Main inductance determination in rotating machines. Analytical and Numerical calculation: A didactical approach
Bargallo, R. (1) , Llaverias, J. (1), De Blas, A(1)., Martín, H. (1), Piqué, R. (2) (1) Electrical Engineering Department. (2) Electronic Engineering Department EUETIB, Politechnical University of Catalonia (UPC) c/Urgell, 187, 08036 Barcelona (Spain) Tel. (34) 934137411, fax: (34) 934137401, ramon.bargallo@upc.edu Abstract.
The accurate determination of saturated magnetizing inductances has been the subject of much research over a long time. These results are necessary for the appropriate adjustment of control regulation loops and for the improvement of transient response and stability of electric drives. Traditionally the analytical calculation involves the determination of some empirical factors, such as the d-axis and q-axis reactance factors. In references [1] to [3] there are many expressions for salient and non-salient pole machines, but these are valid only for the considered pole shape. If possible we should use an expression, or method, independent of the pole shape. Analytical formulation is not adequate for this reason. Now we can use the FE method to calculate this and other parameters. In addition the time devoted today to the design of electrical machines has been reduced and this makes it impossible to use a lot of empirical or graphical methods. The use of FEM provides a way to quickly and accurately calculate the size of an electrical machine and its parameters. This paper has been written to describe this methodology in an educational environment. B. Air gap induction B1 (only considers the fundamental component) C. Total flux per phase D. Main inductance determination: Lm = I The following expression is the result of this process.
Lm = µ0 D L N m g eq p 2 (1) with: m ­ number of phases, D ­ air gap diameter, L ­ length of the machine, geq ­ equivalent air gap (with Carter's and saturation correction), N ­ number of turns per phase, p ­ pole pairs. B. Salient pole machines The calculation is similar, but we found some differences: We calculate the MMF projection over two axis: direct and quadrature axis. Thus we determine the induction create for these two components and determine the fundamental component. Thus we can calculate the flux and the main inductance for every component: Keywords.
Main inductance determination, FE method, cylindrical and salient pole machines. - 1. Analytical calculation of magnetizing inductances.
In the following paragraphs we describe how obtain an analytical expression for the main inductances. This methodology shows how these are function of the pole shape and how explain this in an educational environment. A. Uniform air gap machine The magnetizing inductance of a uniform air gap machine can calculated according to the following procedure: Calculation of: A. MMF created to the 3-phase equilibrate current system Lmd = d I ; Lmq = q I
These process leads to the following expressions: (2) Lmd = k d Lm ; Lmq = k q Lm (3) Where Lm is the magnetizing inductance calculate supposing that the air gap is uniform and kd and kq are coefficients that depended on pole shape. The following table shows these coefficients for different pole shape configuration; the first row is for a classical salient pole synchronous machine and the others are for permanent magnet machines. Table I. direct and quadrature correction factors. For example for the induction machine model with 3 coils in the stator and 3 coils in the rotor, that is:
L [L] = s L rs L as L sr ; [L s ] = L bas Lr L cas
L abs L bs L cbs L acs Salient pole synchronous machine PMSM. Surface magnets PMSM. Inset magnets kd = kq = 1 ( + sin( ) ) 1 2 - sin( ) + cos( ) 3 2 L bcs ; L cs L ar [Lr ] = L bar L car
2 ) 3 L abr L br L cbr L acr L bcr L cr )
) k d = 1; k q = 1
kd = 1 + sin( ) + c g ( - - sin( ) ) [ ] [Lsr ] = [Lrs ]t 1 1 k q = ( - sin( ) ) + ( - + sin( ) ) cg h cg 1 + g L asar cos r 4 = L bsar cos( r + ) 3 2 L csar cos( r + 3 ) L asbr cos( r + L ascr cos( r + L bscr cos( r + 4 3 2 3 L bsbr cos r L csbr cos( r + 4 ) 3 L csc r cos r (7) PMSM. Buried magnets kd = kq = 4 cos 1 - 2 2 = = D - 2 ph 1 ( - sin( ) ) p D With: p- pole pitch, = pole arc/ pole pitch, h - permanent magnet height. 2. Numerical determination of magnetizing inductance.
