Tokamak reactors consist of a toroidal vacuum tube surrounded by a series of magnets. One set of magnets is logically wired in a series of rings around the outside of the tube, but are physically connected through a common conductor in the center. The central column is also normally used to house the
solenoid that forms the inductive loop for the ohmic heating system (and pinch current). The canonical example of the design can be seen in the small tabletop ST device made at Flinders University, which uses a central column made of copper wire wound into a solenoid, return bars for the toroidal field made of vertical copper wires, and a metal ring connecting the two and providing mechanical support to the structure.
Stability Advances in plasma physics in the 1970s and 80s led to a much stronger understanding of stability issues, and this developed into a series of "scaling laws" that can be used to quickly determine rough operational numbers across a wide variety of systems. In particular, Troyon's work on the critical beta of a reactor design is considered one of the great advances in modern plasma physics. Troyon's work provides a beta limit where operational reactors will start to see significant instabilities, and demonstrates how this limit scales with size, layout, magnetic field and current in the plasma. However, Troyon's work did not consider extreme aspect ratios, work that was later carried out by a group at the
Princeton Plasma Physics Laboratory. This starts with a development of a useful beta for a highly asymmetric volume: : \beta=\frac{\mu_{0}p}{\langle B^2\rangle}. Where \langle B^2\rangle is the volume averaged magnetic field \scriptstyle\langle B^2\rangle = \langle B_{\theta}^2 + B_{\rho}^2\rangle (as opposed to Troyon's use of the field in the vacuum outside the plasma, \scriptstyle B_0). Following Freidberg, this beta is then fed into a modified version of the
safety factor: : q_\star= \frac{2\pi B_0 a^2}{\mu_0 R_0 I} \left( \frac{1+\kappa^2}{2} \right). Where \scriptstyle B_0 is the vacuum magnetic field, a is the minor radius, \scriptstyle R the major radius, \scriptstyle I the plasma current, and \scriptstyle \kappa the elongation. In this definition it should be clear that decreasing aspect ratio, \scriptstyle R/a, leads to higher average safety factors. These definitions allowed the Princeton group to develop a more flexible version of Troyon's critical beta: : \beta_\text{crit} = 5\langle B_N\rangle \left( \frac{1+\kappa^2}{2}\right) \frac{\epsilon}{q_\star}. Where \epsilon is the inverse aspect ratio 1/A and \langle B_N\rangle is a constant scaling factor that is about 0.03 for any q_\star greater than 2. Note that the critical beta scales with aspect ratio, although not directly, because q_\star also includes aspect ratio factors. Numerically, it can be shown that \beta_\text{crit} is maximized for: : q_\star = 1 + \left(\frac{3}{4}\right)^{4/5} \approx 1.8. Using this in the critical beta formula above: : \beta_\text{max} = 0.072 \left(\frac{1+\kappa^2}{2}\right)\epsilon. For a spherical tokamak with an elongation \kappa of 2 and an aspect ratio of 1.25: : \beta_\text{max} = 0.072 \left(\frac{1+2^2}{2}\right)\frac{1}{1.25} = 0.14. Now compare this to a traditional tokamak with the same elongation and a major radius of 5 meters and minor radius of 2 meters: : \beta_\text{max} = 0.072 \left(\frac{1+2^2}{2}\right)\frac{1}{5/2} = 0.072. The linearity of \beta_\text{max}\, with aspect ratio is evident.
Power scaling Beta is an important measure of performance, but in the case of a reactor designed to produce electricity, there are other practical issues that have to be considered. Among these is the
power density, which offers an estimate of the size of the machine needed for a given power output. This is, in turn, a function of the plasma pressure, which is in turn a function of beta. At first glance it might seem that the ST's higher betas would naturally lead to higher allowable pressures, and thus higher power density. However, this is only true if the magnetic field remains the same – beta is the ratio of magnetic to plasma density. If one imagines a toroidal confinement area wrapped with ring-shaped magnets, it is clear that the magnetic field is greater on the inside radius than the outside - this is the basic stability problem that the tokamak's electric current addresses. However, the
difference in that field is a function of aspect ratio; an infinitely large toroid would approximate a straight solenoid, while an ST maximizes the difference in field strength. Moreover, as there are certain aspects of reactor design that are fixed in size, the aspect ratio might be forced into certain configurations. For instance, production reactors would use a thick "blanket" containing
lithium around the reactor core in order to capture the high-energy neutrons being released, both to protect the rest of the reactor mass from these neutrons as well as produce
tritium for fuel. The size of the blanket is a function of the neutron's energy, which is 14 MeV in the D-T reaction regardless of the reactor design, Thus the blanket would be the same for a ST or traditional design, about a meter across. In this case further consideration of the overall magnetic field is needed when considering the betas. Working inward through the reactor volume toward the inner surface of the plasma we would encounter the blanket, "first wall" and several empty spaces. As we move away from the magnet, the field reduces in a roughly linear fashion. If we consider these reactor components as a group, we can calculate the magnetic field that remains on the far side of the blanket, at the inner face of the plasma: : B_{0}= ({1 - \epsilon_B - \epsilon}) {B_\text{max}}.\, Now we consider the average plasma pressure that can be generated with this magnetic field. Following Freidberg: : {\langle p \rangle} = \beta_\text{max}\left (1 + \kappa^2\right) \epsilon \left({1 - \epsilon_B - \epsilon}\right)^2 G(\epsilon) \left(B_\text{max}\right)^2. In an ST, where we are attempting to maximize B_0 as a general principle, one can eliminate the blanket on the inside face and leave the central column open to the neutrons. In this case, \epsilon_0 is zero. Considering a central column made of copper, we can fix the maximum field generated in the coil, B_\text{max} to about 7.5 T. Using the ideal numbers from the section above: :{\langle p \rangle} = 0.14 \left(1 + 2^2\right) \left(\frac{1}{1.25}\right) \left(1 - \frac{1}{1.25}\right)^2 2.5 \cdot 7.5^2 = 2.6 \text{ atmospheres}. Now consider the conventional design as above, using superconducting magnets with a B_\text{max} of 15 T, and a blanket of 1.2 meters thickness. First we calculate \epsilon to be 1/(5/2) = 0.4 and \epsilon_b to be 1.5/5 = 0.24, then: :{\langle p \rangle} = 0.072 \left(1 + 2^2\right) \left(\frac{1}{0.4}\right) \left(1 - \frac{1}{0.24} - \frac{1}{0.4}\right)^2 1.2 \cdot 15^2 = 7.7 \text{ atmospheres}. So in spite of the higher beta in the ST, the overall power density is lower, largely due to the use of superconducting magnets in the traditional design. This issue has led to considerable work to see if these scaling laws hold for the ST, and efforts to increase the allowable field strength through a variety of methods. Work on START suggests that the scaling factors are much higher in STs, but this work needs to be replicated at higher powers to better understand the scaling. Research using data from
NSTX and
MAST appears to confirm the supposition that for similar values of field and fusion power, but smaller volume, STs can demonstrate a fusion triple product of up to a factor of three higher and a fusion power gain of an order of magnitude higher than tokamaks.
