Zero-Sequence Current Suppression Strategy for a Common DC Bus OW-FPPMSM with Third-Harmonic Current Injection

Abstract

In the open-winding motor fed by a common DC bus, unbalanced inverter common-mode voltage (CMV), zero-sequence components of the permanent magnet flux linkage, and the PWM dead-time effect can induce a zero-sequence current (ZSC) through the inherent current path. For an open-winding five-phase permanent magnet synchronous motor (OW-FPPMSM) applied in an aerospace rocket starter-generator system, two ZSC suppression strategies based on zero-sequence voltage (ZSV) generation mechanisms are proposed in this paper, which improve motor performance in a simple and efficient manner. In the first strategy, the conventional method is modified to enable asynchronous operation of the two inverters, thereby generating the required ZSV pulses. The switching order and time offset between the two inverters are determined by the reference ZSV. The second strategy employs basic voltage vectors with larger magnitudes, resulting in higher DC bus voltage utilization. By adjusting the switching sequence of the second inverter, the ZSC components at the carrier frequency are eliminated. Both strategies also achieve the injection of the third-harmonic current. Finally, the two strategies are further analyzed in terms of the modulation index and ZSV modulation range. Simulation and experimental results verify the effectiveness of the ZSC suppression strategies.

1. Introduction

An open-winding motor refers to a machine in which the neutral point of armature windings is opened and brought out, and the two ends of each winding are connected to separate voltage-source inverters. To reduce system size and cost, the two inverters can share one DC source, forming a common DC bus topology. However, this topology creates a path for zero-sequence current (ZSC). The ZSC causes additional losses, thermal stress, and torque pulsations, which adversely affect the efficiency and operational performance of the motor [1]. Therefore, effective ZSC suppression strategies are required.

For open-winding motors, two-level inverters are commonly used as power converters [2,3,4]. In addition, five-level stacked inverters have been proposed in [5]. Three-to-five-phase matrix converters have also been employed to feed open-winding induction motors directly from the grid [6]. Regardless of the converter topology, the converter-induced zero-sequence voltage (ZSV) can ideally be eliminated by making the common-mode voltages (CMVs) at both winding ends identical, thereby suppressing the ZSC [7,8]. However, this method only suppresses the ZSC caused by CMV differences between the two inverters. When a zero-sequence component exists in the permanent magnet flux linkage, or dead time exists in the PWM signals applied to the inverter [9], ZSC will still be generated.

The inverter clamping strategy is a widely adopted method for suppressing ZSC. By clamping one inverter to a fixed switching state, different ZSVs are generated while the other inverter switches, thereby compensating for the ZSC [10,11,12]. To balance the workload of the two inverters, they are alternately clamped in odd and even sectors, allowing each inverter to operate at a high switching frequency in turn and thus reducing the failure risk [13,14,15]. In [16,17], the reference voltage vector is decomposed in the stationary reference frame, thereby determining the duty cycle difference between the two switching signals of each phase. When one inverter is clamped, the switching signals of the other inverter can be obtained from this difference. This method is more suitable for carrier-based SPWM schemes. However, the clamping inverter limits the maximum linear modulation range of the control method. To provide a time margin for synthesizing the ZSV, the dwell time of the effective voltage vector is reduced in [18]. Alternatively, flux-weakening control can be employed to decrease the permanent magnet back electromotive force (EMF), thereby reducing the magnitude of the reference voltage vector [19].

For a single inverter, the CMV can be eliminated by using virtual voltage vectors, which are synthesized at the center of two adjacent basic voltage vectors [20,21]. In [22], this virtual vector is combined with an alternating clamping strategy and is applied to a five-phase open-winding induction motor. When all legs of the non-clamped inverter are in the same state, the system generates the maximum ZSV, which can serve as an adjustable term to compensate for the ZSV caused by inherent factors.

In [23], a 3D space for the reference voltage vector is established by introducing the ZSV as a vertical axis to the α-β plane. To avoid the complex tetrahedron identification in 3D space, Ref. [24] clamps the inverter legs based on the polarity of the reference phase voltages, thereby reducing the switching operations of inverters. The computational load of voltage vector selection is also reduced.

