Coupled circuit and magnetic fast model for high-speed permanent-magnet drive design

Transcription

1 IET Electrical Systems in Transportation Research Article Coupled circuit and magnetic fast model for high-speed permanent-magnet drive design ISSN Received on 16th March 2017 Revised 19th September 2017 Accepted on 6th November 2017 doi: /iet-est Mathieu Gerber 1, Adrien Gilson 1,2, Frederic Dubas 2, Christophe Espanet 1 1 Advanced Research and Development Department, Moving Magnet Technologies, Besancon, France 2 ENERGIE Department, FEMTO-ST, CNRS, University of Bourgogne Franche-Comte, Belfort, France mathieu.gerber@movingmagnet.com Abstract: High-speed electrical actuators used in harsh environments for automotive applications have surged in recent years. Examples comprise electrically assisted turbochargers, air compressors for fuel-cells, and waste heat recovery generators. In those kinds of applications, the temperature, size and efficiency are major constraining factors. Thus, the electronic and machine designs must be properly selected and adapted to work reliably in such tough environments. Then, coupled circuit and magnetic simulations must be performed to evaluate the impact of machine control on motor and inverter losses in order to find an optimum system. However, electronic driver and electric motor optimisations based on conventional simulation tools are very time-consuming, since these are based on finite-element methods for magnetic simulations and on Kirchhoff equations resolution for electronic simulations. In this context, this study presents a fast drive system model and its application on a highpower high-speed motor. The electronic and machine performances can rapidly be analysed by changing the electronic and motors parameters and substantially reducing the time required by conventional finite-element analysis software tools. 1 Introduction The growing interest in high-speed electrified systems for automotive applications requires to design electronic drivers and electric motors with new approaches, taking into account the strong coupling between the two parts of the drive. Typical applications include electrically assisted turbochargers, air compressors for fuelcells or waste heat recovery generators. Those kinds of high-speed drives require a significant increase of the energy conversion frequencies, typically up to a few khz for the motor and several tens of khz for the power electronics. This leads to high iron and eddy current losses in both stator and rotor and also causes increased switching losses in the inverter. In addition, system optimisation has to consider both motor and inverter losses while taking into account the designed space and temperature constraints. Indeed, the actuator size is strictly limited in most embedded applications, while the ambient temperature may reach up to 110 C in the particular case of automotive systems. Furthermore, elevated temperature environments imply high constraints for the electronic driver components (usually limited to 175 C for classic automotive metal oxide semicondutor field effect transistor (MOSFETs)), motor magnets (limited to about 220 C for NdFeB and 300 C for SmCo magnets), windings insulation and bearings. Thus, the main system constraints require minimising rotor losses to limit both the permanent magnet temperature rise and also the switching losses to comply with the semiconductor junction temperature limits. In this context, a global study of the system including electronic and magnetic simulations have to be performed to design an efficient adapted mechatronic system. Classic time-based electronic and motor finite-element analysis (FEA) simulations are very timeconsuming, and they depend heavily on the chopping frequency, which defines the minimal time step for controller simulations, as well as the minimal angular step and mesh size for magnetic simulations. At first the controller model will be described. The goal of the presented semi-analytical model is to compute the harmonic currents flowing in the motor by modelling the phase voltages generated by the inverter. Commutation instants, currents, DC-bus voltage and switch parameters then allow calculating the detailed inverter losses. By this harmonic analysis, fast and accurate simulations can be performed with variations of pulse width modulation (PWM) frequencies, PWM strategies for field orientated control (FOC) and block commutation machine control. Secondly, the motor model based on a 2D analytical subdomain model will be described. This model consists in analytically solving the Maxwell equations in different regions and, by applying boundary conditions, to obtain the magnetic vector potential in the machine. It computes static motor parameters (which are then used in the controller model) and all losses generated by the currents injected by the electronic driver into motor phases, i.e. detailed iron, copper, rotor losses including iron and PM losses. Finally, the electronic and magnetic coupling will be presented and applied on three study cases. Three different electronic driver and permanent-magnet synchronous machine (PMSM) will be chosen from three different constraint parameter sets. The model results are presented to highlight the benefit of applying this model. 