50 kW SR machine parameters, adapted from .
An average rated torque estimation for generally saturable switched reluctance (SR) machines based on vector analysis is described. The proposed analytical method enables the switched reluctance machine designers to compute quickly and relatively accurately the rated torque of the machine. This approach offers simplicity, accuracy, and intuitive insight characteristic to analytical solutions of magnetically nonlinear problems otherwise achievable only with time-consuming computer-based numerical simulation tools. The suggested analytical methodology, therefore, offers immediate answers regarding the rated torque performance at the early stages of the machine sizing and design process. In this chapter, the switched reluctance machine rated torque calculation is derived based on the analytically estimated flux-linkage characteristic map and the knowledge of the DC bus voltage of the machine. It is further demonstrated that the proposed analytical rated torque calculations, based on vector analysis, enable construction of highly accurate instantaneous phase current profiles using a graphical method and thus aiding intuition and providing valuable insight into the nonlinear switched reluctance machine operation and control requirements. The proposed method will be found particularly suitable for those studying the nonlinear design and control of switched reluctance machine technologies for electric vehicle traction and industrial applications.
- analytic average torque calculation
- vector analysis
- nonlinear switched reluctance machines
Since the dawn of the popularity of the switched reluctance (SR) motors , many diverse developments of this machine’s technology have resulted, including theoretical analysis, and the volume of the published literature on these topics is enormous. For this reason it would be impossible to encompass the entire range of published methods of analysis and design of SR motors, let alone to effectively utilize them in the machine design process. Therefore the aim of this work is to provide the readers with the essential analytical formulae and methodologies in order to allow them to analyze quickly and relatively accurately a desired SR machine for a particular application.
Since the principle of operation of SR machines does not require any permanent magnets for the production of torque, such machines offer an excellent alternative electric motor technology at this moment in time in the context of the cost and environmental impact uncertainties of permanent magnet materials . Usually, the SR motor will be designed to operate from a fixed DC voltage supply, such as a battery, and the stator windings of the machine will be of the concentrated type. A typical structure of an SR electric motor or generator is shown in Figure 1.
In Figure 1 the depicted machine contains 18 stator poles and 12 rotor poles; therefore this particular SR machine configuration is abbreviated as an 18/12 SR motor. Therefore, with the knowledge of these variables, we are able to compute the number of “steps” the rotor will make in a single revolution as it gets aligned to each of the energized stator poles, as in Eq. (1):
For the SR machine to operate effectively, it requires a capable high power and switching frequency power electronic converter in order to shape the phase current waveforms to be fed into the SR machine phase windings. The most general power electronic converter structure used for SR machines, known as the asymmetric half-bridge converter, is shown diagrammatically in Figure 2.
The power electronic converter for the three-phase 18/12 SR machine is assumed to be connected to a fixed magnitude DC bus voltage
The sequence of operation of the h-bridge converter in terms of phase current profiles as shown in Figure 3—for the motoring mode of the SR machine—is as follows. Assuming the hard chopping voltage switching strategy to begin with, when the rotor pole is furthest away from the stator pole, the two power transistor switches are latched so as to enable the conduction of the current. This action results in the electric current flow from the positive supply terminal, through the phase windings, to the negative terminal, as represented by the red dashed line in Figure 2, with the arrow indicating the direction of the flow. Once the current has reached some predetermined reference value, say peak rated current, the signal from the current sensor
From Figure 3 it may be seen that once the current reaches some specified value, the electronic controller has to check the value of the measured current by the current sensor
It should be noted here that the analytical rated torque calculations presented below are based on the assumption that the correct switching mode of the power electronic converter is assumed first. Furthermore, the actual performance of an SR machine can be greatly affected by the design choice of the power converter topology and the switching action, including the torque production and the energy conversion efficiency of the device . Therefore, the following analytical solutions will be based on the “hard chopping” voltage switching strategy as in Figure 3 for the asymmetric half-bridge converter presented in Figure 2.
The particular SR motor considered in this chapter is taken from . The main reason behind the decision to use a published SR machine design was that it had been extensively optimized, its performance was analyzed, and it was actually built and tested in order to verify the results. Finally, the published work should be of reproducible quality, and therefore the chosen SR machine design makes an excellent choice when comparing the validity of the proposed analytical solution with the solutions obtained from extensive computer simulations and carefully controlled experimental measurements.
