Inductance profile estimation of high-speed Switched Reluctance Machine with rotor eccentricity

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1 Inductance profile estimation of high-speed Switched Reluctance Machine with rotor eccentricity Rajdeep Banerjee, Parthasarathi Sensarma Department of Electrical Engineering IIT Kanpur, India Abstract This paper presents an analytical method to predict the inductance profile of a Switched Reluctance Motor with rotor eccentricity. The percentage eccentricity in high-speed motors which are necessarily smaller is usually larger than their low-speed counterparts. This aggravates the symmetry of the magnetic flux paths, axiomatically assumed during design and analysis, due to the consequent non uniform air-gap. Two correction factors are introduced based on equivalent magnetic circuits to estimate phase inductance from the nominal parameters and rotor eccentricity, for the entire range of rotor positions. Analytical results obtained are shown to be in excellent agreement with the output of benchmark FE software. I. INTRODUCTION For the Switched Reluctance Machine (SRM), its inductance profile, L(θ), is a critical design aspect as the instantaneous electromagnetic torque is directly proportional to its local slope, besides being a quadratic function of the stator current. However, absence of any mutually induced e.m.f causes saturation of the magnetic core, mainly at the stator pole, as stator currents increase. To account for this non-linear behaviour, a complete machine model is usually represented by the current-integral of the inductance profile, obtained at different rotor positions. Shaping these so-called Ψ i curves is probably the chief aspect of the electromagnetic design of SRM. Additionally, these are also critical for operation of SRM drives based on sensor-less position estimation [1], [] [3], [4]. High speed machines, being necessarily smaller than their similarly rated low speed counterparts, have larger percentage tolerance in uniformity of the air-gap length due to inevitable eccentricity in rotor alignment. This paper addresses the effects of this problem on the inductance profile, which directly affects machine performance. Inductance profile is predicted analytically using flux-tube method which is highly accurate for non-eccentric rotor [7]. Specifically, an analytical approach is presented to estimate the effects on inductance profile which does not require recourse to any finite-element (FE) analysis and is based on careful application of magnetic circuit analogy. Estimated results are compared with those obtained from benchmark FE software which show a high degree of accuracy. II. STATIC CHARACTERISTICS Fig.1 shows the static Ψ i characteristics of an SRM with uniform air-gap, obtained from a benchmark FE software (Maxwell). From these plots, Fig. 1. Flux linkage vs current vs rotor position characteristics. co-energy and static torque is calculated using the

2 following equations, [1] W f(θ r, i) = i i 1 Ψ(θ r, i)di (1) T e = dw f(θ r, i) dθ r. () and shown in Fig.. Subsequently, magneto-static 6 4 Fig. 4. D FE analysis: Flux density and magnetic lines of force plot for completely unaligned condition Torque (in Nm) Rotor position (in deg) profile, torque characteristics and hence machine performance. Fig. 5 represents the planar geometry which shows the nominal rotor position (in dotted lines) and the rotor with eccentricity. Here Fig.. Static torque characteristics analysis is done, using the FE software, at rated conditions of machine to calculate flux density for aligned and unaligned rotor positions. The corresponding results are shown in Fig.3 and Fig.4, respectively. It is however clarified that FE analysis is Fig. 5. Diagram of actual rotor position and new rotor position. Fig. 3. D FE analysis: Flux density and magnetic lines of force plot for completely aligned condition not required as a priori information for the analytical approach described later. III. ECCENTRICITY PROBLEM In this section, a major problem in fabrication of machine,i.e. eccentricity problem