Gear efficiency estimation method through finite element and curve fitting

Abstract

The inclusion of high energy density permanent magnet (PM) in MG contributes to the high eddy current loss in magnetic gear and reduces its efficiency. There was limited research done that focused on gear efficiency behavior over a broader range of speed and in different gear ratios. In this paper, the function of gear efficiency concerning gear ratio and rotational speed is proposed. Torque and eddy current loss data were obtained through transient magnetic analysis using finite element software at several rotational ranges and gear ratios. The analytical approach through mathematical substitution was discussed to confirm the finding in the simulation. The result showed that the gear efficiency decreases as the speed increases. Nonetheless, the gear efficiency revealed improvement in efficiency as the gear ratio increases. Finally, gear efficiency behavior was modeled using the curve fitting method. Subsequently, based on the correlation study, an equation was proposed, yielding a 1% error compared to the new simulated data. With this proposed method and equation, the analysis and estimation of gear efficiency behavior over wider speed and gear ratios are simplified, thus reducing the need to perform simulation over different speeds and gear ratios.

Keywords

Magnetic gear machine design gear efficiency eddy current loss curve fitting modeling

1. Introduction

Electric vehicles (EVs) offer promising solutions for a sustainable future. However, their development is hampered by some technological challenges such as limited driving range, high cost, battery reliability and scarcity of the charging infrastructure [1]. The capability range of electric cars is subjected to the load capacity and the energy storage capability, which is directly influenced by the car’s total weight and the efficiency of the propulsion. Thus, a closer look at the drive system should be considered [2]. Gears and gearboxes are used regularly for torque transmission and speed regulation in various applications, including electric vehicles. Mechanical gear has high torque over volume ratio, but it suffers from some inherent problems such as friction, noise and heat, and vibration and reliability problems. In contrast, the magnetic gear (MG) offers exciting features such as low acoustic noise, minimum vibration, maintenance-free, improved reliability, inherent overload protection and no physical contact between the input and output shafts. Besides, MG does not produce debris and does not require lubricant and is, therefore, suitable to be used in harsh environments [3–5].

Three main MG structures were developed in early 2000; concentric, harmonic and planetary, using rare-earth to improve the torque density [6–9]. Figure 1 shows the three distinct MG structures. Ferrite magnet does not acquire enough flux density remnants to be competitive in this industry. Torque density comparison of different rare-earth and Ferrite shows that Neodymium Iron Boron (NdFeB) delivers the highest torque density nearly 15 times that of ferrite [10,11]. If cost-effectiveness is emphasized in CMG, Ferrite may be a better choice. Still, for our EVs application, however, without rare earth design in the design of the magnetic gear, the torque density could not match its mechanical gear counterpart [12,13]. Hence, NdFeB will be used in this study. Among the three structured MGs mentioned earlier, the concentric type structure was most referred to in many publications, mainly because it was successfully manufactured and tested [14–18].

Fig. 1.

Main MG structure: (a) Concentric, (b) harmonic, (c) planetary.

Demand for higher efficiency and smaller size gears has grown the need to accommodate devices that incorporate miniaturization and energy efficiency in their design. To meet this demand, motor designers have to improve their output density and reduce their loss [19]. In general, power losses in the MG system can be split into three main groups; mechanical losses in bearing supports and air drag, eddy current power losses in all electrically conductive materials, and magnetic flux reluctance, eddy current loss, and harmonic distortion effects [20–22]. Loss analysis of motor drive systems for gear efficiency modeling is not a trivial task due to the difficulty of modeling electrical, magnetic, mechanical, and thermal interactions in the motor drive system. The overall loss or efficiency models are highly dependent on the hardware used and are not easily adaptable. An analytical solution is challenging to derive as the number of pole pairs increases to produce a higher gear ratio. Curve fitting is used in the electrical machine to model, estimate and correlate two or more independent variables such as slip-rate versus impedance, efficiency, speed, current and moment of inertia and efficiency map versus loss in [23–25]. In contrast, in [26], curve fitting is used to detect gear ratios based on its linear relationship to the back-emf.

