How to simulate superconducting corrector coils in accelerator magnets

By Chris Riley 

Welcome to this blog on modelling superconducting corrector coils with OperaFEA simulation software.

We will start with a little bit of history.  The Opera software has links back to the Rutherford laboratory’s Computer Applications Group that began developing computer methods for the design of particle accelerator magnets in the late 1960s. The early programs solved integral equations, but finite element methods solving partial differential equations became more competitive as a result of developments in the solutions of sparse systems of equations. Opera’s success is in large part due to continuing development of the founders’ pioneering ideas for numerical software – which made possible the first practical commercial solution for solving complicated, three-dimensional electromagnetic simulations.

The first users for Opera software were almost all at the major physics labs around the world. At these sites, Opera was used to prototype advanced superconducting and resistive magnets for high energy and particle physics research – equipment which was too large and costly to prototype and refine using ‘cut and try’ methods. This user base at the very forefront of global scientific research and advanced engineering still remains a major strand of the company’s business. However, the functionality of Opera and the user-friendliness of the design tools, has evolved almost beyond recognition to offer solutions for a much broader range of design requirements.

Because of Opera’s origins, the software has always been focused on accurate field calculations. In particular, the total and reduced scalar potential formulation, which allows separation of the contributions of coil and magnetization fields in Opera-3d magnetostatics, has proved to be vital in obtaining parts in 104 field accuracy necessary for this application. It allows a fast solution and computational economy, which is still a critical factor even though computer technology has advanced so much.

But Opera has not stood still and the design tools have evolved to model dynamic, time-varying fields where eddy currents are important, multi-physics effects such as temperature rise and can include mechanical stress and complex analyses such as a simulation of superconducting quench – all without compromising speed and accuracy.

The simulation capabilities have been complemented by advanced interactive and macro driven modelling and post-processing tools needed by all designers, and include some features that are very important to magnet designers, in particular the tools for harmonic analysis.

Firstly, just a few examples of some models that have been created over the years. This shows a quadrupole magnet designed at Brookhaven National Lab in New York state.

This is a non-linear magnetostatic simulation and the quality of the gradient field in the aperture is of vital importance in determining its particle beam focussing capability. Below we see another magnetostatic model of an insertion device – a permanent magnet undulator, such as are used in light sources to produce coherent X-ray sources.
The trajectory of a beam of particles through the undulator is also displayed. The pulsed septum magnet modelled by KEK in Japan below has been simulated using the Opera-3d transient electromagnetic solver coupled to an example driving circuit.
The redistribution of currents in the single turn bulk coils is also included within the circuit calculation.

Finally, we see a Quench model created at Jefferson Lab in Virginia where thermal and electromagnetic fields are simulated together, along with the protection circuit.

I mentioned harmonics before and here we see harmonic analysis of the y-component of the field sampled on a 0.1 metre radius circle in the magnet aperture.
The harmonic components tell the designer both the strength of the dipole field (around 1.5 Tesla in this case) from the zero harmonic, and the presence of higher order multipoles – sextupole, decapole etc. – due to the design not producing a perfect dipole. So why do magnet designers use harmonics to compare designs?

In particle accelerator magnets and in MRI / NMR magnets, field quality is expressed as coefficients of harmonics series as this is a very compact notation for comparing designs. In MRI/NMR, the normal practice is to use Associated Legendre polynomial coefficients evaluated in spherical polar space, since field homogeneity in the central spherical volume is important. For accelerator magnet designs, such as quadrupoles, Fourier coefficients are usually used. Fourier series expansion is only valid in two dimensions – the fields in an infinitely long magnet – but it also applies to the integral fields along the length of the magnet. Both types of series are a solution to Laplace’s equation (valid for a source free environment) and the field in all interior space can be determined just by knowing the harmonic expansion on its surface.