The numerical determination of magnetizing inductance involves the realization of Finite Element Analysis (FEA) and the determination of the magnetic energy stored in the air gap. The following paragraphs describe the relations between this and the magnetic inductance. In addition we describe two ways for the calculation of stored energy; the first is by integration of density of energy and the second is by circuit modelling. A. Magnetic energy stores in the air gap (uniform air gap machine). If we consider an ideal machine with sinusoidal distribution of the induction along the air gap, that is,
x ^ B = B sin p
Figure 1. simplified machine. We obtain: W= 1 Lij I i I j 2
+ Las i 2 as + Lbs i 2 bs + Lcs i 2 cs + Lar i 2 ar + Lbr i 2 br + Lcr i 2 cr Labs ias ibs + Lbcs ibs ics + Lcas ics ias + 1 Labr iar ibr + Lbcr ibr icr + Lcar icr iar + W = 2 + 2 + Lasar ias iar + Lasbr ias ibr + Lascr ias icr + Lbsar ibs iar + Lbsbr ibs ibr + Lbscr ibs icr + + Lcsar ics iar + Lcsbr ics ibr + Lcsc r ics icr (8) (We omitted the terms with cos() to simplify the expression). If we consider the following values, corresponding to an instant with: (4) and we calculate the magnetic energy stored in the airgap, we obtain:
W = µ0 m2 2 N D L 2 p g I m eq
2 (5) we obtain: 1^ ibs = ics = - I 2 iar = ibr = icr = 0 ^ ias = I ; (9) If we combine (5) with (1) we can write: W=
(6) m W = Lm I 2 m 2 3 3 Lm I 2 + Ls I 2 2 2 (10) usually m = 3. You can obtain the same expression if you consider the electrical circuit model with coupled coils. Except for the last term, this is the same expression (6). This term is a result to the dispersion effect and will be not considered for the main inductance calculation. B. Magnetic energy stores in the salient pole machine. We can obtain an expression for the magnetic energy stored in the case of the salient pole machine, but this takes longer to determine. We develop an expression based on circuit model approximation. In the salient pole machine we consider the first harmonic approximation for the inductance variation, i.e., 3 3 3 3 W = I 2 L0 + L2 cos(2 er ) + Ls I 2 2 2 2 2 (15) The last term is a result to the dispersion effect and will be not considered for the main inductance calculation. If we consider two selected positions for the rotor, i.e. L L0 + L2 cos 2 er (11) · · cos(2 er ) = 1 Direct field orientation
cos(2 er ) = -1 quadrature field orientation Some after algebraic manipulations, we obtain:
3 2 3 3 3 2 W = 2 I 2 L0 + 2 L2 + 2 Ls I 3 cos(2 er ) = 1 Lmd = [L0 + L2 ] 2 3 2 3 2 W = 2 I Lmd + 2 Ls I (16)
Figure 2. Salient pole machine for the 3-phase synchronous machine we can write: Lbb = Ls + L0 + L2 cos(2 er + 2 / 3) Lcc = Ls + L0 + L2 cos(2 er - 2 / 3) Lac = - L0 / 2 + L2 cos(2 er + 2 / 3) Lbc = - L0 / 2 + L2 cos 2 er
(12) Laa = Ls + L0 + L2 cos 2 er 3 2 3 3 3 2 W = 2 I 2 L0 - 2 L2 + 2 Ls I 3 cos(2 er ) = -1 Lmq = [L0 - L2 ] 2 3 2 3 2 W = 2 I Lmq + 2 Ls I (17) C. Inductance concatenation determination by means of flux Laf = Lsf cos er , Lbf = Lsf cos( er - 2 / 3), Lcf = Lsf cos( er + 2 / 3) Lab = - L0 / 2 + L2 cos(2 er - 2 / 3) The energy stored is: Another technique for the calculation of inductance is by the use of flux concatenation by a coil. If we consider a magnetic field distribution along the air gap, and its first harmonic, we can calculate the flux concatenation and the main inductance: Laa i a + Lbb i b + Lcc i c + L ff i f + 1 W = Lbc ib ic + Lac ia ic + Lab ia ib + (13) 2 2 + Laf ia i f + Lbf ib i f + Lcf ic i f
2 2 2 2 L= =NS I r r × A dS I A dl =N
I r r
(18) (We omitted the terms with cos() to simplify the expression). If we consider the following values, corresponding to an instant with: If we consider a salient pole machine, we use a flux oriented over the direct and quadrature axis respectively, for the determination of direct and quadrature inductances. ^ ia = I ib = ic = - if = 0 ^ I 2 field current )
