Inductively coupled 30 T magnetic field platform for magnetized high-energy-density plasma studies

Over the past decade, there has been considerable interest in magnetized high-energy-density (HED) plasma studies. New and exciting results have been obtained in several HED areas such as direct-drive inertial fusion,^1–4 hohlraum plasma magnetization in indirect-drive inertial fusion,⁵ magnetizing liner-imploded plasma in magnetic liner inertial fusion (MagLIF),⁶ pair-plasma generation,⁷ and the investigation of astrophysical phenomena such as Weibel instability,⁸ magnetic reconnection,⁹ and collisionless shocks.¹⁰ 

To create a high magnetic field for these experiments, a magnetic field platform MIFEDS (magneto-inertial fusion electrical discharge system) has been built at the Laboratory for Laser Energetics (LLE) of the University of Rochester, NY. The original version¹¹ that was in operation from 2007 to 2011 was later updated¹² to a more flexible system with a larger, 1 μF storage capacitor discharged directly through a multi-turn wire coil wound around a 3D-printed plastic frame. While that system has proved valuable for many experiments cited above, the fields typically achieved are 10–15 T, depending on the coil geometry. The field limitation results from several constraints, including the available stored energy, the circuit and the coil inductance, and practical limits on how small a coil can be while still suitable for the experiment.

Measures underway to overcome these limits include increasing the stored energy, decreasing the circuit inductance, and using thinner wires with improved high-voltage insulation. Another approach proposed initially in the work of Furth et al.¹³ and discussed in detail for HED applications in the work of Barnak et al.¹⁴ is to use a coil coupled to the power supply inductively rather than directly.

A schematic of the inductively coupled coil described in this paper is shown in Fig. 1(a). A power supply drives a current pulse through a multi-turn solenoid inserted into the cylindrical cavity of a conductive shell, thus inducing an oppositely directed current in the shell. The slit through the shell forces current to flow around the tip of the shell, thus inducing a strong magnetic field on the axis of the small hole, hereafter the field-coil. Schematically, the device is arranged in the form of a step-up current transformer with a multi-turn solenoid connected to the power supply as its primary and with the shell and the field-coil being its single-turn secondary. The magnetic field on the field-coil axis can be very high because the field-coil current is proportional to the turn ratio N:1 in the primary and the secondary. An attractive feature of this arrangement is that the field-coil is physically separated from the solenoid, while the latter is fully enclosed inside the shell cavity (for clarity, the flanges covering the sides of the cavity are not shown), thus completely eliminating debris in case of the solenoid’s destruction. Finally, the high-current secondary circuit does not contain any electrical contacts and connections that can impede current propagation. A photo of the device is shown in Fig. 1(b).

Inductive coupling can be beneficial when the experimental constraints (e.g., laser beams and/or diagnostic access) require the use of a small magnetic coil with a single-digit number of turns. To create a high magnetic field with such a coil requires a high electric current; thus, a power supply with a high stored energy and a low inductance must be used. In general, the latter can only be achieved by placing the power supply very near the coil, which might not always be possible or desirable. On the other hand, inductive coupling requires only a modest current directly from the power supply. In addition, the requirements to the circuit inductance become much more relaxed, thus making possible the placement of the power supply further away from the coil (e.g., outside of the vacuum chamber of the experiment).

For the experiment, we built a power supply consisting of a storage capacitor (GA 33593 50 μF/20 kV) charged with a regulated power supply (Ultravolt 25C24-P250 25 kV/250 W) and discharged through the solenoid via a triggered spark gap (Excelitas GP-14B). The spark gap is triggered by a 30 kV/100 ns pulse provided by a custom-made NorthStar HV pulse generator. The output of the power supply is connected to the solenoid by a 5 m-long bundle of four parallel high-voltage, low-inductance 60 nH/m cables from Dielectric Sciences, Inc. To keep the current from ringing, a custom-built low-inductive 0.2 Ω shaping resistor (shown in Fig. 2) is inserted in series with the output cables. The resistor is made with a strip of stainless steel foil with a length of 2.2 m, a width of 5 cm, and a thickness of 0.15 mm. The strip is folded multiple times and insulated with 0.1 mm-thick Kapton sheets inserted between the folds. The discharge current is monitored with a Pearson current monitor (PearsonElectronics 1330). The internal resistance of the power supply is 0.22 Ω, and its inductance is 550 nH, including all the circuit elements and the output cables. The power supply is capable of providing a short-circuit current of 67 kA at a capacitor charge of 20 kV.