is discussed. A. Analytical approach A simple analytical method is derived to predict the effect of rotor eccentricity on inductance uni-dimensional eccentricity has been considered, implying that the axes of the nominal and eccentric rotor are parallel. Origin of the co-ordinate system used is the intersection of the stator axis and the plane considered. Here, δl u denotes the displacement between stator and rotor axes and is a measure of eccentricity while θ is the mechanical angle between the stator pole axis and the vector u. Thus for a machine with N s stator poles, θ takes N s discrete values given by θ(k) = θ 0 + k(π/n s ), k [0, N s 1]. (3) The stator pole axis intersects the outer periphery of the nominal and eccentric rotors at points P 1 and P, respectively. For each of these N s pole axes, l(θ) denotes the corresponding distance P 1 P. D is the

3 rotor outer diameter and denotes the nominal airgap length with no rotor eccentricity. Two realistic assumptions are made for this analysis, D δl > δl. The positions of P 1 and P are expressed as P 1 = 0.5D{cos θ + j sin θ} (4) P = 0.5D{(1 + δl/d) cos θ + j sin θ} (5) Therefore, the relation between δl and l(θ) can be expressed as, l(θ) = P P 1 = δl. cos(θ) (6) Now, if F 1 and F are magnetic pull on poles of same phase, resultant magnetic pull is expressed as, F = F 1 F (1) F = 4F δl cos θ (13) ) Effect on magnetic field: Effect of rotor eccentricity on the magnetic field is investigated, considering the equivalent magnetic circuit. This is shown in Fig.6a, which represents the condition for an arbitrary rotor position [1]. Here it is considered that only one phase is excited at a time. In Fig.6a, R 1, R 5 represent reluctances of stator Therefore, modified air gap is expressed as l g = δl. cos(θ). (7) So, from (7), the air gap varies with the rotor position which affect the inductance profile and torque characteristics. 1) Effect on magnetic pull: Under nominal condition of uniform air-gap, the magnetic pull on stator pole and rotor pole at aligned position is balanced. But due to eccentricity, magnetic pull becomes unbalanced which stresses the shaft of the machine and causes vibration [5], [8]. This phenomenon is analytically investigated here and notations used are listed below. F : magnetic pull for one pole, µ 0 : absolute permeability of air, AT g = ampere-turns of excited stator phase. A= stator pole area. Assuming uniform flux density, magnetic pull for one stator pole is expressed as F = B A = µ 0 µ 0 Now, due to unbalanced condition, ( ) ATg A (8) l g = l δl cos θ (9) For SRM two poles of exited phase are mechanically displaced by 180. So l 1 and l are air-gap lengths for the same phase and expressed as, l 1 = δl cos θ (10) l = + δl cos θ (11) (a) Fig. 6. Equivalent magnetic circuit while one phase excited (a) for ideal case (b) for non uniform air-gap. pole, R and R 4 the reluctances of air gap, while R 3 is the equivalent reluctance of rotor and R 6, R 7 denote reluctances of stator back-iron. R m1 and R m are the reluctance of the additional flux paths created due to saturation of the pole face adjoining the shortened air-gap. The stator magneto-motive force (mmf) is expressed as, (b) F = T ph.i (14) where, T ph is number of turns per stator phase and i denotes the phase current. In case of rotor eccentricity, air-gap reluctances are not equal for both the poles of excited phase. Due to this inequality, other flux paths appear which pass through the unexcited stator poles. Therefore, the pole of excited phase adjoining shortened air gap has greater flux linkage than the other pole of that phase. This phenomena causes the saturation of pole