The chosen 50 kW SR motor parameters to be used in the present analysis are given in Table 1.
|Outer diameter [mm]||269|
|Stack length [mm]||135|
|Air gap length [mm]||0.5|
|Iron core material||10JNEX900|
|Wire diameter [mm]||0.6|
|Wire turns [turns]||17|
|Wire parallel turns [turns]||22|
|Slot fill factor [%]||57.0|
|Stator pole arc angle ||10.5|
|Rotor pole arc angle ||11|
|Rated angular speed [rpm]||1200|
|Current, peak [A]||320|
|Max. current density [A/mm2]||33 (33)|
|RMS current (@1200 rpm) [Arms]||204 (206)|
|Max. torque (@1200 rpm) [Nm]||415 (400.4)|
|Number of phases||3|
|DC bus voltage [V]||500|
|Max. power [kW]||50|
By application of the open license finite element method  to the magnetic analysis of the chosen SRM2 design, the aligned and the unaligned rotor positions are analyzed in terms of the magnetic flux distributions in the magnetic circuit of the device, as shown in Figures 4 and 5.
Therefore, if we consider the amount of magnetic flux , measured in units of weber, being established in the stator pole around which the number of conductors
where instead of the intuitive units of weber-turns, we choose to use the volt-seconds (as a derived SI unit) since this unit of measurement will be very handy when we explain the significance of the flux-linkage in the SR machine analysis section. Furthermore, the phase flux-linkage computed in Eq. (2) is multiplied by the number of the coils connected in series, , to obtain the total flux-linkage for that particular phase.
Using the same reasoning as above; Eq. (2) will apply for the unaligned rotor position as well as for the aligned case as shown in Figures 4 and 5. In fact Eq. (2) is used to compute the flux-linkage for any rotor angular position with respect to the stationary stator. One further outcome we are able to accomplish—in addition to varying the rotor angular position to compute the flux-linkage for that rotor position—is to vary the phase current that is being circulated in the concentrated winding of the stator pole. Therefore, if we increase the current from zero to its full rated peak value (see Table 1), for each of the rotor positions from the fully aligned to the fully unaligned, in steps of 1 mechanical degree, the complete flux-linkage map of the 18/12 SR machine will result, as plotted in Figure 6.
Some highly notable observations of Figure 6a are that for the aligned rotor position, the flux-linkage builds up very quickly for the range of currents, after which it levels off considerably; this occurs in the region of 80 (amps) called the magnetic saturation point. The unaligned rotor position flux-linkage is fairly linear for the entire range of the phase current values, especially if compared to the fully aligned position. The intermediate rotor position flux-linkage curves, generally speaking, become highly nonlinear as the rotor tends toward the fully aligned position. If the flux-linkage curves with respect to the rotor position are examined taking the phase current as a parameter, as in Figure 6b, it may be seen that linearity is not present at all, and for a given current value, the flux-linkage value will vary with respect to all rotor positions from the fully unaligned to the fully aligned. This is true irrespective of the rotor rotation in clockwise or counterclockwise direction, as in Figure 6b.
The torque production of the SR machine can be described in terms of the flux-linkage map shown in Figure 6a. If we were to consider only the fully aligned and unaligned rotor position flux-linkage curves, as in Figure 7, then it would follow that as the rotor tends toward the fully aligned position, the current build-up in the phase winding would increase the stored magnetic field energy
As can be seen from the graphical representation, the magnetic co-energy
Thus, generally, if the magnetic field co-energy is created as a result of the rotor moving from the unaligned to the aligned position, while the phase current is kept constant, the torque generated,
Therefore, for a given SR machine, we are interested in maximizing the area
The published and generally accepted phase torque calculations were based on the assumption of a linear machine theory, thus assuming non-saturating SR machine flux-linkage curves and their associated unsaturated aligned and unsaturated unaligned inductances, as shown in Figure 8a. The alternative to the linear theory is the nonlinear flux-linkage curves theory , with the additional saturated aligned inductance , in which
which is the same as Eq. (5) being divided by the rotor angle that is being turned.
