1. Introduction
Besides being simple and robust, the absence of windings and magnets on the rotor in a Switched Reluctance Machine (SRM) makes it suitable for variable speed applications [1]. The SRM is highly non-linear in nature due to its doubly salient structure, magnetic characteristics and coupled relationship among rotor position, phase current and flux linkage [2]. As the linear model of the machine is unsuitable for high performance applications, various techniques are developed to incorporate the non-linearities in the machine.
Since the flux model proposed by A. Radun [3] could not be inverted to express stator current as a function of flux linkage and position, the model was found to be of little use further for real time implementation.
A non-linear analytical model of SRM based on the existing machine geometry and material properties using fourier series was presented by Ahmed Khalil et al [4, 5]. This model involved complex procedure and more information about the material.
Later, Wen Ding and Deliang Liang proposed a simple and rapid non-linear model of integrated SRS/G to characterize the inherent relationship among phase currents, rotor position, flux linkage and torque [6]. In this model, only the first five components of the Fourier Series describing the current in terms of arc tangent function are considered to express the whole flux linkage model ψ(θ,i) of SRS/G.
Analytical model of SRM considering mutual inductance was presented by D.N. Essah et al [7]. This model was developed by considering the mutual inductance between the phases alone. Comparison of different analytical models was done by Shoujun Song et al [8].
Non-linear modeling of SRM using Artificial Intelligence (AI) technique was addressed by T. Lachman et al [9]. The accuracy of the model developed using AI techniques mainly depends on the amount of the given flux linkagecurrent-position data and the prior number of fuzzy rules or the experimental knowledge.
Measurement of flux linkage and inductance of SRM using numerical integration technique reported by Rakesh Saxena et al [10] was proved to be laborious and relatively difficult to implement the control algorithms.
Huijun Zhou, Wen Ding and Zhenmin Yu reported a non-linear model of SRM using fourier series model [13]. To develop this model it is mandatory to use the data obtained either from FEA or from experimental set-up.
While analysis of SRM parameters using different methods have been reported in the past, the performance analysis of SRM considering saturation and fringing effect have not yet been discussed.
Though many methods were discussed for obtaining the inductance profile of the SRM, those techniques involved complex computational procedures as well as requires data either from FEA model or from experimental set-up. It makes the steps involved more cumbersome and consumes more time for computation. More assumptions are made for analysis and model development, therefore accurate results may not be obtained.
In view of this, a Geometry Based Analytical Model (GBAM) is developed from machine geometry and magnetization characteristic of the material used by taking the fringing effect into account. Also a prototype SRM drive model is fabricated and the inductance profile is obtained using a simple technique called Transient Analysis (TA) technique.
This paper is organized into six sections. The various steps involved in developing the GBAM are discussed in section 2. While section 3 briefs the Finite Element Analysis (FEA) model of SRM, section 4 deals with comparison of both GBAM and FEA models. FSM development and its comparison with FEA model is discussed in Section 5. Section 6 explains the implementation of experimental model, results and validation.
2. Geometry Based Analytical Model of SRM
The detailed procedure for developing the GBAM is described below.
2.1 Calculation of aligned inductance
The inductance in a magnetic circuit is calculated using the expression
where N is the number of turns/poleR is the summation of reluctances of different parts of the magnetic circuit.
Reluctance is expressed as
where l is the length of the magnetic pathμ is the permeability of the medium A is the cross sectional area of the flux path.
For air μ = 1.256e-6, but for magnetic material it is not constant and depends on B and H.
Therefore expression for reluctance is expressed as
where B is the flux densityH is the magnetic field intensity.
Values of B and H at each part of the machine are to be known to calculate their reluctances.
According to Ampere’s circuital law,
Phase winding of the SRM is divided into two coils, with each coil having N turns placed on the diametrically opposite poles and carrying a current of I Amperes. Therefore the total MMF of the magnetic circuit is expressed as
The flux path of the machine at the aligned position is shown in Fig. 1.
Fig. 1.Flux pattern at the aligned position
The path followed by the flux lines shown in Fig. 1 is considered for developing the equivalent magnetic circuit at the aligned position and it is shown in Fig. 2
Fig. 2.Equivalent magnetic circuit at the aligned position
Flux lines are closed lines and complete its path through stator yoke, stator pole, air gap, rotor pole and rotor yoke. Flux lines cross the stator pole, air gap and rotor pole twice. The equation to obtain total mmf in terms of magnetic field intensity (H) and length of the flux path (l) is expressed as
where
Hsy, Hsp, Hg, Hrp, Hry are the magnetic field intensities at stator yoke, stator pole, air gap, rotor pole and rotor yoke respectively.
lsy, lsp, lg, lrp, lry are their corresponding lengths.
where Ryir=Inner radius of rotor yokeRyor=Outer radius of rotor yoke
The path of the air gap through which flux completes its path is a complex one and determining length of the air gap is difficult. As Hglg = ΦgRg, instead of using Hglg in the Eq. (6) ΦgRg is used where Φg is the flux through the air gap and Rg is the reluctance of the air gap. Procedure to obtain Rg is explained in section 2.2.
