Identification of magnetostriction vibrations in power transformers
Transkrypt
Identification of magnetostriction vibrations in power transformers
ZESZYTY NAUKOWE POLITECHNIKI ŁÓDZKIEJ Nr xxx ELEKTRYKA, z. 1xx 2006 BEATA WAWRZYNIAK Technical University of Łódź Institute of Mechatronics and Information Systems THE APLICATION OF THE FAMILY OF ORTHOGONAL FUNCTIONS FOR MODELLING THE GENERATORS WITH PERMANENT MAGNETS Reviewer: dr hab. Paweł Witczak prof. PŁ Manuscript received 2007.03.20 Abstract - The paper presents the computing method making possible the replace the inductance in the description of the electric machine, with three dimentional distribution of magnetic flux density in the airgap fot chosen states of work: no-load, armature reaction directed in d and q axes. The input data comes from magnetostatic solution of FE model. The method converts the distributions of the flux density into the space of periodic functions, what allows the similar procedure as in the classic dq method.The presented approach allow to solve the radial and axial flux topology of the machine.The effects of the slot skew is consider using the low-pass filter in spectral domain. Experimental measurements of the existing machine were performed and they were compared with calculations. 1. INTRODUCTION The most popular representation of an AC machine is based on lumped circuit consisting of resistances and inductances converted to dq plane. This approach gives very quick solution, but it is limited to sinusoidal quantities. On the other hand the numerical models in 2D and 3D space are also used. It allows to consider the machine with flux density distribution distorted from sinusoidal shape. Unfortunately, the calculations using for example finite element are time consuming, especially for 3D objects. The aim of the work is to present the combine method, where the inductances will be replaced by flux density distribution in the air-gap obtained from finite element analysis at different load conditions: no-load, armature reaction directed in d and q axes. 2 Beata Wawrzyniak 2. ANALYZED MACHINES Two kind of machines were analyzed – radial flux machine with surface mounted magnets (machine A), radial flux with embedded magnets (machine B) and axial flux machine with surface mounted magnets (machine C). Topologies of the machines are shown in fig.1. machine A machine B machine C Fig.1. Topology of exemplary machines. The machine A is existing one with the fractional, three-phase winding with skewed stator slots, the rest of machines are virtual. Numerical experiment of machine B showed that the contribution of end-winding into flux density distribution is comparable to calculation accuracy and can be neglected. The radial flux machines were analyzed by means of 2D models with the vector potential approximation 3rd order. In case of axial flux machine the 3D model was necessary – the scalar potential approximation 2nd order. The reduction of the models size obtained using adequate boundary and periodicity conditions. The meshes are presented below. Fig.2. Finite element meshes of reduced models of PM generators. 2. PRINCIPLE OF THE METHOD The input flux density distributions were obtained for no-load state, armature reactions of the phase current with given RMS value IN in direct and quadrature axes. From 2D models of radial machines obtained one-dimensional vectors, but The application of the family of orthogonal functions to modelling the generators with permanent magnets 3 for axial machine data was two-dimensional, therefore, to use the same postprocessing algorithm, it had to be averaged in radial direction following equation (1), for the phase belt consisting of k coils (N turns each one) and positioned at angles { 1k, 2k}. α1k Ω t L Ψ(t) B(α, r) r dr dα N k (1) α 2k Ω t 0 where L is the core size in transverse direction to the field movement and is the mechanical angular velocity of the rotor. Afterwards the data was filtered using DFT/iDFT technique to remove these harmonics, which do not fulfil non-dispersive wave condition (2) (2) μ Cν where , are spatial and time harmonic orders, and C is integer multiplier. Fig. 3a presents the exemplary 2D one-sided spectrum comes from timestepping of the radial flux machine at rated load. The selection of slotting and subharmonics components can also be done having only the 1D DFT – fig 3b, because the slotting space harmonics are of order kNs p, where k=1,2,… and Ns is the number of stator slots. This