# A Computationally Effective Method for Iron Loss Estimation in a Synchronous Machine from a Static Field Solution

Aalto University

^{1}23 Aug 2020-

TL;DR: The proposed iron losses computation method showed a fair accuracy and a considerable speed-up of the computations, and can be an excellent alternative for the iron losses estimation in the optimization procedure of synchronous machines, where a considerable amount of finite element solutions needs to be carried out.

Abstract: In this paper, a computationally effective iron loss calculation method for synchronous machines is presented. The method is based on a single static 2D finite element field solution in the machine cross-section, which makes it much faster than the one based on the time-stepping solution. The developed method is applied to a salient pole synchronous machine, and the computational accuracy is validated against the time-stepping method. The proposed iron losses computation method showed a fair accuracy and a considerable speed-up of the computations. It can be an excellent alternative for the iron losses estimation in the optimization procedure of synchronous machines, where a considerable amount of finite element solutions needs to be carried out. Besides the losses comparison, local reconstruction of the time dependency of other quantities such as the magnetic vector potential and the magnetic flux density is reported for a better understanding of the method.

## Summary (3 min read)

Jump to: [Introduction] – [II. METHODOLOGY] – [B. Time Dependence of the Flux Density from the Static Analysis] – [C. Flux Density Waveform Formation in Static Analysis] – [A. Studied Synchronous Machine and Simulation Parameters] – [B. Magnetic Flux Density and Harmonics Analysis] – [D. Computational Cost] – [E. Effect of Damper Windings, Supply Voltage Phase Shift and Rotor Angle Variation on Iron Losses] and [IV. CONCLUSION]

### Introduction

- This reprint may differ from the original in pagination and typographic detail.
- The method is based on a single static 2D finite element field solution in the machine cross-section, which makes it much faster than the one based on the time-stepping solution.
- Moreover, the proposed iron losses computation method is not limited to a particular synchronous machine and applicable to any kind of synchronous machine.

### II. METHODOLOGY

- The iron losses will be computed from the Fourier decomposition of the flux density waveform over one period.
- The empirical iron loss models are fast, easy to implement, and applicable for the iron losses estimation roughly [12].
- Chys and Ced expresses the hysteresis and eddy current loss coefficients, ωs stands for the angular frequency of the supply, Vc is the volume of the stator iron core, and B̂n is the peak flux density value at nth harmonic components.
- Moreover, the minor hysteresis loops were not taken into consideration properly.
- The computation of the excess loss coefficient through the Epstein frame test is difficult as it does not allow us to identify the difference between the eddy current due to the classical loss and excess loss [17].

### B. Time Dependence of the Flux Density from the Static Analysis

- In finite element method, the flux density B values are computed natively in the x-y coordinates; thus, the actual iron losses are calculated from the flux density components Bx and By values.
- In practice, the flux density B is solved by assuming the two-dimensional approximation where the flux density lies only in (x, y) plane in Cartesian coordinates or (r, φ) plane in cylindrical coordinates and does not depend on the z-axis.
- In FCSMEK, the stator finite element mesh is constructed by multiplying the slot pitch mesh.
- The time dependence of the flux density waveform can be achieved by assuming that the flux density waveform is moving with the fundamental angular frequency ω and time t.
- It is intrinsic within the time dependency reconstruction.

### C. Flux Density Waveform Formation in Static Analysis

- In such a case, the number of stator slots and the number of points in the flux density waveform should be equal.
- In such a way, elements to elements flux density waveform are constructed at each slot pitch interval over two pole pitches to calculate the average iron losses from a closed cycle of the flux density waveform using the Equations 1, and 2.
- As one observation point was specified in each element; hence, no elements were missed.
- Such variation over the solution region in Figure 1 provides the half cycle of the flux density waveform.
- For this reason, the total number of elements in this slot pitch is not equal compared to the other slot pitch meshes.

### A. Studied Synchronous Machine and Simulation Parameters

- The proposed method has been applied to a 12.5 MW salient pole synchronous machine, which consists of 90 stator slots and six rotor poles.
- Thus, the iron losses only in the stator core are computed by the proposed method and the conventional method for a fair comparison.
- The non-linearity of the system equations was solved by the Newton-Raphson iteration method.
- It can be seen in Table I that the operating parameters obtained from the static analysis and the dynamic analysis are in relatively good agreement.
- Therefore, the static computation may overestimate or underestimate these parameters slightly.

