Wind or Hydro Homo-Heteropolar Synchronous Generators: Equivalent Magnetic Circuit and FEM Analysis

In an effort to introduce a low cost (PM less), low power electric wind or hydro generators, this paper reports on preliminary design aspects, equivalent magnetic circuit and 3D FEM analysis of a 2.5 KVA, 250-1000 rpm, reactive homo-heteropolar brushless synchronous machine (RHHBSM).


Introduction
One of the main disadvantages of the classic synchronous machines is the armature's excitation winding which determines a great rotor weight and inertia and involves the sliding contacts' existence (brushes and slips rings). Reference [1] presented a new form of heteropolar linear synchronous machine that is capable of providing both thrust and lifting force at relatively high efficiencies and power factor. In [2] is presented a rotary reactive homopolar synchronous machine with stator excitation which removes the disadvantages of the classic synchronous machines.
Conception constraints on electro-technical devices require numerical simulations to be as close as possible to its actual operating conditions. Then, it is necessary to have coupled physical models of devices, especially, for electrical, magnetic and mechanical coupled models which allow the simulation of loaded rotating machines [3].
Finite elements method (F.E.M.) allows such coupling for 2D plan modeling devices. Nevertheless, it requires a lot of calculation time. Its use for three dimensional typical machine has never been done until nowadays and calculation time will be even longer [4].
In order to obtain the best results in designing of the special electric machines, it should be used both the classic methods and the numerical calculation methods. The calculation should be based on a mathematic model as accurate possible. Based on this model are determined by simulation the characteristics of the machine in nonsaturated and saturated regime [2], [5]. The designing particularities of these types of generators are linked to the axial character of the magnetic field distribution. The field calculation in the machine can be achieved by the finite elements method [6], or by field tubes method [2], [5], [7][8][9][10]. Taking into consideration the axial distribution of the machine field, it is necessary a three-dimensional modeling of the machine field. For this three-dimensional model is required a specialized software that needs a performance computer, and the calculations time could be high.

The constructive elements
The reactive homopolar (RHBSM) and homoheteropolar brushless synchronous machines (RHHBSM) which we'll analyze further are rotary machines. In order to understand their constructive elements, in Fig.1 is presenting a longitudinal section. The excitation coil has a ring shape and is placed in the windows of the U or E-shaped laminations stack ( Fig. 1  a, b), and, at passing of the rotor poles, the field is closing, having by this a rectangular variation form. When the rotor pole is not under the laminations stack, the field is practically null.