For air μ = 1.256e-6 is a constant. Therefore reluctance, is found without the knowledge of B and H.
The new equation to calculate total mmf is expressed as
Initially the net flux set up in the machine is assumed as low as 1e-8 and as this is the flux crossing the air gap, Φg=Φ. The flux carried by the different parts of the machine varies. Based on the amount of flux, the flux density at that part is calculated using the formula
Stator yoke carries only half the flux. Therefore flux density at stator yoke (Bsy) is represented as
where Asy= Area of stator yoke and it is given as
where Syt=Stator yoke thicknessL=Stack length
Flux density at stator pole is expressed as
where Asp= Area of stator pole and it is given as
where Spw=Stator pole width
Rotor yoke carries only half the flux. Therefore flux density at rotor yoke (Bry) is represented as
Flux density at rotor pole is expressed as
where Arp= Area of rotor pole and it is given as
where Rpw=Rotor pole width
For the known B, corresponding H is obtained by reverse mapping from the B-H data of M19 and it is shown in Fig. 3.
Fig. 3.B-H curve of M-19 steel
If the obtained B is greater than the data with us, it is limited to the maximum B. Thus H and l for different parts are obtained and ΦgRg is also known. Substituting these values in Eq. (11), left hand side value is compared with right hand side value. The flux is gradually increased till the error is less than 1%.
2.2 Air gap reluctance at the aligned position
The flux bulges in the air gap causing fringing effect and as the difference in the width of stator and rotor pole width is very small the boundary can be considered as circular with diameter equal to the air gap. Top view of the aligned position of stator and rotor pole is as shown in the Fig. 4.
Fig. 4.Top view of the aligned position
To calculate the air gap reluctance, the air gap is split into 100 vertical layers. The area of each layer gradually increases due to increase in the width and height of the air gap.
As shown in the Fig. 5, the increment in the width for any angle Φ will be (lg/2) sinΦ. But two hemispherical regions are present so the total increment along the width or height of the air gap is lgsinΦ. The total angle 1800 is split into 100 partitions of 1.80 each.
Fig. 5.Increment in the width
Width of each layer = wm + lg sinϕ , where wm is mean of stator and rotor pole width.
Area of the individual layer is the product of its width and height.
Reluctance of the individual layer is obtained using the Eq. (2). Total reluctance of the air gap Rg is summation of the reluctance of individual layer.
2.3 Reluctance calculation of iron parts
Reluctance of the Iron parts is calculated as follows.
Reluctance of stator yoke is given by
Reluctance of stator pole is expressed as
Reluctance of rotor pole is written as
Reluctance of rotor yoke is
Substituting the reluctance of various parts in the fundamental expression (1), the resultant expression to obtain reluctance as a function of current is given by
The result obtained by solving the Eq. (25) is shown in Fig. 6. The phase inductance is drooping when the current through the winding increases.
Fig. 6.Inductance Vs Current at aligned position
2.4 Calculation of unaligned inductance
The flux path of the machine in the unaligned position is shown in the Fig. 7.
Fig. 7.Flux pattern at the unaligned position
The path followed by the flux lines shown in Fig. 7 is considered for developing the equivalent magnetic circuit at the unaligned position and it is shown in Fig. 8.
Fig. 8.Equivalent magnetic circuit at the un-aligned position
Equation to obtain total mmf of the magnetic path shown in the equivalent circuit is given as
where Hrpt=Magnetic field intensity at the rotor pole tiplrpt=Length of the rotor pole tip
2.5 Air gap reluctance at the un-aligned position
The flux lines crossing the air gap at unaligned position is given in detail in the Fig. 9. Based on symmetry only half of the total flux is considered for analysis. The air gap is treated as combination of three sections as shown in the Fig. 9. The sections can be treated as ideal geometrical figures without much loss in accuracy.
Fig. 9.Different sections of the air gap at unaligned position
Section 1 – It is modeled as a rectangle, with length equal to the distance between the stator pole tip and the centre of the rotor pole. The centre of rotor pole is at an angle 30 deg from the horizontal and the stator pole tip is at an angle half of the stator pole arc. Considering rotor pole radius as a common radius the length is found using the formula, Arc length = Radius * Arc angle. Divergence of flux lines at the stator pole tip is generally limited to 2mm. Therefore the width is (lg+2mm). Reluctance of the section 1 is computed as follows.
