Sensitivity Analysis on the Voltage Distribution within Windings of Electrical Machines fed by Wide Band Gap Converters

In the last years, Wide Band Gap devices are seeing a significant widespread in electric drives, due to their higher performance compared to conventional semiconductors. However, they also produce higher electric stress due to over-voltages and uneven voltage distributions among winding turns of electrical machines fed by them, which can lead to premature failures and/or reduced lifetimes.This paper presents a sensitivity analysis on the voltage distribution across stator winding turns of an electric motor intended for aerospace applications. The effects of the surge voltage characteristic parameters, such as dv/dt, voltage magnitude and parasitic impedances, are investigated. An equivalent circuit approach, based on the multi-transmission line theory, is developed and implemented in MatLab-Simulink environment, while the relevant circuital parameters are estimated through finite element analysis performed with MagNet and ElecNet software.


I. INTRODUCTION
ARIABLE speed drives (VSDs) have reached an enormous variety of applications, from household appliances to industrial, aerospace [1], automotive, renewable energy generation, etc. VSDs are often equipped with electrical machines fed by Pulse Width Modulation (PWM) converters. These are usually based on conventional switching Silicon devices, but in the last years the number of high switching frequency converters consisting of Wide Band Gap (WBG) semiconductors, such as Silicon Carbide (SiC) or Gallium Nitride (GaN), has grown. WBG devices can provide improved performance than traditional power electronics switches [2]- [4]. The main advantages related to their use are: x lower switching losses due to their shorter rise and fall times; S. Nuzzo is also with the Power Electronics, Machines and Control Group, University of Nottingham, Nottingham, UK (e-mail: stefano.nuzzo@nottingham.ac.uk).
Giacomo Sala is with the Department of Electrical, Electronic and Information Engineering "Guglielmo Marconi", University of Bologna, Italy x the possibility of operating at higher switching frequencies, so that lower current harmonic content can be achieved; x they can bear with higher voltages and temperatures; x they feature a higher power density. However, there also some drawbacks. High overvoltage caused by reflected wave phenomena, already well known for IGBT based drives [5], can occur even with cables of few meters due to their short rise times. According to (1), the machine terminal voltage Vl can reach up to twice the value of the inverter bus voltage Vbus, since the theoretical limit of the reflection coefficient K is 1. However, even higher values can be reached in some cases, for example when double pulsing occur [5]. (1) In addition, the short rise times or, rather, their high dv/dt can trigger also an uneven voltage distribution across winding turns of the electrical machine, causing a higher turn to turn voltage stress [7]- [10]. If no precautions are taken, premature insulation failure can occur, leading to an inevitable stop of the drive [11]. High dv/dt are also related to a high frequency content in the order of a few MHz or more. This can cause unacceptable electromagnetic emissions which have to be reduced [12].
Considering the above, it is clear that, while operations at higher efficiencies and enhanced power density are ever more required especially in the automotive and aerospace fields, on the other hand an electric drive should also meet a certain reliability level [13]. Therefore, its design represents a difficult task and is always the result of a trade-off study aimed at maximizing the above requirements.

A. Motivation
Possible solutions to overcome the main drawbacks of employing WBG devices in electric drives consist in introducing small passive filters for dv/dt and overshoot reduction [14]- [16] or multi-level converters [17]. However, both solutions would increase the overall cost of the drive. In order to achieve a satisfying trade-off in terms of better performance, improved reliability and reduced overall costs, investigations on the impact of over-voltages and voltage distribution in the cabling and machine windings are needed and solutions to reduce the voltage stress must be identified.
The overvoltage phenomenon is well known and has already been largely investigated. A lot of research has focused on the development of models or approaches to predict it, such as in [18]. In addition, while the over-voltage occurring at the machine terminals can be easily measured, a direct access to the turns is needed to check the voltage distribution in the winding. Very fast front transients are known to be the main cause for the non-linear response of the voltage distribution within the winding turns of an electrical machine fed by PWM power converters. These fronts are particularly emphasized when such converters are based on SiC and GaN semiconductors. The uneven voltage distribution can represent a big issue in random-wound windings, where the positions of the turns in each slot are unknown, so that the first turns can result near the last ones, increasing the turn to turn insulation stress as opposed to a form-wound winding.
Some research on the winding voltage distribution has been carried out over the past years [7]- [10], focusing on rather complex modeling approaches which often rely upon FEA calculations. However, a detailed sensitivity analysis on the main parameters affecting such a phenomenon is missing, thus the aim of this paper is to provide an investigation on the effects of rise time, dv/dt, duty cycle, supply frequency, etc., typical of WBG-based electric drives.

