Linear PWM Techniques of Asymmetrical Six-Phase Machine With Optimal Current Ripple Performance

Linear modulation techniques (LMTs) of an asymmetrical six-phase machine (ASPM) with two isolated neutral points synthesize the desired voltage vectors by applying at least five switching states. Different choices of applied voltage vectors, sequences in which they are used, distribution of dwell-times among the redundant switching states give rise to a large number of possible LMTs. It is desirable that these LMTs avoid more than two transitions of a particular inverter leg within a carrier period. Only a subset of existing LMTs of ASPM follows this rule. Through an innovative approach, this article finds a way to account for all possible infinitely many LMTs that follow the rule of at most two transitions per leg. Another important criterion for the selection of an LMT is its current-ripple performance. Therefore, through numerical optimization, this article finds optimal LMTs among the above infinite possible LMTs for all reference voltage vectors in the linear range and for the whole feasible range of a machine parameter. This parameter is related to the leakage inductance of the machine and impacts the current ripple performance of ASPM. An optimal hybrid strategy is proposed with these optimal techniques, which outperforms all existing techniques in terms of the current ripple. The theoretical analysis is validated through simulation in MATLAB and experiments performed up to 3.5 kW on a hardware prototype.

space and supply harmonics [1], [2]. Asymmetrical six-phase machine (ASPM), one of the most common multiphase machines, has two sets of balanced three-phase (3φ) windings with spatial angular difference of 30 • electrical, as shown in Fig. 1. This machine is also known as dual three-phase machine [3], [4], or split-phase machine [5], in the literature.
Vector-space decomposition based modeling of ASPM, as proposed by [3], transforms the variables from original domain to three 2-D orthogonal subspaces, namely, α − β, z 1 − z 2 , and o 1 − o 2 . It is shown that electromagnetic energy transfer occurs only in α − β plane, whereas the remaining two planes do not have coupling between the stator and rotor and hence, are not associated with energy transfer. But, the impedances offered by these two planes are small, so very small excitation in these planes can cause large current and associated copper loss. For two balanced sets of windings with two isolated neutrals, as shown in Fig. 1, o 1 − o 2 plane can not be excited. Linear modulation techniques (LMTs) of ASPM synthesize the desired average voltages in α − β plane and zero average voltage in z 1 − z 2 to avoid the unwanted copper loss. If modulation index (M I ) is defined as the ratio of peak of fundamental line-neutral voltage and dc-bus voltage, the maximum index attainable by LMTs is 0.577. This article focusses on these LMTs and their high-frequency current-ripple performances.
The proposed space-vector based pulsewidth modulation (SVPWM) technique in [3] shows that LMTs of ASPM need to apply at least five distinct vectors. One such set of five vectors, adopted by [3] and [4], is fourth largest active vectors in α − β plane, adjacent to the reference voltage vector, V ref , and one zero vector. By placing multiple redundant switching states, corresponding to the zero-vector, at different positions in the switching sequences, Hadiouche et al. [4] proposed continuous and discontinuous linear SVPWM techniques with different current-ripple performances, C 6φ SVPWM12, D 6φ SVPWM12-A/B1/B2. Hadiouche et al. [4] showed that the current ripple of ASPM is not only function of PWM techniques, but it also depends upon machine parameter, γ, which is defined as the ratio of high-frequency inductances seen by α − β and z 1 − z 2 subspaces. To discuss the disadvantages of the proposed SVPWM12 techniques of [4], two desirable features of a PWM technique, as given by R1 and R2 in the following paragraph, will be discussed first.
R1. Each semiconductor device of the six-phase inverter is allowed to turn-ON and tun-OFFmaximum one time over a carrier cycle.
R2. The switching sequence will be mirror symmetric along carrier half-period.
