Torque density maximization of vernier machine by using series compensation

Abstract

This paper deals with the design of SPM vernier machine with high slots/pole/phase q, to get higher torque density as well as improved power factor by applying the concept of series compensation. The torque density of a vernier machine can be enhanced by increasing the number of slots/pole/phase q, but as q increases the drastic increase in the reactance makes the power factor even worse. Therefore, it is general to choose low q, even if higher torque can be obtained with higher q. In this study, the idea of series compensation is applied to get vernier machine with high q without the low power factor problem. Series compensation is performed by supplying the desired reactive power to the machine from an additional inverter with a floating capacitor. To validate the theoretical analyses, three SPM vernier machines with different q (1 ∼ 3) are designed and then analyzed by using FEM.

Keywords

Floating bridge maximum torque per ampere (MTPA)open-ended winding vernier motor dual-inverter drive (DID)

1. Introduction

In recent years, various modulation flux machines, especially vernier PM (VPM) machines have been extensively researched due to their simple structure and high torque density attributable to the magnetic gear effect or vernier effect [1–3]. Owing to their high torque density, VPM machines are very promising for direct drive applications such as wind power, electric propulsion etc. However, there is a critical problem of the machine, the poor power factor caused by high reactance [4], which leads to a large capacity of power converter with high cost. Besides, the high reactance deteriorates torque characteristics severely in flux weakening area [5].

Furthermore, the slots/pole/phase q influences the back EMF of the vernier machine and hence the torque density, but as slots/pole/phase q increases the power factor gets even worse. So considering the power factor it is general to choose the slots/pole/phase q to be lower than 3 even though the torque is higher as q increases.

As compared to the conventional PM machine vernier machine demands more reactive power due to its low power factor, this demand even goes high as the slots/pole/phase q goes high and consequently the power factor problem of vernier machine becomes even more serious. Recently, it is common to use power-electronics converters to control reactive power. In particular, the DID topology with a floating bridge has been previously utilized to provide reactive power and counteract supply voltage droop [6]. By using the idea of series compensation based on the dual-inverter drive (DID), the increasing demand of the reactive power as slots/pole/phase q increases can be fulfilled.

There is a kind of compromise, that exists among the power factor and torque density of VPM machine, to take over this inherent problem this study proposes the idea of series compensation based on the dual inverter drive. In DID configuration we have an additional inverter connected with a floating capacitor that will perform the series compensation by supplying the necessary reactive power to the machine. By providing the necessary reactive power, the impedance of the machine seen by the primary inverter changes and consequently it is expected to have improved power factor along with better torque characteristics. In this way, the proposed idea allows the design of vernier machine with higher slots/pole/phase q and hence high torque density with the same simple mechanical structure. As a case study, three prototype PM vernier motor q (1 ∼ 3) are investigated to check the validity of the proposed idea.

The characteristic performances are analyzed by using the FEM, FE method coupled with variable capacitance equivalent to the secondary inverter is proposed to save heavy calculation time as well as to get electro-magnetically accurate results. The analysis results demonstrate that the proposed topology not only improves the overall torque characteristics of vernier machine but also elevates the low power factor without increasing the mechanical complexity in design.

2. General characteristics of PM vernier machine

For a non-salient PM machine, assuming the copper and iron losses negligible, both the voltage and current limit leads to the both circle equations in dq-current axes as (1) and (2) which are depicted in Fig. 1 where X_s is the synchronous reactance, E_ph is the back EMF, I_max and V_max are the maximum allowable current and voltage where C is the center of the voltage limit circle given by E_ph∕X_s. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle (V_{max}/X_{s})^{2}=(I_{d}+C)^{2}+I_{q}^{2}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle I_{max}^{2}=I_{d}^{2}+I_{q}^{2}.\end{eqnarray}$ (2)

A well-designed conventional PM motor can be assumed to have the center of its voltage limit circle on the current limit circle, that is, C_conv = I_max. On the other hand, this PM motor in its equivalent vernier configuration needs at least 5 times of the PM pole number [4]. In this case, the vernier motor produces higher back EMF due to magnetic gearing effect, but the reactance goes high due to the increased PM pole number and leads to worse power factor.

Fig. 1.

Circle Diagram Representing the MPTA.

