding Constraints and to the web page titled: Theoretical Spheromak.
SPHEROMAK DIMENSION PARAMETERS:
Let R be the radial distance of a point from the main axis of symmetry of a spheromak.
Let Z be the height of a point above the equatorial plane of a spheromak.
R = 0 is the major axis of symmetry of the spheromak;SPHEROMAK STRUCTURE:
The geometry of a spheromak is symmetric about its central point:
R = 0, Z = 0
and can be characterized by:
its outer radius R = Rs at Z = 0 ,
its inner radius R = Rc at Z = 0
and its height Z = +/- Ho at R = Ro
For the stable spheromak geometry the ratios (Ho / Ro), (Rc / Ro) and (Rs / Ro) are constants irrespective of the value of Ro.
It is shown that the total electromagnetic field energy Ett trapped by a spheromak is proportional to (1 / Lh) where:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2
Lp = 2 Pi Ro
Lt = 2 Pi K Ro = length of one winding turn around the toroidal magnetic path
Current moves along the spheromak closed path length Lh at speed of light C. Hence the natural frequency F of the path length Lh is:
F = C / Lh.
A spheromak's total field energy is Ett.
Ett / F = Ett Lh / C = h
Expressions for Ett typically contain the factor (1 / Lh) so that calculations of (Ett Lh / C) result in the constant h.
Hence:
Hence, Ett is proportional to frequency F.
Ett = h F
The proportionality constant is known as the Planck constant h
Thus spheromaks are the cause of photon energy quantization.
CONSTANT CHARGE FILAMENT CURRENT:
An important issue in spheromak analysis is that the filament current I is the same everywhere on the filament.
FILAMENT GEOMETRY:
Define:
Lt = one purely toroidal filament turn length;
Lp = one average purely poloidal filament turn length;
Np = number of poloidal turns
Nt = number of toroidal turns
When an element of charge has passed through the spheromak core Nt times it has also circled around the main axis of spheromak symmetry Np times, by which time it reaches the point in the closed filament path where it originally started.
There are Nt parallel filament turns that go through the equatorial plane in the central core of the spheromak.
DISCRETE INTEGER SOLUTIONS:
There is a further aspect of charge filaments that is important. In order for a spheromak to be stable over time each circuit of the filament must be identical to every other such circuit. Hence for the spheromak to be stable the number of toroidal turns Nt and the number of poloidal turns Np included in length Lh must both be integers. Furthermore, the filament path length ratio
[(Np Lp) / (Nt Lt)] = Mp / Mt,
where Mp and Mt are integers that have no cmmon factors other than unity.
SPHEROMAK NET CHARGE:
Net charge Qs on the spheromak is given by:
Qs = Qp Nph + Qn Nnh
SPHEROMAK FIELD ENERGY DENSITY FUNCTIONS:
A spheromak has three cylindrically symmetric field energy density functions:
Ut(R,Z) = toroidal magnetic field energy density function
Up(R,Z) = poloidal magnetic field enegry density function
Ue(R,Z) = radial electric field energy density function
The spheromak has two regions which are separated by the thin closed spheromak wall.
Everywhere in the region inside the spheromak wall the field energy density U is given by:
U = Ut(R, Z)
where:
Ut(R, Z) < Up(R, Z) + Ue(R,Z)
Everywhere in the region outside the spheromak wall:
U = Up(R,Z) + Ue(R, Z)
where:
Ut(R, Z) > Up(R, Z) + Ue(R, Z)
Hence the spheromak forms a potential energy well.
Everywhere on the spheromak wall:
Ut(R, Z) = Up(R, Z) + Ue(R,Z).
Ths equation defines the position of the spheromak wall.
SPHEROMAK SYMMETRY:
Spheromaks exhibit mirror symmetry about the equatorial plane. Hence:
Z(R) = - Z(R)
and
U(R, Z) = U(R, - Z)
Hence:
Ett = Integral from Z = - infinity to Z = + infinity of:
Integral from R = 0 to R = infinity of:
U(R, Z) 2 Pi R dR dZ
= Integral from Z = 0 to Z = infinity of:
U(R, Z) 4 Pi R dR dZ
SPHEROMAK WALL POSITION:
The position of the spheromak wall corresponds to the lowest available spheromak energy state. At this state the spheromak energy density distribution is U(R, Z) where:
inside the spheromak wall:
U = Ut(R)
and outside the spheromak wall:
U = Up(R, Z) + Ue(R,Z)
and at the spheromak wall:
Ut(R) = Up(R, Z) + Ue(R,Z)
ISOLATED SPHEROMAKS:
Spheromaks are used by nature to form particles that are local concentrations of charge and electromagnetic field energy. We commonly refer to this locally concentrated electromagnetic field energy as rest mass. Spheromaks are instrumental in nuclear and atomic particle interactions. Spheromaks also have important roles in semi-stable plasmas, chemical binding and thermal radiation.
An isolated spheromak has a unique energy state solution. An isolated spheromak in an external magnetic field has two energy state solutions. When there are multiple interacting spheromaks there is a spectrum of discrete energy state solutions.