Noncompliance of R1 results in increase in instantaneous switching frequency of a particular leg for the same average switching frequency [6], [7]. Carrier based simple implementation is possible if PWM techniques follow rules R1 and R2 [7]. Note, most of the well-known PWM techniques of two-level three-phase inverter, such as SPWM, SVPWM, THIPWMs, and DPWMs, follow these rules for carrier-based implementation [8]. None of the PWM techniques, proposed by [4], follows R1. Marouani et al. [6] proposed 24 sector based continuous and discontinuous PWM techniques, C 6φ SVPWM24, D 6φ SVPWM24-B1/B2, where the furthest of the four largest vectors in α − β is replaced by one small vector. These techniques follow both the rules. The ripple current performance of D 6φ SVPWM24-B2 (the best technique in [6]) is worse than that of D 6φ SVPWM12-B1/B2 techniques for the same average switching frequency at higher M I and higher values of γ. A modified SVPWM24 technique, C 6φ SVPWM24-C, is proposed in [7], which also obeys R1 and R2 but current-ripple performance is not good at higher M I and higher values of γ.
Another group of LMTs were proposed by [9] and [10], where two 3φ inverters, Inverter-1 and Inverter-2 of Fig. 1 [11] showed that this operation is equivalent to synthesizing V ref in α − β and zero in z 1 − z 2 on average. These techniques are termed as two-inverter based technique (TINV) in [11]. Existing TINV techniques are 1) Sine-triangle PWM (STPWM) [5], where zero commonmode voltage is injected between load-neutral and dc-bus midpoint; 2) Double zero-sequence injection PWM (DZIPWM), which is also named as ZS3PWM in [5], where two different zero-sequence signals are used for two 3φ inverters and the modulation signals of both inverters are compared with same carrier signal; these zero-sequence signals are half of the middle values of the corresponding 3φ modulation signals; 3) ZS6PWM [5], where a single common-mode signal is used for both Inverter-1 and Inverter-2 and this is equal to the half of negative summation of maximum and minimum of 6φ modulating waves. These three TINV techniques follow rules R1 and R2. The comparison of current ripple performances as function of M I of these TINV techniques and SVPWM techniques reveals that SVPWM techniques are much better than TINV techniques for the feasible range of γ [5], [12].
Note, existing three TINV techniques, three techniques proposed in [6], and C 6φ SVPWM24-C satisfy the desirable two rules, R1 and R2. In fact, there are infinitely many PWM techniques, not just these seven techniques known in the literature, which satisfy these desirable conditions. This infinite possibility arises due to infinite number of choices of distributions of the dwell-times between two redundant zero states of both 3φ inverters, Inverter-1 and Inverter-2 of Fig. 1, in the sequence. It is known from the two-level 3φ inverter literature that this distribution impacts the current ripple [8], [13], [14]. In this context, this article makes following contributions.
1) First attempts to identify the complete set of possible LMTs, which satisfy rules R1 and R2, in Section II. 2) Then through systematic analysis in Section III, it finds the optimal technique that results into minimum current ripple for a given V ref in α − β, machine parameter, γ, and given average switching frequency. 3) In Section III, it finds all such optimal techniques for all possible V ref in linear range for feasible range of γ. 4) Finally, it proposes a hybrid LMT in Section IV using the optimal techniques obtained in previous step. This LMT outperforms all existing LMTs, which follow R1 and R2, in terms of line current THD. The superiority of the proposed hybrid technique is validated through experiments and simulations in Section V. Finally, Section VI concludes this article. Fig. 1 shows a two-level six-phase (6φ) inverter, which can be seen as two three-phase (3φ) inverters, Inverter-1 and Inverter-2, respectively, connected across the same dc-bus. The pole points of these two 3φ inverters, a, b, c, and a , b , c are directly connected to the terminals of two sets of 3φ windings of an ASPM. These two sets of windings are spatially shifted by 30 • electrical and are connected in star with two isolated neutral points o and o . This 6φ inverter has 64 switching states. An ordered pair denotes these states, (p, q ), where p and q denote the switching states of Inverter-1 and Inverter-2, respectively, according to the standard 3φ inverter notation as given in Table I.

