# Development of Mathematical Models in Explicit Form for Design and Analysis of Axial Flux Permanent Magnet Synchronous Machines

^{*}

## Abstract

**:**

## 1. Introduction

- outer dimensions of the machine and its mass and
- current density in the windings.

## 2. Materials and Methods

#### 2.1. Simplified FEM Calculations

_{z}) along the centerline (shown in Figure 3) with a 55-mm virtual air gap thickness.

_{z_max}and B

_{z_min}, are shown, which represent the value of B

_{z}near the PM (B

_{z_max}) and in the middle of the virtual air gap (B

_{z_min}). Values in the middle of the virtual air gap represent the initial set of input data for development of the mathematical model in explicit form for the calculation of the value of axial component of magnetic flux density in the middle of the virtual air gap and consequently in the middle of the stator.

_{s}≤ 4h

_{TM}). Initial simplified FEM calculations were performed for the 3-phase AFPMSM with the data shown in Table 1 [41] and again repeated for the AFPMSM with the data shown in Table 2.

#### 2.2. Least Square Approximation Method

_{z_min}were used to produce a polynomial using a least square approximation (LSA) method. This is a mathematical procedure that can find a curve that fits best to a known set of given points by minimizing the sum of the squares of the offsets (“the residuals”) of the points from the curve [42]. The sum of the squares of the offsets was used instead of the offset absolute values because this allows the residuals to be treated as a continuous differentiable quantity [43]. Vertical least squares fitting proceeds by finding the sum of the squares of the vertical deviations R

^{2}of a set of n data points [43], which is presented in Equation (1).

_{i}is the data point calculated using simplified FEM, f is the fitting function, x

_{i}is an independent variable of the fitting function, and a

_{1}, a

_{2}, and a

_{n}are coefficients of the fitting function. In our case the polynomial was chosen as a fitting function.

_{z}is the axial component of magnetic flux density in the middle of the stator, d is the thickness of the stator with two air gaps, h

_{PM}is the thickness of the PMs and d

_{Fe}is the thickness of the rotor disks.

_{s}≤ 4 h

_{TM}) and with the PM span selected in such way that it covers 120° electrical. These limitations are selected on the basis of FEM analysis results, which are shown in Section 3.

#### 2.3. Design and Analysis of AFPMSM

_{i}(in addition to the electric and magnetic parameters), where i(t) is phase electrical current, m is the number of phases, e(t) the open circuit-induced voltage (phase value), η the efficiency of the machine, K

_{p}the electric power shape factor, T is one period of induced voltage, and E

_{pk}and I

_{pk}are peak values of the induced voltage and phase current [27].

_{em}is electromagnetic torque, B

_{z}the axial component of magnetic flux density, A

_{i}the average electric load on the radius that matches the inner radius of PMs, R

_{o}the outer PM radius, R

_{i}the inner PM radius and λ the ratio between the inner and outer PM radius.

^{3}).

_{s}is the synchronous speed of the machine, f the network frequency and p the number of pole pairs. It can be seen that the higher number of poles of the machine is required for lower synchronous rotations.

_{w}) is determined by the geometric method, from the ratio between the geometric (e

_{geometric}) and arithmetic (e

_{arithmetic}) value of the induced voltage (Equation (6)) [23].

_{sat}) is determined by Equation (7) [11]:

_{Fe}is the magnetic flux path in the iron, d

_{ag}the air gap thickness (between rotor and stator), d

_{Fe}the rotor disk thickness and μ

_{r}the relative permeability of steel.

_{z}) is determined by Equation (8) [9]:

_{r}is the remanent magnetic flux density, d

_{ag}the air gap thickness (between rotor and stator), d

_{s}the stator thickness, μ

_{rec}the recoil permeability, h

_{PM}the PM thickness and k

_{sat}the saturation factor.

_{i}) is independent of the radius and can be written as Equation (9) [9]:

_{p}is the pole pitch and τ

_{PM}the PM pitch.

