# XYLENE POWER LTD.

## SPHEROMAK COMPRESSION

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

INTRODUCTION:
One of the most remarkable features of plasma spheromaks is that under certain circumstances they can gain energy via adiabatic compression.

During adiabatic compression a spheromak shrinks linearly but gains electromagnetic field energy. This process can be used to produce high energy plasma spheromaks. In a certain sense the word "compression" is a misnomer, because in gases it refers to use of an external pressure.

In the case of an atomic particle spheromak in an externally applied magnetic field the atomic particle absorbs radio frequency electromagnetic energy and shrinks in size. An important related issue is determination of the plasma hose length in a spheromak.

In the case of a plasma spheromak in an appropriate external field the electrostatic interaction of the spheromak with a conical enclosure wall leads to spheromak compression as the spheromak passes down the axis ot the cone.

There are practical limits on the amount of spheromak linear compression that can be practically realized. One of these limits is thermal electron field emission from the plasma injector cone. Another limit is reduced spheromak lifetime due to ionizing collisions betwwen spheromak electrons and neutral gas atoms. Nevertheless, General Fusion has experimentally measured linear spheromak plasma compression of four fold to five fold. Hence linear spheromak compression will almost certainly be important in fusion power systems.

SPHEROMAK SHAPE PRESERVATION:
An important aspect of spheromak compression is that during compression the spheromak shape, as indicated by the shape factor So, remains unchanged. The parameter that changes is the nominal spheromak radius Ro.

EXPERIMENTAL DATA:
A photograph of a typical laboratory spheromak gives:
Rs = 4.2 Rc
Rf = 3.0 Rc
|Hf| = 2.7 Rc
Rx ~ 0.3 Rcc

The data from the spheromak photo indicates that:
Hf = 2.7 Rc
implying that:
So = [(2 Hf + Rc) / Rc]^0.5
= [6.4]^0.5
= 2.53

SPHEROMAK SIZE:
An experimental spheromak formed in a laboratory differs from an ideal spheromak in free spacebecause the experimental spheromak interacts with the enclosure wall.

The spheromak is usually in a long cylindrical metal enclosure that has a wall radius Rw
that may not satisfy the condition:
Rw > 2.71 Rs.
Consequently the electric field energy density outside the spheromak on the equatorial plane is somewhat cylindrical rather than spherical. This change in external electric field energy density causes a reduction in the spheromak outside radius Rs and reduces the position stability of the spheromak outside wall.

Under these circumstances if the spheromak is enclosed within in a long cone and if there is a suitable source of energy the spheromak will absorb energy and shrink as it passes down the axis of the cone. An important issue is the frequency of electromagnetic radiation that the spheromak will readily absorb.

TRAPPED ELECTRON KINETIC ENERGY:
Both theory and experiment indicate that during spheromak compression the trapped electon kinetic energy increases. The corresponding increase in trapped electron velocity increases the probability of a trapped electron undergoing an ionizing collision with a neutral gas molecule occupying the same space. Such ionizing collisions reduce the spheromak lifetime. Hence one of the problems with spheromak compression is a reduction in spheromak lifetime. this issue, together with the quality of the vacuum, limit the amount of spheromak compression that can be usefully used in a fission power reactor.

FREE ELECTRON KINETIC ENERGY IN A PLASMA SPHEROMAK:
Define:
Me = free electron mass

The relationship between Ve and the free electron kinetic energy Eke is:
Eke = (Me Ve^2) / 2
or
- Ve = (2 Eke / Me)^0.5

Recall that the existence requirement for a plasma sheet gives:
|(Ni - Ne) / Ne| = |Ve / C|

Recall that The net charge Qs on a spheromak is given by:
Qs = Q (Ni - Ne)
or
(Ni - Ne) = Qs / Q

Combining these three equations gives:
|(Ni - Ne) / Ne|
= |Ve / C|
= (2 Eke / Me)^0.5 (1 / C)

or
(2 Eke / Me) = (C Qs / Q Ne)^2
or
Eke = (Me / 2) (C Qs / Q Ne)^2

This equation gives the free electron kinetic energy Eke in a plasma spheromak in terms of the physical constants Me, C, Q, the net charge Qs and the number of free electrons Ne.

