Methodology to Generate a Reluctance Network Applied to Permanent Magnet Synchronous Generators

Kasper, João Vitor Gerevini; Kuo-Peng, Patrick; Bazzo, Thiago de Paula Machado; Menezes, Gustavo Garbelini de

doi:10.1590/2179-10742022v21i4268100

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Journal of Microwaves, Optoelectronics and Electromagnetic Applications

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Article • J. Microw. Optoelectron. Electromagn. Appl. 21 (4) • Oct-Dec 2022 • https://doi.org/10.1590/2179-10742022v21i4268100 copy

Methodology to Generate a Reluctance Network Applied to Permanent Magnet Synchronous Generators

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

This paper presents a methodology for the generation of a static reluctance network, with magnetomotive sources only in the rotor, applied in the analysis and design of surface mounted permanent magnet synchronous generators. In the analysis stage, this network can provide the magnetic flux in several points of the generator, enabling an estimate of the generator performance. In the design process, on the other hand, it provides the armature linkage flux and hence the induced EMF can be calculated. It is worth mentioning that the proposed methodology makes use of a good discretization level (high number of elements) on the teeth tips and in the air gap of the generator, in order to get better precision on the flux modeling of these regions. The main contribution of this paper is to present the development process of a static reluctance network applied to surface mounted permanent magnets synchronous generators with different dimensions. The proposed model has proved to be robust and flexible as it can be seen in the results. The reluctance network (RN) developed with the proposed methodology presented in this paper shows low variations of the magnetic flux values, when compared with finite element simulations.

Index Terms
Permanent Magnet Synchronous Generators; Reluctance Network; Finite Elements Method; Electric Machine Modeling

Reluctance	Length (m)	Area (m²)
$T_{t 1.1} T_{t 1.2}$	$\frac{h_{S}}{2}$	$\frac{W_{T}}{2} \cdot L_{A}$
$T_{t 2.1} T_{t 2.2} T_{t 3.1} T_{t 3.2}$	$\frac{h_{S}}{2}$	$W_{T} \cdot L_{A}$
$T_{R 1}$	$\frac{h_{S y}}{2}$	$(\frac{W_{T}}{2} + \frac{W_{S b}}{2}) \cdot L_{A}$
$T_{R 2} T_{R 3}$	$\frac{h_{S y}}{2}$	$(\frac{W_{T}}{2} + \frac{W_{S b}}{2}) \cdot L_{A}$
$S_{y}_{1} S_{y}_{2}$	$(\frac{D_{S t}}{2} + h_{S} + \frac{h_{S y}}{2}) \cdot α_{S}$	$h_{S y} \cdot L_{A}$
$S_{y 3}$	$(\frac{D_{S t}}{2} + h_{S} + \frac{h_{S y}}{2}) \cdot 2 \cdot α_{S}$	$3 \cdot \frac{h_{S y}}{4} \cdot L_{A}$
$S_{y 8}$	$(\frac{D_{S t}}{2} + h_{S} + \frac{h_{S y}}{2}) \cdot (α_{r} - \frac{θ_{d}}{2})$	$\frac{h_{S y}}{4} \cdot L_{A}$
$S_{y 7}$	$W_{T}$	$(h_{S y} + \frac{h_{S}}{4}) \cdot L_{A}$
$T_{t 4.1} T_{t 4.2}$	$\frac{W_{T}}{2} + \frac{W_{S w}}{2}$	$(h_{S w}) \cdot L_{A}$
$T_{t 4.3} T_{t 4.5}$	$h_{S w}$	$h_{S w} \cdot L_{A}$
$T_{t 4.4}$	$h_{S w}$	$W_{T} \cdot L_{A}$
$T_{t 4.6} T_{t 4.9}$	$\frac{W_{S w}}{2}$	$(h_{S o} + \frac{h_{S w}}{2}) \cdot L_{A}$
$T_{t 4.7} T_{t 4.8}$	$\frac{W_{T}}{2} + \frac{W_{S w}}{2}$	$h_{S o} \cdot L_{A}$
$T_{t 4.10} T_{t 4.12}$	$h_{S o}$	$W_{Sw} \cdot L_{A}$
$T_{t 4.11}$	$h_{S o}$	$L_{d} \cdot L_{A}$
$R_{y 1.1} R_{y 1.2} R_{y 1.3} R_{y 1.6} R_{y 1.7} R_{y 1.8}$	$\frac{H_{c r}}{4}$	$\frac{W_{P M}}{6} \cdot L_{A}$
$R_{y 1.4} R_{y 1.5} R_{y 1.9} R_{y 1.10}$	$(\frac{D_{b P M}}{2} - \frac{h_{S w}}{2}) \cdot \frac{α_{P e f}}{4}$	$\frac{h_{S w}}{4} \cdot L_{A}$
$R_{y 1.11}$	$\frac{h_{S w}}{2}$	$\frac{W_{P M}}{3} \cdot L_{A}$
$R_{y 1.12}$	$\frac{h_{S w}}{2}$	$\frac{W_{P M}}{6} \cdot L_{A}$
$R_{y 1.13}$	$(\frac{D_{b P M}}{2} - \frac{h_{S w}}{2}) \cdot \frac{α_{P e f}}{4}$	$\frac{h_{S w}}{2} \cdot L_{A}$

