**4. Pre-design analysis and validation**

This section is devoted to the analysis and validation of the pre-design described in Section 3. This analysis was carried out only from an electrical engineering point of view, correspond‐ ing to an early-stage in the design process: electromagnetics, power electronics, modulation technique, control strategy and power losses. Therefore, other crucial aspects such as mechanical behavior, cooling, EMC, certification testing, series production, quality assur‐ ance, useful life and recyclability are not considered.

#### **4.1. FEM analysis**

The first step consists in analyzing and optimizing the preliminary design by means of FEM simulations. This task is usually divided into two different parts:


For the purposes of this chapter, focus is placed on torque, both in normal and "m-1 phases" mode: average torque, torque ripple and minimum torque to start the vehicle (especially significant in degraded mode, when one of the phases is out of service). For such an analysis, 2D simulations are normally sufficient, especially for machines with a large ratio between magnetic length and rotor outer diameter (L/DR > 1), which is the case analyzed here.

One of the first results that should be obtained by FEM simulations is the instantaneous torque as a function of both rotor position and current. It is sufficient to calculate it for half an electric period, since this function is periodical and asymmetrical with respect to the rotor angle. This period is calculated as follows:

$$
\Delta\theta = \frac{360^{\circ}}{N\_R} \tag{8}
$$

which yields 60° for an 8/6-pole SRM. Results for the 80-kW 8/6-pole SRM are depicted in **Figure 4**; rated current is 360 A. Notice how raising the current increases the saturation level of the machine, which in turn "deforms" the torque pulse. This deformation implies higher torque ripple under constant current profiles, as described later.

**4. Pre-design analysis and validation**

106 Modeling and Simulation for Electric Vehicle Applications

ance, useful life and recyclability are not considered.

simulations. This task is usually divided into two different parts:

number of design variables) to evolutionary algorithms.

**4.1. FEM analysis**

period is calculated as follows:

This section is devoted to the analysis and validation of the pre-design described in Section 3. This analysis was carried out only from an electrical engineering point of view, correspond‐ ing to an early-stage in the design process: electromagnetics, power electronics, modulation technique, control strategy and power losses. Therefore, other crucial aspects such as mechanical behavior, cooling, EMC, certification testing, series production, quality assur‐

The first step consists in analyzing and optimizing the preliminary design by means of FEM

**1.** Verification of the analytical pre-design: The aim is to validate the topology, the main stator/rotor dimensions and the winding properties (e.g., number of turns) of the analytical pre-design. This means checking whether the machine provides rated torque at low speed with the specified current and current density, calculating the back-EMF (electromotive force) and the maximum speed for a given DC voltage, and other verifi‐

**2.** Mild optimization of the geometrical design: FEM simulations are extremely useful to tune certain geometric parameters, such as stator and rotor pole widths and heights. This can be carried out in a very simple but suboptimal manner, or by considering an optimi‐ zation problem which can be solved by conventional methods: from brute force (small

For the purposes of this chapter, focus is placed on torque, both in normal and "m-1 phases" mode: average torque, torque ripple and minimum torque to start the vehicle (especially significant in degraded mode, when one of the phases is out of service). For such an analysis, 2D simulations are normally sufficient, especially for machines with a large ratio between

One of the first results that should be obtained by FEM simulations is the instantaneous torque as a function of both rotor position and current. It is sufficient to calculate it for half an electric period, since this function is periodical and asymmetrical with respect to the rotor angle. This

> 360 *NR*

which yields 60° for an 8/6-pole SRM. Results for the 80-kW 8/6-pole SRM are depicted in **Figure 4**; rated current is 360 A. Notice how raising the current increases the saturation level

° D = (8)

q

magnetic length and rotor outer diameter (L/DR > 1), which is the case analyzed here.

cations such as air gap flux density, magnetic saturation or inductances.

**Figure 4.** (a) Instantaneous torque and (b) flux linkage during an electric period for different current values.

It is also very interesting to calculate the flux linkage of each phase as a function of rotor position and current, as shown in **Figure 4**. This information is extremely useful to build a simulation model of the whole traction drive, as described in Section 4.3. Notice that most SRM models found in the literature neglect cross-coupling between phases, which implies that each phase is supposed to be independent of the rest. This assumption is not valid under operating conditions in which two phases conduct simultaneously. This is especially relevant when torque is generated by two phases (such as phase overlapping in torque-sharing functions to reduce torque ripple).

