Current flow switching device for combined function magnetic field production

An apparatus for creating and controlling a toroidal magnetic field, and a poloidal magnetic field employs a conductive sheet rolled into a spiral with cut outer legs aligned and bent outwardly toward a position approximately adjacent to corresponding other cut outer legs. Such apparatus is formed by forming a conductive sheet; cutting a first plurality of cut outer legs into the sheet; cutting a second plurality of cut outer legs into the sheet; rolling the conductive sheet into a spiral; and bending outwardly each of the cut outer legs. In an alternative, the apparatus has a conductive tube with each of a first plurality cut outer legs bent outwardly to couple to corresponding ones of a second plurality of cut outer legs. Such alternative is made by forming a conductive tube; cutting a plurality cut outer legs at one of the conductive tube; cutting another plurality cut outer legs at another of the conductive tube; bending outwardly the cut outer legs; and coupling ones of the plurality of cut outer leg to corresponding ones of the other plurality of cut outer legs. The above mentioned devices may include a spiral resistive strip or thermally induced resistive spiral in a centerpost region.

BACKGROUND OF THE INVENTION 
The present invention relates to electric current conducting mechanisms for 
controlling a magnetic field in a high magnetic field device. More 
particularly, the present invention relates to current conducting and 
switching mechanisms for conducting axial and toroidal currents for 
controlling a magnetic field in a high magnetic field device. Even more 
particularly, the present invention relates to current conducting and 
switching mechanisms for conducting axial and toroidal currents for 
creating and controlling a toroidal magnetic field in a toroidal region, 
and a time-varying axial or poloidal magnetic field, which in turn 
produces a toroidal electric field, in a high magnetic field device, such 
as, for example, a tokamak fusion reactor (TFR). 
The main elements of a heretofore known tokamak fusion reactor (TFR) 20 are 
shown in FIGS. 1, 2, 3 and 4. The TFR 20 includes a vacuum vessel 21 in 
the shape of a torus. A major axis 22 of a cylindrical coordinate system 
19 (see FIG. 2) is a straight line centered in a hollow center 25 (or 
bore) of the toroidal vacuum vessel 21 (FIG. 1), i.e., inside the 
"doughnut hole", and a minor axis 24 of such coordinate system is a circle 
lying in a plane normal to the major axis 22 and at a center of a core of 
the toroidal vacuum vessel 21. The major axis 22 extends in a z-direction 
and the minor axis 24 is a distance R.sub.o from the major axis 22. Points 
along the minor axis 24 are defined by an angle .phi.. Walls of the 
torodial vacuum vessel 21 are defined with respect to the minor axis 24 by 
another radius r and an angle .theta.. A centerpost region 23, for 
example, in a TFR is defined as a cylindrical volume centered on the major 
axis 22 and extending radially to an innermost cylinder of the vacuum 
vessel 21. 
The present invention relates to a novel and nonobvious approach for 
conducting and controlling currents in this centerpost region 23 or in a 
similar region. 
Referring to FIG. 3, large magnetic field coils 26 (FIG. 1), commonly 
referred to as toroidal field (TF) coils 26 or B-coils 26, envelope the 
minor axis 24 and the toroidal vacuum vessel 21, with part of a current 
I.sub.z carried by the B-coils 26 being in the centerpost region 23. This 
B-coil 26 when energized with the current I.sub.z, produces a strong 
toroidal magnetic field B.sub..phi. that is oriented parallel to the minor 
axis 24 encircling the major axis 22. The strong toroidal magnetic field 
B.sub..phi. is used in, for example, a TFR to contain high-temperature 
deuterium-tritium plasma to promote nuclear fusion of the 
deuterium-tritium mixture. 
Referring to FIG. 4, a toroidal plasma current I.sub.p that flows through 
the high-temperature plasma in a TFR generally flows in a toroidal 
direction (i.e., parallel to the minor axis 24) and is needed to improve 
plasma confinement and provide initial heating of the high-temperature 
plasma. This plasma current I.sub.p is conventionally driven by a 
transformer system. 
In order to produce the plasma current I.sub.p in the transformer system, a 
time-varying current I.sub..phi. flows through an electric field coil 28, 
or E-coil 28, also commonly referred as an ohmic heating coil (or OH 
coil). The E-coil 28 serves as a primary winding of the transformer system 
and serves to induce the plasma current I.sub.p in the plasma, which is 
in-effect the transformer's secondary winding. This plasma current I.sub.p 
is induced by a toroidal electric field E.sub..phi. in the plasma 
generated in response to the time-varying current I.sub..phi.. The E-coil 
28 is positioned principally within the centerpost region 23 and inside 
the bore of the vacuum vessel 21. 
The transformer system thus provides an inductive voltage for generating 
and driving the plasma current I.sub.p. The plasma current I.sub.p in turn 
provides resistive (or ohmic) heating of the plasma. Other systems for 
additional heating of the plasma in a TFR are known in the art. 
The plasma current I.sub.p generates a relatively constant poloidal 
magnetic field B.sub..theta., and, when combined with the toroidal 
magnetic field B.sub..phi., provides a spiral field line geometry located 
on closed toroidal surfaces within the plasma (the surfaces of constant 
magnetic flux are closed about the minor axis 24). 
Referring to FIG. 5, the B-coil 26 (also referred to as the TF coil, or 
toroidal magnetic field coil) and E-coil 30 (also referred to as the OH 
coil, or ohmic heating coil), in accordance with heretofore known 
approaches, are separate coils partially occupying the centerpost region 
23 (FIG. 1) inside the bore of the toroidal vacuum vessel 21 (FIG. 1). 
A combination of the toroidal magnetic field B.sub..phi. with the poloidal 
magnetic field B.sub..theta. provide spiral (or helix-like) magnetic flux 
lines that generally lie on closed nested magnetic surfaces in the 
toroidal vacuum vessel 21 (FIG. 1). 
Ions and electrons within the plasma rotate about the toroidal magnetic 
field B.sub..phi. and flow generally along the minor axis 24 (FIG. 2). As 
a result, the ions and electrons follow a spiraling path around the 
toroidal vacuum vessel 21 (FIG. 21) confining the particles away from the 
wall of the toroidal vessel 21 (FIG. 1). 
The magnetic fields, i.e., the toroidal magnetic field B.sub..phi. and the 
poloidal magnetic field B.sub..theta., within the TFR provide an inward 
force that substantially overcomes outward pressure of the plasma and 
significantly confines the plasma's flow to within the toroidal vacuum 
vessel 21 (FIG. 1). Additionally in heretofore known approaches, plasma 
positioning is accomplished using poloidal field shaping coils 30 (FIG. 
1). 
Referring to FIG. 6, with the constant current I.sub.z flowing in the 
B-coils 26 (FIG. 1), the resulting toroidal magnetic field B.sub..phi. 
interacts with the constant current I.sub.z to produce an approximately 
constant force F.sub.B in the centerpost region 23 (FIG. 1). The constant 
force F.sub.B is directed radially toward the major axis 22 of the TFR. 
Another approximately radial force is directed away from the major axis 22 
of the toroidal vacuum vessel 21 and varies over time. This time-varying 
force F.sub.E is produced by an interaction between the time-varying 
current I.sub..phi. flowing through the E-coil 30 and an axial component 
of the poloidal magnetic field B.sub..theta. produced by the E-coil 30. 
Previous efforts have attempted to configure the E-coil 30 and the B-coil 
26 such that the constant force F.sub.B and the time-varying force F.sub.E 
react at an interface 27 between the E-coil 30 and the B-coil 26 and 
effectively cancel, for certain magnitudes of the constant current I.sub.z 
and the time-varying current I.sub..phi.. 
The size of the TFR is critically dependent on a radial extent of the 
centerpost region 23 (FIG. 1). The size of the centerpost region 23 (FIG. 
