**By R. Belmans**

## Examples of computed models

**In this topic,** the modelling of realistic technical devices will be demonstrated on a selected set of examples. All simplifications applied to the models will be motivated and discussed.

## Synchronous machine excited by permanent magnets

**One of the most popular types of** electrical machine used for servo drives is the permanent magnet-excited synchronous motor. The properties of this type of machine are a high efficiency and dynamic which combined with controlled inverters offer advantages when compared to other drive systems. High-energy permanent magnet material, such as the rare earth grades Samarium-Cobalt (SmCo) and Neodymium-Iron-Boron (NdFeB), enable various designs that have already been discussed in the literature (Fig 10.1, Fig. 10.2).

**Fig. 10.1. Synchronous machine designs with corresponding FEM mesh of a 6-pole machine with inset magnets.**

**The previous topics have shown that** the appropriate choice of model simplifications depends on the aim of the analysis. The manufacturer of the machine might be interested in the local saturation level inside the machine, whereas the application engineer requires the lumped parameters to define the controlled machine’s behaviour in the dynamic drive system.

**Fig. 10.2. Synchronous machine designs with corresponding FEM mesh of a 4-pole machine with buried magnet system.**

**It is assumed here**, that the d/q-axis theory in the analysis of permanent magnet machines is known. Fig. 10.3 shows the phasor diagram of the studied machine.

**Fig. 10.3. Phasor diagram of the studied permanent magnet machine.**

The analysis of permanent magnet excited machines using FEM is performed assuming imposed winding currents. Therefore, end-winding reactances do not have to be considered in a FEM model. Such effects must be taken into account in analytical models which use the lumped parameters that are determined by a FE-analysis.

**In this example,** the magnetic and geometric periodicities allow the model to be reduced to one pole pitch. This is achieved by applying binary constraints (periodic constraints) to connect both sides of the pole structure (Fig. 10.2).

**Static analysis** The simplest analysis procedure for this machine is a single two-dimensional non-linear magnetostatic analysis. The aim is to obtain the torque at a certain instant of time for a given current. Induced currents due to the relative motion of stator and rotor are neglected.

**Since the stator mmf is in alignment** with the winding phase carrying the maximum current, it is convenient to choose the time t – 0 in correspondence with the magnetisation axis of this phase. In this case, the initial phase angleof the stator current is equal to the electrical angle between the stator and rotor mmf. For the steady state conditions at constant load, voltage and frequency, the electrical angleremains constant. Changing the initial angle can simulate different load situations The following data can be derived from the single FEM solution:

**• local field quantities such as flux density, magnetic field strength**

**• the torque (using Maxwell stress tensor method)**

**• the flux linkage with the stator winding at this instant of time**

**• stator core iron-loss.**

### Iron losses

**The iron losses can be approximated** knowing the loss characteristics, the losses per weight at different flux density levels P(B). Such properties are usually available from the material manufacturer. The total iron loss in the stator for a given frequency can be determined by an integration over the stator core volume (10.1).

In a 2D-problem definition this simplifies to a summation over all ‘element losses’ of the core cross section:

withthe length of the device,the area of the element k andthe mass density of the material of element k.

Another possibility for the estimation of the losses at different frequencies is provided by:

with / the synchronous frequency,the factors determining the loss density components (hysteresis losses, eddy current losses). The material manufacturer must supply these factors. Estimated values can also be found in the literature.

### Torque

As the rotor and the stator magnetic field are in synchronous operation, one can assume that the field pattern is the same for each instant of time.

**The relative motion of rotor and stator is not considered**. The torque consists of the synchronous torque, the reluctance torque (if d- and q-axis reactance of the machine are different) and a cogging component caused by the interaction of rotor magnets and stator slots. The cogging torque cannot be analysed by a single solution. It is possible to examine this torque component by a small sequence of steady state problems, where the rotor angle and the initial phase angle are changed simultaneously (using electrical angles) over one slot pitch. Cogging torque analysis is very sensitive to errors in the flux density computation in the air gap region. It is therefore recommended to ensure a very fine discretisation (Fig. 10.4) or/and using higher order elements in the air gap region.

**Fig. 10.4. Equipotential plot (load condition) and discretisation close to and inside the air gap of a buried magnet synchronous motor.**

**The loss computation** using the above equations neglects the local variation of the flux density. The described analysis assumes a constant level of saturation at all times. In reality the saturation is shifted locally with the rotation of the magnetic field as well. Resulting higher harmonic fields may considerably increase the iron losses in the stator but are not considered in this approach.

