# Iron Loss Prediction Using Modified IEM-Formula during the Field Weakening for Permanent Magnet Synchronous Machines

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{d}–i

_{q}currents. The paper has presented the complete method for calculating the iron coefficients from a nonlinear magnetic nodal network of the machine. A detailed study of the local field density waveform and harmonic content in the yoke and teeth was provided for two particular operating points: at maximal power without field weakening and at maximal power at maximal speed [18]. This article investigated mapping of local iron losses coefficients in yoke and teeth, and also the iron losses’ coefficients differences justified per unit volume between yoke and teeth. However, there was no experimental validation present in the article.

## 2. Classic IEM-Formula Evaluation

_{1}, a

_{2}, a

_{3}, a

_{4}, and a

_{5}are the coefficients that will be estimated via nonlinear curve fitting. α is the fitted material parameter, which is found using dc-measurements (quasi-static loss measurements using a field-meter) in a standard Epstein frame, finding the best parameter set describing the hysteresis losses as:

#### 2.1. Skin Effect Consideration

_{2}), which considers the skin effect by accounting for the thickness of the steel used, is:

_{e}) of the soft magnetic material.

#### 2.2. Steady-State Equivalent Circuit with Iron Loss Resistance Consideration

_{s}I

_{d}and R

_{s}I

_{q}of the stator winding resistance are taken into account for the iron loss model based on the equivalent circuit, in which Ref. [23] assumed the winding resistance negligible. However, its ohmic value can be large, especially under the field weakening condition. Therefore, a more accurate iron loss modelling is rooted from both core and winding resistance consideration. The following expressions can be extracted from the equivalent circuit:

_{t}and V

_{y}are the volume of the tooth and yoke in stator.

_{IEM}

_{1}is produced via the no-load fundamental air-gap field density component B

_{m}

_{1}can be rewritten as follows:

_{tf}as the teeth filter constant is defined as:

_{s}is one tooth pitch angle, and k

_{t}(the teeth-width coefficient) base on one tooth pitch (τ

_{s}), and one tooth-pitch (b

_{t}) can be calculated by:

_{y}as the yoke-height coefficient is:

_{p}is the pole-pitch in the air-gap, and b

_{y}is one yoke-pitch.

_{d}= I

_{q}= 0, the term ωλ

_{m}forces an additional current I

_{aq}= −I

_{cq}, which is different from zero. Hence, due to the term −ωL

_{aq}I

_{aq}in the d-axis circuit, a current as I

_{ad}= −I

_{cd}will be increased. Furthermore, the back-EMF term of ωL

_{ad}I

_{ad}occurrs in the q-axis. As a result, the total voltage across R

_{c}(total stator core resistance) is not only equal to ωλ

_{m}, but also back-EMF terms of −ωL

_{aq}I

_{aq}and ωL

_{ad}I

_{ad}should be considered. Despite this, these voltage drops are ignored by the work done in Ref. [23], which causes a considerable error. The modified IEM-Formula, which can be rewritten based on the steady-state equivalent circuit and Equations (4) and (8), is given as:

_{c}) is comprised of eddy-current loss resistance (R

_{ce}) and hysteresis loss resistance (R

_{ch}), where R

_{ce}is eddy-current loss resistance that depends on the type of used material, its dimensions and other machines’ design factors.

## 3. Analytical Concept of the Iron Loss Model with Harmonic Loss Considerations

_{mg}

_{0}and B

_{gr}are the sum of the no-load magnetic field density and the armature reaction air-gap magnetic field density. For non-sinusoidal waveforms, the eddy-current term in Label (1) can be modified (based on [24,25]) to give the following expression:

_{2´}= a

_{2}/(2π

^{2}) is the new eddy-current coefficient, and [dB/dt] is the root mean square (rms-value) of the rate of change of field density over one cycle of the fundamental frequency [29].