1) determines a capacity of the centerpost region 23 (FIG. 1) to carry the 
constant current I.sub.z and time varying current I.sub..phi.. The 
strength of the material in this region and its ability to easily conduct 
current ultimately limit the strength of the fields that can be produced 
in the TFR and thus the ability of the TFR to confine the high-temperature 
plasma. 
Accordingly, there exists a need for improvements in current conducting and 
switching mechanisms for controlling a magnetic field in a high magnetic 
field device, such as a TFR, that, for a given centerpost region size, 
allow greater plasma containment forces to be generated by current flowing 
through the centerpost region than has been heretofore achievable using 
the above-described approaches. 
SUMMARY OF THE INVENTION 
The present invention is directed to current conducting and switching 
mechanisms for conducting axial and toroidal currents for creating and 
controlling a toroidal magnetic field in a toroidal region, and an axial 
or poloidal magnetic field, which if time varying, produces a toroidal 
electric field, in a magnetic field device, such as, for example, a 
tokamak fusion reactor (TFR). 
In one embodiment, the invention can be characterized as an apparatus for 
creating and controlling a toroidal magnetic field, and a poloidal 
magnetic field, in a magnetic field device. The apparatus in one 
embodiment comprises a conductive sheet rolled into a spiral with each of 
a first plurality cut outer legs aligned and bent outwardly toward a 
position approximately adjacent to corresponding ones of a second 
plurality of cut outer legs. 
In another embodiment, the invention can be characterized as an apparatus 
for creating and controlling a toroidal magnetic field, and a poloidal 
magnetic field, in a high magnetic field device, wherein the apparatus 
comprises a conductive tube (which may be normally conducting or 
superconducting, in accordance with variations of the present embodiment) 
with each of a first plurality cut outer legs bent outwardly to couple to 
corresponding ones of a second plurality of cut outer legs. 
In a further embodiment, the invention can be characterized as a method of 
making an apparatus for creating and controlling a toroidal magnetic 
field, and a poloidal magnetic field, in a high magnetic field device. 
Such method involves forming a conductive sheet; cutting a first plurality 
of cut outer legs into a first edge of the conductive sheet; cutting a 
second plurality of cut outer legs into a second edge of the conductive 
sheet, the second edge being opposite the first edge; rolling the 
conductive sheet into a spiral with each of the first plurality cut outer 
legs aligned with another of said first plurality of legs, the one of the 
first plurality of legs and the other of the first plurality of legs being 
at adjacent wraps of the spiral; bending outwardly each of the first 
plurality of legs; and bending outwardly each of the second plurality of 
legs toward a position approximately adjacent to corresponding ones of the 
first plurality of cut outer legs. 
In yet a further embodiment, the invention can be characterized as a method 
of making an apparatus for creating and controlling a toroidal magnetic 
field, and a poloidal magnetic field, in a high magnetic field device. 
This method includes forming a conductive tube; cutting a first plurality 
cut outer legs at one of the conductive tube; cutting a second plurality 
cut outer legs at another of the conductive tube; bending outwardly the 
first plurality of cut outer legs; bending outwardly the second plurality 
of cut outer legs; and coupling ones of the first plurality of cut outer 
legs to corresponding ones of the second plurality of cut outer legs.

Corresponding reference characters indicate corresponding components 
throughout the several views of the drawings. 
Description of Symbols Used 
The following sets forth definitions for symbols used below in the Detailed 
Description of the Preferred Embodiments: 
R=radial component of R, Z, .phi. coordinate system; 
z=axial components of R, Z, .phi. coordinate system; 
.phi.=angular or toroidal component of r, z, .phi. coordinate system; 
R.sub.o =radial distance to center of minor axis; 
.theta.=angular measure from center of minor axis; 
B.sub..phi. =toroidal magnetic field; 
I.sub.z =axial current in centerpost region; 
I.sub.p =toroidal current in plasma; 
E.sub..phi. =toroidal electric field; 
B.sub..theta. =poloidal magnetic field with components B.sub.r & B.sub.z ; 
r=minor radius measured length in radial direction from minor axis and at 
angle .theta.; 
F.sub.B =force in the radial direction in the centerpost region from axial 
current of the B-coil (or TF coil); 
F.sub.E =radial force in the centerpost region from toroidal current of the 
E-coil (or ohmic heating coil); 
T.sub.1, T.sub.2, T.sub.3, T.sub.4, T.sub.5 =different times during 
operation of device; 
V.sub.1, V.sub.2 =voltages at different axial locations of centerpost 
region; 
L=length of centerpost in axial direction; 
E.sub.z =axial electric field; 
J=current density; 
.gamma.=pitch angle; 
.gamma..sub.J =current density pitch angle; 
X=Cartesian coordinate in orthogonal X, Y coordinates; 
Y=Cartesian coordinate in orthogonal x, y coordinates; 
J.sub..phi. =toroidal current density; 
J.sub.z =axial current density; 
.beta.=geometrical pitch.ident.tan .gamma.; 
.beta..sub.J =current density pitch.ident.tan .gamma..sub.J =J.sub..phi. 
/J.sub.z ; 
W.sub.1 =width of conductive region; 
W.sub.2 =width of resistive region; 
.sigma..sub.1 =electrical conductivity of conductive region; 
.sigma..sub.2 =electrical conductivity of resistive region (or strip); 
.beta..sub.jmax =maximum current density pitch; 
.beta..sub.max =geometrical pitch at maximum current density pitch; 
J.sub.1 =current density in the conductive region; 
J.sub.2 =current density in the resistive region; 
B.sub..phi.o =toroidal field at center of minor axis; 
.mu..sub.o =permeability of free space.tbd.4 .pi..times.10.sup.7 T/(A-m); 
R.sub.+ =outer radial extent of centerpost; 
R.sub.- =inner radial extent of centerpost; 
S=average toroidal stress (hoop stress); 
R.sub.E- =inner radius of E-coil (or ohmic heating coil); 
R.sub.E+ =outer radius of E-coil (or ohmic heating coil); 
.DELTA..phi..sub.z =axial or poloidal flux [Weber]; 
N=number of conductor turns; 
R.sub.B- =inner radius of B-coil (or toroidal field coil) in center post 
region; 
R.sub.B+ =outer radius of B coil (or toroidal field coil) in center post 
region; 
T=temperature; 
R=dimensionless conductive tube radius; 
.alpha.=thermal diffusivity=U C.sub.p ; 
C.sub.p =thermal heat capacity; 
T.sub.o =initial temperature; 
T.sub.1 =final temperature. 
Description of the Preferred Embodiments 
The following description is of the best mode presently contemplated for 
carrying out the invention. This description is not to be taken in a 
limiting sense, but is made merely for the purpose of describing the 
general principles of the invention. The scope of the invention should be 
determined with reference to the claims. 
All of the embodiments described herein employ a conductive surface for use 
in a centerpost region of, for example, a tokamak fusion reactor (TFR). 
Numerous uses for this conductive surface are contemplated and thus 
applications beyond TFR applications should be understood to be covered by 
the claims. 
The conductive surface can be, for example, composed of a single conductive 
tube, multiple concentric conductive tubes or a rolled or spiraled sheet. 
The conductive surface conducts both a toroidal field current I.sub.z (or 
relatively constant (axial) current I.sub.z) and an ohmic heating current 
I.sub..phi. (or time-varying (toroidal) current I.sub..phi.), for 
generating a toroidal magnetic field B.sub..phi. and a toroidal electric 
field E.sub..phi., respectively. Note that the terms relatively constant 
(axial) current I.sub.z, and time-varying (toroidal) current I.sub..phi. 
are in reference to how such currents typically function in a TFR. In 
other embodiments, the axial current I.sub.z may be time varying, and the 
toriodal current I.sub..phi. may be constant. The terms "constant" and 
"time-varying" are used herein merely to avoid possible confusion caused 
by referring to the currents by their geometry (because, e.g., their 
related magnetic fields do not have similar geometries, but rather have 
geometries in accordance with the right-hand rule--a magnetic field 
related to one of the currents may have a geometry more similar to the 
geometry of the other current). 