**Figure 5.** (a) Ideal current pulses and (b) corresponding torque profile at 100% load.

Next, torque curves from **Figure 4** can be post-processed to estimate the instantaneous torque profile in steady state for different current levels. Usually, this is carried out under the assumption of ideal current pulses, such as those depicted in **Figure 5**. Of course, this ideal supply is far from being real, but it gives a first estimation without using a full model of the drive, such as the one described later in the chapter. In this ideal case, the activation and deactivation angles *θ*ON and *θ*OFF are given by the intersection of the torque curves corre‐ sponding to different phases, so that there is no overlapping (the previous phase is turned off when the next one is turned on). The torque profile corresponding to these ideal current profiles is shown in **Figure 5** for rated current. Notice how a constant current profile without overlap‐ ping cannot generate a constant torque profile in non-skewed SRMs, even at low current levels (low saturation levels).

Finally, FEM results can also be used to assess the degraded capability of an SRM. **Figure 6** shows the current profile under ideal current pulses when one of the phases is out of service. Compared to normal operation (**Figure 5**), m-1 mode presents lower torque and extremely high-torque ripple under ideal current pulses supply. It is worth mentioning that switching angles should be specifically optimized for degraded operation, so that the control strategy can adapt to the new topology when a phase losses its function. Therefore, the average torque and torque ripple values included in **Figure 6** are not final, as presented later in this chapter.

**Figure 6.** Torque profile in degraded mode of operation (m-1 phases).


**Table 5.** Power electronics losses in steady state at rated and maximum speed, both for normal and degraded modes.

Similarly, it is convenient that the torque dip generated by the missing phase is above 0 at every angle to ensure that the vehicle can start regardless of rotor position (this is not possible with 6/4-pole and 8/6-pole SRMs). Otherwise, special starting routines must be implement‐ ed in the control system, as proposed in **Table 5**. There are many factors that influence the capability of a given SRM to start an EV besides rotor position: inertia, load torque (includ‐ ing the grade of the road α), overloading capability of the traction drive and the control strategy, which can be adapted to the particular conditions in which the vehicle must be started. Further analysis regarding the self-starting capability under degraded operation is presented in section 5.

#### **4.2. Power electronics pre-design**

sponding to different phases, so that there is no overlapping (the previous phase is turned off when the next one is turned on). The torque profile corresponding to these ideal current profiles is shown in **Figure 5** for rated current. Notice how a constant current profile without overlap‐ ping cannot generate a constant torque profile in non-skewed SRMs, even at low current levels

Finally, FEM results can also be used to assess the degraded capability of an SRM. **Figure 6** shows the current profile under ideal current pulses when one of the phases is out of service. Compared to normal operation (**Figure 5**), m-1 mode presents lower torque and extremely high-torque ripple under ideal current pulses supply. It is worth mentioning that switching angles should be specifically optimized for degraded operation, so that the control strategy can adapt to the new topology when a phase losses its function. Therefore, the average torque and torque ripple values included in **Figure 6** are not final, as presented later in this chapter.

**Speed (rpm) Mode Conduction losses (W) Switching losses (W) Total losses (W) Efficiency (%)** 

**Table 5.** Power electronics losses in steady state at rated and maximum speed, both for normal and degraded modes.

Similarly, it is convenient that the torque dip generated by the missing phase is above 0 at every angle to ensure that the vehicle can start regardless of rotor position (this is not possible with 6/4-pole and 8/6-pole SRMs). Otherwise, special starting routines must be implement‐ ed in the control system, as proposed in **Table 5**. There are many factors that influence the capability of a given SRM to start an EV besides rotor position: inertia, load torque (includ‐ ing the grade of the road α), overloading capability of the traction drive and the control strategy, which can be adapted to the particular conditions in which the vehicle must be

 Normal 2254 1161 3415 96.8 Normal 2102 305 2407 97 m-1 phases 1256 771 2050 97.43 m-1 phases 912 368 1280 98.4

(low saturation levels).

108 Modeling and Simulation for Electric Vehicle Applications

**Figure 6.** Torque profile in degraded mode of operation (m-1 phases).

Power electronics to drive the 8/6-pole SRM comprises an asymmetric power converter connecting to the common DC link for all electric phases and a DC filter to improve voltage ripple. The electrical machine is current-controlled by means of a hysteresis band strategy. Other usual alternatives are closed-loop pulse-width-modulation (PWM) [39] and direct torque control (DTC) [40].