From (1), therefore, the center of the vernier machine C_ver lies inside the current limit circle, and the C_ver moves further inside as the slots/pole/phase q increases, as indicated in Fig. 1. For the both ideal and vernier PM motors, up to the base speed, the maximum torque is developed by operating on the point p₀ in Fig. 1. As in case of vernier machine C_ver lies inside the current limit circle, so the control trajectories to get maximum torque of both ideal and vernier PM motors are different.

3. PM vernier with high q using series compensation

As we know in vernier machine, certain condition for slot-pole combination must be satisfied in order to establish the vernier effect, the combination can be determined by using (3), where Z_r is the number of pairs of rotor magnets, p is winding pole pairs and q is slots/pole/phase. $\begin{eqnarray}\displaystyle Z_{r}=p(6q-1). & & \displaystyle\end{eqnarray}$ (3)

To demonstrate the characteristics of back EMF, reactance, and geometrical parameters of vernier machine, three prototype vernier machines with different q (1 ∼ 3) are assumed which have 400 mm of rotor diameter, 2 mm of air gap, 200 mm stack length and 72 total number of series turns. The detailed parameters, such as magnet thickness, and slot open ratio are determined by using previously proposed works [4].

Fig. 2.

VPM machine back EMF as function of q and g_m.

By using the back EMF equation of vernier from previously established analytical method [4], the trend of back EMF characteristics for various magnet thickness under different slots/pole/phase q are calculated as depicted in Fig. 2. It is found that the influence of slots/pole/phase q on the back EMF is significant, it is clear that the back EMF increases as slots/pole/phase q goes high.

To get the vernier effect the vernier condition in (3) must be satisfied, so as we increase the slots/pole/phase q, the rotor pole pairs must be increased accordingly to hold the vernier condition. As the PM pole pair Z_r increases the operating frequency goes high and consequently the reactance goes high. As shown in Fig. 3 the relation between the reactance and slots/pole/phase q, it is clear that as q goes high there is a significant increase in the reactance and from (5) this high reactance leads to even worse power factor. As slots/pole/phase q increases, the proportional increase of reactance to Z_r overtakes the increase in the back EMF, which is the reason why vernier machine design with higher q is avoided. As from the analytical results it is clear that as q increases the back EMF increases but its effects tend to be negligible when q is above 3, but the reactance goes rapidly as q goes high, so we choose vernier motor from q (1 ∼ 3). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle V_{max}=\sqrt{(X_{s}I_{max})^{2}+E_{ph}^{2}}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \cos {\theta}=\left\{\sqrt{1+(I_{q}X_{s}/E_{ph})^{2}}\right\}^{-1}.\end{eqnarray}$ (5)

This high reactance makes the power factor problem even worse, along with poor torque characteristics in the flux weakening region due to the reduced power beyond the base speed. As well-known low power factor demands high voltage for the same maximum current I_max hence increases the inverter cost and size. So considering the power factor it is desirable to choose the slots/pole/phase q to be lower than 3 even though higher torque can be obtained with higher q.

Fig. 3.

Relation between q and X_s.

Fig. 4.

Proposed dual inverter structure.

Table 1

Main design parameters of machines

Parameters		Vernier (q = 1)	Vernier (q = 2)	Vernier (q = 3)
Conductors per slot	N _s	72	36	24
Magnet thickness	g _m	15 mm	7 mm	5 mm
No. of stator slots	Z _s	6	12	18
No. of rotor poles	Z _r	10	22	34
Air gap diameter	D _g	400 mm
Air gap length	g _a	2 mm
Stack length	l _stk	200 mm
Magnet type		NdFe30

Table 2

Back EMF and synchronous reactance of PM vernier machines (100 rpm)

Parameter	Vernier (q = 1)	Vernier (q = 2)	Vernier (q = 3)
Back EMF (E_ph)	76.90 V	113.77 V	135.96 V
Synchronous reactance (X_s)	1.172 Ω	4.117 Ω	7.642 Ω