So, for example, to obtain integral Fourier harmonics in Opera-3d post ­processor we can evaluate the field on a cylindrical surface. In the below, the plot shows the radial component of field. Then, to obtain the main quadrupole integral harmonic component of the field, the post-processor allows us to perform the integration of the radial field multiplied by cosine of twice the physical angle giving a result of around 166 kG-cm2. Similarly, higher order harmonics can also be calculated. To turn to corrector coils, we have already seen one of the reasons why they are needed.

This quadrupole design has got a significant 12-pole integral harmonic (>1% of the main quadrupole) and, even for an optimized design, it isn’t usually possible to eliminate all higher order multipoles. Similarly, in the simulation, perfect symmetry can be assumed but, in practice, manufacturing tolerances mean that coils may not be perfectly aligned, pole shapes may be slightly different and so on. Variation in material properties within the lamination stack will also affect field quality.

All of these things may require a coil or set of coils, as well as passive shims, to correct the field to the quality needed for successful operation of the accelerator – hence, the term corrector coils. Corrector coils are frequently inserted inside the magnet although many accelerators also use additional magnets on the beam path to make corrections. For the remainder of this blog, we will concentrate on the former option. The coils are inserted into the magnets in the position where they oppose the unwanted harmonics. The unwanted harmonics may be off-axis components of the main field – due to manufacturing and material imperfections – or higher order harmonics of the main field that are intrinsic to the design or caused by manufacturing.

In the previous quadrupole model, the coils are modelled to be one degree off axis, which is probably a larger error than manufacturing would ever create but suitable for this demonstration. So we can see that this also produces a sine integral quadrupole harmonic of a little under 1% of the main quadrupole field.

To understand how the corrector coil reduces this sine component, an Opera-2d example is used initially. Next  we examine the radial flux density versus angle at a particular radius around 360 degrees and its Fourier coefficients. If only the required quadrupole field was present, then we would only expect a 2nd order cosine term.

The model is excited with a perfect quadrupole field – defined by the boundary condition shown applied to the outer circumference – but shifted by 4 degrees. The effect in the field map can be clearly seen with the flux line that should be on the Y-axis rotated clockwise by 4 degrees.

The model includes the go and return legs of the corrector coil, but in this model it is not energized. The quadrupole field is, of course, a perfect cosine wave – within numerical accuracy – but as we will see next there are both on and off axis components because of the 4 degree shift.

As can be seen from the Fourier analysis and the attached graph, both 2q sine and cosine components exist. Hence the corrector coil is positioned so that its central axis opposes the unwanted sine component. The corrector coil could, obviously, be made from copper and supplied by a power supply such that it produced the best level of opposing field to the unwanted sine component. So what is the advantage of going to the expense making the corrector from superconducting wire?

The answer to that question lies in the fact that the resistance of the coil is effectively zero. If the end connections of the coil are shorted together (through a superconducting joint), any voltage induced in the coil will also cause a current to flow. From Maxwell’s equation curl E = -dB/dt, we know that the current will flow to oppose the change in the flux linking the coil. Because the resistance is negligible, the time-constant of the shorted superconducting coil is also small and the current will follow the flux linking the coil almost exactly. Note: because we are dealing with superconducting wire which will have rate dependent losses and, consequently, losses that will reduce the correction efficiency or, if the change in field is too fast, cause the coil to quench. Not only will the current change with the field, it will also be persistent in a DC field if the Ohmic losses are very small. The decay time constant is L/R.

So, unlike an externally supplied copper coil, the superconducting corrector will not require any sensor to measure the field or continually require input energy to operate successfully – a great advantage. And the field produced by the corrector coil will oppose the unwanted harmonic component linking it. Any fluctuations in the magnet field during the day will also be followed by the current in the corrector.