^{2}is proportional to the energy of the differential of the field density, which mainly originated via the eddy-currents behavior that is modelled by Equation (16), and specific field density distribution over tooth (B

_{t}) and yoke (B

_{y}) shown in Equation (17), given as:

_{t}) and (B

_{y}) is presented in Figure 3, in which peak values are 1.98 T and 0.56 T, respectively. To simplify the above equation:

_{g}is the airgap magnetic induction between one tooth pitch. The teeth and yoke filter constants (k

_{tf}and k

_{yf}), which are dependent on harmonic order, can be calculated using Equations (9) and (11). b

_{t}= 15 mm, and b

_{y}= 81 mm in this study.

_{syh}), it can be expressed as:

_{s}

_{1}is the fundamental magnetic motive force (MMF) in the stator, k

_{sw}constant is a unit square function and through Fourier series can be developed to:

_{p}is the pole-arc coefficient.

_{h}can thus be rewritten in the following form:

_{ph}as a harmonic constant is employed to include the harmonic magnetic induction range, which can be known from the machine design parameters [23]. In addition, a harmonic voltage U

_{ph}originated from Labels (6), (21) and (22) are defined to model harmonic loss based on equivalent circuit parameters, which is:

_{ph}, the harmonic loss P

_{h}Equation (24) can be simplified into the formula of U

_{ph}and k

_{ph}as:

_{IEM}

_{1}and P

_{h}are the classic IEM-Formula and modified IEM-Formula (which considers iron loss resistance) and harmonic iron loss.

_{m}) and fundamental no load magnetic field density (B

_{m}

_{1}) in Figure 4b, utilized from Labels (21) and (27). The waveforms for a range of 360 θ

_{e}(electrical degree) rotor displacement are shown in Figure 4. The remaining harmonic component waveform of B

_{m}can be calculated through the difference between B

_{m}

_{1}and B

_{m}[28].

## 4. Results and Discussion

## 5. Experimental Verification

_{fe}(total) consists of the no-load iron loss P

_{fe}and the mechanical loss P

_{mech}. The PMSG under testing is fed by a variable-speed frequency converter (ABB ACS600) and loaded by a DC machine (prime mover). The shaft torque is measured by a torque transducer (TORQUEMASTER TM-214). The electrical power (input and output) is measured by a power analyzer (Yokogawa PZ4000). Afterwards, all the data (such as voltage, torque, power, and efficiency) were stored by a reading unit to the laboratories’ computer. The prototype machine is designed particularly for laboratory test use. As the output power is stored by a dynamometer. Thus, the total loss (consists of copper, iron, and mechanical losses) has been obtained by a simple subtraction between input and output powers. The copper loss has been calculated via the measured phase current and resistance, as well as the mechanical loss being provided in the coefficient extracting experiment.