**Figure 7.** SRM power converter topology and switched selection for rated and maximum speed operation.

Motor mode operation requires the use of +*UDC* and 0 V to keep the current within the reference band, using −*UDC* to switch the current off rapidly when the phase is deactivated, as **Figure 7** illustrates. Currentis individually controlledineachmachinephase withanasymmetric bridge topology, allowingtotalindependencebetweenphases.DuringEVbraking,themachineworks as a generator, requiring +*UDC* to increase rapidly the current and then it is kept within the referencebandusing0Vand −*UDC*.Athigher speeds,themachine isdesignedtoachieve singlepulse operation, where just +*UDC* and −*UDC* are sufficient to control the current whilst the commutation frequency is reduced [41].

Since the power level and the DC-link voltage are usually defined, simulation-based analysis can be used to determine the voltage and current levels of the power converter devices. The semiconductor technology is selected to fulfill the following electrical specifications: DC-link voltage, current carrying capacity, maximum switching frequency, load cycle and isolation requirements.

#### **4.3. Drive modeling and analysis**

When analyzing electrical drives, simulation models in the time-domain comprising the electrical machine, the power electronics converter, the control strategy, the switching technique and the equivalent load are extremely useful. When the machine model is built with data extracted from FEM simulations (employing tools such as look-up tables or LUTs), these kind of simulations allow for fast and accurate calculations. In the case of SRDs, such a model is an excellent way to calculate average torque and torque ripple under real current pulses and also to optimize switching angles under different operating conditions and optimization criteria. **Figure 8** shows a schematic of the time-domain simulation model used in this work.

**Figure 8.** SRM-based traction drive model.

SRDs and EVs modeling is outside the scope of this chapter, and therefore, the reader is referred to publications that deal with these aspects, such as [1, 42–45]. For convenience, the most relevant equations are included next, starting with the voltage equation for each machine phase:

$$
\mu\_k = R\_S \cdot i\_k + \frac{d\mathcal{A}\_k}{dt}; \quad k = 1 \ldots m \tag{9}
$$

The flux linkage *λ<sup>x</sup>* of each phase depends on the current of that phase *i <sup>x</sup>*, of the current of the previous/next phase *i <sup>y</sup>* (when simultaneous conduction takes place) and rotor position *θ*. Neglecting cross-coupling yields:

$$
\lambda\_k = f\left(\mathbf{i}\_k, \mathbf{i}\_{k \pm 1}, \boldsymbol{\theta}\right) \approx f\left(\mathbf{i}\_k, \boldsymbol{\theta}\right) \tag{10}
$$

the relationship between flux linkage, current and rotor position being the one given by **Figure 4**, usually implemented via a LUT. Similarly, the torque provided by each phase is given by **Figure 4** and a second LUT, the total torque being the sum of each phase torque (again, crosscoupling is neglected):

$$T\_{em} = T\_k + T\_{k+1}; \qquad T\_k \approx \mathbf{g}\left(i\_k, \boldsymbol{\theta}\right) \quad \text{and} \quad T\_{k+l} \approx \mathbf{g}\left(i\_{k+l}, \boldsymbol{\theta}\right) \tag{11}$$

Notice that the above expression considers up to two phases because it will be the maximum number of phases conducting simultaneously. Finally, the mechanical equation of the drive is as follows:

also to optimize switching angles under different operating conditions and optimization criteria. **Figure 8** shows a schematic of the time-domain simulation model used in this work.

SRDs and EVs modeling is outside the scope of this chapter, and therefore, the reader is referred to publications that deal with these aspects, such as [1, 42–45]. For convenience, the most relevant equations are included next, starting with the voltage equation for each machine

; 1 *<sup>k</sup>*

qq

= × + =¼ (9)

*k kk k* = *fi i fi* ( , <sup>±</sup>1, ) , » ( ) (10)

 q*<sup>k</sup>*+ +*<sup>k</sup>* (11)

*<sup>y</sup>* (when simultaneous conduction takes place) and rotor position *θ*.