### B. Magnetic Flux Density and Harmonics Analysis

- Figure 3 shows the flux contour lines and the flux density distribution of the smallest symmetry section in the static analysis.
- The space distribution of the radial Br and tangential Bφ components of the flux density values in the stator yoke and teeth at given radial positions from the rotor midpoint over one period for both methods are presented in Figure 2.
- The Fourier transformation was performed to analyze the harmonic components present in the flux density waveform in Figure 2.
- Unevenly spaced samples can also be reconstructed as accurately as possible if the average sampling rate follows the Nyquist rule, but the signal may loss one or two samples.
- In the case of dynamic analysis, the most significant harmonic components, i.e., up to 9th are depicted in Figure 4, and all the harmonic components were taken into account for the iron losses calculation.

### D. Computational Cost

- The iron losses computation from a single static field solution is much faster than the time-stepping simulation.
- The iron losses computation from a single static simulation costs 0.512 s; on the other hand, the time-stepping simulation requires 35.07 s for the same number of elements and nodes using an Intel-Xeon 3.4-GHz 16-GB-RAM workstation.
- The computation time for both methods is listed in Table I.

### E. Effect of Damper Windings, Supply Voltage Phase Shift and Rotor Angle Variation on Iron Losses

- The effect of damper windings on the flux density harmonic reduction; consequently, the iron losses computation was studied for the dynamic analysis.
- The conductivity of the damper windings was set close to zero, in the case of dynamic analysis, so that no current can induce in the bars.
- Later, the phase angle φ was varied from 0o to 5o, and the rotor angle θ was varied from 31.9o to 36.9o at the one-degree interval.
- The stator iron losses are decreased slightly at each interval with increasing the supply voltage phase shift φ and the rotor angle θ as illustrated in Figure 7, which is maximum 2.5% compared to the computed iron losses at the initial position, i.e., φ = 0 and θ = 31.9o; hence, no significant effect was found.
- The authors are aware of the importance of computing the iron losses at multiple load points.

### IV. CONCLUSION

- A method of the iron losses computation in a synchronous machine from a static field solution has been proposed in this paper.
- The proposed method has relatively good accuracy compared to the dynamic analysis based method (8.94% maximum difference) and showed a high potentiality of the iron losses computation with a less computation cost over the conventional time-stepping method.
- Moreover, increasing the computational accuracy of the proposed method can substitute the time consuming conventional loss computation method in many applications.
- The accuracy of the developed loss computation technique can be improved by reconstructing the stator mesh, so that, every slot pitch meshes can be taken into account and forming a more densely mesh in order to attain more sampling points, which eventually increase the resolution of the flux density waveform.
- The utmost goal is to improve the accuracy of the Fourier decomposition of the flux density waveform, hence, the iron losses computation, which is their next step.

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Billah, Md Masum; Martin, Floran; Belahcen, Anouar

A computationally effective method for iron loss estimation in a synchronous machine from a

static field solution

Published in:

Proceedings of the 2020 International Conference on Electrical Machines, ICEM 2020

DOI:

10.1109/ICEM49940.2020.9271020

Published: 23/08/2020

Document Version

Peer reviewed version

Please cite the original version:

Billah, M. M., Martin, F., & Belahcen, A. (2020). A computationally effective method for iron loss estimation in a

synchronous machine from a static field solution. In Proceedings of the 2020 International Conference on

Electrical Machines, ICEM 2020 (pp. 751-757). [9271020] (Proceedings (International Conference on Electrical

Machines)). IEEE. https://doi.org/10.1109/ICEM49940.2020.9271020

© 2020 IEEE. This is the author’s version of an article that has been published by IEEE.

