To design an electric machine, different fundamental equations can be used to estimate the size and the geometry of the machine. Therefore, input parameters such as power, speed and voltage have to be provided and assumptions regarding certain machine parameters and boundary conditions have to be specified to start the design process.
Each machine is designed for one specific operating point in the torque speed diagram, i.e., the rated operation, as can be seen in
Figure 6. In this case, this point is set to be the point of rated power
and rated speed
at which the machine’s supply voltage
reaches the maximum output voltage of the inverter
and the maximum continuous torque
is provided. The machine’s induced voltage
which almost matches the supply voltage, is proportional to the magnetic flux
, the speed
, the number of pole pairs
and the number of series turns per phase
. The torque is proportional to the torque-forming part
of the current
. Using the dq-plane, the current
can be displayed as the q-axis current
and the magnetizing d-axis current
[
50]. Since the induced voltage reaches the voltage limit
, for speeds above the rated speed, the current
is increased to provide field weakening and decrease the magnetic flux in the machine. Dividing the current
by the number of parallel branches per phase
, the number of conductors per slot
and the conductors cross sectional area of copper
, the current density
in the slot can be calculated [
18]. Depending on the cooling system, the current density
at which the machine can be operated continuously and the maximum current density
is given. The machine geometry is therefore mainly determined by the magnetic flux
, the current density
, and the maximum voltage
.
The design process can be seen in
Figure 7. First, the main dimensions of the machine such as bore diameter and length are specified. Then the electric quantities and the main parameters of the machine are calculated and determined. From thereon, the geometry of the lamination and the magnets is calculated, based on the magnetic circuit in the stator and the rotor lamination. The machine design is completed, specifying the winding quantities and the slot geometry. The design process of the machine is followed by a general design of the housing. Finally, the volume and mass of the housing and the machine are calculated and the volumetric and gravimetric power density are determined.
3.1.1. Main Dimensions
The first step of the design process is a first assessment of the main dimensions of the machine. The bore diameter and the length determine the provided torque of the machine. The torque
is calculated using the spatial fundamental of the electric loading
and the spatial fundamental of the magnetic flux density of the magnet in the rotor
[
18]. Estimating these two quantities,
and
, the bore volume
can be calculated, if the rated power
and speed
are given. By setting the ratio of length and bore diameter to
, the length and the diameter are determined. This ratio is set accordingly to other traction drives. The electric loading
and magnetic flux density
are chosen according to the maximum speed, (cf.
Table 4). For higher speeds, the electric loading
is chosen smaller. The calculated lengths and diameters of the four machine designs are shown in
Table 5.
The parameters need to be validated concerning their maximum surface velocity
. The PMSM-B1 and PMSM-B2 machine designs have interior magnets. Their surface velocity should not exceed
[
6]. The surface mounted magnets of PMSM-S1 and PMSM-S2 are kept by a bandage. The bandage leads to a bigger magnetic air gap, but allows to increase the maximum surface velocity to
[
5]. The air gap for the designs with interior magnets is assumed to be
. For the PMSM-S1 surface mounted rotor, the bandage height is set to 2 mm and, for the PMSM-S2, the bandage height is set to 4 mm.
The surface velocity of the machine designs
is calculated for an overspeed of 10% of the maximum speed. The results for these machines are listed in
Table 5 and do not exceed the limits of the surface velocity. The PMSM-B1 and PMSMS-S1 machine designs show low mechanical utilization compared to the limits. By increasing the bore radius and though the surface velocity, the machine would become larger and heavier.
3.1.2. Electric Quantities
To distinguish the rated current
of the machine at rated power
, the apparent power
is calculated. The rated phase current
is then determined using the maximal phase voltage
provided by the inverter.
is the number of phases. To determine the current and the apparent power the efficiency
and the power factor
at rated operation must be assumed reasonably. In this case
and
are set to
and
. With the maximum phase voltage of
, the rated current of
is calculated. The maximum current deliverable by the inverter is
.
The maximum operating frequency
is determined based on the number of pole pairs
. The number of pole pairs is kept small to keep the hysteresis and eddy current losses low. The number of stator slots
is chosen to feature a high fundamental winding factor and a good compromise between reasonable slot dimensions and a low harmonic leakage factor. The resulting number of slots per pole and phase is accordingly set to
. The pole pitch
and the stator slot pitch
are then defined. The results are shown in
Table 6.