*<sup>x</sup>*, of the current of the

*<sup>d</sup> u Ri k m dt* l

<sup>1</sup> ( ) 1 1 ; , and ( , ) *T T T T g i T gi em k k* =+ » » <sup>+</sup> *k k* q

the relationship between flux linkage, current and rotor position being the one given by **Figure 4**, usually implemented via a LUT. Similarly, the torque provided by each phase is given by **Figure 4** and a second LUT, the total torque being the sum of each phase torque (again, cross-

*k Sk*

l

The flux linkage *λ<sup>x</sup>* of each phase depends on the current of that phase *i*

**Figure 8.** SRM-based traction drive model.

110 Modeling and Simulation for Electric Vehicle Applications

phase:

previous/next phase *i*

coupling is neglected):

Neglecting cross-coupling yields:

$$T\_{em} - T\_{load} = J \cdot \frac{d\,\alpha\_{mec}}{dt} + B \cdot \alpha\_{mec} \tag{12}$$

The vehicle model represents both the total inertia and the load torque as seen by the motor:

$$J \approx J\_{rotor} + \frac{J\_{wheherls}}{i\_{GEAR}} + \frac{M \cdot ERR}{i\_{GEAR}}^2 \tag{13}$$

$$T\_{load} = \frac{ERR}{i\_{GEAR} \cdot \mu\_{GEAR}} \cdot F\_T \text{ (transition mode)} \tag{14}$$

where *M* is the total mass of the vehicle [kg], *ERR* [m] is the effective rolling radius, *i GEAR* [−] is the transmission gear ratio and *μGEAR* [−] is the transmission energy efficiency (assumed to be constant and equal to 0.95).

The vehicle motion resistances include rolling resistance, aerodynamic drag and gradient resistance; the last one being of active nature (may take negative values and contribute to the movement of the vehicle):

$$F\_T = F\_{aero} + F\_{roll} + F\_{grav} \rightarrow \begin{cases} F\_{aero} = \bigvee\_2 \cdot \rho \cdot A\_F \cdot C\_D \cdot \upsilon^2 \ge 0 \\ F\_{roll} = \mu\_{rod} \cdot M \cdot \mathbf{g} \cdot \cos(\alpha) > 0 \\ F\_{grav} = M \cdot \mathbf{g} \cdot \sin(\alpha) \end{cases} \tag{15}$$

Regarding the control strategy and the switching technique, this work considers hysteresis current control with optimized switching angles as a function of speed and the desired torque, as illustrated in **Figure 9**.

**Figure 9.** Torque control scheme with switching angles controller.

A simulation model such as the one described above is useful to optimize switching angles considering different optimization criteria. In this particular case, a compromise between average torque and torque ripple was chosen, which yields the torque-speed characteristic depicted in **Figure 10** (angle resolution of ±0.25°).

**Figure 10.** (a) Torque-speed characteristic of the 8/6-pole SRM analyzed in this chapter. (b) Corresponding switching angles as a function of speed (all phases are equal but phase-shifted).

Traditionally, one of the biggest disadvantages of SRMs is torque ripple and its consequences (vibrations, efficiency and acoustic noise) with respect to other electrical machines. However, there has been a lot of research on this topic in the last two decades. As a result, torque ripple has improved considerably and it is no longer an issue in many applications that use SRMs. Basically, there are two main approaches to reduce torque ripple (which can be complemen‐ tary): machine design and machine control. The former include techniques such as skewing (see Section 2 for references). These modifications reduce overall performance (torque density and efficiency) and could increase the cost of the machine.

The second alternative consists in improving the control strategy. These methods can be classified into two categories, depending on whether the converter topology is conventional or advanced [44]. Examples of control strategies belonging to the first category are current profiling or current shaping (CP), torque-sharing function (TSF) and direct torque control (DTC). The first two basically consist in adapting the shape of the real current pulses so that torque dips/peaks are minimized [35]. Of course, this adaptation will be feasible depending on the available voltage, the back-EMF of the machine, and the inductance. In other words, the feasibility will decrease with the speed of the machine. This is convenient, given that torque ripple is more detrimental at low speed, both mechanically and from the point of view of comfort. Advanced topologies use a second DC source to increase the available voltage under certain operating conditions, thus allowing for higher *di*/*dt* [44].

**Figure 9.** Torque control scheme with switching angles controller.

112 Modeling and Simulation for Electric Vehicle Applications

depicted in **Figure 10** (angle resolution of ±0.25°).

angles as a function of speed (all phases are equal but phase-shifted).

and efficiency) and could increase the cost of the machine.