Using a conductive surface to conduct both the constant current I.sub.z and 
the time-varying current I.sub..phi. current results in a larger current 
carrying capacity that has heretofore been achievable for a given core 
diameter (i.e., a given radial extent of the centerpost region) and thus 
allows for higher magnetic field strengths for driving, e.g., a plasma 
current I.sub.p. Among other advantages, this larger current carrying 
capacity allows greater plasma containment forces to be generated than 
have been heretofore achievable for a given core diameter. 
Referring to FIGS. 7 through 9, description is provided herein that applies 
generally to the embodiments and variations thereof described hereinbelow. 
Referring to FIG. 7, an embodiment is shown of a centerpost region in which 
combined current flows made up of the relatively constant current I.sub.z 
and the time-varying current I.sub..phi.. Both the constant current 
I.sub.z and the time-varying current I.sub..phi. are induced to flow along 
a generally cylindrical conductive surface 101, 102, 103, 104, 105. The 
cylindrical conductive surface 101, 102, 103, 104, 105 can be configured, 
for example, to reside in the centerpost region of the TFR and to replace, 
in the centerpost region, both the B-coil 26 (FIG. 5) and the E-coil 28 
(FIG. 5) used in prior art approaches. 
By way of example, during a fusion reaction, the constant current I.sub.z 
is held constant, and is thus referred to herein as the constant current 
I.sub.z, while the time-varying current I.sub..phi. is continuously varied 
to induce a toroidal electric field E.sub..phi. to drive and sustain a 
toroidal plasma current I.sub.p within a high-temperature plasma, and is 
thus referred to herein as the time-varying current I.sub..phi.. 
The fusion reaction is substantially maintained during the time between an 
initial start time T.sub.1 and an end time T.sub.5. The cylindrical 
surface 101, 102, 103, 104, 105 is depicted, respectively, in FIG. 7, at 
five points in time, T.sub.1, T.sub.2, T.sub.3, T.sub.4, T.sub.5. 
Referring next to FIG. 8, a graph is shown of the constant current I.sub.z 
and the time-varying current I.sub..phi. flowing through the cylindrical 
surface 101, 102, 103, 104, 105 of FIG. 7 versus time. At a start time 
T.sub.1, the constant current I.sub.z and the time-varying current 
I.sub..phi., each of which flows in a direction that is orthogonal to the 
other, flow on the conductive surface 101, 102, 103, 104, 105 (FIG. 7). 
Their combination, i.e., a combined current, forms a spiral path along the 
cylindrical surface 101, 102, 103, 104, 105. Over time, the time-varying 
varying current I.sub..phi. decreases causing the pitch of the combined 
current's spiraling path to decrease, i.e, to become closer to parallel to 
a major axis of the cylindrical surface 101, 102, 103, 104, 105, such as 
at time T.sub.2, until, at time T.sub.3, the time-varying current 
I.sub..phi. crosses zero. Thus, at time T.sub.3, only the constant current 
I.sub.z is flowing in the cylindrical surface 101, 102, 103, 104, 105. As 
a result, at time T.sub.3, only the toroidal magnetic field B.sub..phi. is 
maintained. The poloidal magnetic field B.sub..theta. (also referred to as 
an axial magnetic field B.sub.z because its flux lines are substantially 
parallel to a major center axis of the centerpost region, in the 
centerpost region) is zero. 
As time progresses to time T.sub.4, the time-varying current I.sub..phi. 
becomes negative, causing the combined current to again follow a spiral 
path down the cylindrical surface 101, 102, 103, 104, 105, with an 
increasing pitch equal and opposite to the pitch at time T.sub.2. 
At time T.sub.5, the pitch of the spiral path is equal and opposite to the 
pitch at time T.sub.1. 
Variation in the time-varying current I.sub..phi. current, and thus in the 
combined current's direction of flow may be effected by an external power 
supply or by internal properties of the cylindrical surface 101, 102, 103, 
104, 105. Both of these approaches are discussed hereinbelow. 
Referring to FIG. 9, a graph is shown of compressive stress (hoop stress) 
induced by the constant current I.sub.z and the time-varying current 
I.sub..phi. through the cylindrical surface 101, 102, 103, 104, 105 
between time T.sub.1 and time T.sub.5. The maximum hoop stress occurs at 
time T.sub.3 when only the constant current I.sub.z is flowing through the 
cylindrical surface 103. 
Referring to FIGS. 10 through 22, various figures are provided relating to 
a rolled conductive sheet embodiment of the present invention, including 
figures depicting one example of a power supply configuration for 
generating an controlling the constant current I.sub.z and the 
time-varying current I.sub..phi. in the centerpost region. 
In principle this embodiment employs a long thin conductive sheet (which 
may be normally conductive, e.g., copper, or superconductive) having slits 
or cuts along its long edges to form cut outer legs. One or both sides of 
the conductive sheet are preferably insulated electrically. The conductive 
sheet is rolled to form a cylinder and the cut outer legs from one end of 
the cylinder are bent outward to meet corresponding legs from another end 
of the cylinder, forming a toroidal region around the cylinder and inside 
the cut outer legs. The cut outer legs are connected to external power 
supplies that control the voltage applied to each cut outer leg. Such 
configuration allows control of the orthogonal, axial, and toroidal 
electric fields in the conductive sheet in a centerpost region. The 
electric field in the conductive sheet causes electric currents to flow, 
which are both axial and toroidal in direction. The external power 
supplies can, by controlling these electric fields, control overall 
current direction and magnitude. The currents in turn produce orthogonal 
magnetic fields (axial and toroidal) that can be used in a device for 
magnetic coupling (by way of example, in a transformer configuration) or 
for magnetic field generation (by way of further example, for plasma 
confinement in a tokamak fusion reactor). 
Referring to FIG. 10, a plan view is shown of a conductive sheet 1000 
having cut outer legs 1002 along its edges, a centerpost region 1004 
thereinbetween, and an insulated surface 1006 for constructing a magnetic 
field coil, in accordance with one embodiment of the present invention. 
In FIG. 11 a perspective view is shown of the conductive sheet 1000 of FIG. 
10 having been rolled with its cut outer legs 1002 aligned. Note that 
while preferred, alignment of the cut outer legs is not required. And in 
FIG. 12, a cross-sectional elevation view is shown of a magnetic field 
coil (also referred to herein as a spiraled or coiled magnetic field coil) 
formed from the conductive sheet 1000 of FIG. 10 with corresponding pairs 
of cut outer legs 1002 at each of its ends bent to form loops 1200 of a 
toroidal field coil that allow connection to a power supply at a mid-plane 
joint 1202. 
In the embodiment of FIGS. 10, 11 and 12, a continuous thin sheet 1000 of 
conducting material is used to form the magnetic field coils in the 
centerpost region, as opposed to the use of two separate coils in the 
prior art (FIGS. 1 through 6). 
In one variation, current direction is controlled using external power 
supplies as shown schematically in FIGS. 20 through 22, as described 
below. In another variation, a resistive spiral section (either with a 
resistive strip or a thermally-induced) is formed at the centerpost region 
to control the current directions through the centerpost region. In these 
variations, above and below the centerpost region, the cut outer legs are 
cut similar to those in the above-described embodiment. 
Unlike on the conductive tubes in the embodiment described below, in which 
each of the cut outer legs of each conductive tube are the same width as 
the remaining cut outer legs on such conductive tube, the spacing between 
cuts along the edges of the conductive sheet 1000 is smaller on one end 
1008 of the sheet than on the other end 1010, increasing from the one end 
to the other end as a function of position. This increasing spacing 
accounts for an increasing radius, and thus circumference, that results 
when the conductive sheet is rolled to form the coiled magnetic field 
coil. The cut outer legs 1002 are spaced so that the cut outer legs 1002 
are aligned with one another, largely for structural reasons, when the 
conductive sheet is rolled into the coiled magnetic field coil. 
