Abstract
The design of low frequency devices relies on the existence of models which can allow a designer to estimate the performance of a proposed device before it is actually constructed. The advances in information processing hardware have enabled more detailed simulations to be developed but these do not provide the interactivity required for a true design process. The paper discusses the needs of the design process and the difficulties of using computer simulations to create real-world solutions to design problems. Several alternate ways in which the increased computational power can be used to enhance the design process are suggested.
Introduction
Since the initial discovery of the links between electricity and magnetism by Oersted [1], and the work of Ampere [2], Faraday [3] and Maxwell [4] plus many others, there has been a constant drive to develop devices which are based on the ability to use electromagnetic fields for energy transmission and conversion. Designing these devices is extremely complex and requires an understanding of the ways in which electromagnetic fields interact with materials and may be modified by environmental conditions such as temperature, mechanical stress, etc.
The advent of the digital computer provided the possibility of improving the simulation of the performance of these devices. As computational capabilities have improved (for a fixed cost for a workstation), simulations have become more and more detailed in what they can represent and nearer and nearer to reality. However, the time taken to complete a state of the art simulation has not decreased. If anything, it has increased and thus is difficult to use in an interactive design process.
The goal of this paper is to review the needs of the design process, examine the development of information processing hardware and discuss the possible uses of the gains in computing power to enhance the design process and thus reduce the overall costs of creating a device to meet a requirement.
Designing to meet specifications
Simple model of an electrical machine with the transfer function of the machine.
Model of the machine with the error feedback loop for design.
Usually, a device is specified in terms of its performance requirements. Figure 1 illustrates a system where the required device or system provides the transfer function necessary to map the inputs to the outputs, and the inputs and outputs are usually given in the specifications. This is a difficult problem to solve and, in general, has a non-unique solution. By definition, this means that the process is iterative and that changes in the transfer function, i.e. the device, need to be driven by the difference between the desired outputs for a given set of inputs and the actual outputs. This is illustrated in Fig. 2. The goal is to reduce the error between the desired and actual performance. To determine how to minimize the error, an effective model of the system is needed. This can be developed in many ways. The first, and maybe most obvious, approach is to construct a physical device and measure its performance. If this does not match the specifications, then the device structure must be changed. This leads to an obvious question – how does the designer know what to change? The second issue is the question of time and cost to do this – or when does the designer terminate the process? Constructing physical models is expensive and time consuming. Hence theories related to the operation of the device are needed and can form the basis of a “virtual” model. This can be used in conjunction with the physical model to provide guidance on how to change the design to move towards the goal. Developing the theoretical models needs considerable experimentation over a reasonable range of devices. In practice, aspects of the device performance are reduced to very simple systems which can be the basis of a series of experiments which, in turn, can lead to basic theoretical models. This, of course, was the approach taken in the nineteenth century to develop the basic laws of electromagnetics and resulting, ultimately, in the laws described by Maxwell’s equations [5]. However, while the physics may be understandable in a general sense, in most situations Maxwell’s equations (or the low frequency subset), are extremely difficult to solve due to the properties of the materials involved and the complex geometries needed to create the desired performances. The result has been the development of several high level models, e.g. equivalent circuit systems, which can act as surrogates for the real device and can be manipulated relatively easily without needing significant computer power. These are the original “virtual models”.
