Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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Magnetic fields and tokamak plasmas Alan Wootton 20 . THE IRON CORE Many tokamaks have an iron core to ensure good coupling between the primary winding and the plasma. This iron core has the additional advantage of keeping 'stray' fields away from diagnostic apparatus. However, it complicates the study of the plasma equilibrium, because the free space expressions for the fields produced by circular conductors are no longer applicable. Here all I want to do is to point out that iron is a pain to deal with. X poloidal field I iron iron Figure 20.1. A straight iron cylinder Figure 20.2. An iron core with an air surrounding a plasma gap, with a linked plasma (current I). The iron core introduces a new boundary condition. It is often stated incorrectly that the lines of force enter a medium of infinite permeability (the iron) perpendicularly, but this is not always true. A simple example of a straight iron cylinder surrounding a straight wire with current shows this not to always be the case: in Figure 20.1 the field lines are tangential to the iron boundary. A correct boundary condition, for the normal component of the induction, is B n ( air) = µ 0 H n ( air) = µH n ( iron) = B n ( iron) 20.1 For permeability u very large we can take H n approaching 0 just inside the iron, so that H n ( iron) = 0 20.2 (from the continuity of B n ) and add the boundary condition H τ is continuous 20.3 144
Magnetic fields and tokamak plasmas Alan Wootton In reality µ is not infinite, but the limit works well. First consider an air gap inductor, as shown in Figure 20.2. We assume that inside the iron there are laminations which ensure no current, so that ∇ × H = 0 20.4 Then H is derived from a single valued potential given by Laplace's equation ∇ 2 ψ = 0 20.5 and the boundary condition on B n (Equation 20.2) is equivalent to ∂ψ ∂n = 0 in the iron. 20.6 The only possible solution of equation 20.5 and equation 20.6 is ψ = constant . Therefore H (but not B) must be zero inside the iron. Continuity of the tangential component of H then shows that the lines of force in the air must be perpendicular to the iron. Therefore in air the magnetic fields are given by the usual equations, together with the boundary condition e n × H = 0 on the iron surface. 20.7 In the iron H = 0, but B is finite. Since ∇xB = 0 in the iron, B(iron) = ∇Φ, and Φ satisfies ∇ 2 Φ = 0 20.8 ∂Φ ∂n = B n( air) 20.9 which is known. contour l iron I 145
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Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

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