II. PROPOSED TWO-INVERTER BASED LMTS
The matrix, T , as given in (1), was proposed by [3] to model ASPM in three 2-D orthogonal subspaces, namely, α − β, z 1 − z 2 , and o 1 − o 2 . This matrix T is used to map the line-neutral voltages generated by 64 states of 6φ inverter into the above six orthogonal planes. It is seen that all 64 states generate zero voltages in o 1 − o 2 plane and therefore, discussion in o 1 − o 2 plane is excluded in the subsequent sections. The voltage vectors generated by 64 switching states in α − β and z 1 − z 2 subspaces can be referred from [3] and [5]. These 64 states generate 24 equivalent sectors in α − β [6]. The modulation in one of these 24 sectors in α − β, which is conventionally called as sector-1 [as shown in Fig. 2(a)], can be discussed with the help of 19 switching states. Fig. 2 shows the mapping of these 19 switching states in α − β and z 1 − z 2 planes and they result into 12 distinct vectors in both the subspaces. In Fig. 2, z denotes zero-state and z ∈ {0, 7}. Sector numbers in Fig. 2(a) are surrounded by oval-shaped curves. Modulation in the remaining sectors will be similar to sector-1.
LMTs of ASPM synthesize 1) the desired reference voltage vector, V ref Δ = v α + jv β , in α − β plane as it is solely responsible for electromagnetic energy transfer; 2) zero average voltage vector in nonenergy transferring z 1 − z 2 plane in order to avoid unwanted average current and associated copper loss resulted due to the low impedance in this plane, i.e., v z 1 + jv z 2 = 0.
Here, bar represents average over a carrier cycle. To generalize the analysis, all lengths in subsequent figures are normalized with respect to dc-bus voltage V DC , and the reference voltage vector is also normalized and denoted by m. Here, m Δ = m α + jm β and m α,β = v α,β V DC . In general, the synthesizable m can lie anywhere within a do-decagon structure obtained by joining the tips of the largest vectors in α − β plane and D 1 of Fig. 2(a) shows part of this do-decagon. But, Paul and Basu [11] showed that synthesizable m of LMTs of ASPM can lie within another smaller do-decagon structure, D 2 , due to the constraint of zero average vector synthesis in z 1 − z 2 . The linear modulation region within sector-1, ΔOAB, is separately shown in Fig. 3, which is a right angle triangle with base length of unity and height of tan 15 • .
Paul and Basu [11] showed that synthesis of m in α − β and zero in z 1 − z 2 is mathematically equivalent to modulating the two 3φ inverters with reference voltage vectors as follows: where m 3φ and m 3φ are the reference voltage vectors of Inverter-1 and Inverter-2, respectively. These 3φ voltage vectors are obtained by applying Clarke's 3φ − 2φ transformation on 3φ average line-neutral voltages, as follows: With respect to this transformation, the mapping of the lineneutral voltages generated by eight switching states (given in Table I) of 3φ inverter is shown in Fig. 3.
Finding all possible LMTs when m ∈ ΔOAB is equivalent to finding all possible ways to synthesize m 3φ and m 3φ in per 3φ inverter basis. The intermediate steps, which are followed to derive these techniques from the given m in sector-1, are discussed as follows.

A. Determine Vector-Sets Per 3φ Inverter and Their Corresponding Duty-Ratios
When m ∈ ΔOAB, 0 • ≤ ∠ m ≤ 15 • . Therefore, according to (2), 0 • ≤ ∠ m 3φ ≤ 15 • and −30 • ≤ ∠ m 3φ ≤ −15 • . Hence, m 3φ and m 3φ lie within sector-I and sector-VI, respectively, in complex plane of 3φ inverter, as shown in Fig. 3. It is known from the modulation techniques of the conventional 3φ inverter that atleast three distinct vectors need to be applied per inverter basis in order to generate these reference voltage vectors. Two The duty-ratios of each of the vectors of these vector-sets can be found by solving the average voltage equations. For example, d z , d 1 , and d 2 , duty-ratios of the vectors of vector-set {z, 1, 2}, can be found by solving the following: d z , d 1 , and d 2 are functions of m α and m β . Similarly, duty-ratios of the vectors of other vector-sets can be found. The expressions of these duty-ratios of the above four vector-sets are given in Table II

B. Determine the Sequences of 3φ Inverters Corresponding to Each Vector-Set
The sequence design will be guided by rules R1 and R2, as given in Section I. R2 ensures that all the vectors within the given vector-set need to be applied atleast once within carrier halfperiod, T s 2 . It can be shown that all possible switching sequences of vector-set {z, 1, 2}, which obey the above two rules, can be generalized as From now onward, we will only consider the vector-sequence within T s 2 as the other half is mirror symmetric. Similarly, vector-set {z, 1, 6 }, {6, 1, 2}, and {6, 1, 2 } can be generalized as 0 − 1 − 6 − 7 , 6 − 1 − 2, and 6 − 1 − 2 , respectively.