## 6. Conclusions

_{2}emissions as a part of green power generation projects.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Steentjes, S.; von Pfingsten, G.; Hombitzer, M.; Hameyer, K. Iron-Loss Model with Consideration of Minor Loops Applied to FE-Simulations of Electrical Machines. IEEE Trans. Magn.
**2013**, 49, 3945–3948. [Google Scholar] [CrossRef] - Eggers, D.; Steentjes, S.; Hameyer, K. Advanced Iron-Loss Estimation for Nonlinear Material Behavior. IEEE Trans. Magn.
**2012**, 48, 3021–3024. [Google Scholar] [CrossRef] - Alatawneh, N.; Rahman, T.; Hussain, S.; Lowther, D.A.; Chromik, R. Accuracy of time domain extension formulae of core losses in non-oriented electrical steel laminations under non-sinusoidal excitation. IET Electr. Power Appl.
**2017**, 11, 1131–1139. [Google Scholar] [CrossRef] - Krings, A.; Soulard, J. Overview and comparison of iron loss models for electrical machines. J. Electr. Eng.
**2010**, 10, 162–169. [Google Scholar] - Ionel, D.M.; Popescu, M.; Dellinger, S.J.; Miller, T.J.E.; Heideman, R.J.; McGilp, M.I. On the variation with flux and frequency of the core loss coefficients in electrical machines. IEEE Trans. Ind. Appl.
**2006**, 42, 658–667. [Google Scholar] [CrossRef] - Ionel, D.M.; Popescu, M.; McGilp, M.I.; Miller, T.J.E.; Dellinger, S.J.; Heideman, R.J. Computation of Core Losses in Electrical Machines Using Improved Models for Laminated Steel. IEEE Trans. Ind. Appl.
**2007**, 43, 1554–1564. [Google Scholar] [CrossRef] - Gerlando, A.D.; Perini, R. Evaluation of the Effects of the Voltage Harmonics on the Extra Iron Losses in the Inverter Fed Electromagnetic Devices. IEEE Trans. Energy Convers.
**1999**, 14, 57–62. [Google Scholar] [CrossRef] - Rasilo, P.; Belahcen, A.; Arkkio, A. Experimental determination and numerical evaluation of core losses in a 150-kVA wound-field synchronous machine. IET Electr. Power Appl.
**2013**, 7, 97–105. [Google Scholar] [CrossRef] - Zhao, H.; Wang, Y.; Zhang, D.; Zhan, Y.; Xu, G.; Luo, Y. Piecewise variable parameter model for precise analysis of iron losses in induction motors. IET Electr. Power Appl.
**2017**, 11. [Google Scholar] [CrossRef] - Han, S.-H.; Soong, W.L.; Jahns, T.M.; Guven, M.K.; Illindala, M.S. Reducing harmonic eddy-current loss in the stator teeth of interior permanent magnet synchronous machines during flux weakening. IEEE Trans. Energy Convers.
**2010**, 25, 441–449. [Google Scholar] [CrossRef] - Li, Q.; Fan, T.; Wen, X.; Ye, L.; Tai, X.; Li, Y. Stator teeth eddy-current loss analysis of interior permanent magnet machine during flux weakening. In Proceedings of the IEEE International Conference on Electrical Machines and Systems (ICEMS), Busan, Korea, 26–29 October 2013; pp. 1226–1230. [Google Scholar] [CrossRef]
- Yokoi, Y.; Higuchi, T.; Miyamoto, Y. General formulation of winding factor for fractional-slot concentrated winding design. IET Electr. Power Appl.
**2015**, 10, 231–239. [Google Scholar] [CrossRef] - Liu, Y.; Pei, Y.; Yu, Y.; Shi, Y.; Chai, F. Increasing the saliency ratio of fractional slot concentrated winding interior permanent magnet synchronous motors. IET Electr. Power Appl.
**2015**, 9, 439–448. [Google Scholar] [CrossRef] - Yamazaki, K. Torque and efficiency calculation of an interior permanent magnet motor considering harmonic iron losses of both the stator and rotor. IEEE Trans. Magn.
**2003**, 39, 1460–1463. [Google Scholar] [CrossRef] - Akatsu, K.; Narita, K.; Sakashita, Y.; Yamada, T. Impact of flux weakening current to the iron loss in an IPMSM including PWM carrier effect. In Proceedings of the Energy Conversion Congress and Exposition, San Jose, CA, USA, 20–24 September 2009; pp. 1927–1932. [Google Scholar]
- Yamazaki, K.; Ishigami, H. Rotor-shape optimization of interior permanent-magnet motors to reduce harmonic iron losses. IEEE Trans. Ind. Electron.