In general, the non-saturating SR machine is rather a special case of the generally saturating SR machine and for this reason will not deliver as high a torque value as is otherwise potentially possible with the saturating SR machine, since the area
These bounds indicate that a fully saturable SR machine having the inductance ratio of less than 6 is considered to be unsuitable as it will not be a torque dense machine. On the other hand, the inductance ratio of 10 or more in Eq. (8) will indicate exceptionally high torque density SR machine capable of high instantaneous torque production, which is most desirable. It follows that our SRM2 design, from Figure 6, is capable of a reasonable torque production since the inductance ratio is about 8. However, achieving the inductance ratio of more than 10 is not a simple task, calling for highly specialized computer-based multi-objective optimization routines that are computationally intensive , and is therefore beyond the scope of the treatment in this chapter.
Going back to the previously stated argument regarding the assumed
2. General analytic theory for saturable switched reluctance machines
In this chapter we set out the proposed nonlinear theory based on vector analysis of quadrilaterals for the SR machines. By presenting the nonlinear theory, it will be demonstrated how to estimate the average rated torque at the rated angular speed of the SR machine taking into account the magnetic nonlinearity of the magnetic circuit of the machine. The nonlinear theory will also be used to construct the instantaneous current profiles at the rated angular speed of the machine as well as to estimate the average phase current and voltage. The proposed nonlinear theory is also applicable to the generator mode of the SR machine, and the presented examples will demonstrate the results. Finally, the nonlinear analytical theory enables the estimation of the power electronic converter volt-ampere requirements of the switched reluctance machine as well as the energy conversion ratio for the machine. All these estimates will be used to construct the estimated speed-torque envelopes for the analyzed saturable SR machine, thus concluding the preliminary design process of the device using the analytical theory.
2.1 Switched reluctance machine energy conversion estimation and flux-linkage map construction
Since the flux-linkage map of an SR machine is the main piece of information to be used for the torque production estimation, the already constructed complete flux-linkage map for the selected SRM2 machine design is redrawn again, this time for the aligned and the unaligned curves only, as shown in Figure 9.
In Figure 9 the flux-linkage curves are first drawn from the finite element analysis-obtained data points of Figure 6a, assuming that. The three linear curves are obtained by splitting the aligned flux-linkage curve in the region of the saturation point, at approximately 80 A current point in this case; however, the two parts of the split aligned flux-linkage curve are overlapping in order to facilitate more accurate linear fit. The linear fit of the unaligned flux-linkage curve also indicates that the linear fit is a very close approximation, as can be inferred from the statistical goodness of fit measure
Having obtained the satisfactory linearization of the curves, as in Figure 9, we now define the following flux-linkage expressions (units of which can be in
Unsaturated unaligned flux-linkage:
Unsaturated aligned flux-linkage:
Saturated aligned flux-linkage:
where is the rated phase current, as given in Table 1, and is equal to 320 A for the selected SRM2 design.
The saturation current,
Moreover, the aligned flux-linkage intercept,, is found from Eq. (11) by substituting the zero current value:
Our next goal is to express mathematically the area
In Figure 10 the phase current is assumed to rise from zero to the full rated phase current value at
The geometric figure formed by
Eq. (15) is only convenient if the flux-linkage and the current quantities are known numerically at the required points on the quadrilateral. Usually it is preferred to work with the inductances
Eq. (16) is not yet the preferred final formula for the area calculation since we wish to eliminate the numerical flux-linkage values , and the saturation current , and leave only the rated phase current values and inductances. Also we note at this stage of derivation that if the rated current were to reduce to such a point so as to coincide with the saturation current in Eq. (16), the resulting area
Figure 11 shows that, generally, the SR machine phase current commutation starts well before the fully aligned position , at point
since the unaligned unsaturated inductance is not affected by the commutation factor
The units of volt-seconds of the flux-linkage difference
the last term in Eq. (18) indicating that the units of result if the rotor rotates by the angle
Since the equality previously considered in Eq. (10) has changed as a result of the early commutation, it is rewritten here for convenience:
since the two linearized curves intersect at point
To obtain the new saturated phase current value , remembering to include the phase commutation factor
Therefore, having obtained Eq. (19) and (22), we are now in the position to express the co-energy
Eq. (23) is the fundamental expression for the computation of the energy conversion
The questions remain as to how to estimate the commutation factor
Assuming that, first of all, the phase current reaches point
Moreover, since the steady angular speed and the DC bus voltage at the point of the commutation are known, the time required for the current to fall from point
since the full wave voltage is applied as in Figure 3.