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A Computationally Effective Method for Iron Loss

Estimation in a Synchronous Machine from a Static

Field Solution

Md Masum Billah, Floran Martin, and Anouar Belahcen, Senior Member, IEEE

Abstract—In this paper, a computationally effective iron loss

calculation method for synchronous machines is presented. The

method is based on a single static 2D ﬁnite element ﬁeld solution

in the machine cross-section, which makes it much faster than

the one based on the time-stepping solution. The developed

method is applied to a salient pole synchronous machine, and the

computational accuracy is validated against the time-stepping

method. The proposed iron losses computation method showed a

fair accuracy and a considerable speed-up of the computations.

It can be an excellent alternative for the iron losses estimation

in the optimization procedure of synchronous machines, where

a considerable amount of ﬁnite element solutions needs to be

carried out. Besides the losses comparison, local reconstruction

of the time dependency of other quantities such as the magnetic

vector potential and the magnetic ﬂux density is reported for a

better understanding of the method.

Index Terms—dynamic ﬁeld solution, ﬁnite element method,

iron losses, synchronous machine, static ﬁeld solution, time-

stepping method.

I. INTRODUCTION

O

WING to the increasing energy demand, highly ef-

ﬁcient synchronous machines play a crucial role in

energy saving by reducing energy consumption. Like other

electrical machines, power losses are a common issue in a

synchronous machine, which increases the temperature and

degrades the performance by affecting the maximum output

power. Besides, extreme temperature rise can lead to insula-

tion failure, consequently, decrease the life expectancy of a

synchronous machine. Power losses can be segregated into

Joule losses in the winding of the machine, the iron losses

in the core of the machines, the frictions, and mechanical

losses, and the permanent magnet losses, which are due to

eddy currents in these parts if any. Usually, the Joule and

friction losses can be estimated from static quantities, such

as the rotational speed and the supply currents, but the iron

and permanent magnet losses require the temporal and spatial

distribution of the magnetic ﬂux density in the corresponding

parts of the machine.

This work was supported by the internal funding from computational

electromechanics group at Aalto University, Finland.

M. M Billah, F. Martin, and A. Belahcen are with Aalto Uni-

versity, Department of Electrical Engineering and Automation, Espoo,

Aalto-00076 Finland (e-mail: md.billah@aalto.ﬁ, ﬂoran.martin@aalto.ﬁ,

anouar.belahcen@aalto.ﬁ)

A. Belahcen is also with Tallinn University of Technology, Tallinn,

Estonia.

The magnetic ﬂux density in the electrical machine is

three-dimensional, and solving a three-dimensional problem

is still laborious. The solution becomes much easier if the

problem domain is simpliﬁed to the two-dimensional model

where the machine geometry and material equations are

independent of the coordinate parallel to the machine shaft,

i.e., z-coordinate. A detailed description of the 2D ﬁnite ele-

ment method and its application on solving two-dimensional

electromagnetic ﬁelds problem is demonstrated in [1] among

others. Moreover, the effects of armature reaction and the

magnetizing ﬁeld, the slotted nature of the machine, and the

highly distorted magnetic ﬁeld appearing from the permanent

magnet cause the non-sinusoidal and time-varying distribu-

tion of the magnetic ﬂux density at different locations of

the machine’s core. As a result, the solution of the magnetic

ﬂux density distribution is usually achieved for each time

step through the time-stepping method [2]. In addition, the

magnetic ﬂux density in the rotating machine is not a single

frequency component anymore and contains higher frequency

components.

A common practice of the iron losses computation is to

obtain the ﬂux density distribution from the time-stepping

ﬁnite element simulation and calculate the losses in the post-

processing stage through the Fourier decomposition of the

ﬂux density waveform over one period [3], [4]. The solution

accuracy of the time-stepping method depends on the size

of the time step, and a smaller step size usually ensures

better accuracy. Moreover, the steady-state characteristics

are often desirable for the iron losses computation, which

requires to run the machine at least for a few electrical

periods. A suitable time resolution selection for the dynamic

analysis and the effect of it on the solution accuracy is a

separate topic, and no exact recommendation was found.

For instance, the number of time steps per period and the

number of electrical periods were used by [5], [6] 300-1000

steps per period and 2-10 electrical periods, respectively.