Reluctance	Length (m)	Area (m²)
$A_{g 4.1} A g_{4.3} A g_{4.6} A g_{4.8}$	$\frac{L_{A G}}{2}$	$W_{S w} \cdot L_{A}$
$A_{g 4.2} A_{g 4.7}$	$\frac{L_{A G}}{2}$	$W_{T} \cdot L_{A}$
$\begin{matrix} A_{g 4.4} A_{g 4.5} A_{g 4.9} \\ A_{g 4.10} \end{matrix}$	$\frac{W_{T}}{2} + \frac{W_{S w}}{2}$	$\frac{L_{A G}}{2} \cdot L_{A}$
$S_{o 3.1} S_{o 3.2} S_{o 3.4} S_{o 3.5}$	$\frac{W_{S w}}{2} + \frac{W_{S o}}{2}$	$\frac{L_{A G}}{2} \cdot L_{A}$
$S_{o 3.3}$	$\frac{L_{A G}}{2}$	$W_{S o} \cdot L_{A}$
$S_{o 3.6}$	$\frac{L_{A G}}{2} + \frac{h_{S o}}{2}$	$h_{S o} \cdot L_{A}$
$S_{o 3.7} S_{o 3.8}$	$\frac{W_{S o}}{2}$	$h_{S o} \cdot L_{A}$
$S_{o 3.9}$	$W_{S o} + W_{S w}$	$h_{S w} \cdot L_{A}$
$\begin{matrix} P_{M 1.1} P_{M 1.4} P_{M 1.5} \\ P_{M 1.8} P_{M 1.9} \end{matrix}$	$(\frac{D_{r e}}{2} - \frac{h_{P M}}{2}) \cdot (\frac{θ_{T}}{2} + \frac{θ_{W_{S w}}}{2})$	$h_{P M} \cdot L_{A}$
$\begin{matrix} P_{M 1.2} P_{M 1.3} P_{M 1.6} \\ P_{M 1.7} P_{M 1.10} P_{M 1.11} \end{matrix}$	$(\frac{D_{r e}}{2} - \frac{h_{P M}}{2}) \cdot (\frac{θ_{W_{S o}}}{2} + \frac{θ_{W_{S w}}}{2})$	$h_{P M} \cdot L_{A}$
$A_{g D 1}$	$(\frac{D_{r e}}{2} - \frac{h_{P M}}{2}) \cdot (α_{P} - α_{P e f})$	$h_{P M} \cdot L_{A}$

Magnetomotive Source	Length (m)	Area (m²)
$F_{1.1} F_{1.13}$	$\frac{h_{P M}}{2}$	$(\frac{W_{P M}}{2}) \cdot (\frac{\frac{θ_{T}}{2}}{\frac{α_{P e f}}{2}}) \cdot L_{A}$
$\begin{array}{l} F_{1.5} & F_{1.9} & F_{1.17} & F_{1.21} \end{array}$	$\frac{h_{P M}}{2}$	$(\frac{W_{P M}}{2}) \cdot (\frac{\frac{θ_{T}}{2}}{\frac{α_{P e f}}{2}}) \cdot L_{A}$
$\begin{matrix} F_{1.2} F_{1.4} F_{1.6} F_{1.8} F_{1.10} F_{1.12} F_{1.14} \\ F_{1.16} F_{1.18} F_{1.20} F_{1.22} F_{1.24} \end{matrix}$	$\frac{h_{P M}}{2}$	$(\frac{W_{P M}}{2}) \cdot (\frac{θ_{W_{S w}}}{\frac{α_{P e f}}{2}}) \cdot L_{A}$
$F_{1.3} F_{1.7} F_{1.11} F_{1.15} F_{1.19} F_{1.23}$	$\frac{h_{P M}}{2}$	$(\frac{W_{P M}}{2}) \cdot (\frac{θ_{W_{S o}}}{\frac{α_{P e f}}{2}}) \cdot L_{A}$