(14) ( without We obtain the following expression: 3. Practical Applications
The following paragraphs show three examples of determination of main inductance. Two of them are compared with experimental results. A. Asynchronous machine: 1.5 kW; 50 Hz; 220 / 380 V; 6.4 / 3.7 A; cos = 0.85; 1420 min-1; F class; J = 0.0105 kgm2; . Connexion. Geometric and electrical data: 36/28 slots; 44 conductors/slot; D = 80 mm; g = 0.375 mm; L = 100 mm. We considered that k c k sat = 1.3 and = 0.955 . Figure 5. FEM model for synchronous machine. Quadrature field orientation Table III. Main inductance for synchronous machine. Method Analytical calculation FEA Experimental results (reduced slip test)
Figure 3. FEM model for asynchronous machine. Only ¼ of the machine has been modeled. Table II. Main inductance for asynchronous machine. Ld (mH) 9.84 10.7 7.42 Lq (mH) 4.25 4.23 5.30 Method Analytical calculation FEA Experimental results Lm (H) 0.310 0.313 0.255 C. Synchronous machine with permanent magnets. 5.1 Nm; 3500 min-1; IN = 2.56 A; F class Geometrical data: D = 80 mm; L = 68.9 mm; 36 slots; 6 pole; 35 conductors per slot; single layer lap winding; = 0.96; permanent magnet height h = 3mm; g = 0.5 mm; = 0.65; k c k sat = 1.3 ; surface permanent magnet. In this case to impose if = 0 we change the PM characteristic from a non-magnetic material with the same magnetic permeability of the PM. This machine is considered as uniform air gap machine due to the value of recoil permeability of the PM (near to 1.0) B. Synchronous machine: 6 kVA; 220 V; 15.8 A; 50 Hz; 1500 min-1; Y connexion. Geometric and electrial data: salient pole with uniform airgap (under the pole) g = 2 mm; D = 304 mm; L = 100 mm; = 0.55; 36 slots; double layer lap winding,; 5 conductors per slot and layer. We considered that k c k sat = 1.3 and = 0.955 . Figure 6. FEM model for synchronous machine. Direct field orientation Figure 4. FEM model for synchronous machine. Direct field orientation Table IV. Main inductance for synchronous machine with PM. Method Analytical calculation FEA (energy) Flux method (FEA) L (mH) 6.55 5.15 6.0 3. Conclusions. · · ·
Figure 7. FEM model for synchronous machine. Quadrature field orientation · For this machine we determined the inductance by the method of flux concatenation. We obtain the magnetic field distribution and harmonic components showed in the figures 8 and 9. We explained some methods to determine the main inductances for alternating current machines in an educational environment. We considered correction factors that are dependents on the pole-shape configuration. FEM is more precise than analytical calculation and is not dependent on an empirical or geometrical factors. Some of these experimental results are discordant with theoretical results due to estimation of some geometrical measures and magnetic characterization. Bibliography
[1] J. Corrales Martín. Cálculo Industrial de Máquinas Eléctricas. Tomo 1. Ed. Marcombo. 1982 [2] B. Chalmers, A. Williamson. AC Machines. Electromagnetics and Design. Ed. Research Studies Press Ltd. 1992. [3] J.F. Gieras, E. Santini, M. Wing. Calculation of synchronous machines of small permanent magnet alternating current motors: comparison of analytical approach and FEM with measurements. IEEE Transactions of Magnetics, vol 34, nº 5, September 1988. [4] M.R. Hassanzadeh, A: Kiyoumarsi. Analytical calculation of magnetizing inductances in interior permanent magnet motors. ICEM, Cracow, September 2004. [5] R. Bargalló. Diseño de Máquinas Eléctricas. Tema 8. Determinación de parámetros. EUETIB. 2004. Figure 8. Magnetic field distribution along the airgap. Figure 9. Harmonic distribution of magnetic field. The following table (IV) shows the calculated values.