However, all these factors lead to high computational cost

for achieving the dynamic solution; consequently, the iron

losses computation from this solution. The difference is more

visible when the iron losses computation from the dynamic

solution required 35.07 s, and from a single static simulation

needed only 0.512 s. Despite the high computational cost,

the time-stepping method is more conventional and still

a popular choice for the iron losses computation. This is

probably because the conventional loss computation method

provides better accuracy and also stays productive as long as

the number of computations remains in a reasonable range.

However, the situation is different when an excessive number

of computations are required, e.g., machine optimization,

which turns this method into unproﬁtable by increasing the

simulation time. In such a case, the computation of the

iron losses from a time-efﬁcient static solution is more than

justiﬁed.

Early on, a series of thirty static ﬁeld solutions were com-

puted in [7] by moving the rotor at one slot pitch interval over

one electrical period, and the resulting ﬁeld solutions were

used to estimate the stator core losses. A similar approach

has been used in [8] with a smaller number of snapshots,

i.e., static ﬁeld solutions than [7]. The method presented in

[7], [8] was extended by [9] in order to take the rotor core

losses into account. A simpliﬁed, i.e., surrogate ﬁnite element

method is introduced in [10] to reduce the requirements of

the number of successive snapshots, i.e., static ﬁeld solu-

tions. However, the method presented in [10] is particularly

developed for the interior permanent magnet machine with

concentrated, i.e., non-overlapping coils. Another snapshot-

based computationally effective ﬁnite element model is devel-

oped in [11] and applied for the optimization process of the

interior permanent magnet machine. The proposed methods

in [10], [11] signiﬁcantly reduced the requirements of the

number of static simulations, i.e., 2-5 solutions required,

respectively for the iron losses computation compared to the

methods presented in [7], [8], [9]. A major drawback of these

snapshot-based methods is that accuracy and precision are

highly inﬂuenced by the number of static simulations taken

into consideration. Adding more static ﬁeld solutions may

make sure better accuracy; however, it also increases the

computational cost. Also, no speciﬁc benchmark is noticed

for the exact requirements of the number of static simulations.

An ultrafast static ﬁeld computation method has been

presented in [5], by coupling the static ﬁeld equations and

space vector model within the same ﬁnite element solution.

The method has given accurate results from a single static

simulation compared to the measurement and time-stepping

method. Modeling a fast and accurate static ﬁeld computation

method is an independent topic of research by itself, which

is out of the scope of this paper. Instead, our main goal is

to develop the iron losses computation method by using the

readily available static ﬁeld computation method presented

in [5], which can be an alternative method of the iron losses

computation from the dynamic ﬁeld solution in many appli-

cations. In this paper, the accuracy and computational cost

of the developed method are validated with the conventional

time-stepping method extensively.

The presented iron losses computation method in this

paper outperforms the previously developed computationally

effective static iron losses calculation methods in [7], [8], [9],

[10], [11] several ways. Here, the iron losses are computed

from a single static simulation, and no rotor motion is taken

into account. Hence, the accuracy of this method is not

restricted to the requirements of the number of static sim-

ulations. Furthermore, this paper solely and comprehensibly

deals with developing an ultrafast iron loss computation tech-

nique, which can overcome the above-mentioned snapshot-

based methods in terms of computational cost and also can

compete with them in terms of accuracy. Moreover, the

proposed iron losses computation method is not limited to

a particular synchronous machine and applicable to any kind

of synchronous machine.

II. M

ETHODOLOGY

A. Iron Loss Model

In this proposed method, the iron losses will be computed

from the Fourier decomposition of the ﬂux density waveform

over one period. A suitable and accurate iron loss model

requires to compute the losses from the peak values of

the ﬂux density at each harmonic component. Moreover,

the selected iron loss model can be easily integrated with

the static or dynamic ﬁnite element solution in the post-

processing stage. The empirical iron loss models are fast, easy

to implement, and applicable for the iron losses estimation

roughly [12]. However, many researchers have extended the

empirical iron loss models and estimated the iron losses

conventionally with relatively good accuracy [13], [14]. The

modiﬁed Jordan loss separation model presented in [15] is

adopted for the iron losses computation

P

hys

=

V

c

(

N

n=1

C

hys

(nω

s

)

ˆ

B

2

n

)dV (1)

