Article: Paper Introduction
Issue 1 Induction Motor Simulation Technology
What kind of impression do you have of induction motors? Do you think of them as traditional motors that have been widely used for decades? The induction motor was invented in the 1880s  and has been applied in a variety of areas, ranging from the industrial to consumer fields. The question, then, is whether the induction motor's design has been completed to the point where there is no longer any room for improvement. As can be seen from the number of cases in the JMAG application notes, permanent magnet synchronous motors, not induction motors, have been the main target of electromagnetic simulation over the last twenty years. One reason for this is that permanent magnet synchronous motor designers need to use electromagnetic simulation to study their designs in detail because this type of motor was developed relatively recently. In addition, electromagnetic field simulation methods like the finite element method, which can handle magnetic saturation with precision, are necessary because the saturation effects in the magnetic circuits of permanent magnet synchronous motors need to be taken into consideration, as even more powerful rare earth permanent magnets have been developed in recent years.
However, we feel that the situation is going to begin to change because of issues with the availability of the rare earth elements used for rare earth permanent magnets. As their name suggests, rare earth elements are distributed unevenly across the globe, and their availability and cost are not stable. This is why induction motors that do not use rare earth materials have begun to garner attention, even in fields that require high output density and efficiency like hybrid/electric vehicles, which used to be dominated by permanent magnet synchronous motors . Because the aforementioned fields require high output density and efficiency, new induction motor designs that differ from those of conventional induction motors are seeing an increase in demand, so the needs for design simulation are rising as well.
The Difficulty of Induction Motor Simulation
The next question is whether or not an electromagnetic field simulation of an induction motor is easier or more difficult than a simulation of a permanent magnet synchronous motor. Though they have many points in common, an induction motor simulation can be more difficult for the following reasons.
(1) A large number of time steps is required in a transient response analysis
This is because:
- A lot of time steps are required to get results in a steady-state when there is a long time constant of the induced current in the secondary side.
- A lot of time steps are required to get results for one period of 'slip frequency' when the slip is small. For example, when a synchronous frequency is 50 Hz and the slip is 0.01, a calculation for 2 seconds is necessary.
While the qualitative characteristics of an induction motor can be calculated in a frequency response analysis, there are problems like the fact that this kind of analysis cannot consider the effects of harmonics, as well as the fact that the analysis accuracy also decreases when the magnetic saturation is strong.
(2) The three-dimensional effects in a magnetic field are strong
- EA squirrel-cage induction motor has an end ring in its secondary conductor, but a two-dimensional analysis cannot consider the effects of end rings.
- It is common to implement skew to reduce the influence of harmonics, and it is necessary to consider the effects of this skew in a simulation.
- The losses in induction motors are increased by 'inter-bar current' which passes between two bars. A two-dimensional analysis cannot consider this phenomenon.
The first reason means that a two-dimensional analysis with a short calculation time is best, but the second reason means that you cannot get accurate results with a two-dimensional analysis. These are the main dilemmas involved.
Approaches to These Problems
I would like to take this opportunity to introduce papers that discuss what kinds of approaches have been carried out so far to solve these problems.
Eliminating Transient Phenomena
The EEC method has been developed as a technique to let transient phenomena converge into steady-states for analyses like induction motors that can have long transient states. This method allows the simulation to reach a steady-state earlier than an actual induction motor would. JMAG has adopted a similar technique. The EEC method can eliminate a transient phenomenon by adjusting the solution vector of the finite element method for every constant period. In paper  , the authors introduce an example that applies it to induction motors and permanent magnet synchronous motors, and they show that, in the case of an induction motor, the analysis time before obtaining a steady-state result can be reduced to less than 10%. In addition, JMAG can use a method of converging a transient phenomenon quickly into a steady-state@by starting a transient response analysis from an initial state that is close to the steady-state given by a frequency response analysis. This method is called 'Steady-State Approximate Transient Analysis.'