_{z}is the axial component of magnetic flux density, p the number of poles, D

_{o}and D

_{i}the outer and inner PM diameter, E the induced voltage, f the network frequency, N

_{1}the number of turns per coil, k

_{w}the winding factor, Φ the magnetic flux, T

_{em}the electromagnetic torque, α

_{i}the PM pitch-to-pole pitch ratio, m the number of phases, and I

_{a}the effective value of stator current.

## 3. Results and Discussions

#### 3.1. Influence of Different Geometrical Parameters on Characteristics of the Machine

_{z}size in the middle of the stator for the machine described in Table 1. This analysis was carried out using simplified FEM.

_{z}size in the middle of the stator for the machine described in Table 1. This analysis was also carried out using the simplified FEM.

#### 3.2. Design and Analysis of AFPMSM

_{PM}= 5 mm and h

_{PM}= 7 mm in comparison with the analytical method. The reason for that can be found in the fact that the analytical method does not consider the PM span, but it works under the assumption that the PM surface and air gap surface are the same.

_{PM}= 10 mm, neither the analytical method nor mathematical model in explicit form produce accurate results compared to the gained FEM results. The reason for this is in the magnetic flux leakage, since the PM thickness is larger than the distance between adjacent PMs on the same rotor disk (h

_{PM}> l

_{PM}). These dimensions are also shown in Figure 9.

_{PM}= 10 mm, the magnetic flux leakage between adjacent PMs on the same rotor disk increases (Path 3). This leakage is also increased by increasing the stator thickness, due to the magnetic flux path enclosing in Path 3 and the decreasing share of magnetic flux in Path 1.

_{s}= 20 mm, which represents twice the PM thickness. For larger stator thicknesses magnetic flux leakage increases and the developed mathematical model in explicit form does not consider it, since it only considers the axial component of magnetic flux density in the middle of the stator.

#### 3.3. Laboratory Measurements

_{z}) in the middle of the distance between PMs on opposite rotor disks we developed a special tool that enables the PMs to stay in their position and measure the values at exact positions. Two tools were modeled and 3D printed (one 17 mm thick and one 22 mm thick). Figure 10a shows the model of the tool with the location of the opening where the measuring probe (Magnetometer KOSHAWA 5 with the magnetic probe) will be inserted and Figure 10b shows the printed tool. The measuring probe is inserted into the opening, which is printed to the middle radius of the PMs.

_{z}) at the middle of the distance between opposite PMs (on Path 1 in Figure 9), with PM thickness h

_{PM}= 5 mm and different rotor disk thicknesses.

_{z}) for different rotor disk thicknesses and two stator thicknesses are shown and compared to the calculated values (using FEM and the developed mathematical model) in Table 10.

## 4. Conclusions

_{z}in the middle of the stator of a double-sided coreless AFPMSM. The developed model simultaneously considers the influence of the thicknesses of rotor disks, PMs and stators along with the nonlinearity of the materials.