The virtual model is crucial to the design process but to be effective, it must achieve particular objectives. The basic goal is to replace the physical model in the transfer function component of Fig. 2. However, the virtual model is an approximation to the real world. As a consequence, it will contain inaccuracies which could result in errors in the design choices made and, as a result, a non-performant physical device. The skill in developing a virtual model thus lies in understanding the levels of accuracy needed. For example, an equivalent circuit model based on the concepts of the magnetic flux in a circuit containing permeable material can be used to generate an airgap flux distribution and the accuracy will depend on the level of detail included in the circuit. The airgap flux might, in turn, be used to estimate the generated torque from the machine and this could be done quite accurately. Similarly, an electric equivalent circuit might be constructed from a concept of energy loss and energy storage. Thus, for an electrical machine, losses happen in resistive components of the circuit (the windings, the core, etc.) and energy is stored in the magnetic components, the steel and air, that make up the device. In this case, the torque can be computed from a power balance related to the circuit of the machine where the mechanical output is just considered to be a resistive loss. However, when considered in the context of the model in Fig. 2, while it may be possible to determine which components of an electrical equivalent circuit need to change to meet performance targets such as the output torque, back emf waveform, efficiency, etc., it is often difficult to relate these back to the materials and geometric structures to be changed to meet the target. The advantage is, however, that they are fast and, if the designer is working by making minor changes to existing designs, the models can be calibrated to make extremely accurate predictions by comparing their predictions with existing measured data. However, if it is necessary to understand the implications of a small physical change on the device performance, or to link with thermal or structural performance, then the forms of model described can be too limited.
An alternate approach to constructing a virtual model is based on solving the field equations themselves subject to the material properties and geometric shapes of the magnetic and electric components. The issue here is that these are partial differential equations and not easy to solve in the general sense. Again, approximations can be made and algebraic solutions to specific structures found. These structures may, in some cases, be the entire problem or, often, a piece of the problem which can be characterized from its field solution. While the possible approaches to the solution of the equations by either discrete representation of the operators (finite differences) or by the use of local basis functions (finite elements), has been understood since the early twentieth century, the solution at a suitable level of detail for an effective model was not possible until the advent of digital computers. These, in turn, generated the need for a new set of approximations, e.g. solutions in two dimensions or three, representing materials as non-linear or linear, the level of discretization, etc. and in most cases the issue is, as always, trading approximation accuracy for time and cost of achieving a solution. If the virtual model is to replace the physical model in Fig. 2, it must be fast enough and sufficiently accurate to allow the iterative design process to work effectively.
Numerical simulation and computer capability
As described in Section 3, the virtual model can be constructed using a discretization system which can result in the values of thousands (or, currently, millions) of variables being computed. Given a suitable computer capability, the main issue is the time it will take to generate the solution. The development of simulation capability in electromagnetics has very much followed the capacity of the available computer hardware. For example, In the late 1970’s processors were becoming cheap enough that it was possible to envisage a desktop computing system available to every design engineer. Up to that point, computer based solutions to the design problem had been handled on mainframe machines with limited access for designers. However, these machines were relatively small (a typical system around 1980 had about 64 kbytes of memory and 720 kbytes of removeable storage). This enabled the solution of about 2000 degree of freedom problems in about an hour. The problems were restricted to two-dimensions largely because the number of degrees of freedom available could not adequately represent a three-dimensional problem. However, the solution time was such that an iterative design process was difficult and the analyses were usually used to confirm a design performance rather than in the process itself.
Gain in processing power for a micro-processor since 1978 in a constant cost ($4000) workstation.
One immediate benefit of even this level of model was that it could indicate areas where losses were likely to be large, or material was being underused.
By the early 1990’s, for about the same cost for the workstation, it was possible to solve around 30,000 degrees of freedom making magnetostatic, three-dimensional solutions possible. While a long way from a true simulation, the restrictions on the virtual model were being removed. However, the time to solve this level of problem was still measured in a few hours, while the 2000 degree of freedom problem for two-dimensions was solving in minutes – making an interactive, iterative design process feasible.
In general, if the progress in processing capability, installed memory sizes and disk sizes for a constant cost are considered, all of these are growing at about 40% per year (Figs 3, 4 and 5). The growth in intercomputer communication speed has been somewhat lower than this and implies that it is taking longer and longer to move a maximal size problem between machines. This is important as networks of processors are constructed to create high performance computing facilities.
This growth has reached a point at which the solution of several million degree of freedom problems including material non-linear effects, time harmonic and time transient excitations, links to thermal and structural systems, etc. have become possible. However, these solutions can take several hours to compute and are consequently used to check designs – i.e. test them in a virtual laboratory following much the same procedures as would be used with the physical device, rather than implement them as part of an iterative design process where the intention is to explore the design space looking for the desired solution.