C. Sequence Grouping of 6φ Inverter
As each of Inverter-1 and Inverter-2 has two sequences to synthesize m 3φ and m 3φ , respectively, m in α − β can be synthesized by using 2 × 2 = 4 different combinations of 3φ sequences. For each of these four combinations, reversal of one sequence to the other gives two different arrangements. Therefore, a total of eight sequence groups can occur, and Table III shows these eight groups. These are labeled as S1-S8. As sequences 0 − 1 − 2 − 7 and 0 − 1 − 6 − 7 can be used to synthesize m within the entire ΔOAB, S1 and S2 can be applied for the entire ΔOAB. Similarly, from our previous discussion, it is obvious that S3 and S4 are applicable in the region within ΔOAB, where condition (5b) is satisfied; S5 and S6 are applicable, where (5a) is satisfied, and S7 and S8 are applicable, where both (5a) and (5b) are satisfied. From 3φ inverter literature, it is known that the current-ripple is a function of zero-vector splitting [8], [13], [14]. Therefore, the zero-vector dwell-times are split as x, (1 − x) (for Inverter-1) and y, (1 − y) (for Inverter-2) between redundant zero states 0 and 7 to study the ripple performances in next section. Here, x and y are nonnegative fractions. Note, x, y ∈ {0, 1} results into conventional bus-clamping PWM of 3φ inverter.
Let us define k f as the ratio of the total number of switching (combining turn-ON and turn-OFF) over T s and the total number of legs (6 in this case). As each change of state is accompanied by one transition, k f = m+n−2 6 when the two 3φ inverters apply m and n number of states, respectively, within T s /2. From this discussion, one can see that k f is function of x and y and this function is shown in Table III. As we will see later that k f plays an important role in current-ripple comparison of different techniques with the same average switching frequency.

D. Determine the Vector Sequence and Corresponding Duty Ratios of 6φ Inverter
The objective of this article is to find the best technique in terms of current-ripple out of the infinite number of possible techniques, grouped as S1-S8, for m ∈ ΔOAB. This infinite possibility arises due to the variability in the choice of x and y (S1-S6). Hadiouche et al. [4] showed that the estimation of current-ripple requires the sequence of application of 6φ switching states (to know the corresponding voltage vectors in both α − β and z 1 − z 2 planes) and their corresponding dwell times. Given any selection, S1-S8, this sequence and the dwell times of the states depend upon the position of m, choices of x and y (in case of S1-S6).
To explain the above, let us consider case S7 over T s /2. As S7 involves vector-sets {6, 1, 2} and {6, 1, 2 }, it can be applied in part of ΔOAB, where both (5a) and (5b) are satisfied. The straight lines with equality in (5a) and (5b) are plotted as CD and ED in Fig. 4. Double-arrow marks on these lines show the part of ΔOAB, where conditions (5a) and (5b) are satisfied and therefore, EDAB is the region where S7 can     Fig. 6. The corresponding duty-ratios of these states with respect to , respectively, where the duty values, dk and dk , k = 1, 2, 6, correspond to vector-sets {6, 1, 2} and {6, 1, 2 }, respectively, of Table II. Similarly, one can find the sequences for other cases of S7, which are tabulated in Table IV. For a given m Δ = m α + jm β , one of these six cases can occur as the duty ratios are defined by m α and m β , as given in Table II. Number of cases for other sequence-groups can also be found in the similar manner and are given in Table V. It is important to mention here that all these possibilities may not arise in case of m ∈ ΔOAB. For example, it is possible to show that only two sequences, cases (c) and (f) of Table IV, can occur within EDAB from the six possible sequences of S7. These two sequences are applied in EDGF and F GAB, respectively, where the equation of F G in Fig. 4

III. DETERMINATION OF TECHNIQUE(S) WITH OPTIMAL CURRENT-RIPPLE PERFORMANCE
To minimize RMS ripple current over a line-cycle, this article proposes to minimize the ripple RMS current at every carrier cycle. The total RMS ripple current of all six phases over a carrier cycle can be derived after following the steps given in [4]. The expression of this RMS current,ĩ RMS , is where RMS ripple current of jth phase,ĩ j,RMS , andĩ RMS are perunitized with respect to base current V DC Tsw L αβ . γ is a machine Fig. 7. Algorithm to find the optimal technique of sequence-groups S1 and S2.