2 Controller model description In automotive applications, voltage source inverters (VSI) with MOSFETs as controllable switches (Fig. 1) are mainly used to drive three-phase PMSM [1]. A VSI fed the motor with sinusoidal or trapezoidal currents according to the selected electronic control mode. For high-speed motors, the electronic driving mode must be chosen to find the best trade-off between inverter and motor losses. Fig. 1 shows a classic three-phase two-level inverter, where V DC is the DC-bus voltage, i DC is the DC-bus current, [T 1, T 3, T 5 ] (respectively T 2, T 4, T 6 ) is the upper (respectively lower) MOSFETs of phases [a, b, c], R is the motor phase resistance, L c is the phase cyclic inductance (equals to the difference between the phase and mutual inductance) which includes the effect of the mutual inductance, [e a, e b, e c ] are the back-electromotive forces (EMFs) of phases [a, b, c], [A, B, C] are the electrical potentials of the phases [a, b, c] connected to the inverter, O is the inverter ground potential and N is the motor neutral potential. Fig. 2 presents the flowchart of the developed semi-analytical model, which is based on harmonic calculation of the electronics driver voltage applied across motor phases. Then, the current is deduced from the controller voltage spectra and motor transfer IET Electr. Syst. Transp. 1

2 We can easily calculate the magnitude and the angle of V abc to create the current I abc flowing in the motor at the electrical radian frequency ω e and with the phase back-emfs E abc. From those calculations, we define the normalised reference voltages v a ref, v b ref, v c ref PWM modulation strategies: The PWM strategies presented here are carrier based. We will create the carrier with the following simple formula to avoid a time-consuming function: carrier = 2 π. arcsin sin 2π f pwmt (2) Fig. 1 Two-level three-phase inverter where carrier is a centre-aligned signal varying between ±1 and f pwm is the PWM frequency. Space vector pulse width modulation (SVPWM) (shown in Fig. 3) is mainly used in vector control. This over-modulation strategy is usually chosen to use the maximum battery voltage [2]. The drawback of this modulation strategy leads to high commutation losses at high PWM frequencies. Another available strategy is the general discontinuous pulse width modulation (GDPWM) (shown in Fig. 3) which allows to reduce commutation losses [2]. Commutation losses are lowered by clamping references applied to the motor. By comparing the references v a ref, v b ref, v c ref.and the PWM carrier, we can create [q 1, q 2, q 3, q 4, q 5, q 6 ] which are the commands of the [T 1, T 2, T 3, T 4, T 5, T 6 ] MOSFETs. Both strategies are implemented in the model FOC phase voltages: The phase voltages can be computed as Fig. 2 Electronic model flowchart v A v B v C = q q 3 V DC (3) q 5 where v A, v B, v C are the [a, b, c] phase voltages. Therefore, since the computed voltages v a, v b, v c are periodic, a fast-fourier transform (FFT) can be used to determine the harmonic contents of the phase voltages. 2.2 Block commutation phase voltages Similar to Section 2.1, the goal of this model is to find an expression of the phase voltages generated by the inverter 120 block commutation mode. Block commutation is a simple control strategy commonly used for low-cost electronic controllers and for high-speed electronic drivers, due to the ease of software integration and the low switching losses resulting from a full-wave six-step control. Fig. 3 SVPWM and GDPWM strategies function. From the currents and commutation instants, the inverter losses are computed. 2.1 FOC voltages model PWM reference calculation: The goal of this part is to compute the fundamental voltage to be generated by the inverter in order to create a current which is collinear with the back-emf. Then, (1) describes the relationship between I is the complex phase voltage, V is the complex phase current, and E is the complex back-emf: V abc ω e = R + jl c ω e I abc ω e + E abc ω e (1) Model description: The model described in the following sections is a slow decay mode with synchronous rectification 120 block commutated machine control. There are three steps to perfectly describe the phase voltages generated by the electronic converter: Two-phase conduction: One phase is connected to the DC-bus voltage, another phase to the inverter ground via the low-side MOSFET, and the third phase is disconnected from the DC-bus, since neither its high-side nor low-side MOSFETs or diodes are conducting. Positive (resp. negative) demagnetisation: At the time of a positive sector change, the currently conducting phase is disconnected and the previously disconnected phase is connected to the DC-bus voltage via its high-side (resp. lowside) MOSFET. The lower (resp. upper) diode of the now disconnected phase turns on as soon as the upper (resp. lower) MOSFET is turned off, until the current flowing in the phase is positive (resp. negative). 2 IET Electr. Syst. Transp.