P

ed

=

V

c

(

N

n=1

C

ed

(nω

s

)

2

ˆ

B

2

n

)dV (2)

where P

hys

and P

ed

represents the hysteresis loss and

eddy current loss, respectively. C

hys

and C

ed

expresses the

hysteresis and eddy current loss coefﬁcients, ω

s

stands for

the angular frequency of the supply, V

c

is the volume of

the stator iron core, and

ˆ

B

n

is the peak ﬂux density value

at n

th

harmonic components. However, the modiﬁed iron

loss model in Equations 1, and 2 also suffers from some

shortcomings. For instance, two-loss coefﬁcients C

hys

and

C

ed

were computed from the Epstein frame test, and no

differentiation was made between the alternating and rota-

tional ﬁelds [16]. Moreover, the minor hysteresis loops were

not taken into consideration properly. In the statistical loss

segregation method, the classical eddy current loss and excess

loss are presented separately. However, the computation of

the excess loss coefﬁcient through the Epstein frame test is

difﬁcult as it does not allow us to identify the difference

between the eddy current due to the classical loss and

excess loss [17]. Alternatively, a two-term iron loss separation

method is developed as shown in Equations 1, and 2 where

the classical eddy current loss and excess loss are combined

and formed a global eddy current loss, P

ed

in Equation 2. It

can be noted that the same iron loss model is also used to

compute the losses through the time-stepping method. An in-

house 2D ﬁnite element solver software FCSMEK has been

developed by the electromechanics research group at Aalto

University and is used to compute the iron losses from the

proposed method and the conventional time-stepping method.

B. Time Dependence of the Flux Density from the Static

Analysis

In ﬁnite element method, the ﬂux density B values are

computed natively in the x-y coordinates; thus, the actual

iron losses are calculated from the ﬂux density components

B

x

and B

y

values. In practice, the ﬂux density B is solved by

assuming the two-dimensional approximation where the ﬂux

density lies only in (x, y) plane in Cartesian coordinates or

(r, φ) plane in cylindrical coordinates and does not depend

on the z-axis. The ﬂux density components B

x

and B

y

can be calculated from the partial derivation of magnetic

vector potential values with respect to the x-y coordinates.

The existence of space harmonic components in case of

sinusoidal voltage supply in the electrical machine affects

the ﬂux density waveform and inﬂuence to depend on time.

However, the obtained ﬂux density B values from a static

ﬁeld solution is independent of time, i.e., static. The time

dependence of the static ﬂux density waveform needs to be

introduced in order to compute the iron losses. In FCSMEK,

the stator ﬁnite element mesh is constructed by multiplying

the slot pitch mesh. Thus, the number of elements in one slot

pitch mesh is repeated to the next slot pitches, and elements

from one slot pitch to another has the same position and

size. Therefore, the space variation of the static ﬂux density

waveform both for the stator yoke and teeth over one period

can be achieved by selecting the elements at one slot pitch

interval. The stator slot pitch is computed as follows

θ

s

=

2π

Q

s

(3)

where θ

s

is the stator slot pitch, and Q

s

denotes the total num-

ber of stator slots. The time dependence of the ﬂux density

waveform can be achieved by assuming that the ﬂux density

waveform is moving with the fundamental angular frequency

ω and time t. The time step for the time dependence is

computed as follows

Δt =

Δθ

s

p

ω

(4)

where p denotes the number of pole pairs, θ

s

is the stator slot

pitch, and ω is the supply angular frequency. At any instant

of angular distance θ and time t, the displacement of the

moving ﬂux density with initial position θ

o

can be determined

as θ = θ

o

+ ωt. With respect to the initial position θ

o

and

time t

o

, the displacement of the ﬂux density waveform can

be written as

B(θ, r, t)=RB(θ − ωt, r, 0) (5)

where θ is the angular position of the point at which the ﬂux

density is estimated and r its radial position from the rotor

midpoint. R is the rotation matrix given in Equation 6.

R =

cos θ − sin θ

sin θ cos θ

(6)

Fig. 1. Element selections for the ﬂux density waveform formation over

the solution region in the static analysis.

It should be noted that the radial dependency does not need

to be solved in Equation 5. It is intrinsic within the time

dependency reconstruction.