The method of equivalent magnetic circuits
A quasi-stationary magnetic field can be divided in field tubes, which are geometrical figures in which all the field lines are perpendicular on its bases and do not intersect the side surface. The plans of the same scalar magnetic potential are perpendicular on the flux lines. Is noted with Γ the specific permeance on the length unit, with λ the permeance, with µ 0 (µ 0 = 4π⋅10 -7 H/m) the void's magnetic permeability and with l the active conductor's length in the slot: In fig. 5 is presenting a conductor into an elementary slot where were made the notations: h the slot depth, w the slot opening towards the air-gap, I the current that circulates through the conductor from the slot, φ the magnetic flux produced by the current that circulates through the conductor and dy the distance element by axis y. The flux element, where B is the magnetic induction in the slot, dA the area element and H the magnetic field's intensity, has the expression: If is neglected the magnetic voltage drop in the iron core comparatively with w from the slot, from the magnetic circuit's law is deduced: By integration is obtained: Is calculated the magnetic reluctance R m and then the specific permeance Γ: . (8) Based on these premises, considering the configuration of the RHBSM or RHHBSM, presented in fig. 1, where λ i,j is the permeance between the stator tooth i and the rotor pole j depending on angle ϑ between the axis of stator tooth i and the axis of the rotor pole j, the permeance in the air-gap has the expression [3]: If D si is the inner stator diameter and D r0 the outer rotor diameter, the term b(ϑ) has the definition relation [11][12][13][14][15]: The graphic representation of b(ϑ) is given in fig. 6. In fig. 7 is presenting the permeance's variation form in the air-gap, with the previous specifications. .
To define the mathematical model's equations, is started from the magnetic circuit's shape with open slots, between the adjacent pockets of laminates. In fig. 8 is presenting the stator slot in which are placed two sides of the coil passed-through by the currents i u and i l and which have the number of windings w u and w l . The number of stator pockets is noted with k and the number of rotor poles with l.
By means of the equivalent magnetic circuits method [11][12][13][14][15] is deduced the equivalent diagram of the magnetic circuit which is presented in fig. 9 [3]. In fig. 9 were made the following notations: R sσathe reluctance of the slot's superior part, R sσu -the reluctance of the stator slot's median part, R sσl -the reluctance of the stator slot's inferior part, R stu and R stlthe superior and inferior reluctances of the stator teeth, φ -the fluxes through air or core in different areas, V mthe magnetic potential and θ -the magnetomotive force. By composing and equivalence is obtained a simplified equivalent diagram presented in fig. 10. The binding relations between the quantities from fig. 10 and the ones from fig. 9 are: Fig. 9. The equivalent diagram of the transversal magnetic circuit for a slot.   Fig. 11. Equivalent diagram of the longitudinal magnetic circuit of a pocket of laminations.
For the E or the U-shaped magnetic circuit of the stator pockets of laminates ( fig. 1), the equivalent magnetic circuits are presented in fig. 11 a and fig. 11 b, where R y is the yoke's reluctance, R z the column's reluctance, R δ the air-gap's reluctance, R p the reluctance of the pole under the pocket of laminates and θ E the source of magnetomotive force due to excitation. The index i refers to the number of the stator pocket of laminates. Solving of the equivalent magnetic circuits is made by determining the magnetic scalar fluxes and potentials in all nodes [11], for given ampere-turns and known permeances. Based on the duality between the electric field and magnetic field, a magnetic circuit contains elements with a unique value of the magnetization curve and it can be equalized with a drycurrent electric circuit with linear or non-linear sources and resistances. If the non-linear reluctances have their magnetization curve strictly monotonous and increasing, then, for a set of initial conditions there is a unique solution if the hysteresis is neglected.
In fig. 12 is presenting the equivalent diagram of the magnetic circuit considering an ampere-turns of equivalent reaction θ 1 for the armature's winding. The values of permeances, fluxes and ampere-turns from fig.  12 have the expressions (23), (24) and (25) [3], where μ Fe and μ 0 are the magnetic permeabilities of the iron core and air, S Fe and S a the sections, l m /2 the field line's length, λ rFe the rotor iron permeance under a pocket of laminates beneath which has entered the rotor pole to a central angle ϑ i -ϑ (ϑ i is the central angle corresponding to a pocket of laminates beneath which has not entered the rotor pole), λ ra the air's rotor permeance under a pocket of laminates corresponding to the central angle ϑ, λ r the rotor equivalent permeance under a pocket of laminates.

3D FEM analysis
The finite element method is used in order to obtain key parameters of the RHHBSM. Since the topology of the machine has is a purely 3 dimensional flux paths one the complete transient characteristics and parameters can not be obtained without an extensive computation effort. Finite element analysis of the machine was done a commercial software platform. When applied to electrical machines, the described problem is usually reduced to cover only one pole or one pole pair with the help of boundary and symmetry conditions in order to reduce the computation time. The windings are in star connections and thus the excitation current shares among phases are I A = I, I B =I C = -I/2. From the 3D finite element we will consider only some key values or verification values which can not be obtained from other means. Since this topology has a 3D field the model required and 3D magnetostatic model solver. In order to simplify    The solution of the finite element model using a 3D finite element solver took about 6 hours per step. The airgap flux density magnitude (the absolute value of B) is presented in Fig 16. Inductance evaluation from finite element solution shows that the machine is not saturated since the maximum airgap value is below 1T. The static torque produced by constant currents and varying the position is presented in Fig. 11 from this value the maximum torque can be obtained equal to 11 (Nm).

Conclusions
The main advantage of the RHBSM and RHHBSM are their improved capability to operate at variable speed for wind or hydro power plants. The design method was validated also by means of 3D FEM model. General performance characteristics of ampere-turns, inductor flux, resulting flux and torque variation are presented.