2. CALCULATION MODEL

The schematic of MID is shown in Figure 1. The coils of the diode magnetic system and a shock induction coil are located inside the chamber with perfectly conducting walls and therefore the applied boundary conditions are ψ = 0.

We solve the task using a discrete grid inside the chamber. The coil current I _θ may be then represented as a current density J _θ defined on the gridpoints. Due to a cylindrical symmetry, the magnetic field can be obtained from the Ampere's law for an azimuthal current density and given as:

(1)$$\nabla \times B = \displaystyle{{4{\rm \pi}} \over c}J_{\rm \theta}. $$

In a cylindrically symmetric system with an azimuthal current, the vector potential A has only an azimuthal component, which may be represented by a scalar flux function ψ(r, z) = rA _θ. The Ampere's law is then:

(2)$$\displaystyle{{\partial ^2 {\rm \psi}} \over {\partial r^2}} - \displaystyle{1 \over r}\displaystyle{{\partial {\rm \psi}} \over {\partial r}} + \displaystyle{{\partial ^2 {\rm \psi}} \over {\partial z^2}} = - \displaystyle{{4{\rm \pi} r} \over c}J_{\rm \theta}. $$

This is the Grad–Shafranov equation. Once it has been solved using suitable boundary conditions, the magnetic field can be obtained from:

(3)$$B_r (r,z) = - \displaystyle{1 \over r}\displaystyle{{\partial {\rm \psi} (r,z)} \over {\partial z}}\quad {\rm and}\quad B_z (r,z) = \displaystyle{1 \over r}\displaystyle{{\partial {\rm \psi} (r,z)} \over {\partial r}}.$$

Equation (2) is an elliptic equation which can be solved using standard methods. Firstly, in Section 3 below, we use a finite-difference method (the Alternating Direction Implicit method) on a rectangular grid neglecting the slots in the anode. The result is used to obtain the boundary conditions for a more limited domain which is centered near the anode slots. In Section 4, we use a finite element method (using the ELLIPT2D Python software package) on an irregular triangular grid to find the magnetic field in the region of the anode slots.

3. TOPOLOGY OF B-FIELD FOR A SOLID ANODE

In the given section, we calculate B-field strengths without taking into account the influence of perturbation of the slots on the anode. The results will be used to obtain the boundary conditions when calculating the magnetic field in the region of the anode slots. The cross-section of the r − z diode is shown in Figure 1. The conical shock coil is located inside the conducting anode with close-lying walls. We assume that the shock coil with internal and external radiuses r _sc1 = 96 and r _sc2 = 130 mm, respectively, is wound in such a way that only azimuthal current component is significant. The internal and external coils of the magnetic system of the diode have 7 and 12 turns, respectively, with 10 kA current per turn. The coils are located in the cathode screens with internal and external radiuses r _c1 = 80 and r _c2 = 120 mm. In this case, the coils of the magnetic system of the diode do not have closely located boundaries with perfect conductors, as in the case with a shock coil except for the screen which is transparent for the magnetic field. As a result, the boundary conditions ψ = 0 for an ideal conductor are specified on the walls of one chamber. In Figure 1, the rightmost and uppermost boundaries on the domain were set at z = 50 and r = 40 cm, respectively. The left boundary is still a conical surface of the anode. The results of the calculation of the topology of the magnetic field for a solid anode and current in a shock coil of I _θ = 22.3 kA are shown in Figure 1.

4. FIELD CLOSE TO THE ANODE SLOTS

To solve Eq. (2), in and around the slots, we used the ELLIPT2D finite element software package, which is well adapted for handling domains with holes in them. A limitation on this package is that it is only effective, if there are <10 000 vertices in the triangular grid. Therefore, to obtain the accurate results around the slots, it is necessary to solve the equation on a reduced domain. We note from Figure 1 that at a medium radius of the anode, at a distance of 1.5 cm from its surface, the field lines are almost parallel to the anode. We therefore construct a reduced domain, shown in Figure 2, which extends 1.5 cm on either side of the anode and centers on a single slot. Our observation that the field lines are approximately parallel to the conical anode allows us to specify simplified boundary conditions on this reduced domain. Along the left and right boundaries, we specify the Dirichlet boundary conditions, using two constant values of ψ, which are obtained from the calculations in Section 3. Along the upper and lower boundaries we used the Neumann boundary conditions. The white rectangles in the domain represent the 2.5 mm brass separations between the 5-mm slots in the 5-mm thick anode. The boundary condition on the white rectangles is ψ = 0.

Fig. 2. The topology of the magnetic field of the B-diode in the domain of the 5-mm wide anode slot. Curves represent the magnetic field line distribution and color represents magnetic field strength. Points A, B, C, D, E, and F are used to specify the B-field distribution and are referred to the next figure. The diode magnetic field coils are on the right side.

In this case, the Dirichlet boundary conditions were calculated using a total current in the shock coil of 25 and 8 kA per turn in the shock coil. The result in Figure 2 represents the field lines superimposed on a color scheme which represents the magnetic field strength. The points labeled A, B, C, D, E, and F are used in Figure 3, where the magnetic field strength is plotted along the straight lines AB, CD, and EF for anode slots with a width of 5 and 3.5 mm.

Fig. 3. Magnetic field strengths along the straight lines AB, CD, and EF from Figure 2. Curves 1 and 2 are the width of the anode slot of 5 and 3.5 mm, respectively.

The main results of the analysis of the obtained dependencies are as follows.

• • The expected difference of the magnetic field strength in the points C and В are related to the reduction of the field with increasing the radius according to the function B = f(1/r).

• • B-field distortions in the acceleration gap of the diode are located in the region of the anode with an extension, which does not exceed the half-width of the anode slot, and decrease with decreasing the slot dimensions. Thus, for the 5-mm slot the field strength at a given distance is ~80% from the undisturbed value on the anode, and for the 3.5-mm slot it is ~93%.

5. OPTIMIZATION OF THE MAGNETIC FIELD STRUCTURE IN THE ACCELERATION GAP

Figure 4 presents the calculated and measured distributions of the magnetic field on the conical surface of the anode of the diode with 8-mm acceleration gaps. The measurements were made using a gauge with a thickness of 1 mm and a size of 2 × 3 mm². The measured values for a solid anode (curve 2 in Fig. 4) reveal the distortion of the magnetic surface in the edge regions of the internal and external cathodes with radiuses of r _k1 = 80 and r _k2 = 111 mm, respectively. These effects often appear in real constructions and depend on many factors, including the values of the acceleration gap that is equal to the distance between the center of gravity of the coils and the anode surface. The impact of the change of the anode profile in the region across the diode B-field coils (Fig. 4) is shown with curve 3 in Figure 4 for the anode with 5-mm slots. The resulting distribution of the B-field with an accuracy of ~15% caused by a deflection into anode slots coincides with the distribution of В = f(1/r) for the specified geometry of the diode.

Fig. 4. Magnetic field induction distribution on the surface of the anode: calculation of B = f(1/r) for a solid anode (1); measured values at a distance of 1 mm from the plate of the solid anode (initial shape) and with the slots of 5 mm (final shape), respectively (2, 3); the geometry of the initial (a) and final (b) shape of the anode; A-anode, K-cathode. r _a1, r _a2; r _c1, r _c2; r _b1, and r _b2 are internal and external radii of the emission surface on the anode, cathode, and break points on the anode.