_{s}, h

_{PM}and d

_{Fe}), so they do not require new long-lasting calculations.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Mirsalim, M.; Yazdanpanah, R.; Hekmati, P. Design and analysis of double-sided slotless axial-flux permanent magnet machines with conventional and new stator core. IET Electr. Power Appl.
**2015**, 9, 193–202. [Google Scholar] [CrossRef] - Caricchi, F.; Crescimbini, F.; Mezzetti, F.; Santini, E. Multistage axial-flux PM machine for wheel direct drive. IEEE Trans. Ind. Appl.
**1996**, 32, 882–888. [Google Scholar] [CrossRef] - Virtic, P.; Pisek, P.; Marcic, T.; Hadziselimovic, M.; Stumberger, B. Analytical Analysis of Magnetic Field and Back Electromotive Force Calculation of an Axial-Flux Permanent Magnet Synchronous Generator with Coreless Stator. IEEE Trans. Magn.
**2008**, 44, 4333–4336. [Google Scholar] [CrossRef] - Lombard, N.; Kamper, M.J. Analysis and performance of an ironless stator axial flux PM machine. IEEE Trans. Energy Convers.
**1999**, 14, 1051–1056. [Google Scholar] [CrossRef] - Daghigh, A.; Javadi, H.; Torkaman, H. Improved design of coreless axial flux permanent magnet synchronous generator with low active material cost. In Proceedings of the 6th Power Electronics, Drive Systems & Technologies Conference (PEDSTC2015), Tehran, Iran, 3–4 February 2015; pp. 532–537. [Google Scholar] [CrossRef]
- Javadi, S.; Mirsalim, M. A Coreless Axial-Flux Permanent-Magnet Generator for Automotive Applications. IEEE Trans. Magn.
**2008**, 44, 4591–4598. [Google Scholar] [CrossRef] - Chen, A.; Nilssen, R.; Nysveen, A. Performance comparisons among radial flux, multi-stage axial flux and three-phase transverse flux PM machines for downhole applications. In Proceedings of the IEEE International Electric Machines and Drives Conference, Miami, FL, USA, 3–6 May 2009; pp. 1010–1017. [Google Scholar]
- Hanselman, D. Brushless Permanent Magnet Motor Design; The Writers’ Collective: Cranston, RI, USA, 2003. [Google Scholar]
- Gieras, J.F.; Wang, R.J.; Kamper, M.J. Axial Flux Permanent Magnet Brushless Machines; Kluwer Academic Publishers: Dordrecht, Germany, 2004. [Google Scholar]
- Mueller, M.A.; McDonald, A.S.; MacPherson, D.E. Structural analysis of low-speed axial-flux permanent-magnet machines. IEE Proc. Electr. Power Appl.
**2005**, 152, 1417–1426. [Google Scholar] [CrossRef] - Jara, W.; Martín, A.; Tapia, J.A. Axial flux PM machine for low wind power generation. In Proceedings of the XIX International Conference on Electrical Machines—ICEM 2010, Rome, Italy, 6–8 September 2010; pp. 1–5. [Google Scholar] [CrossRef]
- Daghigh, A.; Javadi, H.; Javadi, A. Improved Analytical Modeling of Permanent Magnet Leakage Flux in Design of the Coreless Axial Flux Permanent Magnet Generatorx. Can. J. Electr. Comput. Eng.
**2017**, 40, 3–11. [Google Scholar] - Caricchi, F.; Crescimbini, F.; Honorzti, O.; Bianco, G.L.; Santini, E. Performance of coreless-winding axial-flux permanent-magnet generator with power output at 400 Hz-3000 rev/min. In Proceedings of the IAS ’97 Conference Record of the 1997 IEEE Industry Applications Conference Thirty-Second IAS Annual Meeting, New Orleans, LA, USA, 5–9 October 1997; Volume 1, pp. 61–66. [Google Scholar]
- Wang, Y.; Chen, W.X.C.; Dong, Z. A parametric magnetic network model for axial flux permanent magnet machine with coreless stator. In Proceedings of the 17th International Conference on Electrical Machines and Systems (ICEMS), Hangzhou, China, 22–25 October 2014; pp. 1108–1113. [Google Scholar] [CrossRef]
- Chan, T.F.; Lai, L.L.; Xie, S. Field Computation for an Axial Flux Permanent-Magnet Synchronous Generator. IEEE Trans. Energy Convers.