The inclusion of high energy density permanent magnet (PM) in MG contributes to the high eddy current loss in MG and reduces its efficiency. There was limited research done that focused on the behavior of gear efficiency over a wider range of speed and in different gear ratios. In this paper, the function of gear efficiency with respect to gear ratio and rotational speed is proposed. Torque and eddy current loss were obtained through the finite element at several rotational speed. The analytical approach through mathematical substitution was discussed to confirm the finding in the simulation. The result showed that the gear efficiency decreases as the speed increases. Nonetheless, the gear efficiency revealed improvement in efficiency as the gear ratio increases. Gear efficiency data were then tested with three curve-fitting models to determine the best fit to represent the gear efficiency as a function of speed. The coefficients of the fit models were analyzed to determine its correlation against seven gear ratios. Subsequently, based on the correlation study, an equation was proposed. With this proposed method and equation, the analysis and estimation of gear efficiency behavior over wider speed and gear ratios are simplified, thus reduce the need to perform simulation over different speeds and gear ratios.

Although loss iron loss occurred in MG, its ratio to the eddy current loss is less than 5%, hence it was excluded in this study. Mechanical loss from bearing and air drag was also excluded because of the limitation in the simulation capability. Bearing friction can be determined only when its specification is known. It also depends on certain tribological phenomena that occur in the lubricant film specified by the manufacturer.

2. Methodology and scope

In this paper, two main methods are applied to obtain the gear efficiency over speed and gear ratio. The first method is the finite element simulation and the second method is the theoretical derivation and substitution method. The second method is applied to complement and confirm the finding in the finite element result. The next step is to model the gear efficiency behavior by using curve fitting techniques. To correlate gear efficiency and gear ratio, a correlation study is done and formulated in the gear efficiency equation. It is worth mentioning that:

Theoretical derivation can be used to determine the gear efficiency value, as long as the magnetic field density is known.

The dimension of the machine can be scaled up or down evenly without jeopardizing the gear efficiency accuracy.

The loss accounted for in this paper is only coming from the eddy current loss.

This paper’s highest rotational speed is 4900 rpm, near the speed of 50 kW electrical motor use in EVs [27,28].

The model is applicable for gear ratio 4 and below, which is high, considering it only requires a single-stage design.

The method of coefficient correlation in obtaining the model is not limited to gear efficiency study only. It can also be used to model torque across speed and gear ratio as well.

The structure is designed in a conventional way where the magnet arc angle is equals to π divided by the pole pair. Dividing the magnet radially to reduce the eddy current loss is excluded from the analysis to focus on its behavior when the structure is design conventionally.

3. Design specification and simulation setup

Many complicated MG designs inherit the structure and working principle of the concentric MG without the armature coil. Magnetic gear can be realized using a space harmonic method, a harmonic, or a reluctance method [8,29,30]. Some MGs even introduced the armature coil in their design to produce a magnetic gear effect [31,32]. K. Attalah [7] configuration will be used in this paper to study the gear efficiency and modeling. Although surface-mount PM rotor is not robust in a high-speed application, the structure produces the highest torque density and magnetic field utilization among other MG structure, which is essential in eddy current loss study. This study assumes that the PM is firmly attached to the rotor surface. Figure 2 illustrates the magnetic gear having 6 high-speed inner rotor pole pairs, p _i, 16 low-speed outer rotor pole pairs, n _p and 22 pole pieces placed between the rotor, p _o. Seven magnetic gears with different gear ratios were designed according to the application note from the JMAG software [33]. The magnet used in this simulation is Neomax 35AH, while the pole piece, inner rotor yoke and outer rotor yoke use NSSMC 35H210 soft magnetic material. The overall size of the structure and the number of inner pole pair were fixed. The change in ratio was realized by changing the number of the outer pole pair and pole piece. The gear ratios of 2, 2.33, 2.66, 3, 3.33, 3.66 and 4 were evaluated. The gap between the pole piece is as wide as the pole piece itself. An anisotropic radial pattern was set for both inner and outer magnets.

Fig. 2.

Magnetic gear with 2.667 gear ratio.

