**3. SRM under analysis: starting values for the design process**

In this chapter, a SRM is analyzed from a preliminary point of view. For the sake of comparison, this SRM has been designed to replace the original PMSM of a Nissan Leaf, whose main specifications are listed in **Table 1**. Therefore, a SRM has to be pre-designed with the aim of achieving similar performance to that of the PMSM of the Nissan Leaf. This SRM must have the same torque-speed capabilities than the original motor. Besides, both machines should have the same outer diameter, while keeping current density at similar values, in an attempt to keep the comparison as fair as possible. As SRMs have significant lower power density than PMSMs, this means that the proposed SRM will be longer and heavier than the original motor.


**Table 1.** Nissan Leaf motor specifications.

overlapping, increasing average torque and reducing torque ripple. For the same reason, this circuit is very convenient for achieving fault-tolerant SRDs. In terms of cost, the main drawback of this topology is the high number of switching devices in comparison with other alternatives.

The above topology requires two more diodes per phase compared with the most conventional topology used for PMSMs and IMs [30]. As aforementioned, some alternative topologies have been proposed in order to reduce the number of devices, including one that uses the SRM phases as inductive filters to charge the EV battery [31], which adds an additional value to this type of machine over IMs and PMSMs. However, these alternatives reduce the controllability and efficiency of the drive and are generally avoided in applications with demanding torque

SRMs require very precise position determination, since the activation of the phases must be synchronized with rotor position to achieve good performance. Conventionally, SRDs have a measuring device (either an encoder or a resolver) to determine the speed and position of the rotor. These sensors are accurate and relatively inexpensive and have very good perform‐ ance. However, it would be desirable to remove them (without deteriorating performance considerably) in order to make the drive cheaper, less voluminous, less noise-sensitive and more reliable [32, 33]. This can be done by means of a "position sensorless control", also known as "position self-sensing control" given that the external sensor is replaced by the electrical

As the cost and the volume of an external position sensor is not significant when compared to the whole traction system, the main advantage of position self-sensing in SRMs for EVs is the possibility to continue operation in case the external sensor is out of service. This redundancy increases the fault tolerance of the whole drive, which is critical in traction applications and

During the last decades, many self-sensing techniques have been proposed for rotating electrical machines in general [32, 33] and SRMs in particular [35, 36]. Commonly, these methods perform well either at high speed (such as BEF-based, flux linkage-based or induc‐ tance-based techniques) or at low speed/standstill (such as injection-based or di/dt-based techniques). This implies that at least two of these methods are usually combined in the same

**Figure 1.** Asymmetric bridge converter topology for 8/6-pole switched reluctance machine.

machine itself, which acts as its own position sensor [34].

drive, the transition between both being particularly delicate.

one of the main approaches of this chapter.

ripple requirements.

100 Modeling and Simulation for Electric Vehicle Applications

The first decision to make when designing an electrical motor is obviously the machine type and topology (number of phases and number of poles). In this case, a conventional radial-flux inner-rotor non-skewed SRM is selected for consistency with the original PMSM. Regarding the topology, an 8/6-pole configuration is chosen as a compromise between torque density + cost (the less number of phases/poles, the better) and torque ripple + degraded capability (the more phases/poles, the better) [2]. The minimum number of phases that can ensure that the machine will be able to start with m-1 phases available (degraded mode) from any rotor position is four (although a especial starting routine will sometimes be needed, which could also work for 3-phase SRMs; see Section 5). Hence, an 8/6-pole topology makes for a good candidate for fault tolerant EVs.


**Table 2.** Input and output parameters for the analytical pre-design model.

**Figure 2.** (a) Cross section of a 8/6-pole SRM with curved coils. (b) 80-kW SRM considered in this work (after FEM predesign). Rotor position in both figures is 0° (phase AA′ aligned position).