Magnetomotive Source	$\sum θ_{F x . y}$	$θ_{F x . y}$
$\begin{array}{l} F_{1.1} & F_{1.13} & F_{2.1} & F_{2.13} \end{array}$	$0.5 \cdot θ_{T}$	$0.5 \cdot θ_{T}$
$F_{1.2} F_{1.14} F_{2.2} F_{2.14}$	$θ_{W_{S w}} + 0.5 \cdot θ_{T}$	$θ_{W_{S w}}$
$\begin{array}{l} F_{1.3} & F_{1.15} & F_{2.3} & F_{2.15} \end{array}$	$θ_{W_{S o}} + θ_{W_{S w}} + 0.5 \cdot θ_{T}$	$θ_{W_{S o}}$
$\begin{array}{l} F_{1.4} & F_{1.16} & F_{2.4} & F_{2.16} \end{array}$	$θ_{W_{S o}} + 2 \cdot θ_{W_{S w}} + 0.5 \cdot θ_{T}$	$θ_{W_{S w}}$
$\begin{array}{l} F_{1.5} & F_{1.17} & F_{2.5} & F_{2.17} \end{array}$	$θ_{W_{S o}} + 2 \cdot θ_{W_{S w}} + 1.5 \cdot θ_{T}$	$θ_{T}$
$\begin{array}{l} F_{1.6} & F_{1.18} & F_{2.6} & F_{2.18} \end{array}$	$θ_{W_{S o}} + 3 \cdot θ_{W_{S w}} + 1.5 \cdot θ_{T}$	$θ_{W_{S w}}$
$\begin{array}{l} F_{1.7} & F_{1.19} & F_{2.7} & F_{2.19} \end{array}$	$2 \cdot θ_{L_{r a}} + 3 \cdot θ_{W_{S w}} + 1.5 \cdot θ_{T}$	$θ_{W_{S o}}$
$\begin{array}{l} F_{1.8} & F_{1.20} & F_{2.8} & F_{2.20} \end{array}$	$2 \cdot θ_{W_{S o}} + 4 \cdot θ_{W_{S w}} + 1.5 \cdot θ_{T}$	$θ_{W_{S w}}$
$\begin{array}{l} F_{1.9} & F_{1.21} & F_{2.9} & F_{2.21} \end{array}$	$2 \cdot θ_{W_{S o}} + 4 \cdot θ_{W_{S w}} + 2.5 \cdot θ_{T}$	$θ_{T}$
$F_{1.10} F_{1.22} F_{2.10} F_{2.22}$	$2 \cdot θ_{W_{S o}} + 5 \cdot θ_{W_{S W}} + 2.5 \cdot θ_{T}$	$θ_{W_{S w}}$
$F_{1.11} F_{1.23} F_{2.11} F_{2.23}$	$3 \cdot θ_{W_{S o}} + 5 \cdot θ_{W_{S w}} + 2.5 \cdot θ_{T}$	$θ_{W_{S o}}$
$F_{1.12} F_{1.24} F_{2.12} F_{2.24}$	$3 \cdot θ_{W_{S o}} + 6 \cdot θ_{W_{S w}} + 2.5 \cdot θ_{T}$	$θ_{W_{S w}}$

Dimension	PMSG 1 (mm)	PMSG 2 (mm)	PMSG 3 (mm)	PMSG 4 (mm)
$D_{r e}$	287.3	287.3	289.0	287.3
$D_{r i}$	163.9	163.9	165.6	188.4
$L_{A G}$	4.3	4.3	3.4	4.3
$h_{S y}$	66.2	66.2	66.2	49.7
$h_{P M}$	12.7	12.7	12.7	12.7
$h_{S}$	36.6	36.6	36.6	27.5
$h_{S w}$	2.8	2.8	2.8	2.1
$h_{S o}$	2.8	2.8	2.8	2.1
$W_{S o}$	4.2	4.2	4.2	3.0
$W_{S w}$	4.8	4.8	4.8	3.6
$α_{p}$	60.0°	60.0°	60.0°	45.0°
$α_{P e f}$	54.0°	45.0°	54.0°	40.5°
$α_{S}$	10.0°	10.0°	10.0°	7.5°
$θ_{S t}$	5.2°	5.2°	5.2°	3.9°