4 face adjoining the shortened air-gap. Reluctance of each of the air gaps, for an arbitrary rotor position, θ, adjoining the poles of excited phase are R = R 4 = (1 δl cos θ) µ 0 A g (15) (1 + δl cos θ). µ 0 A g (16) From the above equation it is clear that R 4 > R. Accordingly, the magnetic circuit is modified to introduce the reluctances for the additional flux paths indicated before. Fig.6b shows the modified equivalent magnetic circuit, which models the above phenomenon. Since the two coils, associated with each of the poles of any phase, are connected in series, difference in their flux linkage alters the corresponding Ψ i characteristics. To verify the proposed modification in the magnetic circuit, Fig.7 shows the D FE simulation result. It is clearly observed that Fig. 7. D FE simulation: Flux density plot. there is an alternate flux path through an unexcited stator pole as well as the accompanied saturation in the excited pole adjoining minimum air-gap. 3) Effect on air-gap inductance : Flux-linkage and phase inductance for the uniform-gap machine are expressed as, Ψ = T phi R ; L ph = T ph R (17) where, R, L ph represent reluctance and phase inductance respectively. Only air gap reluctances are considered and Fig.8a shows the equivalent magnetic circuit. Due to rotor eccentricity, additional flux paths are introduced through the adjacent poles of active stator pole having less air-gap. Therefore, e- quivalent circuit considering additional parallel path is shown in Fig. 8b. R n is an equivalent additional (a) (b) Fig. 8. Equivalent magnetic circuit only for air-gap with one phase excited. (a) uniform air gap (b) non uniform air-gap. path reluctance parallel to the active phase pole path with less air-gap. Hence the modified inductance associated with the excited phase is ( ) L ph = Tph 1. (18) (R n R ) + R 4 Thus, the ratio, τ, of the phase inductances, in the eccentric and nominal rotor machines, is expressed as where, τ = L ph L ph = R + R 4 R 4 + (R R n ). (19) R n = R m1 R m (0) represents the equivalent reluctance for the additional flux paths created due to rotor eccentricity. For nominal representation, R m1 = R m (1) which lead to a slightly erroneous quantity. However, close inspection shows that these vary with the corresponding effective area A 1 (θ r ) and A (θ r ), where, θ r is the rotor position, defined as the angular position of nearest rotor axis with respect to excited phase axis. R m1 and R m can be expressed as, R m1 = µ 0 A 1 (θ r ) ; R m = µ 0 A (θ r ) ()

5 Therefore, another correction factor β is introduced to include this effect. Hence, R n is modified by β which is expressed as, β = (A n ) A 1 (θ r ).A (θ r ) (3) where, A 1 (θ r ) and A (θ r ) are cross sections of two additional flux paths shown in Fig.6b. A n denotes the nominal cross-section of these additional flux paths, independent of θ r, and is equal to A n = A 1(θ r ) + A (θ r ), 0 θ r 90. (4) Variation of A 1 (θ r ) and A (θ r ) is shown in Fig.9. Hence, β can be expressed as difference in corresponding magnetic circuit geometry, the instantaneous torque and net torque can be expressed as M = 1 i τ dl dθ r ; M net = Σ 1 i jτ j dl j dθ r (8) where j is the index for stator phases. IV. SIMULATION RESULTS In this section FE simulation results are presented to validate the analysis presented. For validation, FE simulations are done for a 6/4 SRM under conditions of uniform air-gap (nominal) and 60% rotor eccentricity. Shift in rotor axis is considered to be along the axis of phase-a. Fig.10 shows the FE results of inductance profile of three phases for the nominal case. Fig.10 shows that three phase Fig. 9. Variation of A 1 (θ r ) and A (θ r ) over rotor position. ( 1 β = 4 β s + β r θ r ) 4 (β s + β r ) θ r (5) where, β s and β r denote stator and rotor pole pitch respectively. Therefore, effect of β on τ is expressed as τ (θ r ) = ( R + R 4 R 4 + {R βr n /(R + βr n )} ) (6) where, τ (θ r ) denotes the correction factor which varies with rotor position. 4) Effect on instantaneous torque: Instantaneous torque (M) is expressed as [] M = 1 dl i. (7) dθ r Mutual inductance is not considered as it is considered that one phase is excited at a time. As the net torque is dependent on inductance profile of each phase, it is obviously affected by rotor eccentricity. Since τ for individual phases is different, due to Fig. 10. Inductance profiles for three phases: uniform air-gap. inductance profile are identical and shifted by 10 o electrical. Maximum inductances (at aligned position) are same for three phases and equal to 441µH. For 60% rotor eccentricity, rotor axis is shifted 0.3 mm (for a 0.5 mm nominal air-gap) along the phase- A axis. Corresponding FE simulation results, for all the three phases is shown in Fig.11. From the Fig.11 Fig. 11. Inductance profiles for three phases: non-uniform air-gap. inductance profiles of three phases are not identical and inductance of phase-a is greater than other