Once the type of machine is fixed, the design process starts with analytical calculations based on some of the methodologies proposed in the literature, such as [35, 37, 38]. The main dimensions of a given SRM can be easily computed using a simple analytical model, which starts from some initial specifications and several simplifications. Obviously, the results need to be checked with more refined tools such as finite element method (FEM) codes. Input parameters to perform the pre-dimensioning and output parameters to be provided by the model are given in **Table 2** and **Figure 2**.

the topology, an 8/6-pole configuration is chosen as a compromise between torque density + cost (the less number of phases/poles, the better) and torque ripple + degraded capability (the more phases/poles, the better) [2]. The minimum number of phases that can ensure that the machine will be able to start with m-1 phases available (degraded mode) from any rotor position is four (although a especial starting routine will sometimes be needed, which could also work for 3-phase SRMs; see Section 5). Hence, an 8/6-pole topology makes for a good

**Magnitude Symbol Magnitude Symbol**  Rated power *Pr* Number of turns per coil *N* Rated speed *ω<sup>r</sup>* Phase current during activation; see **Figure 11a** *I AVG* Number of stator phases *m* Rotor outer radius *RR* Number of stator pairs per pole *Nsp* Active length *L*

Stator outer radius *RS* Stator pole width *WS* =2 · *a*

**Figure 2.** (a) Cross section of a 8/6-pole SRM with curved coils. (b) 80-kW SRM considered in this work (after FEM pre-

Once the type of machine is fixed, the design process starts with analytical calculations based on some of the methodologies proposed in the literature, such as [35, 37, 38]. The main dimensions of a given SRM can be easily computed using a simple analytical model, which

Air gap *g* Stator pole height *HS*

candidate for fault tolerant EVs.

102 Modeling and Simulation for Electric Vehicle Applications

Operational current density *I<sup>ρ</sup>*

**Input parameters Output parameters** 

**Table 2.** Input and output parameters for the analytical pre-design model.

design). Rotor position in both figures is 0° (phase AA′ aligned position).

Regarding the assumptions and simplifications in which the model is based, they can be listed as:

**•** The machine is fully saturated and works at a magnetic field B in the air gap, which leads to use a simple equation for the produced torque T, in the form:

$$T\_{em} = \frac{P\_r}{\alpha\_r} = 2 \cdot K\_T \cdot N \cdot N\_{sp} \cdot I\_{AVG} \cdot B \cdot R\_R \cdot L \tag{1}$$

where *KT* is a torque constant related to the machine saturation level, which varies from 0.5 in non-saturated machines up to 0.8 in heavily saturated ones. Additionally, the voltage *Vk* applied to each phase is:

$$V\_k = 2 \cdot N \cdot N\_{sp} \cdot oo \cdot B \cdot R\_R \cdot L \tag{2}$$

**•** Coils are curved for a better use of the slot between poles. They have the shape shown in **Figure 2**. Under this approach, it must be satisfied that:

$$N \cdot I\_{AVG} = I\_{\rho} \cdot a \cdot H\_S \tag{3}$$

**•** The return yoke width *YS* must be the same as half the pole width in order to work at the same magnetic flux density at the return yoke, without increasing the saturation level. Alternatively, a certain correction empirical factor *Kf* can be introduced to achieve more realistic results. This simplification, together with some geometrical considerations, leads to the following expressions:

$$Y\_S = K\_f \cdot a \tag{4}$$

$$a = \frac{2 \cdot \pi \cdot R\_R}{8 \cdot N\_{sp} \cdot m} \tag{5}$$

$$R\_S = R\_R + H\_S + Y\_S \tag{6}$$

**•** Both coil ends can be considered as a complete circular ring with inner radius *a* and outer radius 2*a*. This assumption allows an easy estimate of the overall length and weight of the coils.

Combining expressions (4)–(6), (3) and (1), one can derive an equation with two unknowns, the rotor radius *RR* and the active length *L* :

$$\begin{aligned} \left(R\_S \cdot R\_R\right)^2 - R\_R \cdot \left(\mathbf{l} + \frac{\boldsymbol{\pi} \cdot \mathbf{K}\_f}{4 \cdot N\_{sp} \cdot \boldsymbol{m}}\right) - \frac{T\_{em} \cdot 2 \cdot \boldsymbol{m}}{\boldsymbol{\pi} \cdot \mathbf{K}\_T \cdot \boldsymbol{B} \cdot \boldsymbol{L} \cdot \boldsymbol{I}\_\rho} &= \\ \begin{array}{l} \text{forling } \boldsymbol{L} \\ \mathbf{0} \rightarrow \begin{array}{l} \text{forking } \boldsymbol{L} \\ \end{array} - \boldsymbol{A} \cdot \boldsymbol{R}\_R \overset{\text{\tiny{3}}}{\text{s}} + \boldsymbol{B} \cdot \boldsymbol{R}\_R \overset{\text{2}}{\text{ }} - \boldsymbol{C} &= \mathbf{0} \end{array} \end{aligned} \tag{7}$$