To overcome the compromising problem between the back EMF and the reactance, this study proposes the idea of series compensation based on the dual inverter structure. As shown in Fig. 4, the primary inverter is connected to the main dc voltage source and one side of the machine windings, while the secondary inverter is linked to a floating capacitor and the opposite end of the machine windings. When the secondary inverter supplies the required reactive power by injecting a compensating voltage V_com which is 90 degree behind to the phase current, this voltage injection emulates as a series capacitance, will change the machine impedance and helps to move the center of voltage limit circle as depicted in Fig. 1, the voltage is given as, $\begin{eqnarray}\displaystyle \mathbf{V }_{com}=jX_{com}(I_{d}+jI_{q})=-X_{com}I_{q}+jX_{com}I_{d} & & \displaystyle\end{eqnarray}$ (6) where X_com is the equivalent reactance provided by the 2nd inverter. Therefore, the Eq. (1) can be modified as, $\begin{eqnarray}\displaystyle \left({\displaystyle \frac{V_{max}}{X_{s}-X_{com}}}\right)^{2}=\left(I_{d}-{\displaystyle \frac{E_{ph}}{X_{s}-X_{com}}}\right)^{2}+I_{q}^{2}. & & \displaystyle\end{eqnarray}$ (7) To make the center of (7) to locate on the current circle, the following (8) should be satisfied. Now one can calculate the necessary reactive power to be supplied from the secondary inverter using (9). $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle C_{ver}^{^{\prime }}=E_{ph}/(X_{s}-X_{com})=I_{max}\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle Q=X_{com}I_{max}^{2}=(X_{s}I_{max}-E_{ph})/I_{max}.\end{eqnarray}$ (9)

Once the secondary inverter provides Q of (9), it is evident that the necessary maximum input voltage reduces and the power factor increases from (4) and (5). In addition, the MTPA control trajectory follows along the current limit circle as the well-designed PM motor in Fig. 1. In other words, by providing the reactive power to the machine, the machine equivalent impedence seen from the side of the primary inverter changes, and then as will be shown in later sections, the torque capability is considerably improved along with the elvation of power factor.

4. Analysis model for case study

As analysis models to verify the proposed ideas, three prototype vernier machines q (1 ∼ 3) are designed, and depicted in Fig. 6. The total number of turns per phase are equal for all the three VPM machines, and thus it is reasonable that the maximum allowable currents of all the three prototypes are set to same with a value of 90 A. By using [4] the optimal magnet thickness for the three vernier machines q (1 ∼ 3) are determined. The more detailed design specifications are given in Table 1.

5. FE-simulations and results

Finite element method is performed to obtain the values of X_s and E_ph as these values are required to estimate V_max by using (4) for the three prototype VPM machines. To estimate no load back EMF E_ph of all the machines are calculated at 100 rpm and shown in Fig. 7 for vernier machine having q =2, which is almost sinusoidal, and the fundamental component is extracted using Fourier expansion. To estimate the reactance X_s, the machines are excited with a current source of I_max, and the fundamental component of the resulting induced voltage along with the no-load back EMF are used to estimate the reactance by using (10). The estimated values are given in Table 2. $\begin{eqnarray}\displaystyle X_{s}=(\mathbf{V }_{ind}-\mathbf{E}_{ph})/j\mathbf{I}_{max}. & & \displaystyle\end{eqnarray}$ (10)

The parameters in Table 2 clearly indicates the advantages and disadvantages of increasing the slots/pole/phase q of vernier motor; Although all the vernier motors q (1 ∼ 3) have the same volume and the same number of turns, the back EMF of the vernier motor increase significantly as slots/pole/phase q increases. On the other hand the increase in the reactance as q goes high overtakes the increase in the back EMF, that is why it is desirable to choose the slots/pole/phase q to be lower than 3 even though the torque is higher as q increases.

Fig. 5.

Equivalent arrangement of proposed DID.

Fig. 6.

Machines structures q (1 ∼ 3).

Fig. 7.

Induced voltage, back EMF and applied current (FEM).

Fig. 8.

Equivalent capacitance and reactive power.

To validate the analytically calculated performance characteristics FE-simulation is carried out. It is extremely time-consuming to simulate the system of Fig. 4 through FEM considering practical switch operation of inverter, so for the sake of FE-simulation, the primary inverter is replaced by a variable frequency voltage source with V_max. In addition, because the secondary inverter plays a role of capacitor, it can be replaced by a variable capacitance, where the equivalent capacitance is obtained by using (11). $\begin{eqnarray}\displaystyle C_{eq}=\{{\omega}(X_{s}-E_{ph}/I_{max})\}^{-1}. & & \displaystyle\end{eqnarray}$ (11) The equivalent capacitance required to provide the necessary reactive power estimated by using (11) is shown in Fig. 8. It is obvious that with increase in speed the required reactive power increases and equivalent capacitance value decreases. Thus, the schematics of FE simulation with DID can be represented in Fig. 5.