**2009**, 24, 1–11. [Google Scholar] [CrossRef] - Bumby, J.R.; Martin, R.; Mueller, M.A.; Spooner, E.; Brown, N.L.; Chalmers, B.J. Electromagnetic design of axial-flux permanent magnet machines. Proc. Inst. Elect. Eng. Elect. Power Appl.
**2004**, 151, 151–160. [Google Scholar] [CrossRef] - Azzouzi, J.; Barakat, G.; Dakyo, B. Quasi-3D analytical modeling of the magnetic field of an axial flux permanent magnet synchronous machine. In Proceedings of the IEEE International Electric Machines and Drives Conference, IEMDC’03, Madison, WI, USA, 1–4 June 2003; IEEE: Piscataway, NJ, USA, 2003; Volume 3, pp. 1941–1947. [Google Scholar] [CrossRef]
- Virtic, P.; Pisek, P.; Hadziselimovic, M.; Marcic, T.; Stumberger, B. Torque Analysis of an Axial Flux Permanent Magnet Synchronous Machine by Using Analytical Magnetic Field Calculation. IEEE Trans. Magn.
**2009**, 45, 1036–1039. [Google Scholar] [CrossRef] - Tiegna, H.; Bellara, A.; Amara, Y.; Barakat, G. Analytical Modeling of the Open-Circuit Magnetic Field in Axial Flux Permanent-Magnet Machines with Semi-Closed Slots. IEEE Trans. Magn.
**2012**, 48, 1212–1226. [Google Scholar] [CrossRef] - Marignetti, F.; Di Stefano, R. Electromagnetic Analysis of Axial-Flux Permanent Magnet Synchronous Machines with Fractional Windings with Experimental Validation. IEEE Trans. Ind. Electron.
**2011**, 59, 2573–2582. [Google Scholar] [CrossRef] - Caricchi, F.; Crescimbini, F.; Honorati, O.; Di Napoli, A.; Santini, E. Compact wheel direct drive for EVs. IEEE Ind. Appl. Mag.
**1996**, 2, 25–32. [Google Scholar] [CrossRef] - Pranjić, F.; Virtič, P. Determination of an Optimum Fictitious Air Gap and Rotor Disk Thickness for a Coreless AFPMM. Teh. Vjesn.
**2018**, 25, 1731–1738. [Google Scholar] [CrossRef] - Pranjic, F.; Virtic, P. Designing Rotor Disks of a Coreless Axial Flux Permanent Magnet Machines by Using Simplified FEM and an Approximation Method. IEEE Trans. Energy Convers.
**2020**, 35, 1505–1512. [Google Scholar] [CrossRef] - Kahourzade, S.; Mahmoudi, A.; Rahim, N.A.; Ping, H.W. Sizing equation and Finite Element Analysis optimum design of axial-flux permanent-magnet motor for electric vehicle direct drive. In Proceedings of the IEEE International Power Engineering and Optimization Conference, Melaka, Malaysia, 6–7 June 2012; pp. 1–6. [Google Scholar] [CrossRef]
- Rallabandi, V.; Taran, N.; Ionel, D.M.; Eastham, J.F. On the feasibility of carbon nanotube windings for electrical machines—Case study for a coreless axial flux motor. In Proceedings of the IEEE Energy Conversion Congress and Exposition (ECCE), Milwaukee, WI, USA, 18–22 September 2016; pp. 1–7. [Google Scholar]
- Kalender, O.; Ege, Y.; Nazlibilek, S. Design and determination of stator geometry for axial flux permanent magnet free rod rotor synchronous motor. Measurement
**2011**, 44, 1753–1760. [Google Scholar] [CrossRef] - Huang, S.; Luo, J.; Leonardi, F.; Lipo, T.A. A comparison of power density for axial flux machines based on general purpose sizing equations. IEEE Trans. Energy Convers.
**1999**, 14, 185–192. [Google Scholar] [CrossRef] - Aydin, M.; Gulec, M.; Demir, Y.; Akyuz, B.; Yolacan, E. Design and validation of a 24-pole coreless axial flux permanent magnet motor for a solar powered vehicle. In Proceedings of the XXII International Conference on Electrical Machines (ICEM), Lausanne, Switzerland, 4–7 September 2016; pp. 1493–1498. [Google Scholar] [CrossRef]
- Xie, H.; Wei, X.; Yang, K. A novel modular multistage axial-flux permanent magnet machine for electric vehicles. In Proceedings of the 17th International Conference on Electrical Machines and Systems (ICEMS), Hangzhou, China, 22–25 October 2014; pp. 206–211. [Google Scholar] [CrossRef]
- Capponi, F.G.; De Donato, G.; Caricchi, F. Recent Advances in Axial-Flux Permanent-Magnet Machine Technology. IEEE Trans. Ind. Appl.