parameter, which is defined as γ Δ = L αβ Lz 1 z 2 ; L αβ and L z 1 z 2 are the high-frequency equivalent circuit inductances in α − β and z 1 − z 2 planes, respectively.λ αβ andλ z 1 z 2 are the ripple RMS fluxes in these two planes. In (6), the factor k f Δ = Ts Tsw is introduced to keep average switching frequency, F sw Δ = 1 Tsw , same for all techniques during comparison. It can be seen thatĩ RMS depends only on modulation technique and γ. The parameter γ is a strong function of winding pitch-factor and geometry of the slot of the machine. The full-pitched winding machine has the smallest γ value, whereas the highest value of γ occurs at 5/6th pitch factor [15]. The full-pitched winding machine used in [4] has γ = 1.58, the machine with 5/6th pitch factor has γ = 11. The full-pitched winding machine, used for experimental verification in this work, has γ = 1.69. Based on these aspects, the feasible range of γ is considered to be 1.5-10 in this work.
With the above formulation, the optimal technique is found numerically rather than analytically due to the complex nature of the problem. The algorithm to find the optimal technique, for a given m Δ = m α + jm β and machine parameter γ, is explained as follows. 1) Step-1: The optimal sequence of a particular sequencegroup is identified, and the correspondingĩ 2 RMS is recorded. A flowchart of this step is given in Fig. 7 for sequence-groups S1 and S2. Here, values of x and y are varied between 0 and 1 with the spacing of 0.1 between two consecutive values. A similar procedure can be followed to find the optimal sequence and corresponding ripple RMS currents of sequence groups S3-S6 (either x or y varies), S7, and S8 (only one sequence).

2)
Step-2: Based on the conditions given in (5), there could be a maximum of eight optimal sequences, one coming from each sequence group, S1-S8. From these eight (or less than 8) sequences, find the sequence for whichĩ 2

RMS
is minimum. For the given m and γ, this sequence is the optimal sequence out of all possible sequences, which follow the rules R1 and R2. The aforementioned algorithm is run for a range of m ∈ ΔOAB and the feasible range of γ. Fig. 8 shows region-wise optimal techniques for three values of γ within ΔOAB. It is interesting to notice that only five techniques, out of so many possibilities, are optimal for most of the region of ΔOAB for 1.5 ≤ γ ≤ 10. These are labeled as optimal linear technique-1/OLT-1-OLT-5. Table VI tabulates 6φ sequences, type of α − β plane vectors used, and the sequence-group and corresponding x, y values of these techniques. Please refer to these states from Fig. 2(a), where the nonzero voltage vectors are of three different lengthslarge (L), medium (M), and small (S). The zero vector, (z, z ), where z ∈ {0, 7}, is denoted by Z. Other optimal techniques, like, S2, x = 0.6, y = 0.5 and S2, x = 0.5, y = 0.4, show optimality for very small region at 1.5 ≤ γ ≤ 2 and are neglected for practical implementation. One should note that OLT-1 is the same as existing C 6φ SVPWM24-C [7]. This is the only optimal technique that exists in the literature. Although 6φSVM3 of [16] uses the active vectors same as OLT-2, zero vectors used by them are different, and so are their current-ripple performances.