Length of the air gap is given as
where Rpr=Rotor pole radiusSarc=Stator pole arc
Area of the air gap is expressed as
where
Reluctance of section 1 is described as
The sides of the Fig. 9 are a, b, c and d which are given as a - distance from centre of stator pole to rotor pole tip, it is expressed as
b and d - distance between rotor yoke and stator pole tip, it is written as
where Rph=Rotor pole heightc - half the distance between two rotor poles, it is given as
Section 2 – It has 3 sides, which are a-c, b and d.
Section 2 is separated in to 100 horizontal layers of uniform area and varying length.
Length of each layer= nb / 100 , where n=1 to 100
Area of each layer = (a − c)L /100
Reluctance of each layer is calculated and the reluctance of section 2 (R2) is parallel combination of hundred layers.
Section 3 – It has sides ‘b’ and ‘c’ and reluctance of this section is described as
Reluctance of 3 sections is in parallel and the total air gap reluctance is given as
The variation of unaligned inductance for the change in current value is very minimum. Therefore it is considered as constant value for particular SRG and it is expressed as
The result obtained by solving Eq. (36) is given in Fig. 10.
Fig.10.Inductance Vs Current at un-aligned position
2.6 Inductance profile of SRM
The phase inductance is drooping when the rotor leaves the aligned position. During falling inductance period, variation of inductance with respect to rotor position is negative. Therefore torque generated is also negative [11]. If the phase is excited in this period, rotor experiences torque which opposes rotation and extracts energy from the turbine. Inductance variation as a function of current and position is given as
where
Complete cycle of inductance variation as a function of current and position is shown in Fig. 11.
Fig. 11.Inductance variation as a function of current and position
2.7 Flux linkage calculation
Flux linkage as a function of current and position is expressed as
Fig. 12 shows ψ-i characteristics of SRM where it is evident that a nonlinearity exists at the aligned position and is considered by making the fourier series coefficients ao(i) and a1(i) dependent on the phase current.
Fig. 12.ψ-i characteristics of SRM
2.8 Torque calculation
The instantaneous torque as a function of current and position is calculated using the formula
The response obtained by solving the Eq. (41) is given in Fig. 13. During falling inductance period, variation of inductance with respect to rotor position is negative. Therefore torque generated is also negative.
Fig. 13.Torque Characteristics
3. FEA Model
Among various numerical methods FEA is most frequently used. There are many types of software available to analyze and compute the electromagnetic field of the machine accurately in two-dimension (2-D). In this paper MagNet 7.1.1 is used to build the model. The FEA model of 4 Phase 8/6 SRM is shown in Fig. 14 (a) for the specifications in Table 4.
Fig. 14.(a) FEA Model of 4 Phase 8/6 SRG; (b) and (c) Mesh pattern at aligned and un- aligned positions
The mesh pattern of the problem region at the aligned and unaligned positions is shown in Fig. 14 (b) and Fig. 14 (c) respectively.
FEA Model shown in Fig.14 is solved to obtain the data like flux linkage, inductance and torque. The variation of flux linkage with respect to current is shown in Fig. 15.
Fig. 15.Flux Linkage vs Current
It is evident from Fig. 16 that inductance is maximum at the aligned position. When the rotor leaves the aligned position inductance decreases and it is minimum at the unaligned position. The inductance profile of the machine is periodic function of rotor position as seen in Fig. 16.
Fig. 16.Inductance Profile
Torque obtained from FEA model for various values of current and position is shown in Fig. 17. It is negative from 0-30 degrees and positive from 30-60 degrees.
Fig. 17.Torque characteristics
4. Comparison of Analytical Model with FEA Model
The response of flux model is compared with ψ-i characteristics of FEA model and is shown in Fig. 18. The comparison is done at aligned and unaligned positions. The results indicate there exists a close match between analytical model and FEA model.
Fig. 18.Flux linkage comparison with FEA
The inductance profile obtained from analytical model is compared with FEA model for a fixed value of current and is shown in Fig. 19. This result also reveals that the Analytical Model and FEA model are alike.
Fig. 19.Inductance comparison with FEA
Comparison of the torque obtained from Analytical model is performed with FEA and is shown in Fig. 20.
Fig. 20.Torque comparison with FEA
5. Fourier Series Model of SRM
The nonlinear model to SRM in terms of fourier series expression of phase self-inductance is calculated analytically, under the assumption of negligible mutual effect between the individual SRM phases and various losses. A fourier series representation is used to express the phase inductance with respect to the rotor position as [13]
where
L(θ,i) – phase inductance L0(i) & Ln(i) – fourier series co-efficients Nr – number of rotor poles θ – angular position of the rotor
To determine the coefficients in the Fourier series, it is necessary to know the inductance at several specific positions. The following relevant assumptions are further made for an 8/6 SRM:
La(i) = L(0,i) - aligned inductance Li(i) = L(10,i) – one third way inductance Lm(i) = L(15,i) – mid way inductance Lj(i) = L(20,i) – two third way inductance Lu(i) = L(30,i) – un-aligned inductance
The general expression of inductance considering the five different positions as stated above is approximated as
where
For different values of current i, La(i), Li(i), Lm(i), Lj(i) and Lu(i) are calculated using the data obtained from either FEA method or experimental measurement and it is shown in Fig. 21.
