Electromagnetic modeling and control of switched reluctance motor using finite elements

Abstract

This paper considered the implementation of a current control method for switched reluctance motors (SRMs) and presented a novel approach to the accurate online modeling of a three phase 6/4 SRM drive. A three phase 6/4 SRM is given theoretical calculation of inductance of the SRM model. The SRM was then tested in a Matlab/Simulink environment and numerically analyzed by using nonlinear 2D look-up tables created from its calculated flux linkage and static torque data. The simulation studied the hysteresis and voltage control strategies. The ideal waveform of stator current under the voltage-current condition and improved shape of rotor were proposed.

Keywords

switched reluctance motor (SRM) / hysteresis / control / finite element analysis

Introduction

Switched reluctance motors (SRMs) are potentially attractive for many industrial applications due to their simple structure, high speed range, low cost, suitable for operating in unfriendly situation, and high torque to weight ratio etc. However, as SRM utilization in industry is still relatively limited compared to other machine types, the practical issues regarding the drive implementation have not yet been fully explored [1–6]. The behavior of the SRM is strongly affected by the nonlinear magnetization characteristics of the constructed magnetic materials. Therefore, the magnetization behaviors of the SRMs are, in particular, important parameters to predict their performances and to the study of advanced control strategies [7,8].

SRMs have good performances such as a high torque/weight ratio and a high reliability [9–11]. The high torque/weight ratio is caused by the large reluctance torque for salient poles of both stator and rotor [12,13].

Previous research has shown that radial attraction forces between the stator and the rotor are the main source of the acoustic noise in SRM drives [14–16]. Also, the magnetic characteristics of these drives are highly nonlinear and difficult to control. Some researchers have used asymmetric converter to excite each phase to counter these problems [17–20]. The torque ripples are sensitive to the size, the mechanical construction and the precision of the switching angles. They are also correlated with the current ripples (in the DC supply) which, in turn, may cause significant AC line harmonics. There are many strategies and methods to reduce or cancel the torque ripples in this type of machines. Essentially, there are two main approaches for the reduction of the oscillations. The first consists of improving the magnetic design of the machine, and the second is based on the usage of an electronic control. The torque is developed by the tendency for the magnetic circuit to adopt a configuration of minimum reluctance or maximize the inductance of the excited coils. Because the SRM uses reluctance torque which comes from the change of reluctance of the magnetic circuit, the accurate estimation of the inductance depending on the rotor position is very important.

The electromagnetic field finite element numerical analysis was an effective tool for analyzing the motor steady-state and dynamic magnetic field distribution. Besides, it was suitable to numerical analysis for electromagnetic for the different tooth and the slots of the motors. So, the finite element method was adopted for analyzing the magnetic distribution and the magnetic performance.

Traditionally, a 6/4 SRM was seldom operated under three phase excitation mode to maintain high torque density. In this paper, a method was presented for calculation of dynamic characteristic of SRM. The SRM was applied to estimate the nonlinear inductance. Then, using the estimated inductance, a dynamic analysis model was constructed using Matlab. The method for determining the dynamic characteristics and simulated results were presented. The SRM model was then tested in a Matlab/Simulink environment, using nonlinear two-dimentional (2D) look-up tables created from its calculated flux linkage and static torque data. The simulation studies for hysteresis and voltage control strategies.

Structure characterization of SRM

Calculus of electromagnetic parameters

The 6/4 SRM presented in Fig. 1 shows the schematic diagram of a six-stator pole and a four-rotor pole. The stator and rotor laminations are assumed to be made of M-19 non-grain-oriented silicon steels (given in Table A1 [21]). The stator windings are chosen to be the vacuum relative permeability of the 6/4 SRM. On the other hand, the stator winding is concentrated and no winding and no brushes exist on the rotor, as illustrated in Figs. 2 and 3, where the details of the construction of this machine are given in Table A1. The rotor has segments which constitute flux guides that serve to bend the flux produced by the current flowing in the coil windings in the stator slots around the slot and back toward the periphery of the rotor.