A simulation model such as the one described above is useful to optimize switching angles considering different optimization criteria. In this particular case, a compromise between average torque and torque ripple was chosen, which yields the torque-speed characteristic

**Figure 10.** (a) Torque-speed characteristic of the 8/6-pole SRM analyzed in this chapter. (b) Corresponding switching

Traditionally, one of the biggest disadvantages of SRMs is torque ripple and its consequences (vibrations, efficiency and acoustic noise) with respect to other electrical machines. However, there has been a lot of research on this topic in the last two decades. As a result, torque ripple has improved considerably and it is no longer an issue in many applications that use SRMs. Basically, there are two main approaches to reduce torque ripple (which can be complemen‐ tary): machine design and machine control. The former include techniques such as skewing (see Section 2 for references). These modifications reduce overall performance (torque density

**Figure 11.** (a) Current pulse considered in this work (no current profiling) and (b) corresponding torque profile, both at 3000 rpm and 100% load.

**Figure 12.** (a) Current in single pulse mode and (b) corresponding torque profile, both at 6000 rpm and 100% load.

For the sake of simplicity, and unless otherwise stated, all the results presented are given for a current control scheme which operates with unshaped current pulses such as that depicted in **Figure 11**, defined by *θ*ON, *θ*OFF, *I*AVG and *I*RIP. Consequently, torque ripple values given in this chapter are noticeably high and should not be considered as reference values for EV applications.

Finally, **Figure 12** contains the same information as **Figure 11** but for a high-speed operating point. As speed increases, the time available for each phase activation reduces and at the same time the phase back-EMF increases, which makes the machine work in single pulse mode (also known as advance angle control or AAC by some authors, although this name could be confusing because turn-on angles are advanced in a wide range of speed, and not only under single pulse operation).

#### **4.4. Power losses analysis**

Power losses in SRDs present some differences when compared to those based in IMs or PMSMs. When a machine works with sinusoidal current supply, simplified loss models based on single-frequency models of the machine are usually sufficient to estimate resistive losses in the electrical circuit and iron losses in the magnetic circuit. However, SRMs have pulsed supply, which invalidates the single-frequency approach.

Resistive losses (also known as Joule or copper losses) depend on three main factors: the current *i*, the DC resistance of the winding *RDC* (defined by its cross-sectional area, its total length and its conductivity) and the skin effect and proximity effect factor *kAC* (which gives the AC resistance of the winding and depends on its geometrical properties and the frequency of the current) [37]. Therefore, the instantaneous power dissipated in each coil is:

$$P\_{Cu,1}(t) = f\left(i, R\_{DC}, k\_{AC}\right) = k\_{AC} \cdot R\_{DC} \cdot i(t)^2 = R\_{AC} \cdot i(t)^2\tag{16}$$

In steady state, average power losses may be estimated analytically using the RMS value of the current and the total number of phases *m*:

$$
\overline{P\_{Cu}} = m \cdot R\_{AC} \cdot I\_{RMS} \stackrel{2}{\approx} m \cdot R\_{AC} \cdot \left(\frac{I\_{AVG}}{\sqrt{m}}\right)^2 = R\_{AC} \cdot I\_{AVG} \tag{17}
$$

In practice, resistive losses will be higher to those estimated by the above equation, since it assumes ideal current pulses and therefore neglects current tails and phase overlapping. A simulation model such as that described in Section 4.3 can prove accurate to estimate resistive losses with real current profiles, provided that the skin and proximity effects factor *kAC* has been properly considered. Notice that such factor depends on the switching frequency, and hence an iterative process may be required. For the SRM analyzed in this work, *kAC* ≈3.

Both skin and proximity effects are caused by eddy currents in the windings of the machine. High-power low-voltage SRMs such as those used for EV applications are usually strongly affected by these phenomena due to the combination of low number of turns, large crosssectional area in the conductors and currents with high-frequency components in the frequen‐ cy domain. As the wire dimensions increase, eddy currents gain importance and neglecting them can lead to significant overestimation of machine performance [46]. As it is well-known, methods such as conductor division and transposition, employing parallel paths or using Roebel bars or Litz wire, help reduce *kAC*.

this chapter are noticeably high and should not be considered as reference values for EV

Finally, **Figure 12** contains the same information as **Figure 11** but for a high-speed operating point. As speed increases, the time available for each phase activation reduces and at the same time the phase back-EMF increases, which makes the machine work in single pulse mode (also known as advance angle control or AAC by some authors, although this name could be confusing because turn-on angles are advanced in a wide range of speed, and not only under