A rare-earth sintered magnet with high energy density was selected for a permanent magnet (PM) use in MG to prioritize the amount of transfer torque. However, since rare earth magnets have high electrical conductivity, eddy current will be generated in the magnetic material and, therefore, cause a decrease in MG’s torque transfer efficiency. The generation of eddy current in the PM is due to the variation in the magnetic flux density of the air gap. However, the inclusion of the pole piece between the magnets produces various harmonic components, and for this reason, the magnetic field analysis can accurately estimate the eddy current loss. The simulation model was performed by setting the motion condition based on the speed shown in Table 1 for each gear ratio. The high-speed inner rotor was set to rotate counterclockwise while the low-speed outer rotor was set to rotate clockwise. The speed pair for each gear ratio is referred to as case 1 to case 4. The output torque data were obtained from each speed pair for all gear ratios. Since the structure is symmetrical, the simulation was run for 1∕4 full cycle to reduce the simulation time.

Table 1

Design specification of concentric magnetic gear

Size (mm)	High speed pole pair (p _i)	Low speed pole pair (p _o)	Pole piece (n _p)	Gear ratio
Overall radius: 90		12	18	2
Inner yoke: 63.5		14	20	2.33
Outer yoke: 80		16	22	2.66
Inner rotor radius: 68.5	6	18	24	3
Outer rotor radius: 75
Stack length: 30		20	26	3.33
Inner air gap: 0.5		22	28	3.66
Outer air gap: 1		24	30	4

Initially, the eddy current in the magnet flows back into the magnet, and the total eddy current that passes through the section becomes zero. To express this phenomenon in a 2D analysis, the eddy current will have to be constrained. The FEM conductor was set for each magnet in the eddy current analysis, as shown in Fig. 3. Opening the terminal on one side of the FEM conductor in the circuit will allow the constraint. Another condition is to set the insulation between the magnets to avoid eddy current being short-circuited to the adjacent magnet. Eddy current loss is expected to increase as the speed increases, but if the gear ratio increases, the PM segmentation can suppress losses due to eddy current [34]. The design specification of the simulation and speed setting for all gear ratio is shown in Tables 1 and 2.

Fig. 3.

Arrangement of conductor coil to constraint eddy current for simulation purpose.

Table 2

Speed setting for all gear ratio

Speed setting
	Case 1		Case 2		Case 3		Case 4
Gear ratio	Inner	Outer	Inner	Outer	Inner	Outer	Inner	Outer
2	600	300	1800	900	3000	1500	4200	2100
2.33	700	300	2100	900	3500	1500	4900	2100
2.66	800	300	2400	900	4000	1500	5600	2100
3	900	300	2700	900	4500	1500	6300	2100
3.33	1000	300	3000	900	5000	1500	7000	2100
3.66	1,100	300	3300	900	5500	1500	7700	2100
4	1200	300	3600	900	6000	1500	8400	2100

4. Finite element result

4.1. Gear efficiency analysis

Figure 4 shows the torque waveform according to the speed setting in Table 1 for the gear ratio of 2.33.

Fig. 4.

Torque waveform for gear ratio 2. (a) Case 1. Inner: 700 rpm; Outer: 300 rpm, (b) Case 2. Inner: 2100 rpm; Outer: 900 rpm, (c) Case 3. Inner: 3500 RPM; Outer: 1500 rpm, (d) Case 4. Inner: 4900 rpm; Outer: 2100 rpm.

In a real setting, the output torque should be positive while the input torque is negative. The setting was reconfigured to allow both waveforms in the negative region to visualize the closeness of the magnitude of the torque. The torque waveform can clearly be seen moving closer to each other as the speed increases from case 1 to case 4. This scenario indicates that the torque ratio is decreasing with the increase in speed.

The average torque for each case is calculated using Eq. (1). $\begin{eqnarray}\displaystyle {\tau}_{avg}=\frac{1}{t}\int _{0}^{t}{\tau}(t)dt & & \displaystyle\end{eqnarray}$ (1) Where 0 and t are the initial and final data points and function τ(t) is the waveform of the torque.