**2012**, 48, 2190–2205. [Google Scholar] [CrossRef] - Seo, J.M.; Jung, I.S.; Jung, H.K.; Ro, J.S. Analysis of Overhang Effect for a Surface-Mounted Permanent Magnet Machine Using a Lumped Magnetic Circuit Model. IEEE Trans. Magn.
**2014**, 50, 1–7. [Google Scholar] [CrossRef] - Mahmoudi, A.; Rahim, N.A.; Hew, W.P. Axial-flux permanent-magnet machine modeling, design, simulation and analysis. Sci. Res. Essays
**2011**, 6, 2525–2549. [Google Scholar] - Taran, N.; Ardebili, M. A novel approach for efficiency and power density optimization of an Axial Flux Permanent Magnet generator through genetic algorithm and finite element analysis. In Proceedings of the IEEE 23rd International Symposium on Industrial Electronics (ISIE), Istanbul, Turkey, 1–4 June 2014; pp. 709–714. [Google Scholar]
- Egea, A.; Almandoz, G.; Poza, J.; Gonzalez, A. Axial flux machines modelling with the combination of 2D FEM and analytic tools. In Proceedings of the XIX International Conference on Electrical Machines—ICEM, Rome, Italy, 6–8 September 2010; pp. 1–6. [Google Scholar]
- Choi, J.Y.; Lee, S.H.; Ko, K.J.; Jang, S.M. Improved Analytical Model for Electromagnetic Analysis of Axial Flux Machines with Double-Sided Permanent Magnet Rotor and Coreless Stator Windings. IEEE Trans. Magn.
**2011**, 47, 2760–2763. [Google Scholar] [CrossRef] - Mohammadi, S.; Mirsalim, M. Analytical Design Framework for Torque and Back-EMF Optimization, and Inductance Calculation in Double-Rotor Radial-Flux Air-Cored Permanent-Magnet Synchronous Machines. IEEE Trans. Magn.
**2013**, 50, 1–16. [Google Scholar] [CrossRef] - Radwan-Pragłowska, N.; Borkowski, D.; Wegiel, T. Model of coreless axial flux permanent magnet generator. In Proceedings of the International Symposium on Electrical Machines (SME), Naleczow, Poland, 18–21 June 2017; pp. 1–6. [Google Scholar]
- Mulyaseputra, P.S.; Hadi, S.P.; Wijaya, F.D.; Suharyanto, S. Analysis of ferrite effect in Axial Flux Permanent Magnet Generator using magnetic circuit approach. In Proceedings of the 3rd International Conference on Science and Technology—Computer (ICST), Yogyakarta, Indonesia, 11–12 July 2017; pp. 45–50. [Google Scholar]
- Mulyaseputra, P.S.; Wijaya, F.D.; Hadi, S.P.; Suharyanto, S. Magnetic field distribution simulation on two permanent magnet rotor discs with twelve pole pairs. In Proceedings of the 3rd International Conference on Instrumentation, Communications, Information Technology and Biomedical Engineering (ICICI-BME), Bandung, Indonesia, 7–8 November 2013; pp. 178–183. [Google Scholar]
- Joss, A.; Randewijk, P.J. Design and optimisation of an Ironless Double-rotor Radial Flux Permanent Magnet machine. In Proceedings of the XXII International Conference on Electrical Machines (ICEM), Lausanne, Switzerland, 4–7 September 2016; pp. 1473–1479. [Google Scholar]
- Virtič, P. Načrtovanje in Analiza Sinhronskih Strojev s Trajnimi Magneti in Aksialnim Magnetnim Pretokom. Ph.D. Thesis, University of Maribor, Maribor, Slovenia, 2009. [Google Scholar]
- Weisstein, E.W. Least Squares Fitting. MathWorld—Wolfram Web Resource. Available online: http://mathworld.wolfram.com/LeastSquaresFitting.html (accessed on 20 March 2019).
- Kenney, J.F.; Keeping, E.S. Linear Regression and Correlation. In Mathematics of Statistics, 3rd ed.; Van Nostrand: Princeton, NJ, USA, 1962; Chapter 15, Part 1; pp. 252–285. [Google Scholar]
- Pyrhonen, J.; Jokinen, T.; Hrabovcova, V. Design of Rotating Electrical Machines; Wiley and Sons: Hoboken, NJ, USA, 2009; Wiley Online. [Google Scholar]
- Spooner, E.; Chalmers, B. ‘TORUS’: A slotless, toroidal-stator, permanent-magnet generator. Proc. Inst. Elect. Eng. Elect. Power Appl.
**1992**, 139, 497. [Google Scholar] [CrossRef][Green Version] - Caricchi, F.; Chalmers, B.; Crescimbini, F.; Spooner, E. Advances in the design of TORUS machines. In Proceedings of the International Conference on Power Electronic Drives and Energy Systems for Industrial Growth, Perth, WA, Australia, 1–3 December 1998; pp. 516–522. [Google Scholar]