II. MODEL DESCRIPTION
Before focusing on the sensitivity study, a detailed modeling approach is developed and implemented. Its description is dealt with in this section.

A. Equivalent circuit
During steady state the voltage distribution across winding turns is uniform. However, this assumption is not valid when fast transients occur. This behavior in the machine windings is related to the high frequency (HF) impedances of the turns, where skin effect, proximity and parasitic couplings through stray capacitances play an important role. Once the HF impedances are estimated, an equivalent circuit of the series connected turns can be built. The equivalent circuit is envisioned following the theory of multi-conductor transmission line model [19]. For the sake of the sensitivity study, only one phase coil is considered. In general, the coil consists of a certain number of turns connected in series. The circuital elements characterizing each turn are its resistance Ri, its self-inductance Li, its capacitive coupling with the other coil turns Cij and towards the ground reference Cig (i.e. the iron parts), as reported in Fig. 1. Each turn can be seen as a small RLC filter, gradually reducing the slope of the voltage waveform. This fact, combined to the different values of R, L and C of the whole coil, depending on their different position in the slot, determine a non-uniform voltage distribution.
In Fig. 1 vfed is the PWM voltage coming from the converter, defined by certain characteristic parameters, including magnitude, dv/dt, PWM frequency, duty cycle. These are the main parameters used for the sensitivity analysis. As an example, the case when 1) the magnitude is 500 V, 2) the PWM frequency is 50 kHz, 3) the dv/dt is 20 kV/μs and 4) the duty cycle is 0.5, is reported in Fig. 2.
Applying the first and second Kirchhoff's laws, as in (2) and (3), to the equivalent circuit of the coil comprising n turns (thus n nodes), it is possible to determine both the node voltages vi and the currents ii entering each node. Equations (2) and (3) are simultaneously solved for each node of the equivalent circuit. The resulting set of equations can be implemented in any dynamic system solver to find the voltage stress between different turns. MatLab-Simulink is selected to do so in this work.

B. FEA Model
As underlined in the previous section, the HF equivalent parameters need to be found. Their estimation has been done via FEA. The capacitive couplings have been calculated through time harmonic simulations using the electrostatic field software ElecNet, while inductances resistances have been found using the time harmonic simulations of the magnetic field software MagNet. Their values are associated to a specific frequency, which, for this case study, is in the order of MHz and is function of the voltage input rise time tr. Assuming that the frequency response of the system can be approximated as a response of a 1 st order system to a step input, the bandwidth of the signal fr is as in (4). Then, the frequency fr is used for the FE calculations. (4) The model for the FEA evaluations is in 2D. This allows to minimize the computation times (as opposed to 3D) for the drawing of the geometry and for the equivalent circuit parameters' calculation, which has been done through MatLab scripting interfacing with the two FEM software. The geometry is based on a single slot of an existing double layer permanent magnet machine for aerospace applications. Fig. 3 shows a field map of the 2D FEA slot model developed, comprising 24 conductors divided in 2 layers. The study is focused on the upper layer only, thus 12 conductors are considered for the sensitivity analysis on the voltage distribution. More observations and assumptions regarding this model can be summarized as follows: x Since the model is in 2D, border effects and end windings are neglected in the analysis. Only the coil active sides are taken into account. x Capacitances calculation is performed with the electrostatic analysis, so it is independent from frequency. This assumption can be considered true with a good approximation. In fact, the influence of frequency over capacitances is very small and it is quite null compared to its influence on resistances and inductances. x Inductive and capacitive couplings between turns placed in different slots can be neglected. At the considered operating frequencies, the ferromagnetic parts act as an electromagnetic shield as it can be seen by the flux lines in Fig. 3. x Each turn is modelled as an equivalent conductor as in [10], meaning that the eventual parallel strands comprising the conductor are not considered. x Inductive and capacitive couplings between the 2 coils belonging to the upper and lower slot layers envisioned in Fig. 3 have negligible effects. All these assumptions enable important computation time saving and simplification of the model.