Increase in size of the installed random access memory since 1984 in a constant cost ($4000) workstation.
Increase in size of the installed disk since 1984 in a constant cost ($4000) workstation.
As computer power has increased, much of the main thrust of development has been to try to improve the simulation of reality, i.e. to increase the accuracy of the simulation in an attempt to exactly predict the performance of the device to be constructed. However, there are other ways in which the gains in computational capability can be leveraged to assist more effectively in the design process. For example, iterative solvers were developed in the 1970’s to overcome the memory requirements of direct solution systems, which seriously limited the problem sizes to something which was too small to be of use, given the capabilities of the computer systems available. With modern memory sizes, it is quite feasible to solve relatively large problems with direct rather than iterative technologies. This can give a significant speed gain for the solution of “smaller” problems. Thus, if a two-dimensional simulation is considered, it is now possible to solve tens of thousands of degrees of freedom in two-dimensions in a few seconds. For many problems in electrical machines, a two-dimensional simulation is sufficient for most of the design process – the third dimension is only important in the end regions and these can often have only a minor influence on many of the machine performance parameters.
This use of the increased computation capabilities may, in fact, often be more effective in the design process by working with lower fidelity models, than trying for highly “accurate” three-dimensional solutions. Unless the relationship between the performance parameters and the design parameters is fully understood, a three-dimensional solution can be over-kill and may not provide the insights needed for effective design.
This, then, leads to the question (at least in the design context) of what level of simulation is needed to sufficiently represent reality? If a design process is being implemented, is an accurate solution needed or is a solution which correctly indicates the trends in performance which result from changes in design parameters more appropriate? Finally, how well does simulation do compared to reality?
There are two reasons for performing simulation. The first is to provide an accurate prediction of the performance of a real device; the second is to correctly predict the trends in performance as a result of changing the design parameters. In other words, correctly identifying the shape of the response surface (or transfer function) relating the inputs in Fig. 2 to the outputs is more important for the design process than the accuracy of the simulation, i.e. it is possible to work with an accuracy level of 5% if the shape is correct to 1% but an accuracy of 3% on average with a response surface shape that does not match the real device can cause instabilities in the design process.
Measured and Simulated torque-speed curves for the TM4 machine.
12 slot, 8 pole permanent magnet machine.
Back emf waveform at 10,000 rpm.
The current level of technology allows the solution of the partial differential equations representing the field distributions with a very small error – many of the issues of numerical instability and round-off problems have been solved with larger memory sizes and word lengths. A solution to the equations to a fraction of a percent is quite feasible. However, the equations represent relationships between field quantities and these are controlled by the material properties which are often inaccurate. Measurements on nominally the same material performed on different production batches may well show differences in the losses and relationships between magnetic flux density and magnetic field of more than 10%. Models exist for representing the non-linear and hysteretic behavior of the material but without the characterization data, they cannot provide accurate performance predictions for the simulation. Similarly, errors in the manufacturing process can mean that unintended changes in geometry and errors in the excitations lead to variations in the flux density waveforms and thus differences in torque and loss calculations between the predicted and measured results. Once all these have been measured and the information included, it is likely that the simulation will match reality but the problem with design is to predict the performance of a device before it is built and measured. The following examples illustrate some of the issues commonly encountered.
All the examples shown were simulated using the same tool set, i.e. MotorSolve and MagNet from Infolytica Corporation [6]. They were solved to the same degree of accuracy using measured material data. The first example is a commercially available (TM4 HSM20–MV80) traction motor [7]. It is a surface mounted, exterior rotor permanent magnet machine. At 10,000 rpm, the line-to-line peak back emf was measured to be 417 volts and the simulation predicted 420 volts. The torque speed curves for both real machine (measured) and the simulation are shown in Fig. 6. As can be seen, the agreement is acceptable given that this is a two-dimensional simulation and thus is ignoring the end effects.
Surface mount 12 slot, 10 pole permanent magnet machine.
Back emf waveform at 3000 rpm for the 12 slot, 10 pole machine.