**Figure 1.**Simplified finite element method (FEM) model: (

**a**) 3D model of the simplified axial flux permanent magnet synchronous machine (AFPMSM), and (

**b**) model of rotor disk with surface-mounted permanent magnets (PMs) with their dimensions and the position of the centerline on which the axial component of magnetic flux density is analyzed.

**Figure 2.**Axial component of magnetic flux density along the centerline (an example is shown at 55 mm virtual air gap thickness, 7 mm rotor disk thickness and 5 mm PM thickness. Values of B

_{z}near the PM (B

_{z_max}) and at the middle of the virtual air gap (B

_{z_min}) are shown).

**Figure 10.**Tool developed for the measurement of the axial component of magnetic flux density: (

**a**) 3D model, (

**b**) printed tool.

**Figure 11.**Measuring procedure for with the developed tool (d = 22 mm) with 5 mm PM thickness and different rotor disk thicknesses: h

_{TM}= 5 mm: (

**a**) d

_{Fe}= 5 mm, (

**b**) d

_{Fe}= 6 mm, (

**c**) d

_{Fe}= 10 mm.

Symbol | Quantity | Value/Unit |
---|---|---|

d_{Fe} | Rotor disk thickness | 7 mm |

h_{PM} | Permanent magnet thickness | 5 mm |

τ_{m} | Magnetic pitch | 25° |

D_{i} | Inner diameter of PM | 80 mm |

D_{o} | Outer diameter of PM | 150 mm |

τ_{p} | Pole pitch | 36° |

I | Electrical current | 2 × 10 A |

Number of windings | 6 | |

d_{s} | Winding thickness | 15 mm |

d_{c} | Coil width | 20 mm |

S_{w} | Copper wire cross section | 1.23 mm^{2} |

d_{ag} | Air gap thickness | 1 mm |

k_{w} | Winding factor | 0.966 |

p | Number of pole pairs | 5 |

Symbol | Quantity | Value/Unit |
---|---|---|

d_{Fe} | Rotor disk thickness | 7 mm |

h_{PM} | Permanent magnet thickness | 5 mm |

τ_{m} | Magnetic pitch | 12° |

D_{i} | Inner diameter of PM | 80 mm |

D_{o} | Outer diameter of PM | 150 mm |

τ_{p} | Pole pitch | 18° |

I | Electrical current | 2 × 10 A |

Number of windings | 12 | |

d_{s} | Winding thickness | 15 mm |

d_{c} | Coil width | 15 mm |

S_{w} | Copper wire cross section | 1.23 mm^{2} |

d_{ag} | Air gap thickness | 1 mm |

k_{w} | Winding factor | 0.966 |

p | Number of pole pairs | 10 |

d_{s}(mm) | h_{PM}(mm) | d_{Fe}(mm) | N_{f} | k_{sat} | B_{z}(T) | Φ (Wb) | T_{em}(Nm) | E (V) |
---|---|---|---|---|---|---|---|---|

15 | 5 | 7 | 107 | 1.011 | 0.4298 | 0.001509746 | 29.80 | 34.67 |

20 | 5 | 7 | 142 | 1.011 | 0.3610 | 0.001268166 | 33.22 | 38.64 |

15 | 7 | 7 | 107 | 1.011 | 0.5274 | 0.00185257 | 36.57 | 42.54 |

20 | 7 | 7 | 142 | 1.011 | 0.4520 | 0.00158751 | 41.59 | 48.37 |

25 | 7 | 7 | 178 | 1.011 | 0.3954 | 0.001388804 | 45.61 | 53.05 |

20 | 10 | 8 | 142 | 1.010 | 0.5574 | 0.001958025 | 51.30 | 59.66 |

30 | 10 | 7 | 214 | 1.011 | 0.4468 | 0.001569545 | 61.97 | 72.08 |

40 | 10 | 7 | 285 | 1.011 | 0.3730 | 0.001310093 | 68.89 | 80.12 |

d_{s}(mm) | h_{PM}(mm) | d_{Fe}(mm) | N_{f} | B_{z}(T) | Φ (Wb) | T_{em}(Nm) | E (V) |
---|---|---|---|---|---|---|---|