Modelling this in Opera-2d and 3d is very simple using a transient electromagnetic simulation with a coupled circuit. In the circuit editor, the coil is shorted through a very small resistance representing the superconducting joint. In practice, this is often a superconducting switch which can be quenched to bring the current in the corrector back to zero, if required. In the simple 2d model, the boundary condition is ramped up linearly for 10 seconds and then maintained at a constant value. As can be seen on the graph, the current in the corrector coil follows this and remains at its persistent value after the ramp is complete.

The effect on the field can be seen. At larger radius than the corrector coil, the field still has an unwanted sine component. But, at radius smaller than the coil, the field is much closer to the required quadrupole. This is most easily seen on the Y-axis where the flux line is now lying fairly close to it. The uncorrected field is also shown again for comparison. This also shows that the next flux line (numbered 2) is more symmetric interior to the corrector coil. The shading of the corrector coil shows that positive current is flowing in the upper leg (coloured purple) and the negative current is in the lower leg (coloured blue).

Visually, the corrector appears to be effective, but it has not completely eliminated the unwanted sine component. With the 60 degree angle between the go and return legs shown here, the sine coefficient reduces from 4.077 to 1.421 mT at radius 2.95. To improve the corrector coil (which was modelled as a single turn but could in fact be a number of turns with small cross-section – the current is really the number of Ampere-turns), the effect of the angle subtended by the go and return legs was investigated.

Angle (deg) None 30 45 60 70 75 87
a2 (mT) -4.077 -3.017 -2.143 -1.421 -1.272 -1.270 -1.749


As can be seen in the table, the best correction is achieved with an angle around 72 – 73 degrees, even though the analytic solution says that 60 degrees will produce the best quadrupole field per unit current. However, the corrector coil itself will also have its own harmonics which feedback into the correction field The Opera-3d quadrupole with the coils shifted 1 degree also showed similar results with a 60 degree single turn coil. The reduction in the sine component integral field is only about 60% from 680 to 280 G-cm2.

However, the correction that can be made by using a number of turns in series at different angles is better, as shown in the below table.

Corrector coil Angles subtended (deg) Integral (G-cm2)
None 679.9
1-turn 60 277.8
3-turns 40, 60, 80 39.7
5-turns 36, 48, 60, 72, 84 5.0

The 5 turn model shown manages a reduction of better than 99%. To see if further improvement can be obtained, the 5 turn model was parameterized so that it could be used in the Opera Optimizer.

5 parameters were introduced in the model with the overlapping ranges shown above. However, using the constraints in the Optimizer prevents any of the coils from physically intersecting each other. There was a single objective function – the integrated sine quadrupole coefficient on a cylinder at 2.9 cm radius.
As there was only a single objective, there is only one design that is the “best” – rank 1 on the Pareto front (highlighted in blue in the left Optimizer results pane). The integrated sine harmonic is further reduced to 3.5 G-cm2 The equivalent subtended angles are: 81, 68.6, 62.8, 48.2 and 41.4 degrees.
The left picture shows the optimized coil set and the right the solution at one operating current for the quadrupole. Of course, even though about 99.5% correction has been achieved, the design can be further improved by introducing more parameters, such as the axial length of the turns and the shape of the end turns. It is worth noting that, in the 3d winding shown above, the current induced in the winding will stop any flux linking it. The correction of the integral field errors may be nearly perfect, but the length of the winding compared to the magnet and the design of the end windings, will strongly affect the local cancellation achieved.

So, to summarize, Opera is widely used in accelerator magnet design with both static and time-varying simulations, including co-simulation with circuits. Many post-processing tools available in the software assist the designer in this application. Opera can be used to design and model corrector coils which reduce imperfections in the field. Transient electromagnetic simulations are available in both 2d and 3d showing how the current in the superconducting corrector can track the off-axis quadrupole component to improve the quality of the main quadrupole field. Finally, there is importance in reducing the harmonics of the corrector coil itself and the Optimizer can be used to obtain a system of 5 series coils that achieves 99.5% correction of the off-axis component.

Thank you for reading, and please visit the link below or browse or website to discover more information that may help you with your design challenge.