2. IDENTIDAD DEL ProgrAMA ACADÉMICo
2.5. Análisis comparativo entre programas de la misma naturaleza
A magnet (or any element thereof) can be represented by an equivalent distribution of linear surface current density K = M × n, together with a volume distribution of current density J = curl M, where M is the magnetization and n is the normal to the magnet surface. For isotropic magnets J = 0 and K = Hca, the apparent coercivity Br/µrecµ0. In the following, Hca is written Hc. The total coercive MMF of
the magnet is Fc = HcLm, where Lm is the magnet length in the direction of magnetization.
At first sight it appears that the stored magnetic field energy can be obtained by calculating the integral
in all regions of the motor, and adding the resulting integrals together. In the air regions, B = µ0H. In
the iron regions, B and H are related by the B/H characteristic of the iron. But within the volume occupied by the magnet,
where Br is the remanence and µrec is the relative recoil permeability. It is not obvious that we can substitute this equation in eqn. (2.130) and get the correct value for the energy in the volume occupied by the magnet. An energy analysis is required, and this must use the principle of virtual work, because a change in stored energy can be brought about only by an exchange of mechanical work when the motor is on open-circuit. Consider a small displacement )2 such that the operating point of the magnet moves from X to Y, as a result of the change in the permeance of the magnetic circuit as the magnet moves past the slot openings. The magnet flux changes by )Mm, and if the change takes place in a time )t, an induced voltage e = )Mm/)t
appears at the terminals of the equivalent coil in Fig. 2.68, if we assume that the coil has one turn. The power at these terminals is just p = e i and the energy exchange during the time interval )t is e i)t = i )Mm.
But i = Fc, the coercive MMF of the magnet. Therefore
the energy exchange with the fictitious excitation coil is Fc)Mm. Because the coil and current source are
fictitious, there is no external electrical exchange of
energy via i and e. Nevertheless, the fictitious coil and current source must be capable of “sourcing” or “sinking” energy. The flux change )Mm causes a change )Wmag in the stored field energy in the space occupied by the magnet, and a further change )Wgap in the remainder of the magnetic circuit, (i.e., the
airgap and the iron). Let Wmag = ½FmagMmag and Wgap = ½FgapMgap, where Fmag and Fgap are the
27See Deodhar RP, Staton DA, Miller TJE and Jahns TM : Prediction of cogging torque using the flux-mmf diagram technique, IEEE Transactions on Industry Applications, Vol. IAS-32, No. 3, May/June 1996, pp. 569!576.
)Wmag ' mmm Vm 1 2 mHdB dVm (2.137) Fig. 2.70 Tcog ' 1 2Fc dMm d2 (2.136) )Wmag ' 1
2( Fmag@ )Mmag % )Fmag@ Mmag)
)Wgap ' 1
2( Fgap@ )Mgap % )Fgap@ Mgap)
(2.132) )Wmag % )Wgap ' 1 2(Fmag % Fgap))Mm % 1 2()Fmag % )Fgap)Mm (2.133) )Wmag % )Wgap ' 1 2Fc)Mm (2.134) )Wm ' )Wmag % )Wgap ! Fc)Mm ' ! 1 2Fc)Mm (2.135) Then
The sum of these is
But )Fmag + )Fgap = )Fc = 0, since Fc is by definition constant and Fmag + Fgap = Fc. Therefore
We can now complete the energy audit using the law of energy conservation. If the movement XY is from a position of higher permeance to one of lower permeance, then mechanical work is supplied
to the motor via the shaft. The stored magnetic energies increase while the flux Mm decreases. Therefore
In order to move the rotor and cause the flux change )Mm, mechanical work )Wm must have been exerted via a torque at the shaft. This torque is the cogging torque Tcog, and if we regard torque exerted
by the motor as positive, then Tcog )2 = ! )Wm and in the limit as )2 tends towards zero,
The shaded triangle )Wm in Fig. 2.69 is equal to
½Fc)Mm, and therefore represents the work of cogging
torque during the rotation from X to Y. Fig. 2.70 shows that the area )Wm is also equal to the triangle OXY.27
The change of energy )Wmag in the volume Vm
occupied by the magnet is suggested by classical theory as
This is represented by the shaded area in Fig. 2.71. The difference between )Wm and )Wmag is the energy
exchanged with the air + iron regions of the motor, denoted as )Wgap. It is important to recognize that )Wm
= ½Fc)Mm includes all the mechanical work, even
though the expression ½Fc)Mm contains only magnet parameters. Indeed B and H do not need to be known in the airgap and iron regions in order to calculate )Wm
(and from it the cogging torque). All the necessary
information about the mechanical energy exchange is included in the expression ½Fc)Mm. This expression also applies to magnetically linear variable-reluctance devices that are supplied from a constant-current source. The factor ½ reflects a partition of energy between mechanical energy conversion and field storage. Evidently permanent magnets have additional storage capability beyond the integral in eqn. (2.137). It is associated with the internal current source in Fig. 2.68 and it is, of course, what distinguishes them as “permanent”.
Fig. 2.71 Fig. 2.72
The equivalent-coil model helps remove doubt about the absolute value of the stored energy in the magnet. Consider a change in the external circuit permeance that takes the operating point up to the point R, where B = Br, the remanence, and H = 0 in the magnet. The mechanical energy exchange along XR is shown as Wmag in Fig. 2.72. For the purposes of calculating cogging torque, with the motor windings open-circuited, the magnet energy at R can be taken to be zero, because any displacement from R requires mechanical work to be provided. Of course, in a motor with a finite airgap, the magnet never operates at R, but approaches closest to it when no external torque is applied.
It follows from Fig. 2.70 that the total energy exchange along XR is OXR = Wm, and therefore the energy exchange with the air and iron regions is Wm ! Wmag , which is shown as Wgap in Fig. 2.72. Fig. 2.72 also
shows the separate components of the coercive MMF Fc : that is, Fgap across the air + iron part of the magnetic circuit and Fmag across the magnet itself. When the magnet is working near the remanence
point R, the total stored energy is small and the magnet energy is smaller than the airgap component. If the rotor is displaced so that the magnet operating point moves close to the coercive point C, the total stored energy is large: an external torque must have been applied to the shaft to make this change, and the energy stored in the magnet is larger than that stored in the gap. The gap energy is low because the flux-density is reduced to a low level, as a result of the reduction in the circuit permeance. The potential energy given to the magnet will be recovered if the shaft is allowed to turn back to its original position. During continuous rotation the cogging torque is generated as the operating point cycles between points such as X and Y. Point X is a point of maximum permeance and point Y is a point of minimum permeance. When the rotation is from X to Y, the stored energy is increasing and the cogging torque is a retarding torque in the direction of rotation. At Y the rotor continues in the same physical direction but the operating point moves towards X, which corresponds to a new maximum permeance position. The angle of rotation between two X’s depends on the number of slots and poles; in an integral- slot motor, it is equal to the slot-pitch.
Fig. 2.70 also shows that the area )Wm used to evaluate the cogging torque can be taken as an
increment in the total stored field energy, or as an equal and opposite increment !)Wm in the
coenergy. The total coenergy is represented by triangle OCX and its components WNgap and WNmag are shown in Fig. 2.72. Since there is no external electrical energy exchange, it does not matter which one is used, and the cogging torque is the total derivative dWm/d2. The PM motor on open-circuit differs
from the electrically excited actuator or motor, where it is necessary to use partial derivatives MWN(i,2)/M2 or !MWf(M,2)/M2 and keep to the rule of “constant MMF” or “constant flux” respectively. In the PM motor on open-circuit there is no distinction between these partial derivatives, except as to their sign.
Fig. 2.73
Even when the magnetic circuit is linear, the coenergy and energy are in general unequal, i.e. OXR is not generally equal to OCX. This is a further difference between the PM motor on open-circuit and the normal electrically-excited device.
The sign of the cogging torque is not particularly important, because it is purely oscillatory when the rotor rotates continuously, and since it cannot sustain any continuous energy conversion, its average value must be zero. Suffice it to say that the cogging torque will be such that mechanical work must be supplied to the motor when the magnetic stored energy is increasing, and mechanical work will be supplied by the rotor when the magnetic stored energy is decreasing.
Finite-element procedure
In a finite-element analysis we must rotate the rotor in small steps (e.g. 1E) and at each position calculate the change in the total stored energy. By evaluating the triangle OCX for each magnet
element, we are automatically including the coenergy stored in the “airgap” (i.e., in the rest of the magnet circuit), and therefore it is not necessary to evaluate Wgap separately by integrating BH/2 in
the air and iron regions. The same is true if we use triangle OXR to evaluate the total stored field energy due to the current magnet element. The required process is as follows: for each magnet element, per metre of stack length,
Coenergy Triangle OCX ½Hc CB × element area
Stored field energy Triangle OXR ½BrCH × element area
TABLE 2.7
The coenergy and energy calculations are shown in Fig. 2.73. Since Br and Hc are constants, the areas require the integration of a linear function over each element. The integral on the right of Fig. 2.73 is the coenergy evaluation that would be done in air regions using ½B H = ½B2/µ0 = ½µ0H2, and it is
quadratic. According to the theory, this is not needed for calculating the cogging torque.
We can now use one of the magnetic energy evaluations in Figs. 2.73. Once the element coenergy integral of Fig. 2.73 or Table 2.7 has been summed over the entire magnet region, it represents the coenergy of the entire machine in the same way that OCX represents this coenergy for a lumped magnet in Fig. 2.70. When the rotor moves from position X to position Y, the change in this coenergy sum will be of the form OCX ! OCY, and this will give the average torque through the interval )2 = XY. On the other hand, if we use the alternative magnetic energy integral in Fig. 2.73, the total energy change from X to Y will be of the form OYR ! OXR. Both methods should produce the same result.
Because of “discretization noise”, the total energy sums should be extracted from the FE analysis and fitted with a cubic spline; then the differentiation to get the cogging torque will be smoother. This method helps avoid the discretization noise which troubles the Maxwell stress method.
Fig. 2.76 Variation of airgap flux-density
Fig. 2.74 Flux calculation
Fig. 2.75 Deodhar’s method: A simple method for evaluating the
triangles OXY treats the entire magnet as a single element. The magnet flux Mm is evaluated the from the difference of vector potential at the magnet edges (multiplied by the stack length), and used together with the magnet MMF obtained from the relation F = Lm/µrecµ0 × (Mm/Am ! Br) to plot the triangle
OXY directly, moving from position to position and calculating the torque at each step. The vector potentials can be evaluated at points shown in Fig. 2.74 on the edges of the magnets, using the expression Mm = (A1 ! A2) × Lstk, where Lstk is the stack length or magnet length in the axial direction. If the centreline of the magnet is a line of symmetry, generally A = 0 there and
A2 = ! A1, so that Mm = 2A1Lstk. An alternative is shown in Fig.
2.75, where Mm is approximately equal to 2(AQ ! AZ) × Lstk. In many cases AZ will be zero.
This method could be made more efficient by recognizing that OXY = ½Fc)Mm where )Mm = MY!MX. Alternatively, OXY = ½Mr(FY ! FX) where FY = HY × Lm and FX = HX × Lm, and HX and
HY are the values of H in the magnet at the positions X and Y respectively, evaluated by H = (B ! Br)/µrecµ0. Together with
the integrals in Table 2.7, evidently there are several simple processes for determining the cogging torque once the FE solution is available.
PC-BDC’s approximate method: The rotor is stepped round in intervals of 1Eelec. At each position the airgap flux-density distribution is calculated including the effect of slotting. In the interests of rapid calculation the slot modulation is very crude. It is estimated using an effective airgap that varies as shown in Fig. 2.76 as the point P sweeps across the slot opening. Mm is evaluated at each rotor position by means of a ratio function Lm/(Lm + gN), where gN is the effective airgap at each position of the point
P. Then the total airgap flux is evaluated by integrating the airgap flux over the pole pitch, and the
rotor is stepped to the next position.
Fractional-slot motors can be accommodated by evaluating the cogging torque for the N and S pole magnets independently, and then adding them together. Both methods allow for skew by summing the effects taken at several axial positions.
Fig. 2.77 Calculation of flux-linkage from vector potential
T/I ' 3 2 2 Br1LstkB1Mdkw1Tph 2 Nm/A (2.139) N ' m A@dl (2.138) T/i ' 4r1BMgLstkTph Nm/A (2.140)