6 two phases as can be concluded from the presented analysis. To verify the analytically derived value of τ, most affected phase i.e. phase-a is considered. Fig.1 shows the corresponding estimated inductance profiles obtained under nominal and eccentric rotor situations. Analytically derived correction factor, τ is 1.08 whereas from FE analysis ratio of inductance of eccentric and nominal rotor is seen to vary between 1.0 and 1.17, depending on rotor position. So, neglecting the effect of rotor position on τ, the Inductnace (uh) Nominal Eccentric rotor Estimated Rotor postion (deg.) Fig. 1. Nominal, eccentric rotor and estimated inductance profile with out considering β. percentage error in estimated inductance varies from 1% to 9%, the larger error being obtained at the unaligned position. For further accuracy, inductance estimation is carried out using the corrected factor τ, as defined in (6). The corresponding results are shown in Fig.13. From Fig.13 it is confirmed that Inductance (uh) Nominal Eccentric rotor estimated Rotor position (deg.) Fig. 13. Nominal, eccentric rotor and estimated inductance profile considering β. is nullified at unaligned position. Also, the error in inductance estimation ranges from 0% to 8%, maximum being obtained at the aligned position. V. CONCLUSION In this paper an analytical method is proposed to estimate the inductance profile of SRM under rotor eccentricity. A correction factor is introduced, based on equivalent magnetic circuit, to facilitate the analytical prediction. Results obtained from the analysis are compared with benchmark FE simulation results. It is observed that with a simplified definition of the correction factor, maximum estimation error could be around 7%. A more precise definition of the correction factor, based on rotor position, is presented, which is shown to reduce the estimation error to around %. Hence the presented approach obviates the use of intensive computational resource for SRM design. This approach could also be used to generate the flux-linkage characteristics for position sensorless operation of the SRM. REFERENCES [1] Krishnan, Ramu, Switched reluctance motor drives: modeling, simulation, analysis, design, and applications,crc press,001. [] Miller, Timothy John Eastham, ed. Electronic control of switched reluctance machines. Newnes, 001. [3] Venkataratnam, K., Sengupta, M., Chattopadhyay, A. K., B- hattacharjee, R.. Development and experimental validation of a novel analytical model for performance prediction of SR motors under saturation. Power Electronics and Drive Systems, 003. PEDS 003. The Fifth International Conference on. Vol.. IEEE, 003. [4] Panda, Debiprasad, and V. Ramanarayanan. Effect of mutual inductance on steady-state performance and position estimation of switched reluctance motor drive. Industry Applications Conference, Thirty-Fourth IAS Annual Meeting. Conference Record of the 1999 IEEE. Vol. 4. IEEE, [5] Ayari, S., et al. Effects of the airgap eccentricity on the SRM vibrations. Electric Machines and Drives, International Conference IEMD 99. IEEE, [6] Torkaman, Hossein, and Ebrahim Afjei. Magnetostatic field analysis regarding the effects of dynamic eccentricity in switched reluctance motor. Progress In Electromagnetics Research M 8 (009): [7] Pohl, R, Theory of pulsating field machines, J. IEE. pp [8] Maruthi, G.S., Hegde, V., Mathematical analysis of unbalanced magnetic pull and detection of mixed air gap eccentricity in induction motor by vibration analysis using MEMS accelerometer, Int. conf on Condition Assesment Techniques in Electrical System,IEEE, 013. use of the corrected ratio (τ ) ensures that the error