Torque ratio is then calculated using Eq. (2). $\begin{eqnarray}\displaystyle {\tau}_{ratio}={\tau}_{avg∼inner}/{\tau}_{avg∼outer}. & & \displaystyle\end{eqnarray}$ (2) Figure 5 shows the torque ratio degradation for this gear ratio when the speed increases.

Fig. 5.

Torque ratio at different case.

Transfer torque efficiency is calculated using formula (3). $\begin{eqnarray}\displaystyle {\tau}_{\mathit{eff}}=100-\frac{{\tau}_{ratio∼ideal}-{\tau}_{ratio}}{{\tau}_{ratio∼ideal}}\times 100\%. & & \displaystyle\end{eqnarray}$ (3) Figure 6 shows the gear efficiency for gear ratio of 2.33.

Fig. 6.

Gear efficiency for gear ratio 2.33.

Torque ripple, τ_r can be caused by the interaction between mmf and space harmonics. τ_r generated by this gear ratio is shown in Fig. 7. τ_r at inner rotor higher than τ_r at the outer rotor and its value slowly decreases as the speed increases whereas outer rotor exhibit opposite trend. Assuming that the inner rotor acts as an input rotor while the outer rotor as an output rotor, 6% τ_r at output is considered acceptable for various application [35]. τ_r can be reduced by modifying the shape of the pole piece with a slight drawback in the output torque [36]. An intensive study was performed to reduce the torque ripple in [37]. The finding concluded that even though pole piece size optimization can minimize the torque ripple, its effect varies significantly from one gear ratio to another.

Fig. 7.

Torque ripple for gear ratio 2.33.

The results of the gear efficiency for gear ratio 2.33, 2.66, 3, 3.33, 3.66 and 4 were compiled and summarized in Table 3. The results show that gear efficiency increase as the gear ratio increases. Gear efficiency at gear ratio 4 produced the highest efficiency of 92.31% while gear ratio 4 produced 98.569% efficiency for case 1. The efficiency demonstrated in Table 3 proves favourable to be used in high speed application. At 2100 rpm outer rotor speed, the gear efficiency has degraded about 15% of its initial value. Assuming that a constant torque motor provides the input rotation connected through the gear shaft, the constant torque motor has to be set to the maximum even at a lower speed so that the output torque and speed can be maintained or else if the input torque is insufficient to overcome the outer rotor opposing torque, the speed of both rotors will decrease.

Table 3

Gear efficiency for all gear ratio and cases

	Gear efficiency (%)
Gear ratio	Case 1	Case 2	Case 3	Case 4
2	96.778	90.873	85.375	80.539
2.33	97.476	92.570	88.432	84.828
2.66	97.603	94.002	90.406	87.047
3	97.407	94.091	91.503	88.836
3.33	98.271	95.069	92.144	89.435
3.66	97.893	94.726	92.136	89.973
4	98.569	95.084	93.330	91.016

4.2. Eddy current loss analysis

For the eddy current analysis, Joule loss was determined for inner and outer magnet only. Although inner yoke, outer yoke and pole piece also generates eddy current loss, its magnitude is minimal, which is less than 1% of the total Eddy current loss magnitude across a wide range of speed. The simulation was run for 1∕4 of full rotation for this test. The integral average of Eddy current loss in Fig. 8 was calculated using Eq. (4) for each magnet piece and case. Then, all these values were summed for the inner rotor and outer rotor denoted by Eq. (5). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{avg}=\frac{1}{t}\int _{0}^{t}P(t)\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{loss,}=\mathop{\sum }_{n-1}^{n}P_{avg}\end{eqnarray}$ (5) where n is the magnet piece number on the rotor. As we can see in Fig. 9, eddy current loss increases exponentially from lower speed to higher speed. Eddy current loss at the inner rotor is smaller because its total volume is lesser than the outer rotor.

Fig. 8.

Eddy current loss obtain from single piece of inner magnet for case 1.

Fig. 9.

Eddy current loss in inner and outer magnet for gear ratio 2.33.

