Superconductivity is a phenomenon of exactly zero electrical resistance, and expulsion of magnetic fields, occurring in certain materials when cooled below a characteristic critical temperature.
Superconductivity was first discovered over 100 years ago by super cooling Mercury to 4.2 K. Mercury is a type I superconductor, as they are classified, at that temperature. The first type II superconductor was discovered nearly 20 years later and the first high temperature variant of a type II was discovered in 1986. High temperature superconductors that can be used in practical applications still have critical temperatures well below zero degrees centigrade (such as the commonly used variants of YBCO and BSCCO which are driving the revived interest in using superconducting materials for AC applications).
Type I, or soft, superconductors are typically metals or metalloids and can be classified as adhering to BCS theory which describes the macroscopic property of superconductivity as a microscopic effect caused by a condensation of Cooper pairs into a boson-like state. Type I superconductors have a sharp transition between superconducting and ‘normal’ or resistive behaviour and in the superconducting state the body will completely expel all incident magnetic fields which can also be termed perfect diamagnetism or the Meissner effect. This assumes that the fields are also below some critical level which can also cause the transition between superconducting and normal states in the material.
So, how would one go about simulating superconductors? The Finite element method is the most commonly used method for virtual prototyping. In finite element analysis techniques a continuous domain of material is broken down into finite regions, or elements, where the variation of physics within the domain may be accurately represented by a system of differential or integral equations.
Any virtual prototyping technique must capture the complete picture of the system or device being designed. This will almost always involve geometry and materials, the physics equations that govern the properties being considered but also other physical attributes of the system. An example to highlight this is the International Muon Ionization Cooling Experiment (or MICE) where the geometry not only includes the magnets that control the particle trajectories but also any active magnetic material in the vicinity – such as the steel walkways, shielding and even the reinforced concrete walls of the building in which it is housed.
Before we look at some applications of superconducting windings modelled with virtual prototyping software, it is worth noting that often in DC systems (typically those that employ low temperature superconducting materials but not exclusively), the fact that the windings are superconducting is not of principle concern, rather the magnetic fields that are generated from currents flowing in the windings are the main design criteria. This means that typically DC systems are much easier to simulate as we will see later.
Finite element simulation is well established for modelling DC superconducting systems and the methods are well understood. In MRI, for example, an accuracy of a few parts per million is required for the central field magnitude and homogeneity in order to obtain good quality images from the device in operation. Obviously, any virtual prototyping tool must be able to match this with solution accuracy for it to be of value, and rarely is the fact that the coils are superconducting taken into account when undertaking field homogeneity calculations. As systems with AC fields gain popularity including the properties of superconducting materials has become more important.
Let’s take a look at some applications that use or have the potential to use superconducting material to great effect. We will start with some of the well-established DC applications and move towards those that are emerging including some that are necessarily AC.
The magnets used in high energy experiments & particle accelerators, which are often one off physics experiments (like the joint European Torus or ITER and large particle accelerators like the large hadron collider at CERN), demonstrate the most successful and varied use of superconducting technologies. A driving factor for this is that they require the enhanced magnetic/electric properties in order to construct a functional system and the development costs of using superconducting materials and the cryogenics systems required are not cost-prohibitive. The use of finite element simulation in the design of these systems is essential as the development cost of creating generations of physical prototypes is prohibitive.
In this area superconducting windings are used in magnets for Bending, Focussing and confining particle beams as well as for conducting experiments and detecting the results. The main simulation requirements are accurate calculation of the magnitude and gradients in the fields produced in both DC and ramping operation and protecting the system against faults due to quenching. Simply put a quench is the transition of the materials in the magnet from superconducting to resistive state during operation. If no care is taken to manage this transition in a controlled way systems can be damaged or even destroyed.
Managing against superconducting quench is also a common consideration in MRI magnets. In MRI systems the superconducting coils typically fall into two categories – those designed to produce the homogeneous central field region, known as the main field coils, and those used to reduce stray fields which can damage nearby equipment or endanger patients in communal areas.
Further evaluation of this system during a quench can be made giving the transient field as the current drops and the temperature distribution in the coil. This in turn can be used to evaluate the stress in the coils at any time which can include contributions from and pre-stress or winding tension, thermal expansion of the coils, Lorentz and other electromagnetic forces acting on the coils.