15 | 5 | 7 | 107 | 0.4319 | 0.001517 | 29.95 | 34.83 |

20 | 5 | 7 | 142 | 0.3628 | 0.0012743 | 33.38 | 38.83 |

15 | 7 | 7 | 107 | 0.5183 | 0.0018205 | 35.94 | 41.80 |

20 | 7 | 7 | 142 | 0.4437 | 0.0015585 | 40.83 | 47.49 |

25 | 7 | 7 | 178 | 0.3827 | 0.0013442 | 44.14 | 51.35 |

20 | 10 | 8 | 142 | 0.5368 | 0.0018855 | 49.40 | 57.45 |

30 | 10 | 7 | 214 | 0.4051 | 0.0014229 | 56.18 | 65.34 |

40 | 10 | 7 | 285 | 0.3168 | 0.0011128 | 58.51 | 68.05 |

Analytical Method | FEM | Analytical Method and Mathematical Models | |||||
---|---|---|---|---|---|---|---|

h_{PM} | d_{s} | T_{em} | E | T_{em} | E | T_{em} | E |

(mm) | (mm) | (Nm) | (V) | (Nm) | (V) | (Nm) | V |

5 | 15 | 29.80 | 34.67 | 29.94 | 33.85 | 29.95 | 34.83 |

20 | 33.22 | 38.64 | 33.43 | 37.69 | 33.38 | 38.83 | |

7 | 15 | 36.57 | 42.54 | 35.96 | 38.21 | 35.94 | 41.80 |

20 | 41.59 | 48.37 | 41.22 | 46.41 | 40.83 | 47.49 | |

25 | 45.61 | 53.05 | 44.56 | 50.5 | 44.14 | 51.35 | |

10 | 20 | 51.30 | 59.66 | 49.69 | 55.27 | 49.40 | 57.45 |

30 | 61.97 | 72.08 | 57.57 | 65.73 | 56.18 | 65.34 | |

40 | 68.89 | 80.12 | 62.63 | 71.29 | 58.51 | 68.05 |

**Table 6.**Results of the design of 20-pole AFPMSM using FEM and comparison of the results of all three methods.

Analytical Method | FEM | Analytical Method and Mathematical Models | Matching (%) | |||
---|---|---|---|---|---|---|

h_{TM} (mm) | d_{s} (mm) | T_{em}(Nm) | T_{em} (Nm) | T_{em}(Nm) | FEM/ Model | FEM/ Analytical |

5 | 15 | 42.78 | 36.78 | 38.87 | 105.68 | 116.31 |

20 | 48.07 | 40.00 | 40.26 | 100.67 | 120.18 | |

7 | 15 | 52.50 | 43.17 | 45.91 | 106.33 | 121.60 |

20 | 60.17 | 47.58 | 48.02 | 100.93 | 126.47 | |

25 | 65.43 | 50.19 | 46.87 | 93.38 | 130.35 | |

10 | 20 | 74.21 | 54.76 | 55.25 | 100.89 | 135.52 |

30 | 88.96 | 60.26 | 51.69 | 85.78 | 147.62 | |

40 | 99.31 | 62.44 | 43.59 | 69.82 | 159.05 | |

Average | 95.43 | 132.14 |

n (min^{−1}) | E_{1} (V) | E_{2} (V) | E_{3} (V) | E_{average} (V) |
---|---|---|---|---|