Once the conductive sheet 1000 has been rolled with the cut outer legs 1002 
properly aligned, the cut outer legs 1002 are brought together at a 
mid-plane joint 1202 and connected to a power supply (not shown), e.g., an 
external power supply such as shown in FIGS. 20 through 22. A thin layer 
of insulation 1006 of the conductive sheet 1000 is between each adjacent 
layer of the coiled magnetic field coil. 
Referring to FIG. 13, a plan view is shown of a variation of the conductive 
sheet 1300 of FIG. 10 having cut long outer legs 1302 along one edge, cut 
short outer legs 1303 along another edge, a centerpost region 1304 
thereinbetween, and an insulated surface 1306 for constructing a magnetic 
field coil, in accordance with a further embodiment of the present 
invention. In FIG. 14, a perspective view is shown of the conductive sheet 
1300 of FIG. 13 having been rolled with its cut outer legs 1302, 1303 
aligned. And in FIG. 15, a cross-sectional elevation view is shown of a 
magnetic field coil (also referred to herein as a spiraled or coiled 
magnetic field coil) formed from the conductive sheet 1300 of FIG. 14 with 
corresponding pairs of cut outer legs 1302, 1303 at each of its ends bent 
to form toroidal loops 1500 of the magnetic field coil connected to a 
current supply at a bottom joint 1502. In other respects this embodiment 
is similar to the embodiment shown in FIGS. 10 through 12. 
FIGS. 16 through 17 are schematic diagrams showing various current 
densities flowing in the centerpost region of the conductive sheets of 
FIGS. 10 through 15. 
Through each corresponding pair of cut outer legs, separate power supplies 
(such as shown in FIGS. 20 through 22) generate an axial or constant 
electric field E.sub.z which drives axial current density J.sub.z between 
the cut outer legs. Other separate power supplies (also shown, by way of 
example, in FIGS. 20 through 22) generate a toroidal electric field 
E.sub..phi. which drives toroidal or time-varying current density 
J.sub..phi.. The constant current density J.sub.z is depicted in FIG. 16 
as it flows through the centerpost region of the coiled magnetic field 
coil (which has been uncoiled for purposed of illustration). 
The time-varying current density J.sub..phi. is depicted in FIG. 17 as it 
flows through the centerpost region of the coiled magnetic field coil 
(which is depicted uncoiled) in a manner to produce toroidal current 
density in the centerpost region. 
With both the constant current density J.sub.z and the time-varying current 
density J.sub..phi. flowing, as is initially the case, the total current 
flowing through the centerpost region of the coiled magnetic field coil is 
approximately as depicted in FIG. 18. 
In operation in accordance with one particular embodiment, with the 
constant current density J.sub.z remaining constant, the time-varying 
current density J.sub..phi. is reduced until the time-varying current 
density J.sub..phi. is zero. When the time-varying current density 
J.sub..phi. is zero, only the constant current density J.sub.z is flowing, 
and thus the total current is approximately as depicted in FIG. 16. 
The polarity of the time-varying current I.sub..phi. is then reversed 
resulting in the approximate total current density flow as depicted in 
FIG. 19. 
In one variation, the configurations shown above in FIGS. 10 through 15 
require external power supplies configured such as shown in FIGS. 20 
through 22 to generate current flow as shown in FIGS. 16 through 19. 
Referring to FIG. 20, a schematic diagram is shown of a toroidal field 
power supply 2000 and an ohmic heating power supply 2002, for generating 
the axial electric field E.sub.z and the toroidal electric field 
E.sub..phi. that generates the combined current density flow shown in FIG. 
18. 
Referring to FIG. 21, a schematic diagram is shown of the toroidal field 
power supply 2000, for generating the constant current density J.sub.z 
shown in FIG. 16. 
Referring to FIG. 22, a schematic diagram is shown of a configuration of 
the toroidal field power supply 2000 and the ohmic heating power supply 
2000, for generating the axial or constant electric field E.sub.z and the 
toroidal or time-varying electric field E.sub..phi., having reversed its 
direction from that of FIG. 20, that form the combined current density 
flow pattern shown in FIG. 19. 
Referring to FIGS. 23 through 41, various figures are provided of a 
conductive tube embodiment of the present invention. Numerous variations 
of such embodiment (which are also applicable to the rolled conductive 
sheet embodiment described above) are presented including, a resistive 
strip variation, and a thermally-induced resistive strip variation (or 
thermal strip variation), both of which may be implemented using normally 
conducting materials, superconductive materials, or a combination thereof. 
In a resistive strip variation, many thin concentric conductive tubes (made 
from normally conducting material, e.g., copper, or superconductive 
material) are employed with ends cut to form cut outer legs. A region 
between the cut outer legs forms a centerpost region and contains a 
helical electrical resistive strip that changes in resistivity relative to 
a base material as a function of temperature. 
One or both sides of the conductive tubes are electrically insulated so 
that the tubes are electrically insulated from one another. The cut outer 
legs from one end of the conductive tube are bent outwardly to meet cut 
outer legs from another end of an adjacent conductive tube, which are 
similarly bent, forming a toroidal region outside the centerpost region 
and inside the cut outer legs having been bent. This arrangement produces 
a multi-turn coil consisting of conductive cylinders connected 
electrically at their cut outer legs in series. An external power supply 
is connected between an unconnected cut outer leg from an inner most 
conductive tube and an unconnected cut outer leg from an outer most 
conductive tube. This arrangement can be used to produce an axial electric 
field in the centerpost regions of the concentric conductive tubes that 
drives current axially in the centerpost regions. The helical resistive 
strip causes the axial current to be redirected onto a spiral path. 
If cryogenic temperatures are used, and the base material is pure copper, 
the resistive strip may be, for example, Beryllium (Be) doped copper, 
which is 10 to 50 times more resistive than pure copper at cryogenic 
temperatures. In this arrangement, the current is redirected from an axial 
flow into a helical path by the spiral resistive strip. As a result, a 
toroidal current component is introduced. 
If the centerpost region is heated to room temperature, the resistivity of 
the Be doped copper is less than twice the resistivity of the pure copper. 
Under this circumstance, the current flows substantially axially, with 
only a slight toroidal component. 
In this way, current direction in the centerpost region is controlled, 
i.e., varied from having a large toroidal component to having a very small 
or zero toroidal component such as shown in FIGS. 7 and 8 from times 
T.sub.1 to T.sub.3. 
In a thermally-induced resistive strip variation (described beginning with 
FIG. 34), an arrangement similar to the resistive strip variation, 
described above, is used, except instead of the resistive strip, a thermal 
gradient within the base material is used to create a thermal-spiral strip 
of greater resistivity than the base material. A helical coolant channel 
(or heating element), for example, can be fabricated into the centerpost 
region of the conductive tube. Coolant in the coolant channel, or thermal 
energy from the heating element, causes the thermal gradient in the base 
material, which in turn produces a helical resistive barrier to the axial 
current flow, thus redirecting the axial current along a helical path, and 
creating a toroidal current component. 
At uniform temperature, i.e., with coolant flow stopped, or the heating 
coil turned off, the centerpost region has a substantially uniform 
resistivity, resulting in substantially axial current flow, i.e., 
approximately zero toroidal component. 
In this way, current flow is controlled in the centerpost region in 
accordance with the sequence shown in FIGS. 7 and 8 from times T.sub.1 to 
T.sub.3. 
Either of the resistive strip and thermal-spiral strip embodiments can be 
produced using superconductive material as the base material. An increase 
in temperative along a helical path will produce a normally conducting 
helical strip in the superconducting base material, thus causing an axial 
current flow to follow a helical path, and thus to have a toroidal 
component. When the normally conducting helical strip is brought back to 
below the critical temperature for the base material, the normally 
conducting helical strip becomes superconducting, and substantially only 
axial current flow is observed. 
Each of these variations is described in further detail below. 
