How to simulate superconducting systems

By Nigel Atkinson

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.

Type II, or hard, superconductors typically have higher critical transition temperatures and consist of metallic compounds and alloys. They generally cannot be explained completely from BCS theory and have a gradual transition from superconducting to resistive states. This makes them generally more tolerant to external magnetic fields as they maintain their superconducting nature despite the penetration of magnetic fields within the volume -which can also be described as imperfect diamagnetism.
Type II superconductors can be further categorised; as low temperature or high temperature. The distinction between low and high being whether they are superconducting below or above the boiling point of liquid nitrogen, at 77 degrees kelvin. This is an important distinction as liquid nitrogen is a much less expensive cryogenic coolant than liquid helium, as used with LTS, it can be produced in-situe and transported easily at atmospheric pressure. This makes it very useful in a wide range of applications including cooling HTS superconductors to the point where they exhibit their properties. HTS materials are further categorised using the manufacturing techniques employed in construction into first, second and third generation.

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.

Care must be taken when ramping the magnet to operational levels so as to not quench the Niobium titanium coils with heat generated by rate dependant losses in the coils themselves or eddy currents produced by the changing fields in support structures. Here we have a demonstration model of an MRI magnet which includes the coils, formers, spacers and coolant pipes being ramped to operational fields over a period of nearly 3 hours showing the current as it is ramped and the resultant maximum temperature rise during this process. In this case the ramping is slow enough for the whole coil to remain below the critical temperature of 9.2 kelvin above which the Niobium titanium would become resistive (quenching to potentially disastrous effect).

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.

The use of superconducting materials in electrical machines is not a new idea. Concepts have been put forward since the late 1970s and test machines have been developed over the last 30 years with reasonable success. However it has only really been in the last 10 to 15 years and the discovery/development of HTS wires, coupled with the drive to use renewable energy that this field has really gained momentum. Aside from the benefits of being able to use cheaper cryogenic systems one of the main benefits of HTS materials is that although in electrical machines it is possible to use superconductors to generate a DC field – with the rotation of the device causing the field modulation – they still must be able to withstand some variation in the field they are exposed to. One example of a real device constructed using a DC superconducting field winding is the prototype developed under the European Union FP6 project named hydrogenie. This is a 1.7 MW hydro generator that was developed in industrial partnership. In this case simulation was able to predict the performance of the device with high accuracy including estimating efficiency to within 0.1 %. The generator has demonstrated that is is possible to reduce the size and weight of a generator for this purpose by up to 70 % with significant improvements to efficiency.
As a quick summary there are also a wide variety of other technologies either already using superconducting materials or with the potential to use them ranging from the purification of water using high field gradients, transportation in the form of superconducting maglev or even in high end computing where the use of superconducting materials has the potential to dramatically reduce the energy usage of data centres which are already overtaking air travel as one of today’s major energy drains.
Whichever virtual prototyping technique is employed a description of the device is required covering its geometry, materials used in its construction the physics that govern its operation and so on. The technique must also provide a means to solve the physics and characterise or review the results, and the most common virtual prototyping technique across any discipline is the finite element method. However even using this method it is still not practical to include a description of every filament in a superconducting wire, or even every turn of wire, in a system level model of a device due to limitations in what can be processed in a realistic time-frame. This statement can be highlighted by looking again at the bizmuth wire used in the high field coil I showed earlier which contains several hundred filaments of superconductor in a supporting matrix. Clearly the number of elements we would require to resolve even a short section of such a wire would be excessive, let alone many coils of many turns, contained within a larger system.


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 key difficulty in simulation of superconducting materials in AC applications is highlighted by this simulation work; the fact that the material has an effective conductivity of 2.7e11 siemens per meter at the critical current density of 2.7e8 A/m^2. This implies a maximum skin depth of approximately 0.14 mm at 50 Hz. When the current is below the critical value, the actual skin depth is far smaller than this. However, there are also substantial volumes of the material which cannot carry more current, and the the surface from which the skin depth exists moves as the external fields vary. This requires that the whole of the strip to be meshed with very small elements throughout its cross section in order to ensure the current flow is resolved accurately. In this instance a maximum element size of 7.5 e-4 mm was necessary to sufficiently resolve the current distribution for an accurate answer. This lead to around 150 thousand elements in the cross sectional model.

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

This brings us back to the discussion regarding homogenization. In DC systems one of the properties (that of diamagnetism or the meissner effect) can be included in a system level simulation as a bulk property simply by setting the permeability of the superconductor to below unity. This approximation is valid assuming the conductor remains superconducting though the fields the conductor is exposed to can no longer contribute to the transition.

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.

Once a quench has begun we have two design issues to deal with, detect that the quench has started and protect the magnet from damage. Here is a typical protection circuit for a multi-coil DC superconducting magnet. As the superconducting material transitions from zero to finite resistance the current will preferentially chose to flow through a small finite resistance rather than the resistive superconductor which now has higher resistance. This reduces the ohmic losses in the conductor protecting the magnet from overheating and allows the cryogenics time to cool the windings. However the reduction of current flowing in one of the windings will cause a rapid change in the fields produced by that coil potentially causing neighbouring coils to quench due to rate dependant losses or losses in other conducting structures that oppose the field changes.
Here is a validation example of an experiment in which comparison is drawn between simulation and experiment for a quench process. The coil set consists of four superconducting coils and an aluminium former. During an initiated quench all of the coils quench despite being electrically and relatively thermally isolated due to the rapid field changes and rate dependant losses. We can also determine and visualise currents induced in the aluminium support structures – perhaps enabling us to make considered design changes to reduce these effects.

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.