3 To compute the conduction times of these diodes, the current at the instant of a sector change of the phase being disconnected is computed in Section From this current, the positive and negative demagnetisation durations are evaluated, then the phase voltages are computed Demagnetisation current computation: The demagnetising current is strictly the same for a negative or positive demagnetisation. To determine the demagnetising current, we assume that the maximal motor power is reached in the middle of the sector. We can then write the following relationship: I p = P 3k t ω e N pp (4) where k t is the motor torque constant and N pp is the number of pole pairs. We write the following semi-average equation driving the current flow in the two phases, phase a is connected to the DC-bus voltage via the high-side MOSFET and phase b is connected to the inverter ground through the low-side MOSFET α V DC = 2R s i a + 2L c d dt i a + e a e c (5) where α is the duty cycle applied on electronic components. From (4) and (5), we compute the peak current I peak with is flowing across the motor phase at the end of the sector: I peak = αv DC 2R 3E where E is the back-emf magnitude. + I P α V DC 3E 2R. e R/L c (π/6ω e ) (6) Demagnetisation duration evaluation: From the equivalent single-phase model, we can write the following equation during demagnetisation times: v AN = Ri a + L c d dt i a + e a (7) From (7), we can write the following semi-average model for positive demagnetisation: d Ri a + L c dt i (2 α) a = V 3 DC e a (8) As described previously, we can write from (8) the following semiaverage model for negative demagnetisation: Ri a + L c d dt i a = α 3 V DC e a (9) The time span t dp is the demagnetisation duration of the lower diode for a positive current and t dn is the demagnetisation duration of the upper diode for a positive current. Both can be evaluated by solving (8) and (9) for t dp and t dn : Phase voltage calculation: Now we intend to compute the phase voltages. To simplify the phase voltage expressions, we define: a nc b nc c nc = 1 q 1 + q 2 + d 1 + d 2 1 q 3 + q 4 + d 3 + d 4 1 q 5 + q 6 + d 5 + d 6 (12) a c b c c c = q 1 + q 2 q 3 + q 4 q 5 + q 6 (13) d c = d 1 + d 2 + d 3 + d 4 + d 5 + d 6 (14) d nc = 1 d c (15) where [q 1, q 2, q 3, q 4, q 5, q 6 ] are the [T 1, T 2, T 3, T 4, T 5, T 6 ] MOSFET commands, [d 1, d 2, d 3, d 4, d 5, d 6 ] represents the conduction of [T 1, T 2, T 3, T 4, T 5, T 6 ] MOSFET body diodes, a nc (resp. b nc, c nc ) and a c (resp. b c, c c ) represent the phase a (resp. b, c) connected or disconnected by the power switches to the DC-Bus, d c represents the three-phase conduction step when a diode conducts during positive or negative demagnetisations and d nc the two-phase conduction step. With these notations, we can write the following equation: v A v B v C = d nc 3.. V DC q 1 + d q 3 + d q 5 + d 5 + d q 1 nc 2. q 3. q 5 a nc b nc c nc. 0 b nc c nc a nc 0 c nc a nc b nc 0 e a e b e c. e a e b e c. V DC (16) These phase voltages are periodic, thus, a discrete FFT can be performed on the signals to determine their harmonic contents. The error from the estimations of positive and negative demagnetisations creates a DC-current in high state phases, then the residual DC current is computed and subtracted from the maximum demagnetising current I peak. 2.3 Motor current harmonic calculation The electronic model is developed for a given operating point in steady state. As soon as the motor phases are balanced, each phase has the same electric behaviour, so that we can use a single-phase model, the calculus is developed for phase a: k = 1 V ak kω e = k = 1 R + jklω e I ak kω e + E ak kω e (17) k = 1 t dp = L ci p + k t N pp 3k t /2 3/π (αv DC /3) t dn = L ci p + k t N pp 3k t /2 (3/π) ((2 α) V dc )/3 (10) (11) where k is the harmonic rank, V ak, I ak, E ak is the kth complex voltage, current and back-emf harmonic. From (17), we can then identify each harmonic member and write: V ak kω e = R + jklω e I ak kω e + E ak kω e (18) The demagnetisation durations are linked to the duty cycle: For full-wave (a = 1) operation, the positive and negative demagnetisation times are identical. We can then express the harmonic current generated in the motor for harmonic voltages created by the electronic driver as I kω e = V kω e E kω e R + jklω e (19) IET Electr. Syst. Transp. 3