Therefore, the angle through which the rotor traverses during this time period can be found, since the rated angular speed is known:
Finally, from the definition of the commutation factor
Also of interest is the
being only an approximate value since the winding resistance voltage is assumed to be unknown at this stage of analysis which would be added to the
Having estimated the commutation factor and the
The total torque at the rated speed of the three-phase,
The analytically calculated rated torque value of the chosen SRM2 SR machine compares favorably with the published measured rated torque value of 400 Nm, being about 8% underestimated. This small discrepancy arises since our calculations in Eq. (29) are influenced by two main factors as a consequence of the simplifications made. First—and this has an overestimating effect—the excursion of the phase current into the decreasing inductance region after the full stator and rotor alignment,
where the stator pole arc angleis in mechanical degrees and and are the number of the stator and the rotor poles, respectively. The answer of Eq. (31) can now be multiplied by the computed torque value of Eq. (30) which makes the discrepancy with respect to the published measured torque of less than about 3% underestimated, which must be considered as relatively very accurate for the analytical method-based design process.
Other factors influencing the accuracy of the analytical torque calculations include the uncertainties in the estimated commutation factor
Finally, the electromechanical power at the rated speed for the SRM2 design can be estimated as
which is an underestimate of about 2% compared to the published power levels at the rated speed . However, the maximum power of the SRM2 design beyond the rated speed was quoted as 54 (kW).
Although Eq. (23) (or Eq. (29)) is rather complex, it represents the most suitable form of Eq. (16), since the numerical evaluation of all flux-linkage values, term by term, at each desired phase current level is no longer required. Even more importantly, Eq. (23) is very convenient to use as the operating parameters that are readily available to machine designers are typically expressed in terms of the inductance estimates, obtained either from finite element analysis or the analytic aligned and unaligned flux-linkage maps. Furthermore, the theory of other electric machines usually relies on inductances, and for this reason, Eq. (23) can be more accessible and intuitive for the control engineers.
The required operating parameters to be used in Eq. (23) for the fundamental energy conversion estimation of the SR machine are collected from Figure 9 and Eqs. (27) and (28) and are summarized in Table 2.
|Saturated aligned inductance, [H]||0.0004948|
|Unsaturated aligned inductance, [H]||0.0071879|
|Unsaturated unaligned inductance, [H]||0.0012072|
|Stator pole arc angle βs [degrees]||10.5|
|Rated angular speed [rad/s]||125.7|
|Rated current, [A]||320|
|DC bus voltage [V]||500|
|Commutation factor ||0.8|
Eq. (29) is further exploited to gain insights into the torque control effectiveness of the saturable SR machines via the rated phase current regulation. The operating parameters from Table 2 were substituted into Eqs. (29) and (30), and the rated phase current was varied from some minimum practical value to the full rated phase current value, as plotted in Figure 13.
Figure 13 reveals critically important results regarding saturable SR machine operation when, at the fixed constant speed, the torque is regulated with the phase current only. If the start of the phase current rise is to be fixed at a certain rotor angle, in Figure 12, and the current decay should start at the commutation angle, , the electromagnetic torque production for the SR machine will be weakly nonlinear and will follow the power law in phase current as in Figure 13. This is significant from the control point of view since—if the linear torque control is to result from the linear increase of the phase current—the current will have to be regulated simultaneously with the turn-on and the commutation angles, as defined in Figure 12. Therefore, for each required torque value at the fixed constant speed, an exhaustive search for the combination of the three control parameters will have to be performed, that is, the turn-on, the commutation, and the phase current magnitude. This type of search is effective only with the help of computer-based design routines; for example, see [8, 10].