A. Preliminary considerations
Once a certain slot geometry and the materials have been chosen, the voltage distribution among winding turns depends only on the input voltage. As mentioned above and shown in Fig. 1, the input voltage's main parameters are switching frequency, duty cycle, amplitude and dv/dt. Once the above parameters have been set, the node voltages of the circuit can be computed. The instantaneous voltage drop Vdi for the ith turn is then calculated as . If the voltage distribution is quite uniform, the peak value of the instantaneous voltage drop across each turn is quite near to the steady-state value. As underlined before, the most stressed turns are the first ones, while the voltage drop waveform tends to become more similar in the subsequent turns. In particular, the first turn will feature the higher value

B. Preliminary simulations -duty cycle and switching frequency
Some parameters have an important impact on the distribution across the turns, while others proved a negligible contribution. Short rise times or high dv/dt are the main causes for the uneven distribution. Also, a higher amplitude of the input voltage would certainly increase the voltage drop across each turn. Therefore, a sensitivity analysis with a variation of these two parameters is worth to be done.
On the other hand, switching frequency and duty cycle of the converter voltage waveform have no impact since they do not affect the HF content of the waveform which determines the voltage distribution. The HF content is in fact mainly related to the rise time of the converter waveform. An example proof is provided in Fig. 4 and Fig. 5, which show the results of two simulations where only the duty cycle has been changed. An uneven voltage distribution appears, since there is great difference from the peak to the steady-state values and even between the two turns (a uniform distribution would result in similar amplitudes in every time instant. The voltage drop is calculated according to (5) for the first turn (indicated as 'Turn 1' in the figures), while (6) is used for the second turn (indicated as 'Turn 2' in the figures). For both the figures, the input voltage has an amplitude of 500 V, the dv/dt is 10 kV/μs and the switching frequency is 50 kHz. The duty cycle is 10% for Fig. 4 and 90% for Fig. 5. Nevertheless, the number of voltage transients in a fixed time depend on the switching frequency, so a higher switching frequency can still negatively affect the insulation lifetime even if it has no influence on the voltage distribution or peak value.

C. Analysis on dv/dt and amplitude
The analysis on the impact of different dv/dt and voltage amplitudes is performed, and the results are shown both in absolute values and in p.u. The voltage amplitude is varied from 300 V to 800 V with a step of 100 V, while the dv/dt ranges from 2 kV/μs to 20 kV/μs with a step of 2 kV/μs, so that a wide spectrum of voltages, dv/dt and consequently rise times is available. The results in p.u have been found using the input voltage amplitude as a reference value, so that a proper comparison could be done also regarding the proportions of voltage distribution with different amplitudes. The reference for the evaluation of the uneven voltage distribution is the peak value of the voltage drop on the first turn. For a certain voltage input amplitude, higher voltage drop across the first turn is associated to a more uneven distribution. Two 3D maps are plotted in Fig. 6 and Fig. 7 These show the trends of the maximum voltage drops against dv/dt and magnitude, respectively considering absolute values and p.u. ones. It can be clearly observed that with increasing dv/dt values the voltage drop increases for all the considered amplitudes. Also, the maximum voltage drop across the first turn increases for higher input voltage amplitudes, as this corresponds to higher rise times. On the contrary, Fig. 7 shows that lower values of the feeding voltage magnitude determine a higher voltage drop in p.u, which means that the distribution is less uniform. Hence when two waveforms of different amplitudes, but same dv/dt, are applied to the winding, the voltage distribution would result to be more uneven for the lower voltage value, while in terms of absolute voltage drop the maximum value is associated to the higher amplitude input. Indeed, this result is not unexpected: an input voltage of 400 V and dv/dt of 20 kV/μs has a lower rise time (16 ns) than an input voltage of 800 V and 20 kV/μs (32 ns). This also means that, according to (4), the 400 V waveform has a bandwidth which is about twice than that at 800 V, thus the voltage distribution is less uniform. Therefore Fig. 7 confirms the direct correlation of rise time and voltage distribution, while the dv/dt needs one more variable, which is the voltage amplitude. Hence, the

Voltage [V]
Voltage [V] general conclusions can be summarized as reported in (7) and (8), being Va the voltage amplitude of vfed.