Power losses in SRDs present some differences when compared to those based in IMs or PMSMs. When a machine works with sinusoidal current supply, simplified loss models based on single-frequency models of the machine are usually sufficient to estimate resistive losses in the electrical circuit and iron losses in the magnetic circuit. However, SRMs have pulsed

Resistive losses (also known as Joule or copper losses) depend on three main factors: the current *i*, the DC resistance of the winding *RDC* (defined by its cross-sectional area, its total length and its conductivity) and the skin effect and proximity effect factor *kAC* (which gives the AC resistance of the winding and depends on its geometrical properties and the frequency of the

( ) 2 2

In steady state, average power losses may be estimated analytically using the RMS value of

2 2 æ ö =× × »× × = × ç ÷

In practice, resistive losses will be higher to those estimated by the above equation, since it assumes ideal current pulses and therefore neglects current tails and phase overlapping. A simulation model such as that described in Section 4.3 can prove accurate to estimate resistive losses with real current profiles, provided that the skin and proximity effects factor *kAC* has been properly considered. Notice that such factor depends on the switching frequency, and hence an iterative process may be required. For the SRM analyzed in this work, *kAC* ≈3.

Both skin and proximity effects are caused by eddy currents in the windings of the machine. High-power low-voltage SRMs such as those used for EV applications are usually strongly affected by these phenomena due to the combination of low number of turns, large crosssectional area in the conductors and currents with high-frequency components in the frequen‐

*Cu AC RMS AC AC AVG*

,1() , , = =×× = × () () *P t f iR k k R it R it Cu DC AC AC DC AC* (16)

2

*<sup>I</sup> P mR I mR R I <sup>m</sup>* (17)

è ø *AVG*

current) [37]. Therefore, the instantaneous power dissipated in each coil is:

applications.

single pulse operation).

**4.4. Power losses analysis**

114 Modeling and Simulation for Electric Vehicle Applications

supply, which invalidates the single-frequency approach.

the current and the total number of phases *m*:

Iron losses are also very relevant in high-speed rotating machines. They depend on the magnetic flux density amplitude and frequency, and thus on the machine current and speed:

$$P\_{Fe} = f\_1(B, f) = f\_2\left(I\_{AVG}, n\right) \tag{18}$$

Again, conventional iron loss estimation techniques based on single-frequency models are not suitable for SRMs, whose current is pulsated and not sinusoidal [47]. In general, SRMs have higher frequencies and higher harmonic content than other machines at comparable speed and power, but less iron volume in high-speed applications [48]. Another distinctive aspect of SRMs is that some iron regions work with bidirectional magnetic flux, while in others the flux is unidirectional. In this sense, the number of flux reversals in the rotor can be minimized using negative current pulses. Hence, designing a SRD with bidirectional current supply can greatly improve rotor iron losses, at the expense of doubling the number of power switches.

Mechanical losses comprise bearing losses, windage losses and air-cooling losses when forced convection is used. Bearing and air-cooling losses in SRMs are comparable to other machines types and they are modeled in the same way. Windage losses, however, are usually higher in SRMs due to their doubly salient nature, and more specifically, to the aerodynamic behavior of the salient pole rotor. This is usually an issue in high-tangential speed applications, in which vacuum chambers or vacuum sleeves are sometimes used for this reason. Windage losses calculations that assume cylindrical rotor shape are not valid for conventional SRMs and hence specific expressions must be used:

$$P\_{wind} = A \cdot \alpha^B; B \in \left[\text{2,3}\right] \tag{19}$$

where *A* and *B* ∈ 2, 3 are coefficients which depend on the rotor geometry, and in the case of *A* also on the operating conditions (pressure, temperature, etc.) [49].

Power electronics losses are also very relevant in electrical drives. They are usually divided into static and switching losses. The former refer to steady state (on- and off-states) while the latter refers to transients between states (switching). Conduction losses are of the first kind, as they correspond to the nonzero on-state voltage of the device *VON* (collector-emitter voltage in an IGBT (Isolated Gate Bipolar Transistor), emitter-collector voltage in a diode) while it carries a current *ION* [50]. The other static losses are those corresponding to the nonzero offstate current of a power semiconductor device when it blocks a voltage *VOFF* . This reverse current *IOFF* is a leakage current which in turn depends on *VOFF* and which is usually very low for the power levels considered in this work. Therefore, blocking losses are conventionally neglected in IGBTs except for high voltages (above 1000 V) and/or high temperatures (above 150°C) [50, 51].