For the described set of results, the solenoid with an outer diameter of 32 mm and a length of 50 mm is formed by 22 turns of a Kapton-insulated wire (Accu-Glass Products, Inc.) with a size of AWG14 (core diameter 1.6 mm) and is tightly fit inside the cavity to maximize the inductive coupling. The field-coil has a diameter of 6.53 mm and a length of 6.35 mm. This geometry can be optimized according to the specific requirements of an experiment.

A simplified circuit schematic is shown in Fig. 2. Figure 3 illustrates a very good agreement between the circuit simulations and measurements.

To further develop predictive capabilities, the coil operation was modeled using the COMSOL Multiphysics® package that allows for the most complete and accurate modeling and includes such important physics as current diffusion (skin effect), heat transport, and temperature-dependent resistivity. A comparison of measured and simulated magnetic field waveforms shown in Fig. 4 illustrates excellent agreement between measurements and simulations. Figure 5 shows the distribution of the field coil current density across the coil cross section. Due to the skin effect, the current is peaked very near the coil inner surface, and the total surface-integrated coil current is I_coil = 220 kA.

The dependence of the field-coil peak magnetic field on the solenoid current is shown in Fig. 6. The magnetic field was measured using a calibrated B-dot probe made of several turns of a thin AWG 18 magnet wire. A peak field of B ≈ 30 T at the field-coil center was obtained at a solenoid current of ≈30 kA. However, the rate at which the field changes with the current falls (the B-vs-I curve deviates from the initially linear dependence), and at a higher solenoid current, it even becomes negative! The reason for this behavior is increased field-coil size resulting from the high magnetic forces. Had the field-coil remained the same size, the field should have been about 40 T as indicated by the extrapolation of the linear fit.

An accurate modeling of the coil expansion is rather complicated given the large electromagnetic forces. The peak of magnetic pressure P_M = B²$$2μ₀ is of the order of 360 MPa at B = 30 T and that exceeds the copper tensile stress of ∼200 MPa. The regime of elastic deformation ends after the tensile stress limit has been reached, and given the copper Young’s module of 100 GPa, it happens very early into the pulse at a small strain of about 0.002 mm$$mm. After that, the coil expansion is governed by very non-linear processes especially if the geometry, material heating, current diffusion, etc., are taken into account.

However, it is instructive to construct an approximate model of the coil expansion to estimate its magnitude and scaling. Given that the profile of the current in the field coil is very narrow and is peaked at the inner coil surface (see Fig. 5), the electromagnetic force action can be approximated by that of a piston applying an outward pressure of P_M = B²$$2μ₀ to the field coil inner surface. An upper bound of the coil expansion can be obtained by neglecting the deformation forces, so the magnetic force is balanced by the coil material acceleration,

and the buildup of the moving coil material is described by an incompressible 1-D “snow-plow” model,

Here, m, $v$ = dx$$dt, and x are the mass (per unit area), velocity, and radial displacement of the moving copper layer, and ρ is the specific density of the material.

Approximating the magnetic field waveform by B = B₀ sin(πt$$T_1/2) (T_1/2 is a half-period of the modulation) results in an expression for the material displacement δ by the end of the current pulse,

For B₀ = 30 T and T_1/2 = 30 μs, δ ≈ 4 mm. These estimates agree, within a factor of 2, with the actual coil expansion, which is quite reasonable given the model approximations.

A possible improvement could be reduction in the pulse duration, replacement of the copper shell material to a material with a higher tensile stress and/or higher density (e.g., BeCu or W), and/or construction of a reinforcing outer shell to contain the shell expansion. Exploration of acceptable methods, as well as a more accurate treatment of the coil expansion, is a subject of future work.

To conclude, a high magnetic field platform based on inductive coupling has been developed and tested. The device is capable of generating a pulse of magnetic field of 30 T and a duration of 10 μs and can be used in many high energy density plasma experiments requiring plasma magnetization.

Currently, the field magnitude appears to be limited by the structural strength of the field-coil material (copper), and future work will explore ways of circumventing this limitation. In addition, even in case of complete solenoid destruction, its debris is fully contained, which makes this coil attractive for debris-sensitive applications.

This work has been supported in part by the University of Michigan research Grant No. U051442 and the U.S. Department of Energy OFES Award No. DE-SC0016258.