Particle accelerators, such as the Large Hadron Collider (LHC) at CERN, make use of dipole magnets to steer the particle beam. The field is essentially unidirectional between the magnet poles and its interaction with the orthogonal particle beam causes the beam to bend. Many of these magnets are operated with dc current to maintain the particle beam in a circular orbit. However, some of these devices are also required to operate in a fast pulse mode to allow injection and extraction of the beam to and from the main accelerator. One type of such a device is known as a septum magnet.
In this note, the operation of a septum magnet, while connected to its drive circuit, is examined using the Opera-3d transient electromagnetic simulation module. This simulation solves the transient diffusion equation in the presence of materials that may be both magnetic and electrically conducting using a time-stepping algorithm and the well-known finite element method. It also allows co-simulation of coil components within the model when they are connected to their electrical circuit. A particularly important feature of Opera’s implementation is that skin and proximity redistribution effects in coils can be captured by the finite element solution as well as those in surrounding structures.
The septum magnet
The geometry of a typical septum magnet is shown in figure 1.
As can be seen, the construction of the magnet is quite simple. The laminated magnetic yoke (in green) is C-shaped and the coils (in blue) sit in the gap between the poles, producing a homogeneous dipole field between the poles. The ends of the coils have been bent upwards to allow the injected or extracted beam tube to pass between the poles, giving the familiar “bedstead” shape. The main particle beam propagates outside the magnet, in the so-called free-field region, where the field from the septum magnet must be minimal. The part of the coil towards the outside of the gap forms the separator – or septum – between the beams from which the magnet derives its name. The overall dimensions of the magnet are 60 x 60 x 40 cm and the gap between the poles is 16 cm.
To allow the field to rise rapidly in the magnet, it is necessary for it to have a low inductance. This is achieved by making each of the upper and lower coils from a single turn of 8 x 2 cm cross-section copper. However, for fast pulses, this will inevitably give some redistribution effects to the current in the coil such that it cannot be assumed to have a uniform density in the cross-section. Consequently, the resistance and inductance of the windings will vary during the pulse – affecting the time constant of the magnet and the rise and fall time of the field.
The upper and lower windings are connected in series and are driven by discharging a capacitor into them. The circuit for the windings and for re-charging the capacitor for the next pulse is shown in figure 2.
The operation of the circuit is such that initially the capacitor, C1, is charged to 200 V and all the switches are open. Note that this is the same voltage as the dc power supply, V1 – that is, switch S3 is assumed to have been closed (while switches S1 and S2 are open) at some time previous to this to allow the capacitor to become fully charged. Switch S1 closes allowing capacitor C1 to discharge through the windings, B1. S1 remains closed for 0.5 ms, after which S1 opens and switch S2 closes. The current in the coil now flows through the dump resistor, R1, and, consequently, decreases as energy is dissipated in R1 and B1.
After another 0.5 ms switch S3 closes allowing capacitor C1 to re-charge. A further 0.5 ms after that, switch S1 closes, the other switches (S2 and S3) open and the operational cycle repeats.
The purpose of this model is to determine how the resistance and inductance of the winding vary during this operation, the consequent variation of power dissipation in both the winding and the dump resistor, and the time profile of the field at the centre of the magnet during the pulsed operation.
Opera-3d Modeller has been used to create the finite element mesh for this simulation. Particular attention has been paid to the mesh in the windings, and Opera’s mosaic meshing facilities have been used to obtain a mesh that is constructed of 8 x 0.25 cm layers. As will be seen, during the initial rise of the pulse, current is largely restricted to the inner part of the winding. However, during the time when the current in the coil is reducing, there is actually some current reversal on the inner part while the main transport current is carried towards the outer part. Consequently, very accurate meshing is required and these effects are better captured using Opera’s hexahedral and prism elements. Most of the remainder of the model is meshed with tetrahedral elements, while pyramid elements are used to join tetrahedral elements to the 4-sided face of a prism or hexahedron. Opera-3d Modeller accomplishes this automatically – the user only defines which parts are to be meshed using hexahedra and prisms.
The geometry of the model exhibits 4-fold symmetry and this can be exploited by applying appropriate boundary conditions to the mesh to imply the reflections. Figures 3, 4 and 5 show the mesh in the coil, the mesh in coil and yoke and the overall mesh including free space.
The mesh contains a total of just over 4 million finite elements. Since the yoke is made from laminations, it has been assumed that its electrical conductivity can be neglected in this simulation. The electrical conductivity in the copper winding is set at 50 MS/m.
Connecting the coil to the circuit
The Opera-3d Modeller includes an interactive tool for creating the schematic of the circuit and defining the operation of the switches etc. For example, switch S1 has been set with a function:
and a condition:
In words, this means the function and condition are setting the switch to be closed during the first 0.5 ms of the simulation or after 1.5 ms. Otherwise, it will be open.
The Opera-3d transient electromagnetic simulation performs a time-stepping solution. At each time-step, the software simultaneously solves Maxwell’s equations in the finite element mesh and the circuit equations. In this model, a fixed time-step of 20 µs is used. Figure 8 shows the variation of the current in both the winding of the magnet and in the dump resistor, while figure 9 shows the voltage across the winding, the dump resistor and the capacitor.