The flux in each leg of the SRM is determined by inverting the reluctance matrix as shown in

(1)\phi ={\displaystyle {\oint }_{\mathrm{\Lambda }}\mathit{{\rm A}}}\mathrm{d}\mathit{l}\mathit{.}

If the phases of the motors are fully decoupled from each other, the induced flux linkage will be related only to the excitation current and flux-inductance. The flux-inductance represents the effective reluctance of the magnetic path as the rotor moves from the unaligned position to aligned position. Further, Eq. (1) can then be rewritten as

(2){\mathit{C}}_{\mathrm{e}}={\displaystyle \frac{\mathrm{1}}{{\mathit{\mu }}_{\mathrm{0}}}}{\mathit{L}}_{\mathrm{n}}{\displaystyle {\int }_{{\mathrm{\Gamma }}_{\mathrm{R}}}\left[\left(\mathit{r}\times \mathbf{B}\right)\left(\mathit{B}\cdot \mathit{n}\right)-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{B}^{\mathrm{2}}\left(\mathit{r}\times \mathit{n}\right)\right]}\mathrm{d}\mathrm{\Gamma },

and

(3)\mathit{B}=\mathbf{r}\mathbf{o}\mathbf{t}\mathit{A},

where B is the magnetic induction, Γ_R is the interface between the rotor and entrefer of the motor, and L_n is the length of the magnetic core of the motor.

The whole mesh when the rotor position angle is θ = 0° and the finite element subdivision graph of the stator one pole are demonstrated in Fig. 2.

Two typical flux distributions of the SRM are depicted in Fig. 3(a) and 3(b) where it is observed that the rotor axis is completely aligned and unaligned with the stator poles respectively. Since the flux distribution plots display the details of the magnetic field over the entire cross-section, locating local saturation and magnetic force distribution by inspection of flux plots becomes straight formation regarding the SRM terminal characteristics.

Flux characteristic

Figure 4 presents the flux linkage for different rotor position and phase currents, revealing the saturation effects. The lowest curve corresponds to the unaligned position and the curve of the top corresponds to the aligned position.

Inductance characteristic

The co-energy the SRM can be determined so long as the flux linkage is known. Figure 5 exhibits the co-energy curves of the rotor positions in [0°–45°] under different test currents. Note that the 45° rotor position means complete alignment of the stator and rotor poles and 0° indicates complete unaligned position of the stator and rotor poles.

Torque characteristics

Figure 6 shows the torque characteristics as a function of rotor position and phase current values and the influence of the nonlinearity of the magnetic on the torque. From Fig. 6, it is apparently seen that there is significant improvement in the torque characteristics of the optimal design when compared with the work of Balaji and Kamaraj [22] who have obtained an average torque of 27.74 N·m with a torque ripple of 22% and confirmed that the finite-element calculation confirms the application of proposed “Novel Strategy Global of our Approach” (NSGA) based optimization to achieve enhanced SRM design.

Flux linkage-current-position and torque-position-current curves are the most important characteristics of switched reluctance machines. Figures 7 and 8 show the correlation of the flux linkage, current and rotor angle of the flux linkage and torque, respectively. The magnetization curves are based on the variation of instantaneous phase of flux linkage and instantaneous phase current with rotor displacement. To identify the symmetry of the phases, the curves of the flux linkage and instantaneous inductance are measured for rotor positions varying from –45° to 0°.

Command dynamic characteristics of a SRM

A dynamic model of a SRM is composed of a set of electrical equations for each phase and equations of the mechanical system [23]. In the typical m-phase SRM, if the negligible phase interaction (no mutual coupling) is assumed, the machine’s voltage applied to the rolling up of a phase of SRM can be expressed as

(4){\mathit{V}}_{\mathit{i}}=\mathit{R}{\mathit{I}}_{\mathit{i}}{\displaystyle \frac{\mathrm{d}{\mathit{\psi }}_{\mathit{i}}\left(\mathit{\theta }\mathit{,}{\mathit{I}}_{\mathit{i}}\right)}{\mathrm{d}\mathit{t}}},\mathit{i}\mathit{=}\{1,2,3\}.