Referring to FIG. 23, a perspective view is shown of a conductive tube 2300 
having a spiral resistive strip 2302 in its centerpost region 2304 and 
having cut outer legs 2306 at each of its ends, in accordance an 
embodiment of the present invention. The centerpost region 2304 is formed 
in the conductive tube 2300 as a cylindrical conductor region that is 
mechanically and electrically coupled to the cut outer legs. 
Referring to FIG. 24, a cross-sectional side view is shown of the 
conductive tube 2300 of the embodiment of FIG. 23, with corresponding 
pairs of cut outer legs 2306 at each end of the conductive tube 2300 bent 
to form loops 2400 of a toroidal field coil. The cut outer legs are 
conveniently cut and bent to form a torus, as shown. 
Referring to FIG. 25 a cross-sectional elevation view is shown of several 
concentrically aligned conductive tubes 2500, 2502, 2504, 2506, 2508, 
2510, such as the conductive tube 2300 in FIGS. 23 and 24, that together 
form a magnetic field coil 2512, for carrying the constant current I.sub.z 
and the time-varying current I.sub..phi. that produce a toroidal magnetic 
field B.sub..phi. and a poloidal magnetic field B.sub..theta. or (B.sub.z) 
in accordance with an embodiment of the present invention. An advantageous 
geometry is shown for the resistive spiral centerpost region variation in 
which conductive centerpost regions of tubes 2500, 2502, 2504, 2506, 2508, 
2510 (or coaxial cylinders 2500, 2502, 2504, 2506, 2508) are connected in 
a series configuration by looping cut outer legs 2514, 2516, 2518, 2520, 
2522, 2524, 2526, 2528, 2530, 2532, 2536. 
In TFR applications, the looped cut outer legs encompass the toroidal 
vacuum vessel 21 (FIG. 1). 
Between each adjacent concentric tube 2500, 2502, 2504, 2506, 2508, 2510, 
and respective looped cut outer legs 2514, 2516, 2518, 2520, 2522, 2524, 
2526, 2528, 2530, 2532, 2534, 2536 is an electrical insulator 2538, 2540, 
2542, 2546, 2548. A cut outer leg 2516 from, e.g., a lower end of one 
concentric tube 2510 is connected to a corresponding cut outer leg 2510 
from an upper end of an adjacent concentric tube 2508. This configuration 
is repeated for each concentric tube 2500, 2502, 2504, 2506, 2508, 2510 to 
effectively form a multiple-turn coil 2512. Current is supplied at an 
unconnected cut outer leg 2514 at an upper end of the outermost conductive 
tube 2500, which serves as a first current supply terminal. (By 
"unconnected" it is meant that such cut outer leg 2514 is not connected to 
a cut outer leg from, e.g., a lower end of an adjacent concentric tube 
2502). In operation, the current follows in a conductive path from the 
unconnected outer leg 2514 through successive layers of the conductive 
tubes 2500, 2502, 2504, 2506, 2508, 2510 until it reaches an unconnected 
inner leg 2536 at the centermost conductive tube 2510. (By "unconnected" 
it is meant that such outer leg 2536 is not connected to an outer leg 
from, e.g., an upper end of the adjacent concentric tube 2508). The 
unconnected outer leg 2536 at the centermost conductive tube 2510 serves 
as a second current supply terminal. 
The cylinders may be configured to allow for coolant channels and to allow 
for the grading of conductive areas to maintain a constant cross sectional 
area for current flow. 
The magnetic coil 114 of the present embodiment is superior to existing 
wedged toroidal field coils used, for example, in TFR's because the 
present embodiment uses the conductive tubes 2500, 2502, 2504, 2506, 2508, 
2510, which are axisymmetric (concentric) within their centerpost region 
2304 and cut outer legs 2514, 2516, 2518, 2520, 2522, 2524, 2526, 2528, 
2530, 2532, 2534, 2436 to form a torus outside the centerpost regions. In 
contrast, conventional wedged toroidal field coils are segmented in a 
toroidal direction, i.e., employ alternating wedges of insulation and 
conductor. Since, in any case, each turn must be insulated, the present 
approach offers a significant advantage in that (by employing axisymmetric 
"turns") insulation is placed between conductive tubes 2500, 2502, 2504, 
2506, 2508, 2510 and in this direction of low stress (i.e., the radial 
direction). In contrast, in conventional wedged toriodal field coils, the 
wedges of insulation are placed across the high stress region, bearing the 
brunt of the toroidal compressive stress (hoop stress). Thus, the wedge 
formation used in the wedged toroidal field coils of existing TFR's is not 
generally the most desirable mechanical or thermal geometry, but it is 
used because multiple toroidal field turns are needed to form the TFR's 
toroidal geometry. Use of conductive tubes 2500, 2502, 2504, 2506, 2508, 
2510 in the present embodiment allows for multiple toroidal field turns 
without requiring the insulation layers to bear the toroidal compressive 
stress (hoop stress). 
The present embodiment is also advantageous because the conductive tubes 
2500, 2502, 2504, 2506, 2508, 2510 can be made of multiple thin sheets of 
material (formed into the conductive tubes 2500, 2502, 2504, 2506, 2508, 
2510), which are generally of higher strength than a similar amount of the 
same material in a thicker form. Furthermore, grading of the conductive 
tubes 2500, 2502, 2504, 2506, 2508, 2510 in a radial direction is natural 
in the present configuration by varying the thickness of adjacent tubes in 
order to assure consistent conductor cross-section throughout the device. 
And, the material properties of each conductive tube 2500, 2502, 2504, 
2506, 2508, 2510 can be tailored in accordance with stresses and thermal 
requirements, both of which can vary as a function of the conductive 
tubes' radii. 
A superconducting coil variation of the present embodiment may be employed 
in which a normally conducting material, e.g., Copper, of the conductive 
tubes 2500, 2502, 2504, 2506, 2508, 2510 is replaced by superconductive 
material, e.g., Niobium-Tin, as described hereinbelow. This 
superconducting material can be made to change the direction of total 
current flow in the centerpost region 2304 similar manners to those in 
which the approaches described herein employing normally conducting 
material change the direction of total current flow. Thus the features of 
the present embodiment work equally well for normally conducting materials 
and for superconducting materials, are easily scalable so as to 
accommodate various sizes of TFR's, and for other applications to which 
the present embodiment can be put. 
Another advantage of the present embodiment is that electrical connections 
are made in the outside areas of the device where relatively low 
mechanical stresses are present, as compared to, for example, the 
centerpost region 2304. 
Also, in TFR applications, the field coils can be made demountable, whereas 
conventional wedged toroidal field coils are constructed as a closed torus 
around the vacuum vessel. Advantageously, the present embodiment allows 
the vacuum vessel to be placed over the centerpost region before the cut 
outer legs 2514, 2518, 2522, 2526, 2530 are bent outward over the vacuum 
vessel and connected to respective other cut outer legs 2516, 2520, 2524, 
2528, 2532, 2536 at their outer extremes. This assembly approach vastly 
improves over conventional wedged toroidal field coils used with TFR's in 
which elaborate and less reliable joints must be fabricated to allow 
insertion of the vacuum vessel. 
Additionally, the present embodiment advantageously provides insulator 
design flexibility because the insulators 2538, 2540, 2542, 2544, 2546, 
2548 are aligned in a direction of minimum stress, i.e., the radial 
direction, and because the stress on the insulators may be significantly 
reduced by configuring the conductive tubes 2500, 2502, 2504, 2506, 2508, 
2510 with sufficient hoop strength to reduce or eliminate the radial 
stresses that are present on the electrical insulators 2538, 2540, 2542, 
2544, 2546, 2548. 
In addition, the insulators 2538, 2540, 2542, 2544, 2546, 2548 in the 
present approach encounter no edges in the centerpost region 2304 that 
inherently result in large electric field density, and create a risk of 
electrical breakdown in the insulators. 