SPHEROMAK COMPRESSION:
An important result of the equation:
|(Ni - Ne) / Ne| = |Ve / C|
is that it gives insight into what happens during spheromak compression. During an ideal spheromak compression the net charge:
Qs = Q (Ni - Ne)
remains constant while Ne and Ni both decrease due to recombination of spheromak electrons and ions. In order for the spheromak existence condition to continue to be met, Ve increases.

Hence, a compressed spheromak has a smaller number of free electrons than an uncompressed spheromak. The neutral gas atoms that are products of recombination tend to reduce the lifetime of a compressed spheromak. This is a major issue in the Plasma Impact Fusion process.

During the process of spheromak compression the recombining electrons and ions lose their kinetic energy by emission of UV radiation. Hence the process of spheromak compression is not perfectly adiabatic. A further complication is that this UV radiation, when it interacts with the enclosure wall, causes emission of electrons. These free electrons, if caught in the radial electric field around a spheromak, will tend to discharge the spheromak. Hence the inside walls of the charge injector need a coating that minimizes UV triggered electron emission. Possibly a high UV reflectivity internal surface can be used to guide most of the UV energy back to the upstream end of the charge injector where it can be absorbed with minimum consequence.

ENCLOSURE WALL PROXIMITY:
FIX THIS SECTION

UNIQUE SPHEROMAK SURFACE POTENTIAL AND SURFACE ELECTRIC FIELD:
At the spheromak plasma sheet surface the potential and electric field values are unique.

At the end of the spheromak the electric field is spherical. Except in the end funnels the plasma sheet is equipotential. Hence a cylindrical solution with respect to the nearby cylindrical enclosure wall for the equatorial surface electric field Es and the equatorial surface potential Phis must yield the same potential and electric field values as a spherical solution with respect to a distant enclosure wall for the corner electric field Ef and the corner surface potential Phif.

Hence, assume that:
Phif = Phis
and
Ef = Es

Then:
Qs / 4 Pi Epsilon (Rf^2 + Hf^2)^0.5 = [Qs Rs / (Epsilon As)][1 + (Rc / Rs)] Ln(Rc / Rs)
and
Qs / 4 Pi Epsilon (Rf^2 + Hf^2) = [Qs / (Epsilon As)][1 + (Rc / Rs)]

Combining these two equations gives:
Qs / 4 Pi Epsilon (Rf^2 + Hf^2)^0.5 = Rs Ln(Rc / Rs) [Qs / 4 Pi Epsilon (Rf^2 + Hf^2)]
or
1 = Rs Ln(Rc / Rs) / (Rf^2 + Hf^2)^0.5
which has a trial solution:
Ln(Rc / Rs) = 1
or
Rc = 2.71828 Rs
which equation is IMPORTANT in quantification of spheromak behaviour.
and
Rs = (Rf^2 + Hf^2)^0.5
or
Rs^2 = Rf^2 + Hf^2
which indicates that the outer wall of a spheromak is spherical. This IMPORTANT equation
exhibits excellent agreement with experimentally observed spheromak geometry.

Substitution of this trial solution into the first equation gives:
Qs / 4 Pi Epsilon Rs = [Qs Rs / (Epsilon As)][1 + (Ra / Rs)]

Substitution of the trial solution into the second equation gives:
Qs / 4 Pi Epsilon Rs^2 = [Qs / (Epsilon As)][1 + (Ra / Rs)]

These equations agree with each other, confirming that the trial solution is the real solution.
These equations further give the result that:
1 / 4 Pi Rs^2 = [1 / As] [1 + (Ra / Rs)]
or
As = [4 Pi Rs^2] [1 + (Ra / Rs)]
This expression for the area As of the spheromak plasma sheet is IMPORTANT in
quantification of spheromak behaviour.

CONDITIONS AT (Rf, Hf):
At R = Rf the spheromak overall length is 2 H = 2 Hf.

For R = Rf outside the spheromak the electric field is entirely spherically radial.