Considering End Rings
The end rings of a squirrel-cage induction motor influence the motor characteristics from the perspectives of secondary resistance and leakage inductance. It is well-known, however, that a two-dimensional analysis does not consider the end rings, meaning that it cannot provide accurate results. It is better to carry out a three-dimensional analysis that models the end rings, but a three-dimensional analysis takes an enormous amount of calculation time. For this reason, various kinds of technique have been examined in order to use a two-dimensional analysis to consider the influence of the end rings.
The simplest methods either revise the electric conductivity of the material characteristic of the bars to match the resistance of the end rings, or treat the cage like an equivalent circuit and the end rings as its lumped parameter (resistance)  . JMAG's 'Cage Macro Component' function can create an equivalent circuit for the squirrel cage that handles the end rings as resistance. These techniques are simple, easy to introduce and are widely used because they allow the value of the revised electric conductivity or the end ring resistance to be obtained easily from the geometry of the end rings.
However, there is a problem that the resistance is underestimated when it is given from a geometry, because the current density in the inside of the end rings has distribution that depends on their shape or its own frequency, and is therefore not uniform.
The leakage inductance from the end rings is not taken into consideration, either.
With paper  , I would like to introduce a technique that considers the current distribution in the end rings and obtains the revision amount of the electrical resistance. This method first evaluates the voltage drop from the end ring resistance in a three-dimensional analysis, and then uses that result in a two-dimensional analysis to determine the revision amount of the electrical resistivity in the bars. The three-dimensional analysis only models the bars and end rings, and it is performed with a linear frequency response analysis. This means that it can derive the revision amount in the two-dimensional analysis without using much time for the calculations.
The 'Multi-Slice' method is known as a technique that uses a two-dimensional analysis to consider the effects of skew. In this method, two-dimensional analyses are carried out on several sections that are perpendicular to the rotation axis of an induction motor. Fig.1 shows an outline of this method. For example, in paper  five slices were taken out. In  , these five two-dimensional analyses are unified into one matrix and solved at the same time. At this time, the circuit equation and the magnetic field matrix are coupled and solved so that the primary current and secondary current flowing through the bars are saved by each sectional analysis. Methods like this make it possible to incorporate the effects of skew in a much shorter calculation time than with a three-dimensional analysis (On the computers of the time it took about 200 seconds). As a result, it was shown that skew can greatly reduce the harmonics in the primary current waveform. However, this method ignores magnetic flux in the axial direction caused by skew.
It is necessary to increase the number of slices in the multi-slice method to express the effects of skew more precisely, but on the other hand, doing this brings about an increased calculation time. To solve this problem, paper  applied the 'Transmission Line Modeling (TLM)' method to prevent an increase in calculation time by parallel computing on multiple PCs. The result was that the calculation time with seven slices hardly changed in comparison with one slice.
Fig.1 Outline of the multi-slice method
Analyzing Inter-bar Current
It is necessary to model electromagnetic steel sheets between bars to analyze the phenomenon of inter-bar current. However, it is almost impossible from the perspective of the calculation time scale to model an induction motor's structure precisely in three dimensions, because several dozen to several hundreds of pieces of electromagnetic steel sheet are usually laminated. Paper  provides an example of an analysis that simulates the inter-bar current in a practical amount of time. First, a two-dimensional transient response analysis is used to calculate the spatial distribution and time variations of the magnetic flux density in an induction motor's air gap. This data is input into a three-dimensional analysis model as a boundary condition for the magnetic flux density, and then the inter-bar current is calculated in a three-dimensional frequency response analysis. Although each of the laminated electromagnetic steel sheets is modeled in the three-dimensional analysis, the calculation time can be reduced by considering the motor's periodicity and modeling only 1 rotor slot pitch, as well as by calculating only the main frequency component given by a Fourier transformation of the results from the two-dimensional transient response analysis. As a result, it was shown that total losses of the induction motor with skew increase slightly compared to a motor without skew because, while skew reduces the copper losses in the squirrel-cage, it increases the losses from the inter-bar current.