$$P\_{COND} = P\_{ON} + P\_{OFF} \approx P\_{ON} = f(V\_{ON}, I\_{ON}, Temp.) \tag{20}$$

Complementary, the energy dissipated during each switching is called either turn-on energy *EON* or turn-off energy *EOFF* . Neglecting parasitic effects, switching losses are given by the following expression [52]:

$$P\_{SW} = P\_{SW,ON} + P\_{SW,OFF} = \left(E\_{ON} + E\_{OFF}\right) \cdot f\_{SW} \tag{21}$$

*f SW* being the switching frequency.Obviously, both *EON* and *EOFF* highlydependonthe current and on the voltage, since they are defined by a power peak that in turn is caused by current and voltage transients. They are also influenced by temperature. The switching frequency meas‐ ures the number of turn-ons and turn-offs per time unit of a power semiconductor. It de‐ pends on the operating point (torque and speed) and on the electrical characteristic of the machine. In a 8/6-pole SRM, each phase is enabled six times per rotor revolution, so that knowing speed in the machine (e.g., 3000 rpm) and the number of commutations per phase activation *nC*, the switching frequency is given as:

$$\text{R } f\_{\text{sw,avg}}(Hz) = \frac{3000 \text{ rpm}}{\text{l min}} \cdot \frac{\text{l min}}{60 \text{ s}} \cdot \frac{6 \text{ activation}}{\text{l revolution}} \cdot \frac{n\_C}{\text{activation}} = 300 \cdot n\_C \tag{22}$$

In single pulse mode, *nC* is reduced to only two commutations per phase activation. As a result, switching losses decrease significantly.

For the sake of illustration, total power losses have been calculated for the SRD studied in this work under rated electrical conditions (*UDC* = 300 V, *I AVG* = 360 A and *n* = 3000 rpm) and maximum speed (*nnom* = 9800) using simulation analysis. Besides, power losses have been calculated in degraded mode of operation (m-1 phases available) as well. **Table 5** shows the results for both modes of operation (normal and degraded).

## **5. Degraded mode (with m-1 phases available)**

As mentioned along the whole chapter, "m-1 phases" operation is particularly interesting in EV applications and one potential advantage of SRMs over its counterparts. Degraded mode was one of the main reasons why a 4-phase topology was chosen in this work, given that performance loss, when one phase losses functionality, is obviously more significant in machines with a low number of phases.

neglected in IGBTs except for high voltages (above 1000 V) and/or high temperatures (above

Complementary, the energy dissipated during each switching is called either turn-on energy *EON* or turn-off energy *EOFF* . Neglecting parasitic effects, switching losses are given by the

*f SW* beingthe switchingfrequency.Obviously, both *EON* and *EOFF* highlydependonthe current and on the voltage, since they are defined by a power peak that in turn is caused by current and voltage transients. They are also influenced by temperature. The switching frequency meas‐ ures the number of turn-ons and turn-offs per time unit of a power semiconductor. It de‐ pends on the operating point (torque and speed) and on the electrical characteristic of the machine. In a 8/6-pole SRM, each phase is enabled six times per rotor revolution, so that knowing speed in the machine (e.g., 3000 rpm) and the number of commutations per phase

> 3000 rpm 1 min 6 activation ( ) · <sup>300</sup> 1 min 60 s 1 revolution activation = × × =× *<sup>C</sup> sw avg C*

In single pulse mode, *nC* is reduced to only two commutations per phase activation. As a result,

For the sake of illustration, total power losses have been calculated for the SRD studied in this work under rated electrical conditions (*UDC* = 300 V, *I AVG* = 360 A and *n* = 3000 rpm) and maximum speed (*nnom* = 9800) using simulation analysis. Besides, power losses have been calculated in degraded mode of operation (m-1 phases available) as well. **Table 5** shows the

As mentioned along the whole chapter, "m-1 phases" operation is particularly interesting in EV applications and one potential advantage of SRMs over its counterparts. Degraded mode was one of the main reasons why a 4-phase topology was chosen in this work, given that

*<sup>n</sup> f Hz <sup>n</sup>* (22)

=+ » = ( , , .) *P P P P f V I Temp COND ON OFF ON ON ON* (20)

*PP P E E f SW SW ON SW OFF ON OFF SW* = + =+ × , , ( ) (21)

150°C) [50, 51].

following expression [52]:

116 Modeling and Simulation for Electric Vehicle Applications

activation *nC*, the switching frequency is given as:

results for both modes of operation (normal and degraded).