Advantages STs have two major advantages over conventional designs. The first is practical. Using the ST layout places the toroidal magnets much closer to the plasma, on average. This greatly reduces the amount of energy needed to power the magnets in order to reach any particular level of magnetic field within the plasma. Smaller magnets cost less, reducing the cost of the reactor. The gains are so great that superconducting magnets may not be required, leading to even greater cost reductions. START placed the secondary magnets inside the vacuum chamber, but in modern machines these have been moved outside and can be superconducting. The other advantages have to do with the stability of the plasma. Since the earliest days of fusion research, the problem in making a useful system has been a number of
plasma instabilities that only appeared as the operating conditions moved ever closer to useful ones for fusion power. In 1954
Edward Teller hosted a meeting exploring some of these issues, and noted that he felt plasmas would be inherently more stable if they were following convex lines of magnetic force, rather than concave. It was not clear at the time if this manifested itself in the real world, but over time the wisdom of these words became apparent. In the tokamak, stellarator and most pinch devices, the plasma is forced to follow helical magnetic lines. This alternately moves the plasma from the outside of the confinement area to the inside. While on the outside, the particles are being pushed inward, following a concave line. As they move to the inside they are being pushed outward, following a convex line. Thus, following Teller's reasoning, the plasma is inherently more stable on the inside section of the reactor. In practice the actual limits are suggested by the "
safety factor",
q, which vary over the volume of the plasma. In a traditional circular cross-section tokamak, the plasma spends about the same time on the inside and the outside of the torus; slightly less on the inside because of the shorter radius. In the advanced tokamak with a D-shaped plasma, the inside surface of the plasma is significantly enlarged and the particles spend more time there. However, in a normal high-A design,
q varies only slightly as the particle moves about, as the relative distance from inside the outside is small compared to the radius of the machine as a whole (the definition of aspect ratio). In an ST machine, the variance from "inside" to "outside" is much larger in relative terms, and the particles spend much more of their time on the "inside". This leads to greatly improved stability. It is possible to build a traditional tokamak that operates at higher betas, through the use of more powerful magnets. To do this, the current in the plasma must be increased in order to generate the toroidal magnetic field of the right magnitude. This drives the plasma ever closer to the Troyon limits where instabilities set in. The ST design, through its mechanical arrangement, has much better
q and thus allows for much more magnetic power before the instabilities appear. Conventional designs hit the Troyon limit around 3.5, whereas START demonstrated operation at 6.
Disadvantages The ST has three distinct disadvantages compared to "conventional" advanced tokamaks with higher aspect ratios. The first issue is that the overall pressure of the plasma in an ST is lower than conventional designs, in spite of higher beta. This is due to the limits of the magnetic field on the inside of the plasma, B_\text{max}. This limit is theoretically the same in the ST and conventional designs, but as the ST has a much lower aspect ratio, the effective field changes more dramatically over the plasma volume. The second issue is both an advantage and disadvantage. The ST is so small, at least in the center, that there is little or no room for superconducting magnets. This is not a deal-breaker for the design, as the field from conventional copper wound magnets is enough for the ST design. However, this means that power dissipation in the central column will be considerable. Engineering studies suggest that the maximum field possible will be about 7.5 T , much lower than is possible with a conventional layout. This places a further limit on the allowable plasma pressures. However, the lack of superconducting magnets greatly lowers the price of the system, potentially offsetting this issue economically. The lack of shielding also means the magnet is directly exposed to the interior of the reactor. It is subject to the full heating flux of the plasma, and the neutrons generated by the fusion reactions. In practice, this means that the column would have to be replaced fairly often, likely on the order of a year, greatly affecting the availability of the reactor. In production settings, the availability is directly related to the cost of electrical production. Experiments are underway to see if the conductor can be replaced by a
z-pinch plasma or liquid metal conductor in its place. Finally, the highly asymmetrical plasma cross sections and tightly wound magnetic fields require very high toroidal currents to maintain them. Normally this would require large amounts of secondary heating systems, like neutral beam injection. These are energetically expensive, so the ST design relies on high
bootstrap currents for economical operation. Luckily, high elongation and triangularity are the features that give rise to these currents, so it is possible that the ST will actually be more economical in this regard. This is an area of active research. ==List of ST machines==