3.1.3. Magnetic Quantities and Geometry of the Magnetic Circuit
The geometry of the stator and the rotor lamination, as well as the magnet geometry, are initially based on the estimation of the magnetic flux density in the air gap
at rated speed and rated power. This flux density is defined by the coupling between the stator and rotor flux and is accordingly higher than the flux density just due to the magnet, that was defined in
Section 3.1.1. This effect is higher for the machines with interior magnets since their air gap is much smaller than for the machines with surface mounted magnets. The assumptions of the flux density for the four designs can be seen in
Table 7. The magnetic flux per pole
is calculated based on this assumption. By knowing the magnetic flux per pole, the tooth width and the height of the stator and the rotor yoke can be calculated by specifying the permissible magnetic flux density in the teeth
, in the stator yoke
and in the rotor yoke
for rated operation. These values are set different for the four machine designs. The machine designs with the lower operating frequency
can be operated with higher flux densities. This is due to eddy-current losses in the iron that depend on the flux density and the frequency. For higher speed, the permissible magnetic flux density should be set lower to limit the eddy-current losses. The chosen permissible magnetic flux is shown in
Table 7. To determine the yoke height
the magnetic flux must be divided by the length of the machine
the permitted magnetic flux density and a factor of 2, since the magnetic flux splits up in the two directions of the yoke [
18]. The index i stands for either stator
or rotor
.
The tooth width is calculated by multiplying the slot pitch with the ratio of the magnetic flux density in the air gap and the allowed magnetic flux density in the teeth as
The results for the four machine designs are shown in
Table 7.
The size of the magnet, i.e., the width
and the height
are set according to the flux density in the air gap and the
-
diagram of the magnet (cf.
Figure 8) [
50]. The flux density of the magnet in rated operation is determined from the estimated flux density in the air gap by
To take into account the drop of magnetic voltage in the iron, in comparison to the drop of voltage over the air gap, a saturation factor of
is defined and set to
. To take into account the slotting of the stator, the carter factor
is used.
is set to 1.1 [
18]. The magnetic motive force of the magnet
is calculated regarding these phenomena. The magnetic field strength of the magnet
is determined from the
-
diagram of the magnet, as depicted in
Figure 9. The height of the magnet is then defined by the magnetic motive force
and the determined magnetic field strength
Since the machine shall be short-circuit-resistant, the height of the magnet needs to be validated after finalizing the winding configuration.
For a PMSM with interior magnets the width of the magnet
is set to 85% of the side length of the equilateral triangle of one rotor pole pitch. For a PMSM with surface mounted magnets the width of the magnet is defined by the pole pitch
The bore diameter defines the magnetic flux in the machine and must be chosen accordingly.
Knowing the size of the magnet and the rotor yoke height, the rotor geometry can be completed. For a surface magnet rotor, the inner radius of the rotor is determined by
If the magnets are interior in a bar shape, the depth is considered by multiplying the magnet height by a factor
to determine the inner radius:
The factor is set to
for the machines with interior magnets. The magnet parameters are shown in
Table 8.
3.1.4. Winding Quantities
The maximum number of series turns per phase is determined for the operating point at rated speed
and rated power
. The magnetic flux per pole
at this point has already been estimated with the magnetic flux density in the air gap (see Equation (9)). Assuming the winding factor
the maximum number of turns per phase
can be calculated [
18]. The number of turns can be achieved with different winding configurations. The number of conductors per slot
, the number of slots per pole and phase
and the number of parallel branches
determine the winding configuration and the winding layout. The number of series turns per phase is set by these winding parameters to
and should be close to the value determined. The chosen values for the winding quantities are listed in
Table 9. This winding configuration also determines the previously estimated spatial fundamental of the electric loading, calculated by
After setting the winding parameters, the geometry of the slot can be determined. The cross-sectional area of the conductors
is calculated from the rated current
and the current density
. The machines are supposed to have a water jacket. The current density is accordingly assumed at
.
Depending on the type of winding and the manufacturing process, the copper fill factor
must be estimated. In this case for an automated round wire winding, it is set to
[
50]. The area of the slot is then determined by
In the case of a round wire winding, the slot can be designed as a trapezium. The width of the slot at the bore radius is set by the width of the tooth and the bore radius to
The width of the slot at the bottom of the slot
and the corresponding slot height
are calculated with the relation
and the width of the trapezium at the bottom of the slot with
Using the height of the slot, the outer radius of the stator is calculated as
Finally, the design needs to be validated to determine if it is short-circuit-resistant. This is done by calculating the magnetic motive force
in case of a sudden short circuit from rated operation occurs [
51]. The calculation is based on the maximum short-circuit current. The winding attempts to keep the magnetic flux constant while the rotor keeps rotating. The short-circuit current is assumed to be 4 times the current at rated power. In the worst case, the magnetic flux is directed opposite to the magnetization direction of the magnet. The magnetic field strength of the winding, in case of the sudden short circuit, leads to a decrease of the magnetic polarization
of the magnet (cf.
Figure 9). If the magnetic motive force of the winding is higher than the product of coercive field strength
and the height of the magnet
the magnet will be demagnetized, leading to the requirement
The resulting heights of the magnets are listed in
Table 10 for the four machine designs. All machine designs are short circuit resistant.
The active parts, i.e., rotor and stator lamination, winding and magnets, of the machine designs are shown in
Figure 10. It can be noticed, that the machine size decreases with increased maximum speed, as expected.