simplification can be applied, when the number of slots per pole and phase is greater than one. In other cases other filtering technique should be used. fundamental, 0.71 T a. b. Fig.3. Spectra of normal flux density component in q axis (a) 2D one-sided DFT and (b) 1D one-sided DFT. Considering the field variation along radius equation (1) can be replaced by (3): α1k Ω t NS (3) Ψ(t) B( ) dα 2 k α 2k Ω t where S denote active surface of the air-gap. If the flux wave is non-dispersive, only the speed component of the electromotive force exists. NSδ (3) e(t) [B(α1k Ω t) B(α 2k Ω t)] t 2 k Using classic Gram-Schmidt algorithm it is possible to convert (3) into set of 4 Beata Wawrzyniak orthogonal basis functions m( t) – details of procedure can be find in [3]. The resultant e.m.f. formula using the basis functions [4] for any load conditions is presented in (4) e(ψ, I, t) NSδ {[b 0 2 (b d0 [b d1CC1 b 0 ) i r CC0 (b q0 b 0 ) i r CS0 ] b q1CS1 ] i r 1 (Ω t) b q2CS2 i r 2 0 (Ω t) (4) (Ω t)} where b0 bm n e0 em , n CC m m ( ), CSm m ( ), m (α ψ) (5) m (α π/2 ψ) and < , > means dot product of basis functions. The load angle =0, when exist the demagnetizing armature reaction only. CC and CS are termed as autocorrelation factors. Coefficient ir is the relative value of the phase current I/IN. a. b. Fig.4. Exemplary set of normalized basis functions for machine A without skew (a) and with skew (b) Knowing the components of electromotive force at any load it is possible to calculate the distribution of phase current following (6) i(ψ, i r , t) ir IN i e (ψ, i r , Ω t) ie (6) To calculate the value of electromagnetic torque TE for basis functions the equation (7) can be used 3 2 (7) TE E k (ψ, i r )I k (ψ, i r ) Ωk0 where Ek and Ik are the orthogonal components respectively of the e.m.f. and current. The results of calculations compared with the measurement for the machine A are presented in fig.5 The application of the family of orthogonal functions to modelling the generators with permanent magnets 5 Fig.5. Torque vs. load angle for machine A (OF – orthogonal functions, FE – finite element) 3. COMMUTATION PHENOMENA Using the rectifier bridge as the load of generator impel to consider the commutation phenomena. The describe of the voltage in selected winding can be done using (8) uV di V dt e0V (t) L (8) Ri V with the sinusoidal quantities assumption it is possible following [1] to calculate (9) the commutation wyprzedzenia angle 0, related with the voltage on resistance: ωt 0 arcsin RI 2 3E 0 RI 6E 0 ν0 (9) where R – sum of the winding resistance and the diode, I – average value of the rectified current. Considering the moment, when phase W took the current from V, it is possible to assume the equalities (10): di V di W (10) i V (t) i W (t) I and dt dt Balancing the voltages for the commutation moment and including (10) obtains: di V (L V dt L W ) 2Ri V e 0V (t ) e 0W (t ) RI (11) It is suggested to apply the elliptical approximation of inductance as follows: L( ) Ld cos 2α Lqsin 2α (12) 6 Beata Wawrzyniak upper envelope W lower envelope U d axis V W 0 t Fig.6. Load angle at immediate commutation, U,V,W denote the phase carrying the current. Based on fig. 6 for t=0 possible to note the equations describing the angle for phase which return current (13) and take over (14) (13) 2 2 t 0 3 2 2 t 0 3 (14) Putting (12), (13) and (14) into (11) we receive the value of current iV(t), which, after insert into formula (8) allows to find the searching voltage curves during commutation. It is worth to mention, that for permanent magnet synchronous generators, there is dependence inductances of the armature reaction from the rotation angle, coming from different magnetic associations in d and q axes. Including this fact it is possible to formulae inducing voltage in one of the phases, during commutation between the others: d (Φ VU dt UV 3ωi U cos(2(ωt Ld Φ VW ) ν0 Lq 2 3 di U sin 2 dt ψ)) ωIsin 2 ωt (15) π 3 ν0 ψ where VU, VW denote the flux induced in phase V by the current flowing respectively in phase U and W. The influence of rotational currents on field dumping depends on properties (resistivity) and position of the permanent magnets. It is possible to consider its influence replacing the inductance Ld by the Ld” [2]. Finally to calculate the value of inductance the equation (16) was used: Ld (bd 0 b0 ) 2 bd21 kd 2 pIN and Lq bq 2 2 pI N kq (15) where b0, bd0, bd1, bq2 means r.m.s. value of the adequate components of e.m.f. For further calculations assumed the value of the coefficient kd”=0,71 for machines with surface mounted magnets and kd”=1 for machine with embedded magnets. The proposed algorithm was used to set the curves of the voltage and current in The application of the family of orthogonal functions to modelling the generators with permanent magnets 7 the analyzed machines. 