Fig. 21.Inductance and fitting curve
The generalized equation for La(i), Li(i), Lm(i), Lj(i) and Lu(i) are obtained using curve fitting technique. Which are expressed as follows
The inductance profile obtained using curve fitting technique for various values of current is compared with FEA model and it is shown in Fig. 22.
Fig. 22.Inductance comparison
As per non-linear inductance model, the torque is described as
The above equation is solved to obtain the torque response of the SRM and it is given in Fig. 23.
Fig. 23.Torque comparison
From the results it is clear that the fourier series model is accurate but it cannot be developed without using the data obtained either from FEA model or from experiments measurement. Many number of data are assumed for analysis and model development. Therefore, GBAM is developed and discussed in detailed manner.
6. Experimental Results
The Parameters like inductance, flux linkage and torque obtained from AM is confirmed with FEA model and found that there is a close match between the parameters. Hardware model [12] is developed to validate the inductance at the aligned and un aligned positions The complete block diagram of the experimental set up is shown in Fig. 24.
Fig.24.Block diagram of the experimental set up
For validation of the inductance values, microcontroller based experimental set up is developed and it is shown in Fig. 25. The stator winding of the SRM is excited by DC supply obtained from a full bridge diode rectifier powered by an Auto-Transformer. MOSFET IRF540 and 1 ohm resistor are connected in series with the phase winding. Triggering signal required for the MOSFET is generated by ATMEGA16 microcontroller and it is supplied to the gate terminal of the MOSFET through Opto-isolator MCT2E.
Fig. 25.Experimental set up
Power circuit for energizing one of the phase winding through MOSFET IRF540 is shown in Fig. 26.
Fig. 26.Power Circuit
Gate signal is obtained from microcontroller and it is supplied to the gate terminal of the MOSFET through driver circuit. The complete circuit diagram of the driver circuit is shown in Fig. 27.
Fig. 27.Driver Circuit
One of the phase winding is excited to bring the rotor to aligned position. Voltage applied to the stator winding is varied to adjust the current flow through the winding and transient response is captured using Digital Storage Oscilloscope (DSO). From the rise time of the transient response, inductance is obtained at aligned position for various values of current. Electrical equivalent circuit of the phase winding is shown in Fig. 28.
Fig. 28.Electrical equivalent of phase winding
The voltage equation corresponding to electrical equivalent of the phase winding for zero initial current [i(0)=0] is expressed as
Solving the non-linear Eq. (42) gives the result as
where
When t=5τ, system reaches steady state and inductance value is obtained from the rise time of the transient response. Transient response of the drive system when the winding carries 1A and 2A are shown in Fig. 29 and Fig. 30 respectively.
Fig. 29.Transient current characteristic at 1A
Fig. 30.Transient current characteristic at 2A
From the transient characteristic of the real time system inductance is obtained for the various values of current and it is compared with the FEA model and Analytical model. The results are tabulated in Table 1.
Table 1.Inductance value comparison at aligned position
Flux linkage at the aligned position for different values of current is calculated from inductance obtained from experimental set up and it is compared with the FEA model and Analytical model. The results are tabulated in Table 2.
Table 2.Flux linkage comparison at aligned position
Un-aligned position is obtained by exciting two phase windings simultaneously and un-aligned inductance is obtained for various values of current. It is found that unaligned inductance is constant irrespective of the current magnitude and it is shown in Table 3.
Table 3.Inductance value comparison at un-aligned position
7. Conclusion
The Geometry Based Analytical Model of SRM considering fringing effect is discussed briefly and the results are compared with the FEA model of a SRM. The results show that a close matches between analytical and FEA models exist. The analytical model presented is very accurate, simple and it consumes less time than the other methods and also gives results close to FEA. Further for justifying the results obtained from GBAM another analytical method called Fourier Series Model is developed with curve fitting technique. The results obtained from FSM is verified with the FEA and found satisfactory but it is mandatory to use the data obtained either from FEA model or from experimental measurements. Many numbers of data are assumed for analysis and model development. Therefore, GBAM is developed and it does not require any data from FEA model/experimental measurements. Implemented microcontroller based SRM drive system comprises of SRM has the same machine geometry which is considered for analytical model. Inductance profile of the machine is obtained for various values of current using Transient Analysis Technique and it is validated with the analytical and FEA models.
Annexure
Table 4.Electrical specifications of SRM
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