The results of Eddy current loss for gear ratio 2.33, 2.66, 3, 3.33, 3.66 and 4 are compiled and summarized in Table 4. Eddy current loss does not increase as the gear ratio increases. The theoretical explanation is that in gear ratio 2, the number of PM on the outer rotor is 24 pieces. As the gear ratio increases to 2.33, the outer rotor’s PM increases to 28 pieces. Electrical conductivity decreases in the smaller piece of the magnet, thereby reducing the Eddy current loss [22,38–40]. This finding correlates to the torque efficiency behaviour presented earlier.

Table 4

Total Eddy current loss at the inner and outer magnet for all gear ratio and cases

	Eddy current loss (Watt)
Gear ratio	Case 1	Case 2	Case 3	Case 4
2.000	71.238	636.201	1747.685	3389.048
2.330	63.300	560.665	1529.239	2940.215
2.660	56.476	497.815	1365.696	2641.317
3.000	51.316	455.057	1243.976	2402.245
3.330	51.540	454.090	1235.962	2379.457
3.660	49.573	435.312	1183.083	2270.495
4.000	44.004	432.435	1085.742	2113.582

5. Theoretical and finite element comparison

The mathematical equations modelled in the next section were based on the finite element method. To verify the accuracy of the data, analytical approach together with the numerical approach, are presented in this section. Experimental verification is not within the scope of this paper mainly because it deals with seven different structure.

5.1. Eddy current loss

The eddy current loss can be written as $\begin{eqnarray}\displaystyle P_{eddy}=K_{e}f^{2}B_{m}^{2}d^{2}V & & \displaystyle\end{eqnarray}$ (6) where P _eddy, K _e, f, B _m and d are the eddy current loss, eddy current loss coefficient, magnetic field frequency in s⁻¹, magnetic field density, the thickness of the material and the material volume respectively [41]. The eddy current loss coefficient can be calculated as follows $\begin{eqnarray}\displaystyle K_{e}=\frac{{\pi}^{2}}{6{\rho}} & & \displaystyle\end{eqnarray}$ (7) where 𝜌 is the material resistivity. In JMAG Designer software, skin effect due to eddy current is included in the calculation [42,43]. Meanwhile, the resistivity of the material changes as the frequency changes as denoted in the following expression $\begin{eqnarray}\displaystyle {\delta}=\frac{1}{\surd {\sigma}{\mu}f{\pi}} & & \displaystyle\end{eqnarray}$ (8) where δ and σ are the skin layer and the conductivity of the material respectively. Thus eddy current loss are

proportional to the square of the magnetic field frequency

proportional to the square of the flux density

proportional to the thickness of the material

inversely proportional to the resistivity of the material.

The eddy current loss produced by a single piece magnet at the inner rotor in 4 cases were calculated for 2.33 gear ratio structure. The values calculated using (6) were plotted together with the result from the finite element in Fig. 10.

Fig. 10.

Eddy current loss produced by a single piece of inner magnet solved using finite element and Eq. (6).

There was less than 15% difference between finite element result and Eq. (6). The difference can be attributed to the finer subdivision of the skin effect by the finite element software for higher accuracy estimation of the value. As the frequency increases, the resistivity due to the skin effect will have a greater influence on the eddy current calculation. Furthermore, the flux density distribution is also assumed to be uniformly distributed across the element when solving the eddy current loss using Eq. (6).