Let's define ψ 2 αβ Δ = k 2 fλ 2 αβ and ψ 2 The analytical closed-from expressions of ψ 2 αβ and ψ 2 z 1 z 2 of all of the above optimal vector-sequences are derived and these are fourth-and third-order polynomials, respectively, of two variables, m α and m β . The generalized analytical expressions of ψ 2 αβ and ψ 2 z 1 z 2 is given in (7) shown at the bottom of the next page, and Table VII lists the coefficients of these polynomial expressions of five optimal techniques. Few important observations of this study are as follows.

IV. PROPOSED HYBRID OPTIMAL TECHNIQUE
Based on the machine parameter, γ, falling in one of the three ranges, one can easily identify the optimal techniques in the entire region of a sector and thereafter, implement a optimal technique at every carrier-cycle for the given m. To know the boundary between the optimal regions of ith and jth optimal techniques, one needs to equate their carrier-cycle RMS ripple current as given in (6), i.e., (ψ 2 αβ + γ 2 ψ 2 z 1 z 2 ) i =  (ψ 2 αβ + γ 2 ψ 2 z 1 z 2 ) j and these expressions are given in (7) and Table VII as the function of m α and m β . The boundary between ith and jth optimal techniques, denoted by C ij , i, j ∈ {1, 2, ..5}, is shown in Fig. 8.
This proposed hybrid optimal technique is compared in Fig which is also per-unitized with respect to base current V DC T sw L αβ . Asĩ RMS denotes switching cycle RMS current combining all six phases, and total ripple RMS current seen by all six phases over a line-cycle is same, factor 1 6 is introduced in (8) to obtainĨ 2 RMS of each phase. For fair comparison, we have chosen five existing techniques, which follow rules R1 and R2; these are 1) D 6φ SVPWM24-B2, the best technique of [6], 2) C 6φ SVPWM24-C, 3) Three TINV based techniques (STPWM, DZIPWM, ZS6PWM). As 6φSVM3 of [16] and C 6φ SVPWM24-C have similar performances for all γ values, 6φSVM3 is not considered   Fig. 9(a)-(c) up to the maximum attainable M I s, which are 0.5 and 0.517 for STPWM and ZS6PWM, respectively, and 0.577 for other cases. The curves shown in these plots correspond to the analytical plots and the asterisk marks correspond to the simulated data-points, which will be referred later. Fig. 9(d)-(f) shows the analytical plots ofĨ RMS as function of γ in the range of 1.5 ≤ γ ≤ 10 for three fixed values of M I ,. viz, 0.4, 0.5, and 0.577, respectively. As the maximum M I s attainable by STPWM and ZS6PWM are less than 0.577, these techniques are absent during comparison at M I = 0.577 in Fig. 9(f). tively. This shows that the proposed technique reduces the ripple current RMS by 68.55 % at M I = 0.577 and γ = 10 compared to the best known existing technique. This improvement results in a reduction in copper loss due to the ripple currents, and hence, the efficiency of the drive improves. The improvement in efficiency is significant in the case of medium-voltage fed high-power drives, where the base current is high due to higher V DC and lower F sw . Please see the Appendix for one such example.
It is discussed at the beginning of this section that one needs to evaluate (ψ 2 αβ + γ 2 ψ 2 z 1 z 2 ) of the OLTs to determine the OLT be implemented in the next carrier-cycle. This is one additional step that the proposed technique requires compared to the other existing methods. This step is calculation intensive.
To show the possible carrier-based implementation of the proposed technique, Fig. 10 shows two 180 • phase-shifted triangular carriers, C + or C − , which vary linearly from 0 to 1, and the comparison of duty signals with these carriers to obtain OLT-2 in sector-1. Here, d j and S j are the duty-ratio and the corresponding gating-signal of the top switch of jth-leg. As S a = S b = S c = 0 and S a = S b = S c = 1 at the starting of the carrier cycle, d a , d b , and d c are compared with C + and d a , d b , d c are compared with C − . Sequences of other OLTs can also be generated using C + and C − in the similar manner. The gating signals of the bottom switches are obtained after complementing the gating-signals of the top switches and incorporating dead-time to it.