4 The VSI model gives the phase voltage from the switch commands and DC-bus voltage. This voltage is directly used to compute the current flowing in the motor phases and generated by the electronic driver. 2.4 VSI loss evaluation From the currents and voltages generated by the power switches, the electronic efficiency can be computed from the switching and conducting parameters of the MOSFETs and diodes. Automotive VSI are mainly designed with MOSFETs for low voltage (12 48 V) DC Bust and with Si/SiC MOSFETs or IGBTS for medium and high voltages ( V) DC-bus. The following section is dedicated to the loss evaluation of VSI including their MOSFETs and diode components. IGBTs losses are not considered in this section MOSFET: MOSFET average conductive state losses P MC are calculated as average losses during an electrical period T e with the following classical formula: P MC = 1 T e 0 T e R DSON I D 2 (t)dt (20) where R DSON is the specific on-state resistance when the MOSFET conducts and I D is the drain current which flows across the power device. MOSFET commutation losses P MSW are created during turn-on t on and turn-off t off of the voltage V D across the MOSFET and the current I D which flows from the drain to the source at the PWM frequency f. Those losses can be approximated by the following formula: P Msw = 1 T e 0 T e V D I D dt (21) Diode: Dead time intervals are required to avoid cross conduction between high-side and low-side MOSFETs. During those idle times, diodes conduct and generate losses. Conduction losses of the diodes P DC result from the current I f and the voltage drop V f across. Those losses are computed as T e V f I f (t)dt (22) 3.1 Analytical model Design of high-speed machines can be extremely challenging since it involves high-frequency phenomena and mechanical considerations that are not commonly considered in conventional machines (e.g. rotor eddy current and rotor stability at high speeds). Mechanical aspects can significantly influence the design and the electromagnetic performance of the machine [3]. The rotor dimensions as well as the necessity to use a retaining sleeve can further limit the design options. This section will especially focus on the electromagnetic modelling of high-speed machines. In order to thoroughly optimise this machine, the need for fast and precise simulation tools has become more and more relevant over the years. Analytical models are an efficient way to tackle this problem. The computation time can be greatly reduced when compared with finite-element methods, and the analytical approach also allows a deeper understanding of the underlying physics. In that category, subdomain models are particularly accurate [4, 5] since they can take into account the slotting effect and the eddy current reaction field [6]. In this paper, a 2D analytical subdomain model such as the one described in [7] is used to evaluate the electromagnetic field in the machine. 3.2 Machine static parameters evaluation This paragraph presents the computation of the static parameters of the machine: The back-emf constant k t, the resistance R and the cyclic inductance L c. First, the resistance of a coil is evaluated using the machine geometrical parameters and the following equation: R coil = ρl avn L k f S slot (25) where ρ is the copper electrical resistivity, l av is the average path of a coil turn, N L is the number of layers, S slot is the slot cross-section and k f is the filling factor. Secondly, in order to compute L c and k t are the flux over each slot for any rotor position is calculated using (26). The method is presented in [8] and applied here to a multilayer winding ϕ i = H stn L S slot Si A i (r, θ) r dr dθ (26) P DC = 1 T e 0 The reverse recovery losses P DRR are created by the internal charge of the diode stored during the conduction. The reverse recovery charge Q rr must be depleted at the PWM frequency f on the DCbus voltage V dc. The corresponding losses are given by the following equation: P DRR = Q rr V DC f (23) By computing (24) during an entire electrical period, an approximation of the converter efficiency on a given operating point and in steady state can be calculated. Those losses can be greatly reduced, typically more than ten times lower, by using a metal interface as used in a Schottky diode. Finally, the total losses of the inverter P el are computed by the following formula: P el = 6 P MC + P MSW + P Dc + P DRR (24) As soon as the voltages and the currents are numerically computed, the integrals are computed numerically too. where S i is the domain of integration (part of the slot that contains the conductors), H st is the stack length and A i is the magnetic vector potential given by the analytical model in the ith slot. The calculated fluxes are then multiplied by a connecting matrix, such as the one described in [8], representing the winding connections and number of turns to obtain the phase fluxes ψ A, ψ B and ψ C. The inductance is calculated according to (27) by putting an arbitrary current in one phase of the machine and the remanent flux density of the magnet to zero L = ψ A I A M = ψ B I A = ψ C I A L c = L M (27) where L and M are the self and mutual inductances, respectively. Using (26) and setting the armature currents to zero, the back-emf constant k t is now derived from (28) by taking the peak value of the fundamental component of the phase back-emfs E i and dividing it by the rotational speed Ω 3 Electrical machine 4 IET Electr. Syst. Transp.