Fig. 9.

Maximum torque characteristics (FEA).

Fig. 10.

Power factor with speed (FEA).

All these VPM motors are analyzed first with single inverter and then with dual inverter under MTPA control with I_max and V_max. For various speeds range to the maximum, torque characteristics are computed using FEM and compared in Fig. 9. First, if we observe the speed torque characteristics of uncompensated machines marked in red color in Fig. 9, it is clear as slots/pole/phase q increases the torque up to the base speed increases but the torque characteristics in the flux-weakening region are getting worse and worse. Similarly, the influence of slots/pole/phase q on power factor is significant, as depicted in Fig. 10 as slots/pole/phase q goes high it makes the power factor problem even more serious. Form q1 to q3 the power factor is diminished by almost 55%.

However, after the application of the proposed idea there is a significant improvement in the torque characteristics, as in Fig. 9 there is an extension of constant torque region by 16.74%, 124.97%, and 239.19% for the three machines q (1 ∼ 3) respectively, while improving the torque characteristics in the flux weakening region with the same maximum voltage applied. The torque up to the base speed of three machines are 1983 Nm, 2933 Nm, 3505 Nm respectively, that is a reasonable improvement in the torque density In vernier machine different slot-pole combinations influences the back EMF and hence the torque density of the machine. Actually different slot-pole combinations results in different gear ratios that comes out with different torque speed characteristics as depicted in Fig. 9).

As shown in Fig. 10, by providing the required reactive power from the secondary inverter, the scale of active power to reactive power is changed, that results in a significant improvement in the power factor characteristics. At the base speed the improvement in the power factor is 6.4%, 55%, and 77% for the three machines q (1 ∼ 3) respectively.

6. Conclusions

In this paper, a PM vernier machine with high slots/pole/phase q using the concept of series compensation is presented for improving both the torque density and power factor. The analytical calculations shows that the back EMF goes high as q increases. But the increase in the back EMF is not as rapid as the increase in reactance, which makes the power factor problem even more serious. This trade off between the back EMF and the power factor is solved by the application of series compensation, by providing required reactive power from an addational inverter. In addition, to save FE simulation time of the vernier machine with the proposed configurations, the secondary inverter with a floating capacitor is replaced with fictious equivalent capacitor which provides the same amount of reactive power as from the inverter. The analytical and 2D-FE results show that the torque density is desirably increased by having a vernier machine with higher q, and the power factor characteristics are considerably improved while keeping the same simple mechanical structure.

Footnotes

Acknowledgements

This work was supported in part by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education under Grant NRF-2016R1A6A1A03013567 and in part by the Human Resources Development Program (grant no. 20194010201800) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP).

Conflict of interest

All the authors have approved the manuscript and agree with the submission to your esteemed journal. There are no conflicts of interest to declare.

References

Toba

and Lipo

T.A.

, Generic torque-maximizing design methodology of surface permanent magnet vernier machines, IEEE Trans. On Industry Applications36(6) (2000), 1539–1546.

Ronghai

Dawei

and Wang

, Relationship between magnetic gears and vernier machines, Int. Conf. on Elect. Mach. Syst. (2011), 1–6.

S.L.

Niu

and Fu

W.N.

, Design and comparison of vernier permanent magnet machines, IEEE Trans. Magn.47(10) (2011), 3280–3283.

Kim

and Lipo

T.A.

, Operation and design principles of a pm vernier motor, IEEE Trans. On Industry Applications46(6) (2014), 3556–3663.

Liu

H.Y.

and Zhu

Z.Q.

, A high power factor vernier machine with coil pitch of two slot pitches, IEEE Trans. Magn.54(11) 2018.

Ewanchuk

Salmon

and Chapelsky

, A method for supply voltage boosting in an open ended induction machine using a dual inverter system with a floating capacitor bridge, IEEE Trans. Power Electron.28 (2012), 1348–1357.