One should note that when a single OLT is followed, the optimal switching sequences never cause multiple switching commutations during the transition from one state to another. When the reference voltage vector, m, moves from the region of one OLT to another OLT, then also it is possible to avoid multiple commutations at the point of transition if the transiting switching states accord to the Table IX.     Fig. 11(b) shows a 2-pole 5 kW asymmetrical 6φ squirrel-cage induction machine, coupled with a dc generator for loading purposes, which is used for experimental validation. The equivalent circuit parameters of this machine are given in Table X. Due to the skin effect of the rotor bars in squirrel-cage motor, L αβ and L z 1 z 2 seen by the switching-frequency ripple current are different from the equivalent 50 Hz values (L αβ =7.42 mH, L z 1 z 2 =3.75 mH at 50 Hz) [17]. The switching frequency inductance values in α − β and z 1 − z 2 planes can be determined from the piecewise constant voltage applied by the PWM technique and the corresponding slope in the current waveforms. These high-frequency inductance values are indicated in Table X. γ of this full-pitched winding machine is 1.69. SKM75GB123D IGBT-based 6φ inverter along with Zynq-7010 based controller card is shown in Fig. 11(a). Simulation is done using Simscape's specialized power system toolbox of MATLAB-Simulink. In the Simulink model, ideal switches are used in the 6φ inverter, and the differential equations of the dynamic model of ASPM, as given in [3], are implemented in Laplace's transformed domain. Bogacki Shampine solver with a fixed step-size of 1 μs is used in the simulation. The simulation scheme is shown in Fig. 12. The load-torque-calculator block in Fig. 12 estimates the load torque, which is equivalent to loading a dc generator with its output dc voltage connected across a fixed value resistive load (45 Ω). When the inverter output frequency is 50 Hz, the output voltage of the dc generator is 400 V. During the experiment, 800 ns dead-time is incorporated between the two complementary switches of one leg of 6φ inverter. Tektronix  spikes, appearing once in every half-cycle of experimental v ab , v a b waveforms, reveals that the width of this voltage pulse is same as dead-time used in the implementation, i.e., 1 μs. This occurs when two legs undergo dead-time period simultaneously, and therefore, the voltage between these two legs is defined by the diodes in conduction (or the current directions in those two legs), which may be of opposite polarity. Fig. 13(i) and (j) shows the frequency spectrum of experimentally obtained phase current, i a , and line-line voltage, v ab , respectively, upto  Fig. 13(i)] is present in phase current, which arises due to machine or converter asymmetry or dead-time of the inverter and is unavoidable in case of ASPM running in open-loop [18]. Therefore, the harmonic distortions present in the voltages and currents are mainly switching-frequency ripple and the total harmonic distortions (THDs) of the experimental i a and v ab are 5.66% and 55.51%, respectively. Fig. 13(k) shows the six pole voltages, v xN , when m = √ 3 × 0.4e j7.5 • lies in sector-1 and the proposed hybrid technique applies OLT-2. Here, x ∈  {a, b, c, a, b, c } and N is dc-bus negative terminal. The switching sequence applied by 6φ inverter can be derived from this diagram, which is (0, 7 ) − (0, 6 ) − (1, 6 ) − (1, 1 ) − (2, 1 ) − (1, 1 ) − (1, 6 ) − (0, 6 ) − (0, 7 ). This sequence matches with the sequence given in Table VI The analytical performance comparison of the proposed technique with existing techniques, as discussed in Section IV, is first validated through simulation at γ = 2, 6, 10. These simulated data-points are shown in Fig. 9(a)-(c), respectively, by the asterisk marks. The carrier frequency, F s = Fsw k f , is varied from one technique to another based on k f values, as given in Table VIII. As the simulated data points are closely following the analytical plots, this study also reaches the same conclusion about the optimality of the proposed technique as discussed in Section IV. Fig. 13(m) shows both the analytical and simulated performance comparison at γ = 1.69 from where it can be seen that D 6φ SVPWM24-B2 and C 6φ SVPWM24-C are two best performing existing techniques at this γ value. Therefore, these two techniques are chosen along with the proposed PWM strategy for experimental validation of performance comparison. Fig. 13(n) shows this ripple-current comparison. At lower M I , C 6φ SVPWM24-C and proposed technique performs in the similar