5 E i = Ω dψ i dθ rs k t = max E i Ω (28) where θ rs is the rotor position, k t, R and L c can now be used as input parameters to evaluate the inverter voltage and current waveforms. 3.3 Machine loss evaluation This subsection aims to describe the different losses that occur in a high-speed surface-mounted permanent magnet machine and how to mitigate them Winding copper: Copper loss can be calculated as the product of the resistance by the squared rms current. However, this equation is only valid when the fundamental frequency is low. At high frequencies the skin effect in the conductor can lead to additional losses. They can be reduced by selecting a wire diameter lower than two skin depths (or much smaller if the effect of the harmonics becomes relevant). Proximity effects caused by the adjacent conductors on each other and eddy currents induced by the permanent magnets of the rotating rotor can also lead to additional losses. Those effects can be mitigated by using Litz wire. In this study, the copper losses are simply calculated using (29), assuming that the wire diameter can be freely selected and that the use of Litz wire is permitted 2 P Cu = 3RI rms (29) where I rms is the rms value of the phase current and R is the phase resistance which is calculated using (25) and the winding configuration Stator iron: Stator iron losses arise from the flux density variation in the stator core. They are often divided into three components: Hysteresis, eddy current and excess. Various equations with different degrees of complexity can be used to evaluate these losses [9]. In high-speed machines, stator iron losses can be reduced by selecting higher grade and thinner steel lamination sheets. High-speed machines tend to have a lower flux density in the stator iron to allow maximising their efficiency. The Bertotti equation as described in [9] is used to compute the iron losses in the laminated stator core P Fe = k hy f B 2 + k ec f 2 B 2 + k ex f 3/2 B 3/2 (30) where k hy, k ec and k ex are, respectively, the hysteresis, eddy current and excess loss coefficients given by the manufacturer, f is the fundamental frequency and B is the peak value of the flux density assuming a sinusoidal waveform Mechanical: In high-speed electric machines, mechanical losses can be divided into two parts: Bearings: These losses are due to the friction in the bearing system. They are not described in more detail here since they depend on the design choice related to the application. As an example, high-speed ball bearings, air bearings or magnetic bearings can be used for this purpose. Windage: These losses are more specific to high-speed machines and are caused by the drag effects in the air gap of the machine. They can be significant when dealing with very high rotation speeds, large rotor diameters and small air gaps. These are described in detail in [10] Rotor eddy current: The main subject of this paper is to propose a global optimisation of both inverter and electric machine. From the machine side, the influence of the inverter topology and control strategy will impact copper, stator iron and IET Electr. Syst. Transp. rotor eddy current losses. This study will particularly focus on the impact of the control strategies on the rotor losses, since a proper evaluation of these losses is essential to avoid demagnetisation and provide an efficient machine design. Rotor losses occur because of spatial and time harmonics of armature currents as well as permeance variation due to the slotting effect. The induced currents in a moving conductive media such as a retaining sleeve, magnet ring or rotor yoke may produce additional losses. These losses are difficult to compute analytically since they involve the resolution of the diffusion equation taking into account the slotting effect. Less complex solutions can be found by using the following simplifications: Neglecting the slotting effect [5]: The armature currents alone are responsible for the losses. Neglecting the diffusion effect [11]: The loss will be overestimated since the skin effect is not considered. This method is commonly known as resistance limited. In this paper, a subdomain analytical model is used to evaluate the rotor eddy current loss. This model is resistance limited and accounts for the