5.2. Gear efficiency

Gear efficiency can be determined when the eddy current loss and the total power are known. The total joule loss at the inner rotor and outer rotor are the summation of all the eddy current loss in all the magnet pieces. Assuming that the magnet piece experienced the same loss at the inner rotor and outer rotor respectively, then the total joule loss at the respective magnet rings are $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle W_{j∼inner}=P_{i∼\mathit{eddy}}\times 2p_{i}\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle W_{j∼outer}=P_{o∼\mathit{eddy}}\times 2p_{o}\end{eqnarray}$ (10) where P _i eddy and P _o eddy are the eddy current losses at the inner magnet piece and outer magnet piece, respectively. Assuming that only eddy current loss is counted, the total power at the inner rotor and outer rotor can be expressed as follows $\begin{eqnarray}\displaystyle {\tau}_{in}v_{in}={\tau}_{out}v_{out}+W_{j∼outer}+W_{j∼inner}. & & \displaystyle\end{eqnarray}$ (11) Through substitution, the gear efficiency can be calculated by $\begin{eqnarray}\displaystyle {\eta}_{G}=1-\left|\frac{W_{j∼inner}+W_{j∼outer}}{{\tau}_{in}v_{in}}\right|. & & \displaystyle\end{eqnarray}$ (12) Table 5 shows the gear efficiency solved using Eq. (4) for 2.33 gear ratio structure in 4 cases and from the finite element (FE) results. The input torque value was extracted from the simulation.

Table 5

Comparison between gear efficiency resulted from finite element software versus Eq. (6)

FE	Eq. ((6))	% diff
0.97	0.97	0.73
0.93	0.92	1.07
0.88	0.87	1.02
0.85	0.84	0.71

There was a slight difference in values between finite element results and Eq. (6). These differences were less than 10% and were contributed by the assumption that flux density was distributed evenly and each magnet piece generated the same losses. These values confirm the acceptable difference between simulation and manual calculation.

5.3. Machine volume influence on the gear efficiency

The relatively short length of the gear unit may influence the magnitude of the total eddy current loss of the machine, as shown in Eq. (6). The selected length described in Table 1 earlier was 30 mm. If the length were to be increased by three times, the volume would also increase accordingly. Hence the same goes to the eddy current loss [44]. However, gear efficiency will remain the same. This case can be expressed in torque and gear efficiency equation when variable l is included as in (13) and (14) respectively. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\tau}_{in}=\frac{l}{{\mu}_{o}}\int B_{r}B_{{\theta}}rds\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\eta}_{G}=1-\left|\frac{l(W_{j∼inner}-W_{j∼outer})}{{\tau}_{in}v_{in}}\right|\end{eqnarray}$ (14) where l is the length of the machine in mm. μ_o, B _r, B _θ and r are the permeability of the free space, radial field density, tangential field density and radius of the air gap [45,46]. Finite element result confirmed that the gear efficiency did not change when the machine length changed.

5.4. Magnet piece size in high gear ratio

In (6), it was demonstrated that the eddy current loss is proportional to the magnet size. In this paper, the size of the outer magnet pieces differs according to the gear ratio, while the sum of all the magnet size is equal across all seven structures. In Table 3, the gear efficiency increases as the gear ratio increases. This case can be explained by grouping the non-size related variable as a constant K _v and representing (6) as thickness and volume of the magnet piece. $\begin{eqnarray}\displaystyle \text{} & \text{}K_{v}=K_{e}f^{2}B_{m}^{2} & \displaystyle\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle \text{} & \text{}P_{eddy}=K_{v}d^{2}V. & \displaystyle\end{eqnarray}$ (16) As in (10), the factor of [d ² ⋅ V], [2 ⋅ p _o] and its combination [2p _o ⋅ d ² ⋅ V] are drawn based on gear ratio in Fig. 11.

Fig. 11.

Behaviour of the size related parameters and number of magnet pieces versus gear ratio.

This graph displays that sizerelated parameters [d ² ⋅ V] have a more significant influence on the eddy current loss if compared to the number of magnet pieces 2 ⋅ p _o. The behaviour of the product of these parameters [2p _o ⋅ d ² ⋅ V] finally follows the direction of the sizerelated parameters instead of the number of magnet pieces 2 ⋅ p _o. This finding agrees with the simulation result.

6. Curve fitting modelling and discussion

6.1. Gear efficiency modelling

Curve fitting is the process of constructing a mathematical function that has the best fit to a series of data points. The gear efficiency obtained in the earlier section is tested with 1st order polynomial, 2nd order and rational polynomial. The accuracy of the fit model is measured based on SSE, R-square, adjusted R-square and RMSE. Figure 12 shows three fit models of gear efficiency against outer rotor speed for gear ratio 2. Table 6 summarized the accuracy of the fit of gear efficiency calculated through Matlab for three fit models and seven gear ratios.