**2010**, 57, 61–69. [Google Scholar] [CrossRef] - Yamazaki, K.; Kumagai, M.; Ikemi, T.; Ohki, S. A novel rotor design of interior permanent-magnet synchronous motors to cope with both maximum torque and iron-loss reduction. IEEE Trans. Ind. Appl.
**2013**, 49, 2478–2486. [Google Scholar] [CrossRef] - Kuttler, S.; El KadriBenkara, K.; Friedrich, G.; Abdelli, A.; Vangraefschepe, F. Fast iron losses model of stator taking into account the flux weakening mode for the optimal sizing of high speed permanent internal magnet synchronous machine. Math. Comput. Simul.
**2017**, 131, 328–343. [Google Scholar] [CrossRef] - Tessarolo, A.; Mezzarobba, M.; Menis, R. Modeling, Analysis, and Testing of a Novel Spoke-Type Interior Permanent Magnet Motor With Improved Flux Weakening Capability. IEEE Trans. Magn.
**2015**, 51, 1–9. [Google Scholar] [CrossRef] - Atiq, S.; Kwon, B. Susceptibility of the winding switching technique for flux weakening to harmonics and the choice of a suitable drive topology. Int. J. Electr. Power Energy Syst.
**2017**, 85, 22–31. [Google Scholar] [CrossRef] - Rekik, M.; Besbes, M.; Marchand, C.; Multon, B.; Loudot, S.; Lhotellier, D. Improvement in the field-weakening performance of switched reluctance machine with continuous mode. IET Electr. Power Appl.
**2015**, 9, 439–448. [Google Scholar] [CrossRef] - Vaez-Zadeh, S.; Zahedi, B. Modeling and analysis of variable speed single phase induction motors with iron loss. Energy Convers. Manag.
**2009**, 50, 2747–2753. [Google Scholar] [CrossRef] - Li, Q.; Fan, T.; Wen, X. Characterization of Iron Loss for Integral-Slot Interior Permanent Magnet Synchronous Machine during Flux Weakening. IEEE Trans. Magn.
**2017**, 53, 1–7. [Google Scholar] [CrossRef] - Basic, M.; Vukadinović, D.; Petrović, G. Dynamic and pole-zero analysis of self-excited induction generator using a novel model with iron losses. Int. J. Electr. Power Energy Syst.
**2015**, 42, 105–118. [Google Scholar] [CrossRef] - Li, Q.; Fan, T.; Wen, X. Armature-reaction magnetic field analysis for interior permanent magnet motor based on winding function theory. IEEE Trans. Magn.
**2013**, 49, 1193–1201. [Google Scholar] [CrossRef] - Saavedra, H.; Urresty, J.; Riba, J.; Romeral, L. Detection of inter turn faults in PMSMs with different winding configurations. Energy Convers. Manag.
**2014**, 79, 534–542. [Google Scholar] [CrossRef] - Donolo, P.; Bossio, G.; Angelo, C. Analysis of voltage unbalance effects on induction motors with open and closed slots. Energy Convers. Manag.
**2011**, 52, 2024–2030. [Google Scholar] [CrossRef] - Ueda, Y.; Ohta, H.; Uenosono, C. Instrument for real-time measurements of airgap flux distribution of on-load synchronous generators. IEE Proc. A Phys.
**1987**, 134, 331–334. [Google Scholar] [CrossRef] - Hendershot, J.R.; Miller, T.J.E. Design and Performance of Brushless Permanent-Magnet Motors; Oxford University Press: London, UK, 1994. [Google Scholar]
- Zivotic-Kukolj, V.; Soong, W.L.; Ertugrul, N. Iron loss reduction in an interior PM automotive alternator. IEEE Trans. Ind. Appl.
**2006**, 42, 1478–1486. [Google Scholar] [CrossRef] - Lasdon, L.S.; Waren, A.D.; Jain, A.; Ratner, M.W. Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Optimization; AD-A009-402; Case Western Reserve University, National Technical Information Service U. S. Department of Commerce (NTIS): Stanford University, CA, USA, 1975; pp. 1–45. [Google Scholar]