Figure 13 also shows the torque obtained from Eq. (5) where the linear non-saturating SR machine is assumed for all rated current values. The linear theory suggests that the torque production follows the quadratic power law with the phase current, as is also evident from Eq. (5) itself. The torque values in the region below the saturation current are similar using both methods; however, once the current is increased beyond the saturation level, the linear theory overestimates the torque value and becomes invalid. To conclude, the general torque of the saturable SR machine can be computed relatively quickly and accurately with Eq. (23), even if the rated phase current value is reduced below the saturation current value.
Further discussion regarding Eq. (23) is as follows. It should be understood that the commutation factor
The first term in Eq. (23), the “voltage” term,
indicates that the energy conversion part of this term is dominated by the rated current and the commutation factor; however, even with zero commutation factor, the machine will produce some useful torque, since the full wave DC bus voltage is still utilized, while with the
The second term in Eq. (23), the “inductance” term,
contributes significantly to the energy conversion process and is dominated by the square of the rated current in the phase winding. Therefore, it is desirable to maximize this value in order to operate at the highest possible torque, yet giving proper consideration to the cooling requirements of the SR machine, as at very high phase current values, the cooling of the windings becomes very challenging. It therefore appears that in order to arrive at a torque dense SR machine design, as the main goal, it is important to drive as large a current through the motor windings as is practicable. Furthermore, the inductance difference should be made as large as possible. This can be achieved in practice if the saturated aligned inductance is minimized with respect to the unsaturated unaligned inductance .
The third term in Eq. (23), the “saturation quality” penalty term,
is a relatively large negative quantity and is dominated by the denominator defined by the inductance difference. Therefore, to reduce this “penalty,” it would be desirable to maximize the unsaturated aligned inductance, while the saturated aligned inductanceshould be kept as low as possible in relation to the unsaturated aligned inductance. This effect can only be achieved with the highest-quality electric steels, having very high saturation flux density values, which must therefore be used to make the core of the SR stator and rotor .
2.2 Saturable switched reluctance machine energy conversion ratio estimation
Since the SR machine requires a DC bus capacitor if the DC supply voltage is to be kept undisturbed by the returning magnetic field energy from the phase windings, as discussed and shown in Figure 2, the estimation of the magnitude of such returning energy is an important consideration when designing the h-bridge power electronic converter for the SR machine. As was shown in Figure 10, which for convenience is replotted here as Figure 14, the magnetic field energy
The amount of energy (in joules) being created as a result of the phase current (and the magnetic field in the windings) build-up is represented by the area bounded by the entire aligned flux-linkage curve, the horizontal line at point
Knowing the magnitude of the stored magnetic field energy
In order to estimate the energy conversion ratio, it is necessary to estimate the stored magnetic field energy
Eq. (37), being somewhat less complex than Eq. (23), can be interpreted as follows: maximizing the rated current and the saturated aligned inductance,, will greatly increase the stored magnetic field energy. Furthermore, minimizing the difference between the inductances in the identified “saturation penalty” term in Eq. (37) will help to minimize the stored magnetic field energy, which is a desirable effect and can be achieved by selecting the electric steel with high saturation flux density values for the construction of the SR machine.
Substituting the known operating parameters of the SRM2 design from Table 2, the estimated field energy is found as
Using Eq. (37) for the operating parameters from Table 2, with the phase current magnitude varied from some lower practical limit to the full rated current, the energy conversion ratios were obtained with Eq. (36) as shown in Figure 15.
From Figure 15 it is seen that the amount of the energy converted by the SRM2 design is rather small at the low phase current values, but it improves quickly as the phase current increases, in effect encompassing more of the co-energy area
2.3 Saturable switched reluctance machine average rated current and voltage estimation
Having successfully obtained the rated torque values for the selected SRM2 machine design using Eq. (29), we are now in a position to extend the analytical treatment for the estimation of the average rated phase current as well as the voltage. These quantities are very important during the preliminary SR machine design process in order to estimate the demand from the available DC bus supply, such as a battery of an electric vehicle. Furthermore, the knowledge of the rated current is required for the estimation of the heat dissipation requirements for the designed SR machine.