slotting effect. The rotor is made of a ferromagnetic yoke with the hypothesis of an infinite magnetic permeability, a ring magnet and a non-conductive sleeve. According to these assumptions, eddy currents are only induced in the magnet ring and can be calculated as J(r, θ, t) = σ A ma t + C(t) (31) where σ is the electrical conductivity of the magnet, A ma is the magnetic vector potential in the magnet annular region and C(t) is a constant that must be calculated to set the total current to zero in the magnet. Then, the rotor losses are evaluated using: P(t) = H r ma 2π st σ J 2 (r, θ, t)ds (32) rry 0 where r ry and r ma are, respectively, the inner and outer radius of the magnet. Finally, the rotor losses at the rated speed are given by the average value: P rot = Ω 2π 0 (2π)/Ω P(t)dt (33) A comparison of the calculated PM loss between the model and FEA is carried out to ensure that the skin effect is negligible in the studied case. Even if the eddy currents generated in the rotor are 3D, 2D computations are still approximate, but sufficient for comparisons between different motor architectures. For the ideal case of sinusoidal currents, the calculated magnet eddy currents for a 60 krpm machine given in Fig. 4 are in good accordance with the FEA results. 4 Electronic and magnetic coupled simulation 4.1 Electronic and magnetic coupled procedure and design parameters As shown in Fig. 5, the electronic and magnetic models are combined to obtain a global one that is able to evaluate the system performance according to the required specifications. The design variables include: The motor command mode (i.e. FOC or 120 block commutation). MOSFET and diode characteristics. PWM frequency and strategy for the inverter. The geometry, materials and winding for motor. All parameters are chosen and their respective impacts on the overall system are quickly studied. From all parameter sets, the 5

6 Fig. 4 Rotor eddy currents induced by sinusoidal currents in the magnet ring at 60 krpm (a) Analytical model, (b) FEA (Flux2D ) Table 1 Dimensions Parameter M 1 M 2 M 3 Unit air gap mm stack length mm outer stator radius 40 mm rotor radius 12 mm stator yoke thickness 8 mm slot opening 30 deg magnet thickness 5 mm Fig. 5 Design flowchart of the high-speed drive design systems electronic model computes the currents flowing in the machine from terminal motor parameters. Then the influence of the chopping current on copper, iron and magnet losses is computed. Finally, the overall system efficiency can be studied. 4.2 Model results and limitations From the coupled model presented in the previous section, the following losses are computed: Detailed motor losses, including copper, iron and magnet losses from motor geometry and material characteristics. Detailed electronic losses, including diode, MOSFET, DC filter losses from power switches and passive element characteristics, and PWM strategy and frequency. Even if the model is rather accurate, there are a few limitations due to the hypothesis used to create it: The motor model is done with linear materials, thus, no saturation is taken into account. If saturation must be analysed, machine current and performance have to be checked using different methods, the easiest one is probably to include FEA simulations. The eddy currents computed in the magnet are intrinsically 3D, so, those losses have to be computed with a 3D FEA to give a better evaluation. The MOSFETs and diodes are considered ideal in the model, so the calculations of the electronic driver could be refined using a conventional electronic analysis software. Apart from the mentioned limitations, this model is useful to make a first optimisation and to select possible solutions. 