Fig. 12.

Fit models for gear efficiency for gear ratio 2.

Table 6

Accuracy of gear efficiency for Fit 1, 2 and 3 for all gear ratio

Ratio	2.000	2.330	2.660	3.000	3.330	3.660	4.000
	Fit 1: f (x) = s1 * x + s2
SSE	0.289	0.432	0.017	0.141	0.063	0.246	0.613
R-square	0.998	0.995	1.000	0.997	0.999	0.993	0.980
Adjusted R-square	0.997	0.993	1.000	0.995	0.998	0.989	0.970
RMSE	0.380	0.465	0.092	0.265	0.177	0.351	0.553
	Fit 2: f (x) = s1 * x ² + s2 * x + s3
SSE	0.003	0.003	0.002	0.032	0.000	0.001	0.265
R-square	1.000	1.000	1.000	0.999	1.000	1.000	0.991
Adjusted R-square	1.000	1.000	1.000	0.998	1.000	1.000	0.974
RMSE	0.051	0.051	0.049	0.179	0.009	0.034	0.514
	Fit 3: f (x) = (s1)∕(x + q1)
SSE	0.023	0.071	0.025	0.056	0.004	0.121	0.477
R-square	1.000	0.999	1.000	0.999	1.000	0.997	0.984
Adjusted R-square	1.000	0.999	0.999	0.998	1.000	0.995	0.976
RMSE	0.106	0.189	0.112	0.167	0.046	0.246	0.488

In SSE, a value closer to 0 indicates that the model has a smaller random error component and that the fit is more useful for prediction. R-square can take on any value between 0 and 1, with a value closer to 0 indicating that a lesser proportion of variance is accounted for by the model. The adjusted R-square value can take on any value less than or equal to 1, with a value closer to 1 indicating a better fit. Just as with SSE, RMSE value closer to 0 indicates a fit that is more useful for prediction [47]. Fit 1 shows consistent SSE and RMSE across different ratios, good fit quality in R-square and adjusted R-square statistics. On the other hand, Fit 2 reveals consistent fit prediction values in all indicators. Fit 3 shows acceptable predictability parameters in all aspects.

6.2. Fit model validation

Additional data points of gear efficiency were simulated for each gear ratio to validate the fit model. The selected outer rotor speeds are 4200 rpm and 6300 rpm. The inner rotor speed setting is simply the product of the outer rotor speed and the gear ratio. The validation reveals that Fit 3 is the best fit model to represent the gear efficiency. Fit 1 and 2 predictions fail to predict gear efficiency beyond 4200 rpm. Figure 13 shows the validation data points on Fit 3 for gear ratio 2.33.

Fig. 13.

Validation of data points plotted on Fit model 3 of gear efficiency for gear ratio 2.33.

6.3. Correlation between fit model at a different gear ratio

Equation (17) represents the fit model for gear efficiency with coefficients of 95% confidence bounds which were shown in Fit 3 in Fig. 12. $\begin{eqnarray}\displaystyle {\eta}_{G}(v)=s_{1}/(v+q_{1}) & & \displaystyle\end{eqnarray}$ (17) where s ₁ and q ₁ are the coefficients of the fit model 3 for gear efficiency while v is the outer rotor speed. In order to represent the gear efficiency across different gear ratios, each coefficient of the equation generated from the fit model is examined. This analysis also helps to reduce the number of coefficients to be presented in the final equation. Fit model coefficients analysis shows that gear efficiency yields a good correlation. Correlation analysis between the coefficients is plotted in Fig. 14.

Fig. 14.

Gear efficiency Fit 3 model coefficients versus gear ratio.