Because of the double salience construction of the SRM and the magnetic saturation effects, the flux linked in an SRM phase varies as a function of the rotor position θ and the phase current, Eq. (1) can be expanded as

(5)\mathit{V}=\mathit{R}\mathit{I}+{\displaystyle \frac{\mathrm{d}\mathit{\psi }}{\mathrm{d}\mathit{I}}}{\displaystyle \frac{\mathrm{d}\mathit{I}}{\mathrm{d}\mathit{t}}}+{\displaystyle \frac{\mathrm{d}\mathit{\psi }}{\mathrm{d}\mathit{\theta }}}{\displaystyle \frac{\mathrm{d}\mathit{\theta }}{\mathrm{d}\mathit{t}}},

where {\displaystyle \frac{\mathrm{d}\mathit{\psi }}{\mathrm{d}\mathit{I}}}, defined as L(θ, I), is the instantaneous inductance; {\displaystyle \frac{\mathrm{d}\mathit{\psi }}{\mathrm{d}\mathit{\theta }}}{\displaystyle \frac{\mathrm{d}\mathit{\theta }}{\mathrm{d}\mathit{t}}} is the instantaneous back electromotive force (EMF).

Whatever the vectors Ψ and I are, the fructuous of co-energy, verify inequality (6),

(6)\overline{\mathit{W}}\left(\mathit{I},\mathit{\theta }\right)+\mathit{W}\left(\mathit{\psi },\mathit{\theta }\right)\ge {\mathit{\psi }}^{\mathrm{T}}\mathit{I}.

The partial derivative of the energy function in relation to the rotor position gives the torque of the machine as

(7){\mathit{C}}_{\mathrm{e}}\left({\mathit{\psi }}_{\mathrm{1}},{\mathit{\psi }}_{\mathrm{2}},{\mathit{\psi }}_{\mathrm{3}},\mathit{\theta }\right)={\displaystyle \frac{\partial \mathit{W}}{\partial \mathit{\theta }}}\left({\mathit{\psi }}_{\mathrm{1}},{\mathit{\psi }}_{\mathrm{2}},{\mathit{\psi }}_{\mathrm{3}},\mathit{\theta }\right).

When one energizes one phase, the torque appears so that the rotor evolves in the direction where the inductance increases. Therefore, the torque will be in the direction of the nearest aligned position.

The motor total torque is obtained by

(8){\mathit{C}}_{\mathrm{e}}={\displaystyle \sum _{\mathit{i}=\mathrm{1}}^{\mathrm{3}}{\mathit{C}}_{{\mathrm{e}}_{\mathit{i}}}}\left({\mathit{i}}_{\mathit{i}},{\mathit{\theta }}_{\mathit{i}}\right).

The mechanical equations are

(9)\mathit{J}{\displaystyle \frac{\mathrm{d}\mathit{\omega }}{\mathrm{d}\mathit{t}}}={\mathit{C}}_{\mathrm{e}}-{\mathit{C}}_{\mathrm{r}}-\mathit{f}\mathit{\omega }\mathit{,}

and

(10){\displaystyle \frac{\mathrm{d}\mathit{\theta }}{\mathrm{d}\mathit{t}}}=\mathit{\omega }\mathit{.}

The key to achieving a good simulation of an SRM is to use a methodology that permits the nonlinearity of its magnetic characteristic with minimizing the simulation time. The procedure adopted in Matlab-Simulink is to avoid all the partial derivatives which are sources of errors. The technique was to utilize a look-up table.