At the spheromak end (Rf, Hf) the surface electric field Ef is given by:
Ef = Qs / 4 Pi Epsilon (Rf^2 + Hf^2)

Define:
Uf^2 = (Rf^2 + Hf^2)
Then the surface potential Phif is given by:
Phif = Integral from U = Uf to U = infinity of E dU
= Qs / 4 Pi Epsilon Uf
= Qs / 4 Pi Epsilon (Rf^2 + Hf^2)^0.5

Recall that at the spheromak equator where:
R = Rs
H = 0
the spheromak surface electric field Es is given by:
Es = [Qs / (Epsilon As)][1 + (Ra / Rs)]
????????? and the surface potential Phis is given by:
Phis = [Qs Rs / (Epsilon As)][1 + (Rc / Rs)] Ln(Rc / Rs)
?????????

FREE ELECTRON KINETIC ENERGY:
As shown on the web page titled PLASMA SHEET PROPERTIES within the plasma sheet the free electron kinetic energy Eke is given by:
Eke = (Me / 2) Ve^2
= (Me / 2) C^2 [(Ni - Ne) / Ne]^2

where:
Me = electron mass;
C = speed of light;
Ne = number of free electrons in the spheromak;
Ni = number of ions in the spheromak.

NUMBER OF FREE ELECTRONS:
The total number of free electrons Ne in a spheromak is important for the design of Plasma Impact Fusion (PIF) or Magnetic Target Fusion (MTF) energy systems.

The free electrons in a spheromak are concentrated in the plasma sheet. Hence the free electron volumetric density in a spheromak is nonuniform, being high in the plasma sheet and low elsewhere.

SPHEROMAK INTERNAL ENERGY:
Define:
Qs = spheromak net charge;
Qsa = initial value of Qs before spheromak compression;
Qsb = final value of Qs after spheromak compression;
Pi = 3.1415928;
Epsilon = 8.85 X 10^-12 coul^2 / Nt-m^2 = permittivity of free space;
Rs = equatorial radius of a spheromak;
Rw = inside radius of a spherical enclosure, where Rw > Rs;
Rwa = initial value of Rw;
Rwb = final value of Rw;
Eo = (Qs^2) / (8 Pi Epsilon Rs) = nominal spheromak electric field energy;
Eoa = initial value of Eo;
Eob = final value of Eo;
Em = total spheromak magnetic field energy;
Ema = initial value of Em;
Emb = final value of Em;
Eeo = electric field energy outside the spheromak;
Eeoa = initial value of Eeo;
Eeob = final value of Eeo;
Ba = axial magnetic field strength inside the spheromak straight core;
Mu = 4 Pi X 10^-7 T^2 m^3 / J = permiability of free space;
Es = external radial electric field at the spheromak surface where R = Rs;
Esa = initial value of Es;
Esb = final value of Es;
Eca = electric field at the enclosure inside wall when the spheromak is at its initial position;
Ecb = electric field at the enclosure inside wall when the spheromak is in the throat of the plasma injector;
Eke = electron kinetic energy.

SIGNIFICANCE OF Eo:
Let Eo be the electric field energy outside a sphere of radius Rs that has a charge Qs and a surface electric field Es.

Basic electromagnetic theory gives:
Qs = Es 4 Pi Rs^2 Epsilon
or
Es = Qs / 4 Pi Rs^2 Epsilon

The electric field energy Eo is given by:
Eo = Integral from R = Rs to R = infinity of:
(Epsilon / 2) [Qs / (4 Pi Rs^2 Epsilon)]^2 4 Pi R^2 dR

= Qs^2 / (8 Pi Epsilon Rs)

RATIO RELATIONSHIP SUMMARY:
G = Rsa / Rsb = Eca / Ecb

(Rsa / Raa) = (Rsb / Rab)

Neb^2 Ekeb = Nea^2 Ekea

Bab Rsb^2 = Baa Rsa^2

(Veb / Vea) = (Nea /Neb)

Hence:
(Bab / Baa) = (Rsa / Rsb)^2 = G^2
and
(Ekeb / Ekea) = (Veb / Vea)^2 = G^2

FREE ELECTRON DENSITY:
The average free electron/ion density in a spheromak is really almost meaningless. The parameter that is important is Ne which is the total number of free electrons in the spheromak.