5 Case studies 5.1 Case definitions The system studied to highlight the benefits of the presented model is a 9 kw 400 V compressor for hybrid vehicles as presented in [12]. The operating point is a power of 9 kw at a rotation speed up to 60,000 rpm and supplied by an inverter connected to a 400 V DC-bus, capable of working with FOC and 120 block commutation strategies. The following optimisation goals (Goals #1 3) are chosen: Goal #1: Low rotor losses by keeping a high electronic efficiency (typically high-temperature constraints on the rotor). Goal #2: High electronic efficiency and small motor volume (i.e. reduced length), typically resulting in high thermal stress within the electronic driver. Goal #3: High electronic efficiency and low rotor losses with a minimal cost of the system. For those studies, a set of three PMSM machines with six slots, a two-pole NdFeB ring magnet and concentrated windings has been selected to demonstrate the capabilities of the model. This design allows easier control of the machine (low rotor pole number) as well as a simplified manufacturing process (low slot number and offline tooth-coil winding). Of course the model can be applied to other topologies of PM motor. The external stator radius, the power and the speed are defined by the application. The electronic driver works with FOC and the PWM frequency is fixed between 10 and 70 khz with SVPWM and GDPWM strategies. As an alternative, with block commutation and electronic driver in full-wave operation, the electronic losses will be a function of the electrical frequency. For the three machines (M 1, M 2, M 3 ), Tables 1 3, respectively, show the dimensions, the magnet properties and the static parameters calculated using the analytical model. On the same motor basis, a comparative study between the three presented motors driven by FOC and block commutation is performed. For each presented goal, an optimal design will be chosen. 5.2 Model results and system selections The efficiencies of all motors increase with higher PWM frequencies. With FOC, current harmonics generated in the motor 6 IET Electr. Syst. Transp.

7 Table 2 Magnet properties Parameter M 1 3 Unit remanent flux density 1.18 T relative permeability 1.05 electrical resistivity 1.5 μω M Table 3 Static parameters Parameter M 1 M 2 M 3 Unit back-emf constant mv/(rad s 1) resistance mω inductance µh Fig. 7 Inverter efficiency for SVPWM, GDPWM and full-wave 120 block commutation Fig. 6 Rotor losses with FOC SVPWM, GDPWM and 120 block commutation (noted 120 BC) (a) For 3, (b) For 2, (c) For 1 are directly linked with the PWM frequency of the electronic driver. With 120 block commutation, no PWM is applied in our study case. This method is widely used for high-speed drives because of the low cost of the solution and its simple software implementation which requires small microcontroller and one current sensors. When the control is working in full-wave mode, this method is really suitable for high-frequency motors since we have only few switching losses. Obviously, its harmonic current content mainly contains lower harmonics (typically ranks 5, 7, 11, 13), which generate higher rotor losses (Fig. 6). Each MOSFET is only switching two times per electrical period. In our case, the selected motor architecture induces long demagnetisation times, then high diode losses, which ineluctably induced a low inverter efficiency. IET Electr. Syst. Transp. This method is rather good for low inductance machines as slotless PM machines. One offers the highest efficiency and reaches 93.4% from about 30 khz for SVPWM. Fig. 6 also shows the influence of the air gap on the rotor losses. One which has the widest air gap and the longest active length offers remarkably low rotor losses (52 W when driven by a 30 khz PWM), while three with the smallest air gap and active length, has the higher rotor losses but the smallest volume. Then, we study the influence of the PWM strategy on the motor and overall efficiencies providing the best trade-offs for goals #1 through #3. Goal #1: With classic FOC method working with SVPWM, the best motor efficiency can be reached. Nevertheless, GDPWM substantially reduces the electronic driver losses (Fig. 7) and thus increases the overall efficiency. Fig. 8c shows that for the problem studied there, the maximum efficiency is obtained for a large air gap motor (1) and a 20 khz GDPWM. The motor efficiency is 93% and the electronic efficiency is 98.8%, pushing the system to an overall efficiency of 91.9%. Goal #2: 120 block commutation offers low cost and low losses in power electronics driver and, therefore chosen for this case, along with three, having the smallest volume. If the rotor heat dissipation does not allow using this small motor length (Fig. 6a), the design will change and a longer motor with a larger air gap reduces the rotor losses (shown in Fig. 8b. The motor efficiency is 86.5% and the electronic efficiency is 98.4% pushing the system to an overall efficiency of 85.5% (Fig. 8a). Goal #3: 120 block commutation offers low losses in power electronics. As shown in Fig. 6c, one has the lowest rotor losses with 120 block commutation and can be chosen if the resulting losses are acceptable. The cost of the controller is reduced by using 120 BC. If not, we can imagine to choose FOC with a GDPWM at 10 khz which generates less rotor losses and reasonably low electronic losses. The motor efficiency is 90% and the electronic 7