The next machines considered were designed to be relatively low cost motors intended for use in university laboratories and were produced by MotorSolver [9].
Figure 7 shows a 12 slot, 8 pole permanent magnet machine. Again, the back emf waveform was measured and also simulated using exactly the same settings in MotorSolve as the machine in the previous example. In this case, the error in the peak value of the back emf is nearer to 15% and the waveform shapes do not match. A similar problem is seen in Figs 9 and 10 – a surface mount permanent magnet machine with 12 slots and 10 poles was measured and simulated and the results show similar errors.
12 slot, 4 pole interior permanent magnet machine.
Measured and simulated back emf waveforms for the 12 slot, 4 pole interior permanent magnet machine.
Finally, Figs 11 and 12 show the measurements and predictions for a 12 slot, 4 pole interior permanent magnet machine (IPM). In this case, the waveform is much more complex in terms of harmonics but the measurements and simulation match to better than about 7%. Significantly better than the previous two machines. If all the simulations used the same software, the original design drawings, the same data from the materials database, and were solved with the same accuracies, why are some good and others bad? Can the simulations be trusted in the design process?
The issue here is that the machines that were built did not match the specifications given in the materials databases and the drawings – there were manufacturing errors which could not be included in the design process. In this case, the permanent magnets were not magnetized either with the strength that the materials datasheets had suggested or in the directions that the design had specified. For the machine performance shown in Fig. 6, the machine was constructed carefully and the magnetization and performance of the magnets was controlled, as were the properties of the magnetic materials and the winding shapes. In general, if the manufacturing process is controlled, then the simulation results achieved in the design process will accurately predict the performance of the device and using the increase in computational capability to increase the accuracy of the simulation is appropriate. However, if the machine is built to a minimum cost and with less control on the manufacturing process, then the use of the increased computational capability to improve simulation accuracy is probably not appropriate. However, using the computational power to explore the design space using less accurate solutions could allow the design process to proceed in a truly interactive manner. In addition, providing the manufacturing processes are stable and the type of machine being designed is always similar, then “calibrating” the software to incorporate the errors might be appropriate.
Fractional slot concentrated winding IPM.
Variations of torque with airgap and magnet dimensions for 500 possible designs.
Clearly there exist many challenges in simulation. Designing an electromagnetic device is a complex problem and the design approach taken varies depending on the goal of the process. Consequently, if the goal is to produce a “one-off” design for an extremely expensive device, e.g. a generator transformer, then accurate simulations are needed and all aspects of the device operation must be considered. However, if the goal is to generate a design for a mass-produced device, then the questions to be answered are likely to involve the manufacturing process and the uncertainties involved in it. So, is the real simulation challenge to produce ever more accurate simulations or to develop systems which can allow for the stochastic variations in the manufactured product, or both?
The area of “robust design” requires that the increased computational power be used to consider parameter variations in the simulation. Approaches include the concept of stochastic finite elements [10], worst case analysis [11] or, simply, an exhaustive search of the design space followed by a statistical analysis on the data obtained.
In developing simulation systems, three questions are crucial. First, can the problem be solved to obtain sufficiently accurate answers for the design process? Second, can the problem be solved more efficiently? Third, how does the solution being generated fit in the design process?
One example of using the computational power to explore the design space is related to the development of high performance computing where multiple workstations can be connected into a network and it becomes possible to consider thousands of design variations of an electrical machine in the space of a few hours. The data extracted from this process can provide a design engineer with significant information on how to guide the design process. Figures 13 and 14 show an example of a parameter space exploration for a fractional slot concentrated winding IPM. This involved varying 7 input parameters and generating the torque and efficiency of the machine for each of 500 devices.
Conclusions
The developments in information processing hardware over the past four decades have enabled highly detailed simulations of electromagnetic devices to be developed. However, there are other possible ways to use the increased power in the implementation of an effective design process. The paper has discussed some of the possibilities and has indicated possible areas for future research and development.
Footnotes
Acknowledgments
The work was supported by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada.