3.1. Upper Side
The above conditions (a)–(g) lead to the definition of five independent variables related to the pole shoes’ coordinates and to the PM sizes. This condition would require too wide an exploration, especially if more performance parameters are to be evaluated with and without excited coils. Therefore, the design of the AHMB upper side has been developed in two steps. The former determines the PM geometry to achieve a predefined air-gap flux density, which fulfills the force target and avoids excessive magnetic saturation in the core yokes. The latter performs a parametric analysis to determine the more convenient widths of the core poles to optimize some performances related to the axial forces, as well as to the winding inductances.
The preliminary FEAs vary the PM sizes in the ranges {2 mm, 10 mm} and {150 mm, 190 mm} to achieve the suspension force target at the rated air-gap and operating temperature . The coil section is fixed considering limits related to the maximum current density , winding inductance increase (more relevant with higher height-to-width ratio), and overall geometric dimensions. The other variables are simply adjusted to fulfill the condition (f).
Table 2 gives the main data of the most favorable configuration for the flux density distribution.
Figure 5 shows the flux density in the cores and in the air-gap, respectively.
Figure 5a shows the nearly uniform distribution in the cores with values low enough to avoid high saturation when UCs are excited. Even the air-gap flux density in front of the pole shoes meets the target value (
Figure 5b). It is worth noting that a remarkable contribution to the magnetic pressure is also present in front of the slots, which can hardly be evaluated by analytical formulations.
The selection of the current polarity in the two coils is carried out by calculating the force-current characteristic at an augmented air-gap length
. This approach aims to evaluate the current density
when the winding exerts the maximum upward force. In this case, with
as the coil net section, the resultant current density must fulfill the maximum value
.
Figure 6 shows the four possible cases. The flux direction highlighted in the figure denotes that concordant current polarities tend to strengthen the PM flux on the outer poles (A) or on the inner ones (B); differently, discordant current polarities (C, D) generate a mixed condition of the preceding ones.
Figure 7 shows the resulting axial force
as a function of the current density. The supply strategy A is clearly the most efficient, as it allows for the target force achievement at the lowest current density (
, corresponding to a total ampere-turns
460 A for each coil).
Furthermore, the force profile is almost linear, which can be useful for the design of the control system. The strategies C and D behave alike, with unacceptable current densities (≅15 A/mm2). It is worth noting that strategy B provides a demagnetizing effect with a minimum for 8 A/mm2.
Though B is clearly not convenient for the augmented air-gap, the switching from A to B by the coil current inversion could be profitable to weaken attractive force if the flywheel should approach the AHMB upper side
. In order to verify such a possibility, the force values are recalculated using the strategy B at the minimum upper air-gap value
(conf. B1). The results (strategy B1 in
Figure 6) evidence that the AHMB would develop values too high to balance the suspended mass whatever the current density, therefore making inapplicable such a supply strategy.
As for the design of the magnetic core, the pole shoe geometries remarkably affect both the air-gap flux density profile and the value of the coil inductance that in turn determines the voltage rating and the promptness of the supply circuit. Larger poles provide a lower magnet volume and reduced magnetic saturation, but they generally increase the leakage coil flux. Therefore, a more extensive parametric analysis is needed to explore different combinations of the pole geometric variables, keeping unvaried PM sizes and coil sections.
With reference to the geometric parametrization shown in
Figure 4a, the radial coordinates of the pole shoes are expressed by the per unit (p.u.) variables
. First, the radial coordinates of the endpoints are defined, by parametrizing
as a function of
:
Then, the following conditions are derived by imposing that the annular areas are equal and that no overlapping occurs between the middle pole shoes:
with
and
. By the elaboration of (4), the quantity
is therefore obtained as:
The remaining parameters to define the pole shoe geometry are finally calculated as:
The quantities considered for the performance evaluation are:
the suspension force at the air-gap due to the only PM;
the suspension forces and with augmented air-gap due to the PM only and in the presence of current excitation (current density ), respectively;
the coil per-turn inductance in the same condition as is calculated; because of the series connection and the chosen current polarity, it is , with being the inner and outer coil inductances, respectively, and being the mutual inductance coefficient.
For the purpose of obtaining a homogenous comparison, the following p.u. quantities are defined for each
-th configuration (
):
The chosen formulation aims at finding the optimal configuration as the one minimizing all the quantities in (7). For instance, the parameter is related to the force with coil excitation, which must be as high as possible for a given current.
Figure 8a compares the quantities (7) for
126 configurations defined in the ranges
and
, considering as reference configuration the one obtained from the preliminary analysis. It is worth remarking the very limited improvement in the p.u. forces, while the p.u. inductance has a large variation. As no configuration approaches simultaneously the minimum value for all the quantities, a performance index
is defined to achieve a global ranking by its minimization. This index is expressed as a weighted function, which integrates the performances described by (7):
The coefficients ( 1, 2, 3) give the desired importance to each performance, fulfilling the condition 1. The subscript denotes the values for the reference configuration, in order to have 1 as the reference one.
The function (8) also includes a penalty for the configurations which do not fulfill the static force reference; in this case, the unit value is increased by a contribution depending on the force deficit weighted by the coefficient .
The index
is evaluated for the same
126 configurations assuming the coefficient set
, which is considered a good trade-off among the different performances.
Figure 8b highlights a limited reduction, as expected, of the index
because of the very small variations of the p.u. forces. The optimal configuration (#72) improves
only by about 2% with respect to the reference one (#60), however, confirming the effectiveness of the preliminary design. As shown in
Table 3, the most evident improvement concerns the reduction of
by
%, obtained mainly by increasing the slot opening of the inner core. It is worth remarking that for
,
and
; the difference between the inner and outer coil parameters would make the force control troublesome if a parallel connection were used.
Figure 9 shows the resulting PM force with no excitation as a function of the air-gap length
. The nonlinear dependence on
with a steeper increase as
can be noticed. This behavior will result in higher loading for the LC since it must generate a balancing force
with the air-gap variation
being the same. However, this is not an issue because the AHMB lower side allows for an easier coil placement as no PM is present.
3.2. Lower side
The electromagnetic configuration of the AHMB lower side must be sized to provide a suitable downward attraction force on the flywheel when a disturbance tends to lower the upper air-gap below the rated value
. As observed from
Figure 9, the LC loading condition is higher than the UC one for the same air-gap variation
At the PM temperature
80 °C and
, the rated force for the AHMB lower part is
−580 N, with
as the PM force at
; for the AHM upper part, at
, it is
330 N.
The design procedure has been developed investigating the sensitivity of some relevant geometric parameters on the axial force and the coil inductance, as well as on the flux density distribution in the core legs. Indeed, it is worth remarking that heavy saturation needs to drain more current from the electrical supply for a given winding arrangement, therefore affecting the supply ratings and the winding thermal stress. Therefore, both the coil shape and the core widths are examined as well. In order to restrict the number of variables, the design assumptions a–d are considered, including two further conditions:
constant coil section with variable aspect ratio ; it follows that and ;
variable pole shoe height + with fixed value for .
The variable definition develops like the upper side procedure. With reference to the quantities in
Figure 4c, first, the radial sizes are calculated:
where
is a p.u. variable describing the variation of the inner pole shoe width. By elaborating the expression of the leg cross sections as:
the core leg widths can be obtained by imposing the equality of the annular areas:
Therefore, by combining (9)–(11), it can be derived that the parameter set enablesone to define uniquely the geometric configuration.
A preliminary analysis proves that the target force is achieved with the set and current density 6 A/mm2 (total ampere-turns 720 A). Moreover, the force characteristic is approximately linear, and the coil inductance is almost constant for up to 8 A/mm2 because of the limited magnetic saturation. Then, the net axial force and the per-turn coil inductance as functions of and are examined, imposing that 750 A and keeping unvaried the remaining parameters.
Figure 10a shows the appreciable decrease of
as
increases, becoming nearly constant for
4. On the contrary, the force varies very little, even if a tendency to decrease appears for
2.
Figure 10b highlights the importance of the parameter
. Its value must be high enough to balance the PM attraction force (
0.8), however, leading to a noticeable
increase.
As a final investigation, the parameters
and
are jointly varied to examine possible trade-offs between force and inductance increase, as well to guarantee uniform flux density distribution inside the core legs. For such a purpose, the standard deviations
and
are calculated from the axial flux density sampled on the lines
and
, drawn in
Figure 4c. The parameters vary within the ranges of 140 mm
180 mm and 5 mm
12 mm, mainly established to deal with geometrical limitations.
Table 4 reports the quantities kept constant during the parametric sweep. For the sake of comparison, besides
, the p.u. quantities
(defined as in (7), replacing
with
),
, and
are evaluated. The latter two quantities are expressed as ratios of
and
with respect to their minimum value among all the examined configurations.
Figure 11 shows the progress of the selected quantities. As
increases up to a certain threshold (≈156 mm, configuration #24), there is a general improvement, achieving
0 as well as low and similar flux density distributions for the two core legs. The value
0 means that the balancing condition is achieved with a lower current than in the initial configuration. From a general overview, configuration #28 represents an acceptable performance trade-off. Its design parameters are summarized in
Table 4.