The present embodiment also allows coolant channels to be placed on the 
outside of mating tubes during fabrication of the conductive tubes 2500, 
2502, 2504, 2506, 2508, 2510. Joining of two mating tubes can produce a 
water tight coolant channel. 
Referring to FIG. 26, a schematic view is shown of an approach for use in 
coupling together the cut outer legs 2514, 2516, 2518, 2520, 2522, 2524, 
2526, 2528, 2530, 2532, 2534, 2536 of the conductive tubes 2500, 2502, 
2504, 2506, 2508, 2510 of FIG. 25. Cut outer legs 2516, 2520, 2524, 2528, 
2532 from the lower portion of the conductive tubes 2500, 2502, 2504, 
2506, 2508, 2510 are mechanically and electrically connected to cut outer 
legs from 2518, 2522, 2526, 2530, 2534, 2536, the upper allowing current 
(depicted with a dashed line) to flow through adjacent conductive tubes 
placing each tube in series with remaining tubes. The inside and outside 
cut outer legs 2514, 2536 not connected to cut outer legs from adjacent 
conductive tubes but are connected to a power supply 2600. 
Referring to FIG. 27, a schematic diagram is shown of the resistive spiral 
centerpost region 2700 of the embodiment of the conductive tube 2300 of 
FIG. 23. The operation of the resistive spiral centerpost region in the 
conductive tube embodiment is described. Shown is a pair of spiral 
resistive strips 2302, pure copper regions interposed between the two 
resistive spiral strips, a resistive spiral strip pitch angle y, and a 
current density pith angle .gamma..sub.J. Also shown is a lower voltage 
V.sub.1, an upper voltage V.sub.2, a length L, a radius R, and a major 
axis Z. These parameters are used in modeling currents though the 
resistive spiral centerpost region. 
An axial electric field E.sub.z is produced by external means, with it's 
magnitude given by 
EQU [(V.sub.2 -V.sub.1)/L]=E.sub.z 
(Note that use of the cut outer legs described above is not essential to 
the workings of the centerpost region 2700, but rather merely serve, in a 
basic embodiment, as a series current path. It is contemplated that any 
appropriate current path can be used in combination with the described 
centerpost region in order to achieve the current control features 
described herein, i.e., in order to control the direction of current flow 
in the centerpost region.) 
FIG. 28 (along with FIG. 27) is a schematic diagram of current density 
pitch angle .gamma..sub.J relative to physical resistive spiral strip 
pitch angle .gamma. of spiral resistive strip(s) 2302 on the centerpost 
region of the conductive tube of FIG. 23. The spiral resistive strips 2302 
can be used to control the current density pitch angle .gamma..sub.j, 
allowing for variation of the time-varying current I.sub..phi. (described 
above), while maintaining the constant current I.sub.z, a thrust allowing 
for variation in total current density J. .gamma. and .gamma..sub.J can be 
related to the local material thicknesses of the conductive region and the 
resistive region. 
Referring to FIG. 29, a schematic diagram is shown of a spiral wound 
resistive layer in the conductive tube of FIG. 23 with graphical 
definitions of parameters used in analyzing current flow through the 
conductive tube. x and y are orthogonal Cartesian coordinate axis in a 
plane parallel and perpendicular to the local resistive strip. The 
conductive region 1 has width W.sub.1 and conductivity .sigma..sub.1. The 
resistive strip region 2 has width W.sub.2 and conductivity .sigma..sub.2. 
Local electric field E.sub.x and E.sub.y and current density J.sub.x and 
J.sub.y are shown and related to their respective components in a Z-.phi. 
plane. 
The geometry and parameters shown in FIGS. 27 through 29 are used below to 
define the current switching capability of the spiral resistive strip. 
To study the magnetic fields generated by the currents flowing through the 
centerpost region of a conductive tube, initially, assume that the pitch 
of the spiral conductive path cannot be changed, i.e., the pitch of the 
conductive region between resistive spirals. Further assume that a 
constant difference in electrical potential (i.e., voltage) is applied 
between the bottom of the centerpost region of the conductive tube and the 
top of the centerpost region, which drives current, preferably in an axial 
direction. 
The spiral resistive strips causes the current to flow in a helical 
(spiral) path due to the resistance (lack of conductivity) in the spiral 
resistive strips. In this configuration, geometrical pitch .beta. and 
physical pitch angle y are related by the formula .beta.=tan .gamma.. The 
spiral resistive strips, however, are not pure insulators and, as shown in 
FIG. 28, the current density pitch angle .gamma..sub.1 is less than the 
geometrical pitch angle .gamma. because some current "leaks" through the 
spiral resistive strips. Thus, the current density pitch .beta..sub.J is 
related to the current density pitch angle .gamma..sub.J by the formula: 
.beta..sub.J =tan .gamma..sub.J =J.sub..phi. /J.sub.z ; where J.sub..phi. 
and J.sub.z are the current densities in the toroidal and axial 
directions, respectively. The local (total) current density J in the 
centerpost region is given by the following formula: 
EQU J=J.sub.z (1+.beta..sub.J.sup.2).sup.1/2. 
Because of the current that leaks through the spiral resistive strips, the 
current density pitch .beta..sub.J is less than the geometrical pitch 
.beta. of the spiral resistive strips. The current density pitch 
.beta..sub.J is related to (1) a spiral resistive strip to conductive 
region width ratio W.sub.2 /W.sub.1, where W.sub.1 is the width of the 
conductive regions, and W.sub.2 is the width of the spiral resistive 
strips; (2) a material conductivity ratio .rho..sub.2 /.sigma..sub.1 
between the conductive regions, e.g., pure copper, and the resistive 
region, where .sigma..sub.1 is conductance in the conductive region, and 
.sigma..sub.2 is conductance in the resistive regions; and (3) the 
geometrical pitch .beta.. 
Thus, the current density pitch .beta..sub.J is represented by the formula: 
.beta..sub.J =J.sub..phi. /J.sub.z =tan .gamma..sub.J =.beta.G; where G=G 
(.beta.; W.sub.2 /W.sub.1 ; .sigma..sub.2 /.sigma..sub.1).ltoreq.1. For a 
thin conductive tube, and neglecting the effects of current entry at the 
ends of the conductive tube, the dimensionless parameter G can be 
expressed as: 
##EQU1## 
The expression for the current density pitch angle has a maximum value of: 
##EQU2## 
The geometrical pitch at this maximum current pitch angle .beta..sub.Jmax 
is defined as: 
##EQU3## 
Referring to FIG. 30, a graph is shown of resistivity versus temperature 
for copper of differing purities and dopings, such as may be used in the 
conductive tube of FIGS. 23 and 24. The resistive spiral centerpost region 
of the conductive tube of FIGS. 23 and 24 is formed by a relatively high 
resistance spiral resistive strip 2302 around the centerpost region of the 
conductive tube. The resistive spiral centerpost region is produced by 
forming a thin spiral strip having a relatively high resistivity, which in 
some variations includes a base material of high purity copper or a 
similar high-conductivity material. The spiral resistive strip can be 
formed by adding an impurity to the base material during fabrication, ion 
implantation of impurities, or residual stress introduced during 
fabrication. The embodiment of FIGS. 23 and 24 takes advantage of the fact 
that, at cryogenic temperatures, copper with even a small amount of an 
impurity or residual strain has a much higher electrical resistivity than 
pure copper. The resistivity of copper with various levels of impurities 
is shown in FIG. 30. 