The Future of Induction Motor Analysis
When induction motors expand into fields where permanent magnet synchronous motors are currently widely adopted, such as hybrid or electric vehicles drives, what kinds of analysis technology will become necessary? There are several possibilities.
(1) Magnetic saturation becomes severe
When induction motors begin to be applied to fields that until now have been dominated by permanent magnet synchronous motors, they will be required to have output density equivalent to a permanent magnet synchronous motor. This density, however, could potentially cause magnetic saturation in the magnetic circuit in an induction motor. Until now, the main approach has been keeping the influence of magnetic saturation confined in the equivalent circuit constants of an induction motor that is expressed in an equivalent circuit like a T-type circuit. For example, in , the starting calculation of an induction motor was performed after considering the influence of magnetic saturation on leakage reactance (Leakage reactance usually decreases when magnetic saturation occurs). The authors said that it is important to consider magnetic saturation because line starting causes a high current to flow at start-up.
In contrast, it does not appear that there has been sufficient testing to utilize electromagnetic field analysis effectively on the machine design side, or to optimize a magnetic circuit design that considers magnetic saturation. These tests, however, are being performed with permanent magnet synchronous motors. Will the role of electromagnetic field analysis grow larger in the future as induction motors attain higher output density?
(2) Vibration, noise analysis become important
Motors in hybrid/electric vehicles require a level of quietness that is higher than in motors for industrial or home appliances, so techniques to analyze and reduce the vibration and noise of induction motors will likely become increasingly important. Like with a permanent magnet synchronous motor, vibration and noise in an induction motor are thought to be caused primarily by resonance between the time variation of the magnetic attractive force on the stator and the mechanical eigenmode. The kind of frequency or mode shape in an induction motor's magnetic attractive force has always been inspected theoretically, or by using simulation. For example, in , the frequency and mode shape of the magnetic attractive force that can occur in a small induction motor were expressed using a numerical formula. Then, the authors measured the state of the vibrations in an actual induction motor that was running, and compared the sets of results to consider the sources of the vibration as well as the relationship between the magnetic attractive force and the mechanical eigenmode. As a result, the following became clear.
·Vibration at high frequencies is generated by time variations of the magnetic attractive force, caused by the interaction of the slots in the stator and the rotor.
·When vibration occurs because of resonance with a mechanical eigenmode, the induction motor vibrates with the number of modes of the eigenmode, regardless of the number of modes of the magnetic attractive force.
In addition,  shows an example of investigating the influence of skew on the magnetic attractive force by using the multi-slice method introduced in the `Considering Skew' section.
As indicated here, there are many examples of using a magnetic field analysis to calculate the magnetic attractive force or to measure eigenmodes and vibration, but there are not yet many examples that use a magnetic field-structural coupled analysis to examine the vibration and noise of induction motors.  shows a case in which a two-dimensional transient response magnetic field analysis was coupled with a modal method vibration analysis. After that, the actual vibrations were measured and compared with the analysis results at every frequency. Although there are places where the magnitude relationship of the vibration does not match the measured values because a two-dimensional analysis was used, the frequency where vibration was activated agrees well with the measured frequency. In addition, the mode shape of vibration that was activated by resonance is similar to the mechanical eigenmode, as shown in .
In this issue, I examined papers about induction motor simulation, and felt that this is a field where electromagnetic field analysis is not being utilized as much as it is with permanent magnet synchronous motors. I hope that in the future, electromagnetic field analysis will be utilized for the design of new induction motors with high performance.
 H. Katagiri, "Improvement of Convergence Characteristic for Steady-State Analysis of Motor With Simplified Singularity Decomposition-Explicit Error Correction Method," IEEE Tran. on Magn., vol. 47, no.6, 1786-1789, 2011
 S.L. Ho, "A comprehensive approach to the solution of direct-coupled multislice model of skewed rotor induction motors using time-stepping eddy-current finite element method," IEEE Tran. on Magn., vol. 33, no.3, 2265-2273, 1997