**5. Degraded mode (with m-1 phases available)**

,

switching losses decrease significantly.

Performance priorities under these circumstances are different from those in normal operation. Degrade mode will be used only in case of emergency: it will allow the EV to keep driving, but the owner should take it to a repair workshop as soon as possible for proper fixing (similar to an emergency spare tire). This consideration has many implications. For instance, switching angles optimization should probably prioritize average torque over torque ripple, which will be huge anyway, or even over energy efficiency or IGBT (Isolated Gate Bipolar Transistor) aging.

Degraded mode is implemented by detecting that one phase cannot operate anymore (severe fault detection), by disabling it, and by adapting the control strategy to the new situation so that performance loss is minimized. Fault detection may be carried out in different ways, such as [53–55]. Faults that can be surpassed by the aforementioned m-1 phases mode include single coil faults, single IGBT faults and single converter branch faults, which are the most frequent. Notice that those faults that affect more than one phase simultaneously will generally prevent operation in degraded mode, unless the number of phases is high (five or more). In any case, grave faults such as DC-link short circuits will completely prevent operation.

**Figure 13.** (a) Torque-speed characteristic of the 8/6-pole SRM in m-1 phases mode. (b) Modification of switching an‐ gles with respect to the normal mode.

Performance analysis under degraded mode can be carried out with the same methodology presented in Section 4.3. For instance, **Figure 13** shows the torque-speed characteristic of the 8/6-pole SRM in m-1 phases mode. Two curves are given: one corresponding to default switching angles (those used in normal mode) and a second obtained by optimizing the angles specifically for m-1 phases operation, thus increasing torque but also current loading in both the power converter and the machine. In this last case, one of the phases is disabled, and consequently the adjacent phases are used in a wider range of angular position, as depicted in **Figure 6** for an ideal case. Namely, the previous phase is turned off later than usual, while the next phase is activated before, as shown in **Figure 13**.

Complementary, **Figure 14** shows the current pulses in steady state in "m-1 phases" mode. The torque profile is also depicted. It is worth mentioning that increasing the conduction period of the adjacent phases implies overloading both the corresponding power electronics and some of the machine coils from the thermal point of view. Therefore, results from **Figures 13** and **14** could be invalid for long periods of operation (in this sense, angles corresponding to normal mode are a safer choice). However, in this work, current was kept constant for the sake of comparison.

**Figure 14.** (a) Current pulses in m-1 phases mode (phase D is disabled) and (b) corresponding torque profile, both at 3000 rpm and 100% current.

It is important to notice that degraded mode should be able not only to keep the EV running, but also to start it from any rotor position and from different conditions, such as different road grade values. In this sense, when rotor position implies that a rotor pole is aligned with phase A in **Figure 2** (i.e., *θ* =0° ± *k* ⋅60° ; see **Figure 6**), there is no available torque. In practice, torque is very low in a zone of ±5° around these zero-torque positions (see **Figure 14**), which consti‐ tute the worst situation to start the vehicle. There are at least three possible ways to proceed in such a case, depending on the road grade:


speed of the vehicle is not enough to go through the whole low-torque region, the vehicle will stop due to the gravitational force. This would require repeating the process, but making the linear displacement in step 1 larger, so that higher speed can be achieved before facing the low-torque zone.

**•** Flat road (there is no gravitational force): the main problem here is that releasing the mechanical brakes will not start the vehicle. However, a two-step starting process similar to that described above could suffice: First, the phase opposite to the faulty one (if the activation sequence is A–B–C–D and D is the faulty phase, then B is the phase opposite to the faulty one), which has high negative torque capability precisely when the rotor is in the low-torque region, is used to reverse the vehicle for up to 7 mm. For this, it is very convenient that the SRM has an even number of phases, so that the phase opposite to the faulty one can provide high torque (although more than one phase can be used otherwise, as usually in SRMs). The second step is exactly the same as in the case of positive grade, but without the gravitational force opposing acceleration, which highly simplifies the starting process.

As an example, a starting process such as that described above is shown in **Figure 15**, corre‐ sponding to a flat road situation. As can be seen, the results suggest that the proposed starting protocol would successfully start a Nissan Leaf with the 8/6-pole SRM studied in this work.

**Figure 15.** Vehicle start-up from the worst possible rotor position (none of the healthy phases produce torque) and un‐ der the worst possible conditions (flat terrain).