Thus, Eq. (17) expressed as a single coefficient in (18). $\begin{eqnarray}\displaystyle {\eta}_{G}(v)=s_{1}/(v+s_{1}/100). & & \displaystyle\end{eqnarray}$ (18) The linear relationship between s ₁ and gear ratio shows good representation with R-square of 0.94. Coefficient s ₁ and the gear ratio is expressed in Eq. (19). $\begin{eqnarray}\displaystyle s_{1}=6.386\times 10^{5}\ast Gr-3.02\times 10^{5}. & & \displaystyle\end{eqnarray}$ (19) Substitute (19) into (18) yield Eq. (20) $\begin{eqnarray}\displaystyle {\eta}_{G}(G_{r},v)=\frac{6.386\times 10^{5}\ast Gr-3.02\times 10^{5}}{v+6.386\times 10^{3}\ast Gr-3.02\times 10^{3}} & & \displaystyle\end{eqnarray}$ (20) where 2.0 ≥ Gr ≥ 4.0.

Equation (20) is validated against new sets of simulation data and presented in Table 7 for two outer rotor speeds setting. The error distribution for validation of Eq. (20) is shown in Fig. 15.

Table 7

Gear efficiency validation of Eq. (20) against simulation for outer rotor speeds of 4200 rpm and 6300 rpm

		Gear ratio
Speed		2	2.33	2.66	3	3.33	3.66	4
4200 rpm	Simulation	66.696	73.830	77.631	80.278	81.349	81.751	83.423
	Eq. (10)	69.897	73.847	76.881	79.349	81.288	82.894	84.284
	Error (%)	4.799	0.023	0.967	1.157	0.075	1.398	1.032
6300 rpm	Simulation	56.816	65.724	70.153	73.414	74.924	74.944	77.369
	Eq. (10)	60.753	65.307	68.915	71.923	74.333	76.363	78.143
	Error (%)	6.929	0.634	1.766	2.032	0.789	1.893	1.000

Fig. 15.

Equation (20) error percentage distribution against simulation data for outer rotor speed setting of 150, 450, 750, 4200 and 6300 rpm.

There were many designs proposed to minimize the eddy current loss in CMG. Researchers can choose to use lower energy density magnetic material such as Ferrite to lessen the loss due to eddy current. However, the torque produce will not be strong enough to be used in EVs. By applying insulation around PM can also help to suppress the eddy current from penetrating the PM. This method would also increase the reluctance, thus decrease the air gap flux density. One of the most effective ways to achieve it is through magnet segmentation. Magnet segmentation is used in many PM machine design to reduce the eddy current loss in the magnet. According to the author, the eddy current loss had reduced about 13% when the four-segment PM was applied [48]. Even though this method is effective, it is not practical to implement this strategy as magnets need to be cut, insulated and re-glued, which is a laborious and costly process [39,49]. An alternative to this structure is by replacing the PM with the DC coil, similar to a Vernier machine. This structure requires a bigger machine to cater for slot area and external power source to function.

7. Conclusion

Until the writing of this paper, there is no publication found that analyse gear efficiency at different speed on different gear ratio in MG. The information delivered in this paper facilitate the researcher in many ways especially in predicting gear efficiency over wider speed range and multiple gear ratios. Standard mechanical gear efficiency is proportional to speed in the range of 65% to 90%. In this study, we found that as the outer speed increases towards 2100 rpm, the gear efficiency reduced between 10–15%.

The relationship between torque transfer efficiency, speed and gear ratio was shown through the 1st order, 2nd order and rational polynomial curve fitting techniques. Based on SSE, R-square, adjusted R-square and RMSE, fit 3 model showed high predictability for gear efficiency. However, variation between coefficients needed to be low and linear against gear ratio to predict gear efficiency across gear ratio. Hence Fit 3 models were selected for estimating gear efficiency behaviour. Equation (20) were found to represent torque efficiency in terms of outer rotor speed and gear ratio with low error percentage of 1%.

Footnotes

Acknowledgements

The authors would like to thank the Centre for Research and Innovation Management, Universiti Teknikal Malaysia Melaka (UTeM) for the technical and financial support provided for this research. The authors would like to thank the Research Management Centre (Research Fund E15216), Universiti Tun Hussein Onn. The authors also would like to thank Shahril Yahaya and Azhan Ab Rahman for proofreading the manuscript and Syed Muhammad Naufal Syed Othman for guidance in familiarizing with the software.

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