Since by using Eqs. (24) to (28) we were able to evaluate the commutation factor
To find the time it takes for the phase current to rise to the rated value, using Eq. (25),
For the rotor to traverse from point
where the saturated aligned flux-linkage at
Next we estimate the time needed for the phase current to decay from point
Finally, the time from
Thus the area bound by the quadrilateral
The triangular area
Now the areas are added and divided by the total rotor traverse time from
However, by revisiting the argument regarding the overlapping phase operation of the selected SRM2 machine at the rated speed, we make use of the overlap ratio
Likewise, all of the information needed to compute the average phase voltage from the known instantaneous voltage waveform of Figure 16 is readily available. The computed value found for the single phase at the rated speed is
Therefore, at the rated speed, the SRM2 machine will be demanding the electric power:
The result in Eq. (51) does not need to include the power factor at the rated speed since the phase currents and voltages are averaged over the full electrical cycle in Eqs. (49) and (50). Finally, knowing the demanded electric power and the generated electromagnetic power, the efficiency of the SR machine at the rated speed can be readily found which for this particular case is:
which compares favorably with the published efficiency value of around 80% at the rated speed and torque .
The above estimated electric power demand compares well with the electromagnetic power production as found in Eq. (32). Therefore, the relatively accurate SR machine design process is accomplished.
2.4 Saturable switched reluctance machine converter volt-ampere requirement estimation
As was shown in Figure 2, the asymmetric h-bridge SR machine power electronic converter requires two active devices (transistor switches) and two passive devices (the power diodes per single phase) in order to realize the most flexible SR machine operation in a motoring mode or a generating mode . Consequently, we can define the metric of the volt-ampere product that these power electronic devices have to be rated at in order to deliver the required phase current at the required DC bus voltage.
For a three-phase,
Eq. (53) indicates that the stated maximum of the current will have to be delivered by the transistor switches having the stated voltage rating. The quantity obtainable with Eq. (53) is in units of kVA, and in this case, it is permissible to form a ratio of two quantities not having the same units  as long as both units are explicitly retained, and thus the converter volt-ampere value in Eq. (53) is divided by the produced electromagnetic power, as computed with Eq. (32), to obtain a measure of economic utilization of the converter in terms of kVA/kW:
where the volt-ampere quantity
Figure 17 shows the h-bridge converter volt-ampere requirement for the rated angular speed of 1200 (rpm) with the phase current as a parameter. The curve in Figure 17 indicates that approximately 25 (kVA/kW) rating is required at the power electronic converter end of the SR machine drive in order to deliver the rated torque, as found in the previous section using Eqs. (29) and (30). As the phase current magnitude is reduced in order to reduce the torque level, while the angular speed is kept constant, the
2.5 Saturable switched reluctance machine generator mode energy conversion estimation
Since SR machines are able to operate in the motoring as well as the generating mode, the energy conversion performance of an SR generator will also be quite accurately predicted with the Varignon parallelogram theorem as was the case for the motoring mode given by Eq. (23).
Consider Figure 10 where for the SR machine-generating mode the locus of the phase current is reversed. Therefore the phase current will form the quadrilateral
Having the co-energy relation for the generator, we then make use of Eq. (19) and Eq. (22) to express the generator co-energy in Eq. (55) in terms of the inductances, rated phase current, and
Eq. (56) is the fundamental expression for the computation of the energy conversion
Having successfully found the rated torque and power production capability of the analyzed SR machine design, we are now in the position to summarize the advantages of the proposed analytic method based on Varignon parallelogram and vector-based computations.
The estimated rated torque and power of a general SR machine are the two most important quantities a machine designer is seeking at early stages of the machine design process. Once these two estimates are known for the required rated speed, the speed-torque and power-torque characteristics for the SR machine can be constructed as in Figure 18.
The particular characteristics in Figure 18 are assumed to obey the constant-torque and constant-power operation of the SR machine with respect to the speed of rotation. Such specific characteristics are most desirable from the accurate torque and power control point of view and are used for electric vehicle propulsion applications or the general industrial motor drive applications.
The convenient analytical form of Eq. (23) enables the SR machine design engineer to estimate relatively accurately the electromagnetic torque of the machine at the assumed required rated speed, at point