**Sequential analysis** Next to the single analysis, a sequence of steady-state analyses is suggested. This approach is not equal to a transient analysis including motion. The type of analysis remains the same, non-linear and magneto static.

**The mesh used in the single steady**-state analysis remains unchanged for this approach. A sequence of analyses has to be prepared, in which the rotor position and the initial phase angle are varied simultaneously (in electrical angles). The current density (according to a defined load situation and instant of time) is applied to the stator winding regions. To generate a sequence of meshes, the following tools can, if available, be used:

**• parameterisation** of both geometry and excitation with automated re-meshing of the model

•** the use of special air gap elements** (elements with overlapping shape functions, sliding boundaries) to avoid the re-meshing of the model for every new rotor position

**• BEM/FEM coupling**; the boundary elements are in the air gap, and the non-linear ferromagnetic parts of the machine are modelled by the FEM

**• scripting facilities,** which allow creation of new models using combinations of commands normally entered by the user.

As a sequence of analyses is carried out, it is highly recommended that the post-processing is automated as well.

### Induced voltage at no-load operation

**To evaluate the induced winding voltage,** generate a sequence of models with different rotor positions and the permanent magnet excitation only. The offset in rotor positions must be chosen smaller than the slot pitch to consider the slotting effects. Compute the flux linkage with each phase at each rotor position using (5.344) or (5.345). From this result, the induced voltage in the stationary winding is given by:

is the flux linked with the entire phase winding, considering all poles. If the FEM model consists only of one pole (Fig. 10.2), this must be taken into account. The rms value of the induced voltagemay be determined by integrating the induced voltage time form over one electrical period. The induced voltage may be compared to a no-load generator experiment. The machine is driven by another motor at constant speed and the terminal voltage is measured. The induced voltage from the permanent magnets at no-load is different from under load conditions. The saturation of the flux paths is different in this case. This voltage may therefore not be used to determine the d-axis reactance under load conditions.

### Flux linkage

**The flux linkage with the stator winding** over one electrical period under load condition can be evaluated by generating a sequence of models which covers half a period of the stator current excitation. For the computation of the induced voltage time form apply (5.344), (5.345) and (10.4). The inner torque angle is given by:

where the phase angleis determined from the phase difference between the induced voltage and the given current (Fig. 10.3). As illustrated in the phasor diagram, the d- and q-axis components of the induced voltage are defined by:

where E and I are the effective values of voltage and current. Whereas obtainingfrom (10.7) is straightforward,under load condition cannot be assumed to be equal toat no-load. This assumption leads to very inaccurate results.

### Skewing effects

**The models described above neglect skewing**. The influence of skewed stator slots can be considered in the following way. However, at the expense of higher computational cost, skewing can be taken into account by a rather simple concept:

**Analyse a sequence of** models that have the equal stator current excitation, but the rotor position is changed over fractions of the skewing angle. Each model serves as a partial model of the entire machine with lengthwith n the number of models generated per skewing angle.

### Cogging torque

**While the sequence of models under load conditions is evaluated,** the torque may be computed as well. The resulting torque time form allows the evaluation of the cogging torque, provided that the steps between two models in the sequence are chosen smaller than the slot pitch.

### Iron losses

**The actual iron losses depend on** the local change of the flux density. The results given by (10.1) and (10.2) assume a single frequent sinusoidal change of the flux density with respect to the time. The realistic local change of the flux density however is different. Whereas the vector of the flux density in the tooth head describes an elliptic path over one electrical period, the flux density in the tooth shaft is always parallel with the tooth but changing its direction. This behaviour can be examined by reporting local flux density vectors (describing selected sub-volumes of the iron core) over a sequence of loaded models. The hysteresis losses may then be computed by the summation of the prescribed area in the hysteresis characteristic for all the taken sub-volumes.

**Loading method** The above procedure for the determination of the lumped parameter model of the studied permanent magnet machine is computationally expensive. The saturation-dependent value of the induced voltage generated by the magnets (10.6) cannot be determined accurately.

**Therefore**, the combination of the finite element method with the analytical calculation of this type of machine is an interesting concept. This method is called the loading method, as all parameters are determined under load conditions, considering the mutual influence between the direct and the quadrature axis fields.