Referring to FIG. 31, a graph is shown of variation in current density 
pitch angle versus resistive strip pitch angle, at a resistive region 
width to conductive region width ratio (W.sub.2 /W.sub.1) of 0.01, for a 
number of different material conductivity ratios .sigma..sub.2 
/.sigma..sub.1, useful in analyzing current flow through a variation of 
the conductive tube of FIGS. 23 and 24. Peaks in different conductivity 
ratio curves of FIG. 35 represent optimal geometrical pitch angles for 
given spiral resistive strip to conductive region ratios .sigma..sub.2 
/.sigma..sub.1. Even for a relatively low strip-thickness ratio of, for 
example, 0.01, relatively high current density pitch angle is achieved by 
using a relatively high conductivity ratio. (At an infinite conductivity 
ratio, i.e., zero conductivity in the resistive strips, or zero resistance 
in the pure copper portions of the centerpost region as is the case for a 
superconductor, the current density pitch angle is the same as the 
physical pitch of the strip.) Referring to FIG. 32, a graph is shown of 
maximum current density pitch angle versus conductivity ratio 
.sigma..sub.2 /.sigma..sub.1 for different spiral resistive strip to 
conductive region width ratios W.sub.2 /W.sub.1, useful in analyzing 
current flow through a variation of the conductive tube of FIGS. 23 and 
24. For the centerpost region of a magnetic field coil used, e.g., in a 
TFR, the geometrical pitch angle is selected to maximize the current 
density pitch angle after selecting the thickness ratio and the 
conductivity ratio. 
The following is a description of the major design parameters and a 
comparison of the use of the invention when compared to prior art in a 
TFR. 
The toroidal magnetic field B.sub..phi. at a plasma center radius R.sub.o 
is: B.sub..phi.o =.mu..sub.o J.sub.z [R.sub.+.sup.2 -R.sub.-.sup.2 
]/2R.sub.o ; where R.sub.- and R.sub.+ are the inner radius of the 
centerpost and the outer radius of the centerpost, respectively. The 
poloidal magnetic flux .PHI..sub.z is: .PHI..sub.z =2 .pi.B.sub..phi.o 
.beta..sub.J [R.sub.+.sup.3 -R.sub.-.sup.3 ]/3[R.sub.+.sup.2 
-R.sub.-.sup.2 ]. 
The Ampere-turn ratio (NI) for a conductive tube of length L and radius R 
is: .beta..sub.I =NI.sub..phi. /NI.sub.z =.beta..sub.J L/2 .pi.R. 
Use the above formula, the current density pitch angle variation can be 
calculated when the conductive tube of FIGS. 23 and 24 has spiral 
resistive strips as shown in FIG. 27, and when the conductive tube is 
operated between two different temperatures. 
Referring to FIG. 33, a graph is shown of conductivity versus temperature 
for an impurity controlled (Beryllium (Be) doped) resistive strip showing 
different operating points of a variation of the conductive tube of FIGS. 
25 and 26. To compare the performance of a heretofore known DIII-D TFR, 
such as shown in FIG. 1, with a TFR having a magnetic field coil based on 
the embodiments described herein, the existing B-coil 26 (FIG. 1) and 
E-coil 28 (FIG. 1) are replaced with the magnetic field coil of the 
present embodiment. The cylindrical surfaces are formed using 1/2 hard 
OHFC copper (material 1) with a Beryllium (Be) spiral strip (material 2) 
of varying thickness ratio. FIG. 43 shows the resistivity at two operating 
points of the coil. The first point is cryogenic operation at 30 K as 
indicated by points A1 and A2 for materials 1 and 2, respectively. The 
second point is room temperature operation at 400 K, as indicated by 
points B1 and B2. The performance of this embodiment of the invention is 
compared with the performance of an existing B-coil/E-coil based on the 
generated toroidal field B.sub..phi.o a center radius R.sub.o of the 
plasma and a magnetic flux capability .DELTA..PHI..sub.z. 
By way of example, for a heretofore known TFR such as the TFR produced by 
General Atomics of San Diego, and referred to as the DIII-D TFR, the 
center post coils have the following parameters: 
##EQU4## 
A centerpost region using conductive tubes in accordance with one 
embodiment described herein has the following parameters: 
R.sub.- =0.250 m; R.sub.+ =0.787 m 
J.sub.z =18.4 MA/m.sup.2 
NI=32.2 MA 
B.sub..phi.o =3.85 T @ R.sub.o =1.69 m 
.sigma..sub.2 /.sigma..sub.1 =0.00875 @ T=30 K 
.sigma..sub.2 /.sigma..sub.1 =0.309 @ T=400 K 
Exemplary flux output comparison results for a magnetic field coil in 
accordance with one variation of the embodiments described herein based on 
the geometry of the DIII-D TFR are shown in Table 1 below: 
TABLE 1 
______________________________________ 
W.sub.2 /W.sub.1 0.10 
.gamma..sub.max 72.7 
.gamma..sub.Jmax @ 30.degree. K. 
55.4 
.gamma..sub.J @ 400.degree. K. 
2.06 
.DELTA..beta..sub.J 1.41 
Flux: Invention 16.3 V.sub.S 
Flux: DIII-D 10 V.sub.S (.+-.I.sub.swing) 
______________________________________ 
The flux output is shown to be 63% higher for the same toroidal field in 
both cases and shows the performance of the magnetic field coil of the 
present embodiments is superior to conventional B-coil/E-coil approaches. 
The present embodiments are even more advantageous for low aspect ratio 
(LAR) devices, which have no room for poloidal flux generation and in 
which any flux output from a centerpost coil is an improvement. (Aspect 
ratio is defined as a ratio of the plasma's major radius R.sub.0 to its 
minor radius, a or AR=R.sup.0 /a). 
In another embodiment of the invention a resistive spiral strip can be 
induced in the centerpost region by heating a thermal spiral strip in a 
pure material conductive tube, such as made from copper, to higher 
temperature. Referring to FIG. 34, a perspective view is shown of an 
assembly of a centerpost region 3400 of first and second tubes 3401, 3402 
of a conductive tube 2403. The first tube 3401 includes one or more 
spiraled channels 3404 for use in heating or cooling the conductive tube 
3403 along a spiral path. These channels 1304 can be used to produce a hot 
helical strip in the centerpost region 3400 to cause a more resistive 
region than in the remaining base material, causing the current to flow in 
a spiral similar to the previously described spiral resistive strip 
embodiment in which different materials with different resistivities are 
used to produce the spiral current flow. 
Referring to FIG. 35, a perspective view is shown of a partially assembled 
centerpost region 3400 of the conductive tube 3403 of FIG. 34 
incorporating the one or more spiraled channels 3404 for use in heating or 
cooling. In practice, as shown, the second tube 3402 is placed over the 
first tube 3401 and the assembly brazed or otherwise assembled. 
Referring to FIG. 36, a partial perspective view is shown of a portion of 
the first tube 3401 of FIGS. 34 and 35 with a heating element 3600 in one 
of the channels 3404. This heating element 3600 allows heating of a thin 
spiral strip on the conducting tube 3403 creating a barrier to flow in a 
purely axial direction. 
Referring to FIG. 37, a partial perspective view is shown of a portion of 
the first tube 3401 of FIG. 35 with a cooling water conduit 3700 in one of 
the channels 3401. This coolant water channel 3700 can carry hot fluid to 
produce similar results to the heating element or can carry cooling fluid 
to achieve an opposite result. 
Referring to FIG. 38, a schematic diagram is shown of geometry for another 
embodiment employing spiral thermal gradients to produce current pitch 
angle switching. A spiral-thermal strip 3800 that creates a 
temperature-induced resistive barrier to current flow, in accordance with 
a variation of the conductive tube of FIGS. 25 and 26 is shown. In 
accordance with the variation shown, the spiral-thermal strip 3800 on the 
centerpost region is generated using thermal gradients in a pure copper 
conductive tube as shown for example in FIGS. 34 and 35. 
The spiral thermal strip 3800 can be induced in the pure copper (or 
superconductor) using a spiral wound cooling channel or a spiral wound 
heating element such as shown in FIGS. 34 through 37. This concept is much 
simpler to implement than other embodiments described herein because the 
basic conductive surface is pure copper and only an adjacent layer, i.e., 
coolant tubes or heating elements, have a spiral configuration. 