C. Flux Density Waveform Formation in Static Analysis

The process of element selections in the stator yoke at

each slot pitch interval is depicted in Figure 1. In such a

case, the number of stator slots and the number of points in

the ﬂux density waveform should be equal. Similarly, the ﬂux

density waveform in the stator teeth is formed by choosing

the elements at one slot pitch interval. In such a way, elements

to elements ﬂux density waveform are constructed at each slot

pitch interval over two pole pitches to calculate the average

iron losses from a closed cycle of the ﬂux density waveform

using the Equations 1, and 2. The same size element in each

stator slot pitch meshes from 2 to 15, i.e., represents full slot

pitch is found at a distance of one slot pitch interval. On the

other hand, the element in positions 1, and 16 represent half

of the slot pitch. The ﬂux density waveform as depicted in

Figure 2 was formed by specifying an observation point at

one element, i.e., position 1, and varying by one slot pitch

from 1 to 16 as shown in Figure 1. As one observation

point was speciﬁed in each element; hence, no elements were

missed. As a result, the ﬂux density points are evenly spaced

in Figure 2 even though the element in positions 1, and

16 are different compared to other selected elements. Such

variation over the solution region in Figure 1 provides the half

cycle of the ﬂux density waveform. A complete cycle of ﬂux

density waveform was formed in Figure 2 by mirroring the

obtained half cycle of the ﬂux density waveform due to the

symmetry of the solution region. A problem was identiﬁed

during the computation as the element in positions 1, and

16 have different sizes compared to other slot pitch meshes,

i.e., 2 to 15. For this reason, the total number of elements in

this slot pitch is not equal compared to the other slot pitch

meshes. However, the same number of elements in each slot

pitch mesh is required when the elements to elements ﬂux

##### References

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TL;DR: In this article, the authors proposed a new approach for predicting iron losses in soft magnetic materials with any voltage supply, starting from the knowledge of the iron losses with a sinusoidal or pulsewidth modulation supply.

Abstract: We propose a new approach for predicting iron losses in soft magnetic materials with any voltage supply, starting from the knowledge of the iron losses with a sinusoidal or pulsewidth modulation supply. The model is based on the separation of the loss contributions due to hysteresis, eddy currents, and excess losses with the two supplies. Since any contribution depends on the voltage supply characteristics, it is possible to find a direct mathematical relationship between the iron loss contribution and the voltage supply characteristics. As a consequence, an iron loss prediction can be obtained with any voltage supply if it does not produce a hysteresis minor loop. The energetic model is based on coefficients that depend on the magnetic material characteristic. We performed an accurate analysis of the model on eight magnetic materials used for electrical machine construction, of different thicknesses and alloy compositions. In this way, we found the main coefficients for a large spread of magnetic materials. As a consequence, our approach can be a useful support for electrical machine designers when the energetic performance of a magnetic material has to be predicted for a voltage supply different from the sinusoidal one.

282 citations

••

TL;DR: In this article, the authors investigated the iron loss of interior permanent magnet motors driven by PWM inverters from both results of the experiments and the finite-element analysis, and clarified that the largest component at low speed condition under maximum torque control, whereas the loss caused by the harmonic magnetomotive forces of the permanent magnet remarkably increase at high-speed condition under flux-weakening control.

Abstract: In this paper, the authors investigate the iron loss of interior permanent magnet motors driven by pulsewidth modulation (PWM) inverters from both results of the experiments and the finite-element analysis. In the analysis, the iron loss of the motor is decomposed into several components due to their origins, for instance, the fundamental field, carrier of the PWM inverter, slot ripples, and harmonic magnetomotive forces of the permanent magnet in order to clarify the main loss factors. The Fourier transformation and the finite-element method considering the carrier harmonics are applied to this calculation. The calculated iron loss is compared with the measurement at each driving condition. The measured and the calculated results agree well. It is clarified that the iron loss caused by the carrier of the PWM inverter is the largest component at low-speed condition under the maximum torque control, whereas the loss caused by the harmonic magnetomotive forces of the permanent magnet remarkably increase at high-speed condition under the flux-weakening control

226 citations