However, for completeness the above equations can be expressed in terms of free electron density as follows:
(Neb / Nea)^2 (Ekeb / Ekea) = 1
or
(Neb / Nea)^2 = (Ekea / Ekeb)
or
[(Neb Rsa^3) / (Nea Rsb^3)]^2 = (Ekea / Ekeb)(Rsa / Rsb)^6
= (1 / G)^2 (G)^6
= G^4

Hence:
[(Neb Rsa^3) / (Nea Rsb^3)] = G^2

Thus the theoretical ratio:
(compressed spheromak average free electron density) / (uncompressed spheromak average free electron density)
= [(Neb Rsa^3) / (Nea Rsb^3)]
= G^2
= 25

General Fusion claims to have experimentally measured the free electron density in both compressed and uncompressed spheromaks. According to General Fusion:
(compressed spheromak average free electron density) / (uncompressed spheromak average free electron density)
= (2 X 10^16 / cm^3) / (2 X 10^14 / cm^3)
= 100
.

In this particle density ratio there is a four fold disagreement between theory and experiment.

Much worse, the free electron/ion density experimentally measured by General Fusion seems to be inconsistent with the other mutually consistent data regarding spheromak shape, size, magnetic field, free electron kinetic energy and lifetime.

The total number of spheromak free electrons/ions calculated from the mutually consistent experimental data indicates that the average free electron/ion density is much less the free electron/ion density that General Fusion claims to have measured using ion probe and Thomson scattering/ion doppler methodology.

The good agreement between MFI theory and experiment with respect to spheromak shape, spheromak linear size, spheromak free electron kinetic energy, spheromak poloidal magnetic field and spheromak lifetime strongly suggests that the MFI theoretical predictions with respect to spheromak total free electron/ion content are correct and that there are problems in the General Fusion experimental determination of spheromak free electron/ion density. This author suspects that at the root of these particle density measurement problems are implicit assumptions with respect to random particle movement and uniform particle density that are not valid for a spheromak.

The MFI Spheromak model predicts spheromak lifetimes that are in general conformance with experimentally observed spheromak lifetimes whereas if the General Fusion measurement of free electron density (Ne / Rs^3) was valid the spheromak lifetimes would be much smaller than are typically observed.

This matter of uncertainty in the experimental determination of total spheromak free electron/ion content has yet to be totally resolved to this author's satisfaction. However, on the web page titled PLASMA SHEET PROPERTIES this author has set out how an analysis of Ion Probe, Thomson Scattering and Ion Doppler data using normal instrument calibrations, which implicitly assume a random plasma, could easily lead to General Fusion's free electron/ion density claims.

AVERAGE PARTICLE DENSITY MEASUREMENT ISSUES:
Spacial Nonuniformity:
The MFI mathematical model of a spheromak indicates that the free electron/ion density in a spheromak is highly non-uniform, being relatively high in the plasma sheet and relatively low elsewhere. Hence the free electron/ion density is strongly dependent on position in the spheromak. Any method of measuring free electron/ion density that senses the particle density in only a small portion of the spheromak is subject to major error, high or low, depending on whether or not the sampled volume contains a portion of the spheromak's plasma sheet.

Orientation:
The particles move in the plane of the plasma sheet with a combination of toroidal and poloidal motion. Particle density sensing methods that rely on the doppler effect will give an orientation dependent signal depending on the orientation of the laser beam and the observer with respect to the net particle motion in the plane of the plasma sheet.

Ion Probes:
Unlike ordinary plasmas, a spheromak plasma is surrounded by a high radial electric field. Measurements of spheromak ion density via ion probes are subject to major errors due to extremely high electric fields at the ion probe surface. Unless extraordinary measures are taken to shield the ion probe from this radial electric field, the probe current due to field emission electrons flowing away from the probe will swamp the probe current due to ions flowing toward the probe.

Thomson Scattering/Ion Doppler:
Practical Thomson scattering and ion doppler instruments sense only a small volume near the center of the spheromak and hence are subject to errors due to laser beam orientation with respect to the particle motion in the plasma sheet and due to particle density non-uniformity.