manner, which is better than D 6φ SVPWM24-B2. As M I increases, the performance of D 6φ SVPWM24-B2 becomes better than C 6φ SVPWM24-C, and the proposed hybrid optimal technique is superior compared to both of these techniques. To calculateĨ RMS of the experimental and simulated waveforms, sampled data-point at a regular interval of 1 μs of a phase current is stored and then processed in the MATLAB script file. The experimental phase current contains 1) fundamental component; 2) few lower-order harmonics due to converter or machine asymmetry or dead-time, as explained before; 3) ripple components at switching frequency and its multiples; and 4) high-frequency transient noise occurring due to transition in six-phase inverter legs. The magnitudes of low-frequency components are negligible after 20th harmonics (maximum 1 kHz). It has been verified through simulation, where the high-frequency switching transient noise is absent in current waveforms that the amplitudes beyond 100 kHz are negligible. Therefore, during evaluation ofĨ RMS of the experimental and simulated current waveforms, RMS currents due to 1 to 100 kHz are considered (F sw = 8.33 kHz) so that only the ripple components are accounted. The maximum error between the experimental and the analytical values of Fig. 13(n) is 6% and the root-mean-square deviation (RMSD) of the experimental results of C 6φ SVPWM24-C, D 6φ SVPWM24-B2 and the proposed strategy are 3.3e-4, 6.8e-4, and 4.8e-4, respectively.

VI. CONCLUSION
This article accounts for an exhaustive set of LMTs of the ASPM, which avoids a major disadvantage present in a few existing LMTs-turning ON and turning OFFa device more than one time within a carrier period. It increases the instantaneous switching frequency of the device for a given average switching frequency. This article first identifies the sequences per three-phase (3φ) inverter basis, where the above limitation was overcome. After that, combining the sequences of two 3φ inverters, this article forms eight sequence-groups. The number of possible techniques in six sequence-groups was infinite because the zero-vector dwell-times of two 3φ inverters can be distributed between the redundant zero states in infinite possible ways. This distribution impacts the current ripple of LMTs. After comparing the current-ripple performances of the techniques of all eight sequence-groups, this article identifies five techniques, which show the optimality within the entire linear region for the whole feasible range of the machine parameter, γ. Only one of these five techniques exists in the literature. After that, an optimal hybrid strategy was proposed, which applies the best of these five techniques in each carrier cycle depending upon the position of the reference voltage vector. For any given γ within the feasible range, the proposed hybrid optimal strategy shows the best ripple current performance in the entire range of modulation indices compared with the existing best-performing techniques, which do not have the disadvantage mentioned above. The proposed strategy shows 68.55% improvement in the ripple performance compared to the existing best-known PWM technique at maximum modulation index for γ = 10. The implementation of this PWM technique involves evaluation of current-ripples of a few optimal methods. This step was calculation-intensive. Simplification of this step and after that finding carrier-based implementation is the possible future work. The theoretical analysis was verified through simulation and experiment on an ASPM with γ = 1.69 up to 3.5 kW power level.

APPENDIX EXAMPLE OF EFFICIENCY IMPROVEMENT IN MEDIUM-VOLTAGE HIGH-POWER MACHINE
Let us consider a 150 h.p. (110 kW), 2 kV, 3φ, 50 Hz, 600 RPM, slip-ring induction motor, B.S.S. 168-1936 machine, whose design is given in [19]. The required V DC to drive this machine in ASPM configuration is 1.5 kV. The machine parameters in 3φ configuration are derived in [19] from which the approximate equivalent parameters in 6φ configuration are deducted. The short-circuit resistance, L αβ are 0.675 Ω and 4.23 mH; let the γ of the machine is 10. Let, F sw of this high-power drive is limited to 1.5 kHz. Therefore, the base current, V DC Tsw L αβ , is 236.41 A. The expected performance for γ = 10 is given in Fig. 9(c), from where the copper losses occur at M I = 0.577 due to the proposed hybrid technique and the existing best technique (D 6φ SVPWM24-B2) are estimated. These values are 565.8 W and 2264 W, respectively. Therefore, the proposed technique improves the efficiency by 1.54% compared to the best state-of-the-art technique.