Spheromaks gain or lose energy by absorption or emission of electromagnetic radiation.
BOUNDARY CONDITIONS:
For an isolated charged particle spheromak in a vacuum the electric field inside the spheromak wall is zero.
For an isolated spheromak the field energy density outside the wall is the sum of the energy densities caused by the electric field due to fixed charge Qs and by the polodal magnetic field arising from current circulation. The field energy density inside the spheromak wall is caused only by the toroidal magnetic field that arises from toroidal current circulation in the spheromak wall.
Thus, from basic electromagnetic theory inside the spheromak wall the toroidal magnetic field is given by:
Bt = Muo Nt I / 2 Pi R
so inside the spheromak wall the toroidal magnetic field energy density is given by:
Ut = Bt^2 / 2 Muo
= [Muo Nt I / 2 Pi R]^2 / 2 Muo
= Muo Nt^2 I^2 / 8 Pi^2 R^2
where:
Muo = permiability of free space = 4 Pi X 10^-7____
Pi = 3.14159265
Nt = number of toroidal turns
I = circulating current
R = the radial distance from the main axis of spheromak symmetry, herein referred to as the Z axis.
This region inside the spheromak wall exists within a larger region outside the spheromak wall where the magnetic and electric field energy densities are approximately equal to the magnetic and electric fields produced by a current ring located at R = Ro, Z = 0 carrying a current (Np I) and having net charge Qs.
SPHEROMAK STABILITY:
Spheromak stability arises from prime number constraints that affect Np and Nt as well as from formation of a potential energy well inside the spheromak wall. Due to a combination of electromagnetic field theory, spheromak geometry and prime number theory only certain values of Nt and Np can exist in a stable spheromak.
The integers Np and Nt cannot be equal and cannot have a common factor other than unity. As a result of Np and Nt having no common factors the filament current path (winding) has only one layer with no cross overs. The filament only intersects itself at the current path closure point. Spheromak stability suggests that small errors in Np or Nt should lead to self correction rather than spheromak collapse.
Apart from the constraints on Np and Nt detailed analysis shows that spheromaks are stable due to formation of an energy well inside the spheromak wall.
The optimum spheromak geometry is stable over a wide range of spheromak energy. The spheromak wall relative gepmetry remains constant and can be geometrically characterized by the size parameters Ro. For a stable spheromak the ratios: (Ho / Ro), (Rc / Ro) and (Rs / Ro) remain constant independent of the spheromak's total field energy.
The stable spheromak structure enables the existence of isolated charged particles and semi-stable plasmas. Particles with rest mass are stable packets of charge and energy that are non-propagating solutions to electromagnetic equations. Hence atoms and charged atomic particles embody spheromaks.
SPHEROMAK WINDING:
The spheromak winding has Npo poloidal turns and Nto toroidal turns.
The numbers Np and Nt are positive integers with various special mathematical properties.
The winding length Lh is given by:
Lh^2 = (Np Lp)^2 + (Nt Lt)^2
= [Np 2 Pi Ro]^2 + [Nt Lt]^2
where:
Lp = 2 Pi Ro
is the length of a single poloidal filament winding turn and
Lt =
is the length of a filament winding turn that encircles the toroidal magnetic flux once.
The spheromak inner wall and the spheromak outer wall have different radii with respect to the main axis of symmetry. Hence, at the inner wall the current filament to current filament distance is less than at the outer wall. Since the net charge is evenly distributed along the filament this issue causes the surface charge per unit area at the outer wall to be less than the surface charge per unit area at the inner wall.
The current filament turns occur in a single layer which contains no cross overs. The current path only intersects itself at the current path closure point. The spheromak must remain stable in the presence of small geometric disturbances that can cause either Np or Nt to increment or decrement from their nominal Np = Npo and Nt = Nto values. Hence, in a stable spheromak, Npo and Nto are not equal and do not share a common integer factors other than 1. Similarly there is no common factor sharing between: Npo + 2, Nto - 1; between Npo - 2 and Nto + 1.
Spheromak stability is enhanced if a small energy disturbance does not lead to spheromak collapse. Then if there is a disturbance that increments/decraments only Np or only Nt the spheromak remains stable.
Any point on the spheromak winding can be identified by its Phi and Theta values. Phi is the angle about the major axis of symmetry measured with respect to a radial line from the minor axis to the orgin. Theta is the angle about the minor axis at R = Ro, Z = 0 measured with respect to the same radial line. Hence at the current path closure point:
Phi = 0, Theta = 0 or Phi = 2 Pi Np, Theta = 2 Pi Nt
The range of Phi is:
0 < Phi < 2 Pi Npo radians;
The range of Theta is:
0 < Theta < 2 Pi Nto radians
The average value of dTheta / dPhi is given by:
dTheta / dPhi = 2 Pi Nto / 2 Pi Npo
= Nto / Npo.
However, in a spheromak dTheta / dPhi varies slightly along the winding and is a function of R.
Symmetry indicates that at the center of the spheromak core the net electric field is zero.
This web page last updated on October 16, 2026.
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