Normal Thomson scattering and ion doppler methods of measuring free electron density or ion density involve an assumption that the particle motion in the plasma is random in both direction and magnitude, whereas in reality in a spheromak the particle motion is not random in either direction or magnitude. With a spheromak the data recorded using the Thomson scattering or ion doppler methods will strongly depend on the laser and receiver: position, orientation and optical filter transfer function. A narrow band optical filter will give an apparently larger doppler shifted signal from a spheroamk than from a random plasma containing an equal number of free electrons/ions in the sense region. This increased signal strength is easily misinterpreted as indicating a much higher average particle density in the spheromak than actually exists.

SPHEROMAK FREE ELECTRON KINETIC ENERGY REQUIREMENTS:
In order for the initially formed spheromak to have a long life its free electrons must not have enough kinetic energy to cause impact ionization of neutral gas molecules. The ionization energy of hydrogen is 13.6 eV. Hence to maximize the spheromak lifetime it is a good idea to limit the initial spheromak electron kinetic energy to about 13.5 eV.

The converse is also true. When it is desired to rapidly transfer energy from a spheromak to
the surrounding gas or plasma it is desireable for the spheromak to have a high electron kinetic energy
so that the field energy in the spheromak is discharged to the surrounding gas or plasma as quickly as possible.

UNCOMPRESSED SPHEROMAK GEOMETRY:
The spheromak shown in the photograph on the General Fusion website has the following ratios:
Hf = 2.7 Rc
Rx = 0.3 Rc
Rf = 3.0 Rc
Rs = 4.2 Rc
2.0 < (Ha / Ra) < 2.3

The MFI mathematical model computes the spheromak end shape from the boundary conditions, and in the region:
Ra < R < Rf
gives:
H = [((R / Rf) (Rf^2 + Hf^2)) - R^2]^0.5

Evaluation of this equation at R = Ra, H = Ha gives:
Ha = [((Ra / Rf) (Rf^2 + Hf^2)) - Ra^2]^0.5
or
(Ha / Ra) = [(Ra / Rf) ((Rf / Ra)^2 + (Hf / Ra)^2) - 1]^0.5
= [(1 / 3.0) ((3.0)^2 + (2.7)^2) - 1]^0.5
= 2.26
indicating reasonable agreement between the MFI mathematical model and the experimental results.

General Fusion has experimentally measured the axial magnetic field strength of this spheromak and found it to be:
Ba = 0.12 T
for a free electron kinetic energy in the range 20 eV to 25 eV.

Recall that:
Ln(Rw / Rs) = 1
or
(Rw / Rs) = 2.71.

FIELD ENERGY FOR UNCOMPRESSED SPHEROMAK:
Recall that:
Eo = (Qs)^2 / (8 Pi Epsilon Rs)
and
Ba = (Rs / Ra) (Es / C)
and
Es = Qs / (4 Pi Epsilon Rs^2)

Combining these equations gives:
Eo = (Qs)^2 / (8 Pi Epsilon Rs)
= (4 Pi Epsilon Rs^2 Es)^2 / (8 Pi Epsilon Rs)
= (2 Pi Epsilon Rs^3 Es^2)
= (2 Pi Epsilon Rs^3 (C Ba Rc / Rs)^2
= (2 Pi Epsilon Rs C^2 Ba^2 Rc^2)
= (4 Pi Rs Rc^2) (Ba^2 / 2 Mu)

This equation indicates that for a spheromak with equatorial radius Rs and with axial core magnetic field strength Ba the electric field energy Eo is given by:
Eo = (4 Pi Rs Rc^2) (Ba^2 / 2 Mu)

= (4 Pi Rs^3) (Rac / Rs)^2 (Ba^2 / 2 Mu)

The MFI model for an uncompressed spheromak, which numerically integrates the magnetic field energy over the spheromak volume, gives:
Em = (Ba^2 / 2 Mu)(Pi Rs^3) (.2306)

DETAIL THIS NUMERICAL INTEGRATION!!!

