Electromagnetic, thermal and stress analyses of RF cavities in Opera-3d

Introduction

Many high frequency devices operate at high power, and material losses can lead to significant rise in temperature if left unchecked. In such devices, it is important to know the temperature distribution to determine the performance impact of changes in dimensions and material properties, to ensure that the permitted maximum operating temperature and stresses of the material are not exceeded, and if required, to enable a thermal management system to be designed. Clearly, the simulation of these effects requires the ability to perform electromagnetic, thermal and mechanical analyses on the same physical model. For some time, Opera-3d has been able to perform such Multiphysics simulations, a feature of the tool being that all relevant results are passed from one stage of the analysis to subsequent stages without user intervention. This includes the deformation resulting from the thermal and stress analyses – allowing the effect of dimensional changes to be evaluated immediately. Recent developments have added features that are aimed specifically at accounting for effects that are relevant to high frequency applications.

RF devices often contain various materials of low and high conductivity. The former might be, for example, intentionally lossy dielectric intended to act as a load; the latter includes metallic structures whose loss is unavoidable. The evaluation of thermal loss requires knowledge of the distribution of current within the material. The current decays at a characteristic rate with depth, dependent on the frequency and material properties – leading to the concept of skin depth. Since Opera-3d analyses use finite elements, the normal requirement is that the mesh element size perpendicular to the surface must be small enough to resolve the skin depth.

For materials of low conductivity, less than a few 10’s S/m, the skin depth is of the order of millimetres, even at GHz frequencies. Such distances are typically commensurate with the physical dimensions of RF components, and the current distribution may be captured relatively easily with a practicable mesh size. Even when the physical dimensions are considerably larger than the skin depth, Opera-3d includes features, such as mesh layering and hexahedral/prism elements, that enable these structures to be meshed with high aspect ratio volume elements. However, as the conductivity increases towards that of typical metals, the skin depth becomes so small that these tools are unable to provide adequate resolution of the current distribution, and a different approach must be used.

In Opera-3d, the approach taken is to represent the material properties by a surface impedance boundary condition (SIBC). In this, the surface current density is calculated, and the loss power density then evaluated from the known decay of the current with depth; where the skin depth is much smaller than the physical dimensions, this is an excellent engineering approximation.

The example shown below illustrates Opera’s ability to evaluate the dissipation in devices containing both high and low conductivity materials, and to couple high frequency EM results to thermal and stress analyses. The example chosen is an RF quadrupole (RFQ), as often used in low energy particle accelerators.

RF Quadrupole

Magnetostatic and electrostatic quadrupole structures are often used as focussing elements in particle beam devices and as the analysing component in spectrometers. A modification to the electrostatic quadrupole – introducing a ripple on the pole-pieces – gives the ability to focus, bunch and accelerate particles when driven at the correct frequency. However, many examples of such devices operate at high power, and the design must mitigate the effects of losses on the temperature and deformation of the device.

The cut-away view in figure 1 shows the structure of the RF quadrupole (RFQ) used in this example. It comprises a set of shaped poles and RF feed probes sitting within a vacuum chamber; cooling tubes are attached to the outside of the vacuum chamber. Most of the structure is made from high conductivity material, and the SIBC representation is used for these in the EM analyses. The dielectric in the coaxial feed probes is meshed.

Figure 1  RFQ geometry

Figure 1 RFQ geometry

In the first part of the analysis, the correct modal frequency is determined using Opera’s EM eigenvalue solver, known as Modal HF. The model is then driven at this frequency in the Harmonic HF solver, followed by thermal and stress simulations. These three analyses are performed using Opera Multiphysics. Figure 2 shows the electric field amplitude when the original geometry is driven harmonically. The power dissipation from losses in the components is automatically passed to the Opera thermal solver, and results in the temperature profile, also shown in figure 2. Opera Multiphysics then passes the temperature profile to the stress solver, where stresses and deformation are calculated. The deformation is apparent in Figure 3, where the deformed geometry – exaggerated and depicted in mesh outline – is overlaid on the original geometry. In addition to deformation, the components of stress and strain are also available.

The deformed geometry from the stress analysis is available for use in subsequent analyses to assess its effect on performance.

Figure 2  Electric field amplitude and temperature profile

Figure 2 Electric field amplitude and temperature profile

Figure 3  Deformation of the geometry from thermally induced stresses (shown as the mesh outline)

Figure 3 Deformation of the geometry from thermally induced stresses (shown as the mesh outline)