D. Further Analysis
The voltage distribution depends on the input voltage for a fixed geometry of the winding, however wire location and geometry (e.g. different cross-section areas or shapes) can determine a different impact. Inductances and resistances values are frequency dependent, whereas the capacitances are mainly determined by the location of each turn with respect to the slot wall (i.e. the tooth) and to the rest of the slot conductors. For example, considering fixed slot and wire geometries, different scenarios could be found in a randomwound winding as simplistically illustrated in Fig. 8. This figure shows 4 possible configurations which would result in 4 different capacitance matrices, both in terms of turn to ground and turn to turn values. For these reasons, further analyses can be done varying the main turn circuital parameters, i.e. inductances, capacitances and resistances. This analysis has been performed with a fixed voltage of 500 V and a dv/dt of 10 kV/μs. The impact of the impedance parameters has been estimated again with the evaluation of the peak voltage drop across the first turn of the coil. In particular, the analysis has been carried out by multiplying for different coefficients (from 0.1 to 1) the resistance, capacitance and inductance matrices.
The results are shown in Fig. 9 and Fig. 10. It can be seen that both the capacitances and the resistances have a    significant impact, while the impact of the inductances is very low. For capacitances and resistances, the trend is the same. Bigger values correspond to a worse voltage distribution. Hence, with a reduction of the parasitic capacitances or of the conductors' resistances, a more uniform voltage distribution can be achieved. The FE evaluation showed that resistance values are mainly affected by proximity effects rather than the skin effect. Fig. 11 shows the instantaneous voltage drop for the first three turns of the coil when the resistance matrix coefficient is 0.1 (i.e. the resistances are 0.1 p.u. of the values evaluated with FEA for the reference system). The peak values of the voltage drops are not very different. In addition, they are not very distant from the steady-state values, which means that the distribution is only slightly uneven. Considering all the above, it can be concluded that the parameters mostly affecting the voltage distribution within windings of electrical machines fed by fast switching converters are the input voltage amplitudes and gradients (dv/dt), or the rise times, the turns' series resistances and the capacitances. Table I provides an example of the impact on the voltage distribution given by the three principal factors when they are modified by a factor equal to 2. The voltage reference value is 500 V.

IV. CONCLUSIONS
With the spreading of WBG devices, higher performance drives can be realized, but increased challenges related to reliability aspects, such as early insulation failure, need also to be addressed. Among them, the uneven voltage distribution associated to fast rise times represents a serious source of electric stress. This paper has provided an analysis on the impact of the main parameters which might affect the voltage distribution among winding turns. The sensitivity study carried out in this paper highlighted that this phenomenon is mainly influenced by the applied waveform coming from the power converter and by the characteristic impedances of the turns. The most important waveform parameter is certainly the rise time, which can trigger a significantly non uniform voltage distribution of across the machine coil turns. On the other hand, the dv/dt alone is not enough to determine with precision the voltage distribution. However, in the range of low voltage applications, dv/dt values in the order of a few kV/μs can be enough to determine a highly non uniform voltage distribution. The investigation also highlighted that the winding structure can influence this high frequency phenomenon. In particular, lowering values of turn resistances and capacitances can mitigate the electric stress due to the winding voltage distribution, with potential benefits in terms of insulation lifetime. Contrarily, inductances seem to have a much lower impact. The future work will focus on the experimental validation of this paper's findings.

V. ACKNOWLEDGMENT
The project RAISE has received funding from the Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation program under grant agreement No. 785513.