Thus:
Eo / Em = 4 (Ra / Rs)^2 / (.2306)
= 4 (1 / 4.2)^2 / (.2306)
= .9833

Thus the expressions for electric field energy and magnetic field energy give identical results within measurement errors. Hence:
Em = Eo
and the spheromak's total field energy Et is given by:
Et = 2 Eo

For a typical uncompressed spheromak:
Assume:
Rsa = 1 m / 2.71828
and
Ba = 0.12 T
giving:
Eoa = (4 Pi Rs^3) (Rc / Rs)^2 (Ba^2 / 2 Mu)

= 4 X Pi X (1 m / 2.71828)^3 X (1 / 4.2)^2 (0.12 T)^2 / (2 X 4 Pi X 10^-7 T^2 m^3 /J)

= (.049787) X (.056689) X (.0144) X 10^7 / 2

= 203.21 J

SPHEROMAK COMPRESSION:
A situation of great practical importance is what happens if the channel radius Rw and the spheromak equatorial radius Rs are both simultaneously gradually reduced while keeping the net charge Qs constant and keeping the ratio:
Rc / Rs = constant.

Let subscript "a" indicate initial values before compression.
Let subscript "b" indicate final values after compression.

Let Ekaa = core electron kinetic energy before size reduction;
Let Ekab = core electron energy after size reduction;
Let Rca = Rc value before size reduction;
Let Rcb = Rc value after size reduction;
Let Rfa = Rf value before size reduction;
Let Rfb = Rf value after size reduction;
Let Hfa = Hf value before size reduction;
Let Hfb = Hf value after size reduction.

LINEAR COMPRESSION:
General Fusion indicates that it has observed apparent spheromak linear compression in proportion to the decrease in plasma injector radius Rc.

EXPERIMENTAL DATA RELATING TO Eka AND LINEAR SHRINKAGE:
Data provided by General fusion indicates that:
Rsa / 5 < Rsb < Rsa / 4
or
Hfa / 5 < Hfb < Hfa / 4
or
Rfa / 5 < Rfb < Rfa / 4
There is no clarity as to which parameter(s) General Fusion used as a spheromak linear dimension indicator.

General Fusion further indicates that:
Ekab = 400 eV to 500 eV
Ekea = 20 eV to 25 eV
Bab = 2.4 T to 3.0 T
Baa = 0.12 T

Numerical substitution gives:

Experimentally:
General Fusion reports that:
0.2 < (Rsb / Rsa) < 0.25
and
20 < (Bab / Baa) < 25
and
16 < (Ekeb / Ekea) < 25
which indicates that:
(Ekeb / Ekea) ~ (Bab / Baa)

SPHEROMAK COMPRESSION:
Assume that a plasma sheet forms the winding on a toroidal solenoid.
Define:
La = magnetic length of solenoid
Ra = radius of solenoid
Ba = magnetic field within solenoid
Then:
Ba = Mu Q Nea Vea / (2 Pi Ra La)

Now suppose that the toroid is linearly compressed by a factor G so that:
Veb = G Vea.

General Fusion's experimental result was:
Ekeb / Ekea = (La / Lb)^2 = (Ra / Rb)^2 = (Bb / Ba) = G^2

After compression:
Neb = Nea / G
Veb = G Vea
Rb = Ra / G
Lb = La / G

Hence:
Bb = Mu Neb Veb / (2 Pi Rb Lb)

= Mu (Nea / G) (G Vea) / [(2 Pi (Ra / G) (La / G)]

= Mu Nea Vea G^2 / (2 Pi Ra La)

= G^2 Ba

Hence:
(Bb / Ba) = G^2 = (Veb / Vea)^2
which agrees with the experimental observations by General Fusion.

Hence plasma sheet theory is consistent with almost all available experimental data relating to spheromak compression.

PLASMA INJECTOR METHODOLOGY:
Recall that:
Eo = (Qs)^2 / (8 Pi Epsilon Rs)

The plasma injector operates by reducing the spheromak radius Rs by a factor of 5 while holding the spheromak net charge Qs constant. Hence:
(Emb / Ema) ~ (Eob / Eoa)
= (Qsb / Qsa)^2 (Rsa /Rsb)
= 1^2 (5)
= 5

Thus the plasma injector realizes a spheromak magnetic field energy gain of about 5.

However, General Fusion does not measure the spheromak field energy. General Fusion measures the spheromak free electron kinetic energy which increases by a factor of about 25. When the spheromak randomizes the field energy release greatly exceeds the free electron kinetic energy release.