**Figure 1.**Comparison of five coefficients institute of electrical machines formula (IEM-Formula) with measurements under 50 up to 700 (Hz) frequencies from the Epstein test, where solid lines denote analytical data and markers indicate measured data.

**Figure 2.**Steady-state equivalent d–q circuits of permanent magnet synchronous machine (PMSM) with iron loss resistance, listed as: (

**a**) d-axis equivalent circuit; (

**b**) q-axis equivalent circuit.

**Figure 3.**Magnetic flux density distribution using Equations (17) and (18) for tooth and yoke of stator.

**Figure 4.**Comparison of no-load air-gap magnetic field density waveforms using Equations (21) and (27), for (

**a**) fundamental no-load magnetic induction (B

_{m}

_{1}), and (

**b**) no-load airgap magnetic induction (B

_{m}).

**Figure 6.**Predicted iron losses using modified IEM-Formula as function of torque and frequency, where: (

**a**) the hysteresis loss; (

**b**) the eddy-current loss; (

**c**) excess loss contribution; and (

**d**) saturation loss.

**Figure 7.**Harmonic spectra and total iron loss prediction using classic and modified IEM-Formula during flux weakening operation time (FWOT), where: (

**a**) existing dominant harmonics on total iron loss and specially eddy current loss; (

**b**) presentation of the torque-frequency-power loss by the modified IEM-Formula.

**Figure 8.**Experimental investigation. (

**a**) the proposed PMSM with non-assembled parts under operation; (

**b**) experimental setup.

**Figure 9.**Comparison of classic IEM-Formula with modified IEM-Formula during FWOT, where: (

**a**) shows total iron loss evaluation; and (

**b**) presents efficiency computation.

Parameters | Values | Units |
---|---|---|

Stator outer/inner diameters | 209/115 | mm |

Rotor outer/inner diameters | 230/217 | mm |

Axial length | 100 | mm |

Slots/poles = SP | 36/40 = 0.9 | |

Air-gap length | 1.0 | mm |

Magnet thickness | 8.0 | mm |

Magnet pole-arc | 100 | ºe |

Rated power | 6.0 | kW |

Rated speed | 200 | rpm |

Dirrect current (DC) link voltage | 320 | V |

Steel sheet’s type | M400-50A | |

Lamination length | 95 | mm |

**Table 2.**Coefficients calculation using the modified institute of electrical machine formula (IEM-Formula) by curve fitting.

Coefficients | Values | Units |
---|---|---|

k_{tf} (1) | 7.0439 × 10^{−1} | |

k_{yf} (1) | 7.7938 × 10^{−1} | |

k_{t} | 0.4567 | |

k_{y} | 0.3031 | |

k_{pe} (1) | 0.3991 | |

k_{U} | 0.4586 | |

a_{1} | 398.0363203 | W/m^{3} |

a_{2} | 2.3821 × 10^{−2} | W/m^{3} |

α | 1.705944 | |

a_{3} | 11.74239805 | W/m^{3} |

a_{4} | 8.27 × 10^{−2} | |

a_{5} | 1.3617 × 10^{−9} | W/m^{3} |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Asef, P.; Bargallo, R.; Lapthorn, A. Iron Loss Prediction Using Modified IEM-Formula during the Field Weakening for Permanent Magnet Synchronous Machines. *Machines* **2017**, *5*, 30.
https://doi.org/10.3390/machines5040030

**AMA Style**

Asef P, Bargallo R, Lapthorn A. Iron Loss Prediction Using Modified IEM-Formula during the Field Weakening for Permanent Magnet Synchronous Machines. *Machines*. 2017; 5(4):30.
https://doi.org/10.3390/machines5040030

**Chicago/Turabian Style**

Asef, Pedram, Ramon Bargallo, and Andrew Lapthorn. 2017. "Iron Loss Prediction Using Modified IEM-Formula during the Field Weakening for Permanent Magnet Synchronous Machines" *Machines* 5, no. 4: 30.
https://doi.org/10.3390/machines5040030