The procedure to dimension the machine consists in fixing the length *L* , and then finding the zeros of the polynomial function given by the above equation to calculate *RR*. If the value chosen for *L* is too low, there will be no real solutions for the rotor radius *RR*, meaning that for a given stator radius *RS* there is no machine able to provide the required torque. Beyond that limit, there will be two main options: one with a small rotor radius and a large coil with many ampere-turns and another with a large rotor radius and a smaller coil. The optimum solution in terms of the shortest machine will be the one for which both rotor radiuses are the same. Alternatively, the criteria for finding the optimum machine could be to select the lightest one. In that case, the weight of the coil ends should be taken into account.


As an example, a pre-dimensioning of a SRM with the input specifications listed in **Table 3** has been performed.

**Table 3.** 8/6-pole SRM pre-design specifications.

**Figure 3** shows the results obtained using the previous model, particularly the calculation of the optimum active length with its corresponding overall weight, for different stator outer radiuses. It can be seen that, for this simple approach, there is an optimum stator radius which minimizes the overall weight. Nevertheless, further and deeper analysis should be performed, since this solution can be far from being the optimum in terms of efficiency, for instance. In this regard, the evaluation of the coil resistance and inductance, end-winding included, is essential to estimate the behavior of the machine, including its overall efficiency.

**Figure 3.** Calculation of active length and overall weight for the proposed SRM.

Combining expressions (4)–(6), (3) and (1), one can derive an equation with two unknowns,

<sup>2</sup> <sup>1</sup>

The procedure to dimension the machine consists in fixing the length *L* , and then finding the zeros of the polynomial function given by the above equation to calculate *RR*. If the value chosen for *L* is too low, there will be no real solutions for the rotor radius *RR*, meaning that for a given stator radius *RS* there is no machine able to provide the required torque. Beyond that limit, there will be two main options: one with a small rotor radius and a large coil with many ampere-turns and another with a large rotor radius and a smaller coil. The optimum solution in terms of the shortest machine will be the one for which both rotor radiuses are the same. Alternatively, the criteria for finding the optimum machine could be to select the lightest

As an example, a pre-dimensioning of a SRM with the input specifications listed in **Table 3**

**Figure 3** shows the results obtained using the previous model, particularly the calculation of the optimum active length with its corresponding overall weight, for different stator outer radiuses. It can be seen that, for this simple approach, there is an optimum stator radius which minimizes the overall weight. Nevertheless, further and deeper analysis should be performed, since this solution can be far from being the optimum in terms of efficiency, for instance. In this regard, the evaluation of the coil resistance and inductance, end-winding included, is

essential to estimate the behavior of the machine, including its overall efficiency.

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p

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*sp T*

r

(7)

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*R R*

one. In that case, the weight of the coil ends should be taken into account.

**Magnitude Value (units) Comments**  Type of machine 8/6-pole SRM *m* = 4, *Nsp* = 1

Power (rated) 80 kW *Tem* = 254 Nm

Speed (rated) 3000 rpm Current density 7 A/mm2 Air gap magnetic field 1.8 T Pole to return Yoke factor 1

**Table 3.** 8/6-pole SRM pre-design specifications.

*AR BR C*

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the rotor radius *RR* and the active length *L* :

104 Modeling and Simulation for Electric Vehicle Applications

*fixing L*

has been performed.

2 3


**Table 4.** Comparison between two possible analytical solutions for the proposed SRM and the tuned solution after the FEM simulations described in Section 4.

**Table 4** provides the calculated output parameters for the minimum weight machine, along with those computed for a 250-mm outer stator diameter, which is the one designed to replace the original PMSM in the Nissan Leaf. **Table 4** also includes the parameters corresponding to the final pre-design, obtained after the design tuning achieved by FEM simulations and timedomain simulations such as those described in next section. This last design is the one that will be analyzed in detail for the rest of the chapter.