The plasma injector cone is shaped to keep the quantity:
[(Rc - Rs) / Rc] approximately constant along the plasma injector axis. Hence:
[(Rca - Rsa) / Rca] = [(Rcb - Rsb) / Rcb]

Recall that:
Rc = Ex Rs

In the General Fusion succesful plasma injector design:
Rca = 5 Rcb
and
Rsa = 5 Rsb

INTERNAL FIELD ENERGY OF COMPRESSED SPHEROMAK:
Recall that the field energy of a spheromak is given by:
Eo = (4 Pi Rs Ra^2) (Ba^2 / 2 Mu)

Hence:
(Eob / Eoa) = (Rsb / Rsa) (Rcb / Rca)^2 (Bab / Baa)^2

Recall that:
Eo = (Qs)^2 / (8 Pi Epsilon Rs)

Hence: (Eob / Eoa) = (Qsb / Qsa)^2 (Rsa / Rsb)

Combining these equations gives:
(Qsb / Qsa)^2 = (Rsb / Rsa)^2 (Rcb / Rca)^2 (Bab / Baa)^2

ASSUMPTION OF CONSTANT Qs:
Now assume that the spheromak net charge Qs remains constant as the spheromak moves down the plasma injector axis. Thus:
Qsa = Qsb
giving:
(Eob / Eoa) = (Rsa / Rsb)
and
Bab^2 Rsb^2 Rcb^2 = Baa^2 Rsa^2 Rca^2 or
Bab Rsb Rcb = Baa Rsa Rca
which is an important equality that was experimentally observed by General Fusion.

MAGNETIC FIELD VERSUS SPHEROMAK LINEAR SIZE:
Assume that the experimentally measured magnetic field is the poloidal magnetic field in the spheromak core and hence equals Ba in the MFI spheromak mathematical model. Assume that the initial spheromak equatorial radius is Rsa. Then the experimental data gives the initial value of Ba Rs^2 as:
Baa Rsa^2 = (.12 T) (Rsa)^2 = .12 T Rsa^2

The experimental data indicates that the final value of Ba Rs^2 lies in the range:
(2.4T) (Rsa / 5)^2 < Ba Rs^2 < (3.0 T) (Rsa / 4)^2
or
.096 T Rsa^2 < Ba Rs^2 < .1875 T Rsa^2

Combining the above expressions gives:
.096 T Rsa^2 < .12 T Rsa^2 < .1875 T Rsa^2
which is a valid inequality chain.

Thus the General Fusion experimental data with respect to the spheromak core magnetic field ratio and the spheromak linear size reduction ratio indicates that:
Baa Rsa^2 ~ Bab Rsb^2

MAGNETIC FIELD ENERGY GAIN:
Note that we have shown above that according to the MFI mathematical model:
(Eob / Eoa) = (Rsa / Rsb) ~ 5
This is an extremely important theoretical result from the perspective of energy system development. It can be used to calculate the magnetic field energy realized when the spheromak passes through the plasma injector.

This result is confirmed by General Fusion's experimental data. Recall that from Maxwells's equations the magnetic field energy density is:
(Ba^2 / 2 Mu).

Recall that under the circumstances of linear compression spheromak volume Vs is proportional to a linear side R cubed. Hence the spheromak magnetic energy gain through the plasma injector is given by:
Eob / Eoa = (Bab^2 Vsb / Baa^2 Vsa)
= (Bab / Baa)^2 (Rb / Ra)^3

However, experimentallly General Fusion found that:
Bab / Baa = Ra^2 / Rb^2
Hence:
Eob / Eoa = (Bab / Baa)^2 (Rb / Ra)^3
= (Ra^2 / Rb^2)^2 (Rb / Ra)^3
= (Ra / Rb)

Thus the MFI theory with respect to the magnetic field energy gain:
(Eob / Eoa)
is consistent with the experimental measurements by General Fusion.

SPHEROMAK MAGNETIC FIELD ENERGY AT PLASMA INJECTOR DISCHARGE:
One of the most important aspects of the MFI mathematical model is the the spheromak field
energy gain through the plasma injector which is given by:
(Eob / Eoa) = (Rsa / Rsb) ~ 5

ELECTRIC FIELD ENERGY AT PLASMA INJECTOR DISCHARGE:
The external electric field energy Eo is about:
Eo ~ Em

Hence:
Et = Ee + Em

This web page last updated April 5, 2015.