**Equation (10.6)** is underdefmed, asare unknown. The idea consists of a linearisation around the operating point at a given load. A small change is applied to the load (the stator current), assuming that a small change does not influence the saturation level of the machine. With this assumption the reactance and the induced voltage generated by the magnets do not change. A second equation can be obtained from this linear solution:

**From (10.6) and (10.8),**can be calculated. Combining this approach with the sequential approach above, E, and S, can be derived from the time form of the induced voltage and the given current time form. Two static problems per time instant have to be prepared for the analysis at a given load. The first one is a non-linear static model, equal to the models prepared for the single steady-state analysis. The accurate alignment of the rotor and stator magnetic axes is very important. The problem is solved and the solution is saved for later post-processing. At the same time, the local values of the permeability (reluctivity) in all elements are saved, A new linear problem is defined, with a small change in the stator current (around 5%). The model is solved using the (fixed) permeabilities from the non-linear solution.

**The combination with** the sequence approach increases the computational cost. At each instant of time, two FEM models must be solved. However,can be determined in a faster way. Both values can be determined by using a Fourier series of the vector potential A along a contour inside the air gap (near the stator surface).

with v the ordinal number of the harmonic, m the maximum harmonic considered (higher than the slot harmonics) and tp the pole pitch in degrees. Due to the symmetric design of the machine, the vanishes. The angular dependency can be transformed into the time domain considering the synchronous speed of the machine. With the alignment of rotor and stator magnetic axis, the cosine term coefficient in (10.9) represents the quantity of half the q-axis flux per pole and unit depth. The sine term coefficientrepresents the quantity of half the d-axis flux per pole and unit depth.

The following data can be extracted from both solutions. The resultant flux per pole is given by:

**The inner torque angle is:**

**The effective value of the induced voltage can now be evaluated using the analytical expression:**

withthe synchronous frequency, N the number of series turns per phase andthe stator winding factor for the fundamental harmonic (product of winding distribution factor and pitch factor). It should be noted thatis a magnetic quantity and represents a magnitude value, but £ is an electric quantity representing an effective value. Skewing can be considered by an additional skewing factor that is multiplied with the winding factor.

**The d-axis reactance and induced voltage** from the magnets under load conditions can be determined by applying (10.9) to (10.12) to both FEM solutions. Inserting the known values into (10.6) and (10.8) yields the value forThe leakage reactance of the end-winding may be determined analytically or with a 3D end-winding model.

The q-axis reactance is determined by using (10.7) and adding the end-winding leakage reactance.

**The entire input power for the m-phase machine is given by:**

withthe ohrnic resistance of the stator phase winding at operating temperature.

**The electromagnetic power can be obtained by employing:**

and the electromagnetic torque can be calculated by:

with p the number of pole pairs.

**Ohmic losses are considered in the input power (10.13)** whereas the iron losses are neglected there. They can be estimated using (10.2) and added to (10.13). Rotor losses are neglected.

It must be noted that this method can not be applied for the overall motor characteristics.can not be evaluated whenrespectively are zero. The underlying equations would require a division by zero.

A harmonic analysis can be performed by introducing harmonic ordinal numbers and dependencies (winding factors for the harmonics). Combining the loading method with the sequencing approach requires a different error discussion as the application of the loading method alone.

**Combining the loading method and sequencing** approach requires that the post-processing for the loading method is applied after the sequence of solutions is obtained. The computation of the time form of the induced voltage requires the differentiation of the time form of the linked flux. Effects of induced currents are neglected in the analysis.

**By applying the loading method only,** the quantitative influence of the relative motion of rotor and stator can not be considered because the parameters are determined from one rotor-stator alignment only. However, as the slot harmonics are present in the Fourier-series expression of the potential, they will be present as well in the voltage time form derived from this expression. The exact value of the derived quantities will be erroneous.

**The linearisation around the working point,** which is the underlying idea of this method, imposes an error. Imposing the permeabilities to the state at the given load decreases this error. The changes in the stator current for the linear model should not exceed 10% of the load current.

**The leakage field crossing the slots is not considered** using the introduced approach. It could be examined along contours inside the slots, integrating the penetrating flux through the contours. This flux has to be subtracted from the flux computed with equation (10.10). The advantage of the loading method is its low computational expense when compared to the sequence method or to a transient analysis.