4. EXPERIMENTAL VERIFICATION OF THE THEORY The experimental verification of the method was done using the permanent magnet machine with rated data TN=38 Nm, nN=2000 rpm, p=4. The machine has winding with fractional number of slots per pole and phase. The air-gap field was measured by the search coils - number 1, 2, 3 in fig.7– follow the scheme of phase winding, coil s was placed on single tooth. Fig.7. Outlook of investigated permanent magnet machine. The measurement of electromotive force for machine loaded by the r.m.s. current I=13,9 A was done and compared with the results of calculations – fig.8 search coil EMF, [V] 1.5 measured 1 calculated 0.5 0 1 -0.5 -1 -1.5 time Fig.8. Measured and calculated electromotive force in search coil under rated load. Table 1. Comparison of main parameters at generator duty, n=1500 rpm (symmertical load) item measurement calculation maximum torque [Nm] rated torque [Nm] rated r.m.s. voltage [V] 50 38 109 49.8 36.2 102.7 One of analyzed states of work was the machine loaded by rectifier bridge. The results of experiment in this case are presented below. 8 Beata Wawrzyniak U [V] 200 I [A] 12 calculations measurements calculations measurements 8 100 4 T 0 T 0 -4 -100 -8 a. -12 b. Fig.9. Phase voltage (a) and phase current (b)for machine loaded by 6D rectifier bridge. -200 Table 2. Comparison of measurements and calculation for machine loaded by rectifierr bridge. current [A] 14.2 6.4 item measurement calculation P [W] U [V] P [W] U [V] 4402 111.3 2246 124.4 4706 113.0 2291 124.9 6. CONCLUSIONS The method presented in this paper enables the computation of PMS machines by means of the family of orthogonal functions. The main advantage of proposed algorithm is the independence on the shape of the field in the air-gap. This approach allow to include the influence coming from different in inductances in d and q axes. It is worth to mention that also diverse topologies of machine can be analyzed using the same procedure. The possibility of expand the family of basis functions according to necessaries (include the magnetic saturation effects), make this method very useful to more precise analyses. The advantage against time stepping follow the availability of interim results showing the influence of many geometric and construction factors. REFERENCES [1] Jabłoński M.: Przekształtniki, PŁ, Łódź 1978. [2] Paszek W: Dynamika maszyn elektrycznych prądu przemiennego, HELION, Warszawa 1986 [3] Wawrzyniak B.: Zastosowanie rodziny funkcji ortogonalnych do modelowania The application of the family of orthogonal functions to modelling the generators with permanent magnets 9 prądnic z magnesami trwałymi współpracujących z turbinami wiatrowymi, rozprawa doktorska, Politechnika Łódzka, Łódź 2007 [4] Witczak P., Wawrzyniak B.: Determination of exploitation parameters of permanent magnet synchronous machine following the finite element solutions, CEM, pp. 177178, Aachen 2006 . ZASTOSOWANIE RODZINY FUNKCJI ORTOGONALNYCH DO MODELOWANIA PRĄDNIC Z MAGNESAMI TRWAŁYMI Streszczenie Praca dotyczy metody obliczeniowej, pozwalającej na zastąpienie indukcyjności w opisie maszyny elektrycznej przestrzennymi rozkładami indukcji magnetycznej w szczelinie wyznaczonymi w wybranych stanach pracy – stan jałowy, reakcja twornika w osi q i d. Proponowane podejście pozwala na uwzględnienie rozmieszczenia cewek uzwojeń fazowych, skosu żłóbków stojana oraz dopuszcza analizę maszyn o strumieniu radialnym i osiowym za pomocą tego samego algorytmu. Dodatkową zaletą jest możliwość rozszerzenia rodziny funkcji bazowych o kolejne składniki odwzorowujące np. wpływ prądów wirowych. W pracy zaproponowano również połączenie metody funkcji ortogonalnych z obwodowymi równaniami opisującymi zjawiska zachodzące w mostku prostowniczym, przez co możliwe jest również analizowanie wpływu niesymetrycznego obciążenia na parametry maszyny. Dokonano porównania wyników obliczeń z pomiarami rzeczywistego obiektu. Promotor dr hab. inż. Paweł Witczak, prof. PŁ Recenzenci pracy doktorskiej: 1. prof. dr hab. inż. Marian Łukaniszyn – Politechnika Opolska 2. dr hab. inż. Adam Pelikant, prof. PŁ – Politechnika Łódzka