As can be seen, the current in the winding does not completely discharge through the dump resistor during the period 0.5 to 1.5 ms. Consequently, when the capacitor is re-connected at 1.5 ms the current in the winding returns to a higher value in the subsequent 0.5 ms capacitor discharge. It would be necessary to run several more cycles so that the steady state operation can be achieved.
The result of this is that the flux density in the magnet is also higher at the end of the second current ramp. Figure 10 shows the y-component of flux density at the centre of the magnet gap as a function of time during the ramping / discharge / ramping process. It follows the same trajectory as the current in the winding since the magnet is operating in the linear part of its magnetic characteristic.
Ideally, the winding current and flux density would have exhibited periodic behaviour even after the first cycle – that is the current in the winding would effectively reduce to zero by 1.5 ms. This could be easily achieved, of course, by making the dump resistor a higher resistance to reduce the time-constant of the circuit. However, as can be seen in figure 11, the power dissipated in the dump resistor already peaks at about 5.5 MW and nearly all of the 1180 joules of stored energy at 0.5 ms. is dissipated in the dump between 0.5 and 1.5 ms. (Compare this with the amount of power dissipated in the winding – also shown in figure 11. The winding dissipation is calculated from the Opera-3d Post-processor ENERGY command.) Using a higher resistance will cause the energy to be dissipated in the resistor even faster, , requiring more highly thermally rated components or a better cooling system for the dump resistor.
The power dissipated in the winding is examined in more detail in figure 12, as this shows how important it is to consider the redistribution of the current in the winding. The power dissipated including redistributive effects is compared with the power that would be expected if the same current flowed through the dc resistance of the winding (42.1 µ). When the voltage is applied to the winding at the start of the simulation, the current is not able to flow across the entire cross-section of the winding due to skin effect. Hence, the resistance seen by the current is higher. This can be seen in figure 13, which shows both the actual resistance of the winding and the dc value. This time-varying resistance, R(t), is evaluated in the Opera-3d post processor as:
where W(t) is the power dissipated in the winding and iB1(t) is the current through the winding. Similarly, the inductance of the winding (as shown in figure 14) can be determined as:
where vB1(t) is the voltage across the winding.
As can be seen, the effective resistance of the winding varies from nearly 14 times the dc resistance at 0.02 ms to only twice the dc value at around 0.7 ms but then reaches almost 16 times the dc value at 1.5 ms. As mentioned earlier, this is due to skin and other eddy current effects and can be clearly seen in figure 15, which shows the current density in the outer limb of the winding at various times in the transient. The graphs show the variation of axial (beam direction) current density on a line defined by the intersection of the vertical and horizontal symmetry planes (which is the X-axis in the Opera-3d model). Distance = 0 cm corresponds to the inside of the limb (that is, the gap side) and distance = 2 cm corresponds to the exterior of the limb.
When the capacitor is reconnected to the winding and the current starts to increase again (1.5 ≤ time ≤ 2.0 ms), the effect of the circulating eddy current is to reduce the current density at the interior of the winding – compare the curves at 1.6, 1.8 and 2.0 ms with those at 0.1, 0.3 and 0.5 ms respectively – even though the transport current is actually higher (as shown in figure 8).
Although the resistance of the winding varies considerably during the simulation because of the redistribution effects, the inductance is relatively stable – varying only between 3.85 and 4.31 µH. Because the magnet yoke is operating in the linear (unsaturated) part of the magnetic characteristic, the inductance is dominated by the air-gap flux. Figure 16 shows a flux density map in the axial mid-plane of the air-gap at 1.0 ms while figure 17 shows the same map at 2.0 ms. As can be seen, the patterns are quite similar even though the operating current is very different. Figure 18 shows the variation of the ratio of flux density to winding current with distance across the gap on the X-axis. Distance = 0 cm corresponds to the inside of the inner limb while distance = 26 cm is the inside of the outer limb. Since inductance is defined as flux linked per ampere, it can be seen that the ratio is fairly constant at all operating currents. The effects of the redistribution of the current are more significant near the outer limb than at the inner limb. Hence, the stability seen in the inductance.
The Opera-3d model of the pulsed septum magnet demonstrates the importance of including the current redistribution effects in “massive” conductors when they are operated within fast transient regimes. Opera-3d includes a transient solution of the electromagnetic diffusion equation which can be coupled to Opera’s circuit simulation. Massive conductors used as windings for magnets and other electromagnetic devices represented by volumes of the finite element model can also be included in the circuit equations. Derived values such as resistance and inductance of the winding can be determined as well as important design parameters such as power dissipation in the winding and other circuit components.
The designer also has access to the field simulation results throughout the model, such as flux and current density. This gives insight into the circuit results, relating values such as effective resistance to physical phenomena, and would enable an improved design to be developed.
Other applications where current redistribution in windings can be significant include:
- Bus-bars in high power electrical supply systems such as in metal production
- High frequency transformers
- Fusion energy experiments
“Coupling eddy current and circuit equations – a finite element implementation”, A. Venskus, M. Hook, J. Simkin, C.P Riley, UK Pulsed Power Symposium, Loughborough University, March 2014