Current flow analysis of this spiral thermal strip embodiment is more 
complex than for the spiral resistive strip described above, because of 
the time dependency of thermal diffusion. In the present embodiment, heat 
is introduced along an infinitesimally small spiral strip at time t=0. The 
spiral thermal path is denoted a "wall" boundary condition as is 
conventional in thermal diffusion problems. Initially, the wall 
temperature is T.sub.w and the conductive tube temperature is T.sub.o. A 
quantity of heat Q.sub.o is input along the spiral strip with a pitch 
angle of .gamma.. Constant heat input is assumed which corresponds to a 
constant resistive heat input from an external heating element. 
At time t=t.sub.o, temperature is at a predetermined maximum value at the 
"wall" and monotonically decreases away from the wall. At this time, 
supply voltage is applied to the conductive tube and current 
preferentially flows in the low-temperature spiral region and not through 
the higher temperature spiral strip of the thermal wall. However, the heat 
input along the thermal wall eventually diffuses into the low-temperature 
copper, and as the low-temperature copper heats up, the conductivity of 
the two regions becomes similar. The decreased conductivity, or increased 
resistivity, of the conductive tube's copper, in combination with the 
large currents through the conductive tube, results in additional ohmic 
heating (I.sup.2 R heating) of the conductive tube. Eventually, the ohmic 
heating heats the entire conductive tube until the conductive tube's 
temperature reaches a uniform equilibrium. With the entire conductive tube 
at an equilibrium, the current flows in the axial direction only. 
The problem can be modeled using a one dimensional transient heat 
conduction equation. Numerical integration of this equation results in an 
expression for the current density pitch ratio of: 
##EQU5## 
where the dimensionless conductive tube radius ratio is: 
##EQU6## 
and the conductivity ratio .sigma..sub.o /.sigma..sub.1 is the ratio of 
initial conductivity to final conductivity. N is the number of spiral 
layers, .alpha. is the thermal diffusivity of the material and t is time. 
Although a number of free parameters exist in the equations for the 
above-described thermally induced current path embodiment, a universal set 
of curves can be established for a pure copper cylinder based on 
dimensionless numbers: geometrical pitch, temperature ratio, and radius 
normalized by a diffusion length scale. 
Referring to FIG. 39, a graph is shown of resistivity versus temperature 
for pure copper, such as may be used in the conductive tube of FIGS. 23 
and 24, showing different operating points. Because the resistivity of 
copper is highly dependent on temperature, as shown, the spiral thermal 
region shown in FIG. 38 induces current to flow in a spiral path between 
spiral-thermal strips. 
Referring to FIG. 40, a graph is shown of current density pitch angle 
versus thermal strip pitch angle, at various conductive tube radius 
parameters, for a thermally-induced spiral resistive layer in a pure 
copper conductive tube, such as may be used in the conductive tube of 
FIGS. 34 and 35, operating between an initial temperature of 30 K and a 
final temperature of 400 K. 
FIG. 40 shows numerically generated solutions to the thermal equations 
based on operating copper between 30 K and 400 K. As before, the current 
density pitch is less than the geometrical pitch and the maximum current 
density pitch angle for each radius parameter is a unique function of the 
temperature range specified. The maximum pitch angle for each radius 
parameter in FIG. 40 can be determined by the functional minimization of 
the governing equation. 
Referring to FIG. 41, a graph is shown of maximum current density pitch 
angle versus initial temperature for a pure copper conductive tube, such 
as may be used in the conductive tube of FIGS. 34 and 35, that has a 
thermally-induced spiral resistive layer and that is heated to a final 
temperature of 400 K using a constant heat input. The maximum current 
density pitch angle depends very strongly on initial temperatures in the 
range between 10 K and 100 K. 
As with the resistive path embodiment, the overall performance of the 
thermal path embodiment as shown substantially in FIG. 38 is predicted 
using the above equations based on a design that is within, for example, 
the DIII-D TFR geometry. The parameters needed to determine the maximum 
performance are provided by FIG. 41. The calculated results are shown in 
Table 2 below. Using an initial start temperature T.sub.o of 30 K, the 
magnet flux output is greater than nominal DIII-D TFR magnetic flux values 
by 46%. As mentioned before, use of this embodiment in a low aspect ratio 
design would essentially provide poloidal flux with little or no decrease 
in overall toroidal field performance. 
Table 2: Comparisons flux output for thermal gradient spiral concept based 
on DIII-D TFR geometry operating between T=30 K and 400 K showing a 46% 
improvement in flux output. 
______________________________________ 
Initial Temperature T.sub.0 = 30 K 
Final Temperature T.sub.1 = 400 K 
.gamma..sub.max 70.8 
.gamma..sub.J max 51.7 
.gamma..sub.J @ 400K 0 
.DELTA..beta..sub.j 1.26 
Flux: .DELTA..PHI. 14.6 V.sub.S 
Flux DIII-D 10 V.sub.S 
______________________________________ 
In another embodiment, the conductive surface is formed of superconducting 
materials. An advantage of using superconducting materials is that the 
conductivity ratio is infinite between the two regions of differing 
conductivity and, therefore, the geometrical pitch and current density 
pitch will be close to equal. In this embodiment the resistive spiral is 
formed of a superconductor having a higher critical temperature than a 
base superconductor's critical temperature. At high temperatures, e.g., 10 
K, the spiral superconductor is in a normal conducting state while the 
base region (at 4.7 K) is a superconducting state. After the current is 
flowing through the superconducting portion of the conductive surface to 
generate both the poloidal magnetic field B.sub..theta. and the toroidal 
magnetic field B.sub..phi., the normally conducting strips's temperature 
is lowered, which causes the normal conducting zone between spirals of the 
spiral superconductor to also become superconductive such that the current 
through the conductive tube switches from a spiral direction to an axial 
direction. Thus, the conductive tube's temperature is decreased with time 
rather then increased. Alternatively, heat may be introduced into a strip 
on a superconducting tube. Local increase in temperature cause the strip 
to become normal conducting and thus change the current direction from 
axial to spiral. 
In the foregoing description, a single directed pitch is used to describe a 
technique for directing current from toroidal to axial. Using a 2nd pitch 
with opposite direction, the current can be made to traverse from a 
positive toroidal direction, to cross through zero and to reverse 
direction as described in the time sequence shown in FIGS. 7 and 8. 
While the invention herein disclosed has been described by means of 
specific embodiments and applications thereof, numerous modifications and 
variations could be made thereto by those skilled in the art without 
departing from the scope of the invention set forth in the claims. 
For example, magnetically soft material (like pure iron) or magnetically 
hard materials (such as permanent magnets of carbon steel) can be used in 
regions of magnetic field production, such as in a transformer, to improve 
performance. Magnetic material can be inside the centerpost region, inside 
the toroidal region created by the bent cut outer legs, and connecting 
these regions to enhance the transfer of magnetic energy from and between 
these regions. 
Other embodiments of the present invention include a variable transformer 
in which voltage in different cut outer legs are used for voltage increase 
or decrease depending on the electrical connections at peripheral 
connections of the variable transformer, or through modifications of these 
connections, or through modifications if a helical resistive strip in the 
centerpost region of such device. 
By way of further example, an energy transfer device embodiment allows 
magnetic energy to be stored in a toroidal field region to be transferred 
to a primarily cylindrical field region for use, for example, in a rail 
gun to accelerate a projectile in the cylindrical field from energy stored 
in the toroidal field. A converse flow of energy can also be employed to 
transfer energy from a cylindrical field region to a toroidal field 
region, such as for confining plasma in a tokamak fusion reactor. 
In yet a further example, a variable inductor device embodiment features a 
variable inductance controllable by connection of the cut outer legs 
through appropriate switching or through modification of the helical 
resistive strip in the centerpost region to change both resistance and 
inductance of such device. 
As in an additional example, an energy storage device embodiment uses a 
single conductive tube with a 45 degree resistive strip pitch angle in the 
centerpost region to store magnetic energy in both toroidal and axial 
regions of space as for example in the storage of electrical energy in a 
superconducting magnetic energy storage (SMES) device.