Although the fields are relatively low when compared with self-destructive or even the latest non-destructive pulsed magnets that can achieve a field close around 100 Tesla, one example of a high field magnet is a project undertaken by Oxford instruments in producing continuous high central fields. This example uses three separate superconducting coils each made from different materials to make best use of their properties to get a central field of just over 22 tesla. Although this is still a DC magnet operating at liquid helium temperatures the inner most coil is made from an HTS material. Rather than making use specifically of the high temperature properties of the HTS material here they are using the material because of its ability to withstand high fields. One further point of note relating to this system is that simulation is able to highlight the progress of a quench without needing to risk damage of a real device. Simulation shows that during a quench process this coil is subjected to field variations as high as 4 tesla a second – changes easily fast enough to cause the transition of low temperature materials from superconductive to resistive states due to induced currents exceeding the critical currents.
Instead, in system modelling, we typically use a bulk/homogenised approximation for the coil as a whole, based on the relative fractions of the component materials in the coil. We can use expressions from theory, or tabulated data taken from measurement. Here we look at the thermal properties which are very anisotropic, so the bulk approximation needs to accurately represent the winding, not just as a single bulk value but typically in both the direction of current flow and perpendicular directions. This means that some method of describing the current direction is required so that the anisotropic material properties can be aligned with this local orientation, rather than within the global system, or some simple transformation of the global coordinate system.
One important characteristic when considering a potential fault in the system is that of the critical current. This highly non-linear property varies with both temperature and the flux density the winding is exposed to, and it defines the point at which the transition between normal and superconducting state occurs. Clearly in a homogenised model where the current flowing in the superconducting filaments is distributed equally in the volume fraction of the superconductor and matrix, the critical current density must also be diluted using this volume fraction to enable the bulk to transition correctly.
So let’s look at a validation of finite element methods for the most challenging of applications; the simulation of an HTS material in an AC application operating in the mixed state where external oscillating magnetic fields are driving the induced current flow into operation above critical current density. We are considering this in a 2d approximation to enable us to simulate only a short piece 1 element thick of the material – the remainder of the infinite strip being implied by the boundary conditions applied on either side. The external H field oscillates at 50 Hz with a magnitude of 12 kA/m. It induces currents within the strip that attempt to screen the superconductor from the external fields. With finite element methods we can observe the field quantities as they vary during a cycle, and here we show a snapshot of them at the position of peak external magnetic field.
The results of this analysis were verified against the theoretical predictions of Brandt and Indebohm. A case study describing this work in more detail is available through our website operaFEA.com.
Another method of including the meissner effect suitable for DC systems is one proposed at the applied superconductivity conference in Houston in 2002 where the available screening current and the incident fields on the superconducting material are used to determine an equivalent magnetization for the bulk material which determines the magnitude of the diamagnetism. Here the mixed state behaviour is better approximated as this method enables the inclusion of the hysteretic/irreversable behaviour of the filament magnetization which contribute to the fields. Again this is only suitable for an effective DC system as the contribution to the transition between normal and superconducting state of the screening currents is not considered correctly.
In AC systems similar to the filament level simulation we have already reviewed where the filament magnetization and hysteretic losses, along with the relationship of transport and screening currents and the transition from purely superconducting to a mixed state behaviour is important to obtain an accurate description of the fields produced, work is still ongoing in determining a method suitable for rapid prototyping of such systems. Approximations again include homogenization of the materials constituting the winding and lattice however now with the inclusion of insulation boundaries between individual turns or highly anisotropic conductivities to restrict the path of current flow between turns in the coil. This implies it is still necessary to resolve the current flow in individual homogenised turns explicitly as it is not uniform as in DC systems. One caveat to this is that recent advances in parallel processing are enabling more complex problems to be tackled.
At this point in this section discussing the modelling of superconducting materials we have looked at only the electromagnetic properties of the coils however in superconducting systems often the thermal and structural properties are also of interest. One such situation as described earlier is that of superconducting quench. Here we are simulating the transition from superconducting to resistive states driven by the propagation and generation of heat in the device. In an AC system it is much more difficult to detect and hence protect against due to the slower/smoother transition from purely superconducting, through mixed to resistive states.
The heat that triggers a quench event can be from a variety of sources. In a DC system typically it will be due to a failure with the cryogenic system, ramping the system too quickly or in test situations can be introduced deliberately. In simulation we can include this heat as a surface or volume property or through rate dependant, ohmic or hysteresis losses in materials due to current flowing or fields in them. In this instance as described earlier we have significant anisotropy in the material properties as thermal conductivity is dominant along the winding direction.
Thank you for reading, I hope that you have found this useful. More information about superconductivity can be found at the link below, or contact us for further details.