200 | 11.31 | 11.52 | 11.48 | 11.33 |

300 | 16.8 | 17.12 | 17.1 | 16.99 |

400 | 22.5 | 22.88 | 22.67 | 22.65 |

500 | 28.14 | 28.47 | 28.34 | 28.31 |

600 | 33.84 | 34.18 | 33.96 | 33.97 |

700 | 39.44 | 39.83 | 39.6 | 39.62 |

800 | 45.04 | 45.53 | 45.23 | 45.27 |

900 | 50.68 | 51.22 | 50.89 | 50.92 |

n | E_{average} | f | E_{model} | E_{average}/E_{model} |
---|---|---|---|---|

(min^{−1}) | (V) | (Hz) | (V) | (%) |

200 | 11.33 | 16.67 | 10.85 | 104.42 |

300 | 16.99 | 25 | 16.28 | 104.36 |

400 | 22.65 | 33.33 | 21.7 | 104.38 |

500 | 28.31 | 41.67 | 27.13 | 104.35 |

600 | 33.97 | 50 | 32.55 | 104.36 |

700 | 39.62 | 58.33 | 37.98 | 104.32 |

800 | 45.27 | 66.67 | 43.41 | 104.28 |

900 | 50.92 | 75 | 48.83 | 104.28 |

I | T_{measured} | T_{model} | T_{measured}/T_{model} |
---|---|---|---|

(A) | (Nm) | (Nm) | (%) |

10.21 | 16.72 | 15.29 | 109.35 |

11.00 | 17.97 | 16.48 | 109.04 |

12.03 | 19.52 | 18.01 | 108.38 |

13.06 | 21.05 | 19.55 | 107.67 |

13.98 | 22.41 | 20.93 | 107.07 |

15.08 | 23.94 | 22.57 | 106.07 |

16.14 | 25.48 | 24.16 | 105.46 |

17.18 | 26.91 | 25.72 | 104.63 |

18.16 | 28.27 | 27.19 | 103.97 |

19.3 | 29.8 | 28.89 | 103.15 |

20.19 | 30.97 | 30.22 | 102.48 |

h_{TM}(mm) | d (d_{s} + 2 d_{ag})(mm) | d_{Fe}(mm) | B_{z_FEM}(T) | B_{z_Model}(T) | B_{z_Meas}(T) | B_{z_FEM}/B_{z_Meas}(%) | B_{z_Model}/B_{z_Meas}(%) |
---|---|---|---|---|---|---|---|

5 | 17 | 5 | 0.4401 | 0.4367 | 0.4490 | 98.02 | 97.27 |

5 | 17 | 6 | 0.4581 | 0.4551 | 0.4720 | 97.06 | 96.41 |

5 | 17 | 10 | 0.4671 | 0.4700 | 0.4790 | 97.52 | 98.12 |

5 | 17 | 11 | 0.4676 | 0.4687 | 0.4800 | 97.41 | 97.64 |

5 | 17 | 15 | 0.4686 | 0.4718 | 0.4830 | 97.02 | 97.68 |

5 | 22 | 5 | 0.3758 | 0.3737 | 0.3870 | 97.10 | 96.57 |

5 | 22 | 6 | 0.3810 | 0.3842 | 0.3950 | 96.45 | 97.27 |

5 | 22 | 10 | 0.3851 | 0.3838 | 0.4020 | 95.79 | 95.46 |

5 | 22 | 11 | 0.3856 | 0.3820 | 0.4020 | 95.92 | 95.02 |

5 | 22 | 15 | 0.3861 | 0.3929 | 0.4020 | 96.04 | 97.74 |

Average | 96.83 | 96.92 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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**

Pranjić, F.; Virtič, P.
Development of Mathematical Models in Explicit Form for Design and Analysis of Axial Flux Permanent Magnet Synchronous Machines. *Appl. Sci.* **2020**, *10*, 7695.
https://doi.org/10.3390/app10217695

**AMA Style**

Pranjić F, Virtič P.
Development of Mathematical Models in Explicit Form for Design and Analysis of Axial Flux Permanent Magnet Synchronous Machines. *Applied Sciences*. 2020; 10(21):7695.
https://doi.org/10.3390/app10217695

**Chicago/Turabian Style**

Pranjić, Franjo, and Peter Virtič.
2020. "Development of Mathematical Models in Explicit Form for Design and Analysis of Axial Flux Permanent Magnet Synchronous Machines" *Applied Sciences* 10, no. 21: 7695.
https://doi.org/10.3390/app10217695