Generator	PMSG 1 (mWb)			PMSG 2 (mWb)
Region	RN	FEM	Difference	RN	FEM	Difference
$ϕ_{T_{b 1}}$	2.06	2.04	0.96%	2.06	2.04	0.97%
$ϕ_{T_{b 2}}$	4.11	4.08	0.82%	4.08	4.05	0.87%
$ϕ_{T_{b 3}}$	4.01	3.94	1.88%	3.08	3.07	0.53%
$ϕ_{S_{y}}$	10.19	10.02	1.67%	9.23	9.12	1.15%
$ϕ_{R_{y}}$	11.15	11.29	-1.21%	9.29	9.34	-0.49%
$ϕ_{A_{g 1}}$	1.85	1.79	3.53%	1.85	1.79	3.44%
$ϕ_{A_{g 2}}$	3.70	3.58	3.42%	3.70	3.58	3.35%
$ϕ_{A_{g 3}}$	3.63	3.50	3.76%	2.83	2.75	2.95%
$ϕ_{P_{M 1}}$	11.15	11.30	-1.37%	9.23	9.12	1.15%

Generator	PMSG 3 (mWb)			PMSG 4 (mWb)
Region	RN	FEM	Difference	RN	FEM	Difference
$ϕ_{T_{b 1}}$	2.20	2.14	2.68%	1.55	1.54	0.54%
$ϕ_{T_{b 2}}$	4.37	4.27	2.47%	3.08	3.06	0.53%
$ϕ_{T_{b 3}}$	4.28	4.14	3.34%	2.97	2.90	2.25%
$ϕ_{S_{y}}$	10.85	10.51	3.18%	7.59	7.48	1.53%
$ϕ_{R_{y}}$	11.87	11.81	0.53%	8.36	8.59	-2.62%
$ϕ_{A_{g 1}}$	1.98	1.88	5.02%	1.39	1.35	2.91%
$ϕ_{A_{g 2}}$	3.95	3.76	4.90%	2.78	2.70	2.74%
$ϕ_{A_{g 3}}$	3.89	3.71	4.70%	2.69	2.59	4.14%
$ϕ_{P_{M 1}}$	11.87	11.82	0.40%	8.36	8.57	-2.47%

Induced Internal Voltage (V)			Computational Time (s)
RN	FEM	Difference	RN	FEM	Ratio
312.576	312.12	0.14%	0.09	1.80	20

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[TFN1] _t_7.1 and _t_7.2 equal to _t_1.1; _t_5.1, _t_5.2, _t_6.1 and _t_6.2 equal to _t_2.1; _R₅ and _R₆ equal to _R₂; _R₇ equal to _R₁; _y₄and _y₅ equal _R₅ to _y₁; _y₆ equal to _y₈; _y_2.1, _y_2.2, _y_2.3, _y_2.6, _y_2.7and _y_2.8 equal to _y_1.1; _y_2.4, _y_2.5, _y_2.9 and _y_2.10 equal to _y_1.4; _y_2.11 equal to _y_1.11; _y_2.12 equal to _y_1.12; _y_2.13 equal to _y_1.13.

Permeance Coefficient	2	2.2	2.4	2.6	2.8	3	3.2	3.4	3.6	3.8
Operative Induction (T)	0.7920	0.8180	0.8410	0.8620	0.8800	0.8970	0.9120	0.9260	0.9390	0.9500
Permeance Coefficient	4	4.2	4.4	4.6	4.8	5	5.2	5.4	5.6	5.8
Operative Induction (T)	0.9610	0.9710	0.9800	0.9890	0.9960	1.0040	1.0110	1.0170	1.0240	1.0290
Permeance Coefficient	6	6.2	6.4	6.6	6.8	7	7.2	7.4	7.6	7.8
Operative Induction (T)	1.0350	1.0400	1.0450	1.0490	1.0540	1.0580	1.0620	1.0660	1.0690	1.0730
Permeance Coefficient	8	8.2	8.4	8.6	8.8	9	9.2	9.4	9.6	9.8
Operative Induction (T)	1.0760	1.0790	1.0820	1.0850	1.0880	1.0900	1.0930	1.0950	1.0980	1.1000
Permeance Coefficient	10	10.5	11	11.5	12	–	–	–	–	–
Operative Induction (T)	1.1020	1.1080	1.1120	1.1170	1.1531	–	–	–	–	–