The proposed SRM dynamic model utilizes the look-up table of calculated characteristics (from the previous subsection) to estimate the phase current and phase torque when the input voltage and the rotor position are known. Figure 9 shows the schematic diagram of the one-phase dynamic model. It can be noticed that the flux linkage, as input to the current look-up table, can be found from the solution of

(11){\mathit{\psi }}_{\mathit{i}}={\displaystyle \int ({\mathit{V}}_{\mathit{i}}-{\mathit{R}}_{\mathit{i}}{\mathit{I}}_{\mathit{i}})}\mathrm{d}\mathit{t}\mathit{.}

When all three models are integrated and the mechanical subsystem is included, a complete nonlinear dynamic model of the SRM can be formed (see Fig. 10).

Strategy–voltage source

Figure 11 shows a second set of simulation resultants using θ_on = 0° and θ_off = 30° with the motor functioning without load applied. It can be seen from Fig. 11(a) that the θ_off angle value is now sufficient enough to avoid the current starting to grow when the aligned position is reached. As expected, it can be noticed from Fig. 11(b) that the phase current produces a very small negative torque. However, the total torque is always positive, as shown in Fig. 11(c).

It is interesting to note the flux linkage current trajectory of the phase winding as shown. The operating point for one phase current impulse is plotted while the rotor is rotating as compared to Fig. 12. The area circled out by the trajectory equals the co-energy variant for the cycle. The average torque production of SRM then can be computed from the co-energy variation of the phase winding with respect to the incremental rotor angle.

Hysteresis current control

The dynamic behavior of the SRM is illustrated in the employment case of the ordering of the current by hysteresis.

The results shown in Fig. 13 were achieved from θ_on = 0° and θ_off = 30°, with reference current I_ref = 8 A and without torque load. Figure 13(a) shows the influence of hysteresis current control on the shape of the current. The influence of this hysteresis current control on the torque of the phase can also be observed from Fig. 13(b). The ripple of total torque has high amplitude for the values of θ_on = 0° and θ_off = 30°, as shown in Fig. 13(d), and by consequence of the oscillation speed, as show in Fig. 13(c).

To decrease the oscillations speed, it is necessary to produce more torque. Consequently, the vale urn angle θ_off of 30° was adjusted to 38°.

The new dynamics of the SRM are given in Fig. 14, which reveals that this ripple of the torque was decreased. However, it can be noticed from Fig. 14 that one phase now produces the most negative torque because of the new value of the angle θ_off. The negative torque produced by one phase does not have an impact on the total torque since it is compensated by the other torque signals generated in the other two phase.

Fig. 15 shows the flow according to the current, in which the area of contour shows the co-energy, with on= 0° and off= 38°.

For a better general view, the mean torque variance was plotted as a function of I_ref and θ_off. It can be noticed that for different values of reference current, there is always a θ_off value that maximizes the mean (see Fig. 16.)

The mean couple is calculated by

(12){\overline{\mathit{C}}}_{\mathrm{e}}={\displaystyle \frac{\mathrm{1}}{\mathit{T}}}{\displaystyle \int {\mathit{C}}_{\mathrm{e}}}\mathrm{d}\mathit{t}\mathit{.}

Conclusions

An effective nonlinear dynamic of a 1-hp 3-phase 6/4 SRM was developed using current and torque look-up generated from its measured flux linkage and static torque characteristics. Theoretical investigations of the inductance of the SRM reveal that the current has driven the stator under constant voltage. The torque characteristic driven by the proposed current waveform was calculated by the two-dimensional finite element method in a Matlab/Simulink environment. Several simulations were performed to study the dynamic behavior of the SRM. The influence of the turn-off angle θ_off on its dynamic behavior was mainly verified. It was proved to be dependent on the operating point of the machine. A θ_off value was shown to exist, which enables torque ripple reduction. For model validation, two strategies of order, that is, phase current hysteresis and voltage control strategies were simulated. The results obtained are acceptable as a whole.

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