8 6 Conclusion In this paper, a coupled electronic and electromagnetic model was described and applied to design high-speed, high-power systems with different optimisation goals. This alternative method offers fast responses to multiphysics problems which classically require time-consuming simulations. This model allows to vary significant motor features (geometry, air gap, windings) and electronic parameters (motor control mode, MOSFET and diode characteristics, PWM frequency and strategy) to analyse the system behaviour. Due to the short calculation times, a true optimisation of highpower, high-speed mechatronic systems can be performed in a rapid and straightforward way. A few examples of optimisation are presented in this paper. Based on different environment constraints, fast optimisations are performed and thus, several motor and electronic driver combinations are chosen. However, even if the presented paper presents a tool to pre-design the drive system, special efforts have to be performed on the thermal analysis when the mechanics is built which can modify the optimal design founded. The rotor losses have to be computed by a 3D transient simulation. 7 References [1] Lusignani, D., Barater, D., Franceschini, G., et al.: A high-speed electric drive for the more electric engine. IEEE Energy Conversion Congress and Exposition (ECCE), 2015, pp [2] Hava, A.M., Kerkman, R.J., Lipo, T.A.: Carrier-based PWM-VSI overmodulation strategies: analysis, comparison, and design, IEEE Trans. Power Electron., 1998, 13, (4), pp [3] Riemer, B., Leßmann, M., Hameyer, K.: Rotor design of a high-speed permanent magnet synchronous machine rating 100'000 rpm at 10 kw. IEEE Energy Conversion Congress and Exposition, 2010, pp [4] Dubas, F., Espanet, C.: Analytical solution of the magnetic field in permanent-magnet motors taking into account slotting effect: no-load vector potential and flux density calculation, IEEE Trans. Magn., 2009, 45, (5), pp [5] Dubas, F., Rahideh, A.: Two-dimensional analytical permanent-magnet eddycurrent loss calculations in slotless PMSM equipped with surface-inset magnets, IEEE Trans. Magn., 2014, 50, (3), pp [6] Pfister, P.D., Yin, X., Fang, Y.: Slotted permanent-magnet machines: general analytical model of magnetic fields, torque, eddy currents, and permanentmagnet power losses including the diffusion effect, IEEE Trans. Magn., 2016, 52, (5), pp [7] Gilson, A., Tavernier, S., Dubas, F., et al.: 2-D analytical subdomain model for high-speed permanent-magnet machines. Presented at the Int. Conf. Electrical Machines and Systems (ICEMS), Pattaya, Thailand, 2015, pp [8] Lubin, T., Mezani, S., Rezzoug, A.: 2-D exact analytical model for surfacemounted permanent-magnet motors with semi-closed slots, IEEE Trans. Magn., 2011, 47, (2), pp [9] Bertotti, G.: General properties of power losses in soft ferromagnetic materials, IEEE Trans. Magn., 1988, 24, (1), pp [10] Borisavljevic, A.: Limits, modeling and design of high-speed permanent magnet machines (University of Belgrade, Serbia, 2011) [11] Wu, L.J., Zhu, Z.Q., Staton, D., et al.: Analytical model for predicting magnet loss of surface-mounted permanent magnet machines accounting for slotting effect and load, IEEE Trans. Magn., 2012, 48, (1), pp. 1 2 [12] Gerber, M., Gilson, A., Depernet, D., et al.: Coupled electronic and magnetic fast simulation for high-speed permanent-magnet drive design IEEE Vehicle Power and Propulsion Conf. (VPPC), 2016, pp. 1 6 Fig. 8 System efficiencies with FOC SVPWM, GDPWM and 120 block commutation (noted 120 BC) (a) For 3, (b) For 2, (c) For 1 efficiency is 98.8% (Fig. 7) pushing the system to an overall efficiency of 88.9% (Fig. 8b). 8 IET Electr. Syst. Transp.