Vector control for multiphase machines typically requires multiple PI controllers. To suppress the ZSC, additional controllers for AC signal regulation must also be introduced [25,26,27,28]. To simplify the control structure, model predictive control (MPC) techniques have been widely applied to open-winding motor systems [29,30,31]. MPC relies on an accurate dynamic model, treating the motor and inverters as an integrated system to establish the discretized state equations. However, inevitable simplifications in modeling may cause dynamic or parameter errors, which affect the prediction performance. Furthermore, the method brings a considerable computational load and, therefore, requires higher processing capability from the digital processor.

To reduce model dependency and enhance system robustness, several advanced nonlinear control methods have been introduced to motor drive applications. For instance, a composite adaptive super-twisting sliding mode control employs the super-twisting algorithm to achieve finite-time convergence of current errors, while imposing constraints on system states by incorporating a barrier function [32,33]. This provides a promising approach for ZSC control in open-winding motor systems. In addition, a super-twisting-like fractional controller uses the memory effect of fractional-order operators to improve the dynamic current response in a three-phase PMSM system [34]. It also enhances steady-state accuracy under parameter variations. These methods offer clear advantages in handling nonlinearity and uncertainties. However, their control laws are typically complex, leading to a relatively high implementation effort in multiphase open-winding systems. Consequently, further exploration of control strategies that balance high performance with implementation feasibility remains of significant importance for ZSC suppression.

For open-winding motors, especially multiphase open-winding systems, the dual inverters provide significantly more switching combinations than a single inverter, resulting in a large number of basic voltage vectors. Consequently, the main challenge in suppressing the ZSC lies in optimizing the control method under limited computational and hardware resources.

This paper analyzes the ZSC path in an open-winding five-phase permanent magnet synchronous motor (OW-FPPMSM) system with a common DC bus topology. The motor is used in an integrated starter–generator system for an aerospace rocket and is directly connected to the engine by a common shaft. Accordingly, two SVPWM modulation strategies with closed-loop ZSC control are proposed. Based on newly developed ZSV generation logics, these strategies achieve effective ZSC suppression while improving motor performance in a simple and efficient way. Instead of passive suppression or indirect compensation used in conventional methods, they enable active generation and control of the ZSV.

The main contributions are as follows:

• The first strategy is modified from the conventional ZSV-free SVPWM strategy. It generates the required ZSV pulses by introducing a switching offset between the two inverters. The switching order and time offset are determined by the magnitude and direction of the reference ZSV;
• The second strategy employs another set of basic voltage vectors with larger magnitudes. Thus, it achieves a higher modulation index. To suppress the high-frequency component corresponding to the switching period in ZSV, the switching sequence of the second inverter is further adjusted;
• Both strategies enable the injection of the third-harmonic current. The modulation index and the ZSV modulation range of the two strategies are analyzed.

The rest of this paper is organized as follows. Section 2 introduces the electromagnetic structure of the OW-FPPMSM and analyzes the inherent ZSC path. Section 3 proposes an improved ZSV-free SVPWM strategy. Section 4 presents an asymmetric-CMV SVPWM strategy using larger basic voltage vectors. Section 5 analyzes the modulation indices and ZSV modulation ranges of the two methods. Section 6 shows the experimental platform and presents the results. Finally, Section 7 concludes the paper.

2. ZSC Path in OW-FPPMSM System with Common DC Bus

This study focuses on a 10-slot 12-pole fractional-slot concentrated-winding OW-FPPMSM, whose structure is shown in Figure 1. Each phase winding consists of two series-connected coils wound in opposite directions. The armature reaction magnetomotive forces (MMFs) produced by these two coils cancel each other in the air gaps of other phases, and therefore, no armature reaction mutual inductance exists among the phase windings. To further minimize slot leakage mutual inductance, small teeth are added at the center of the common slots between two adjacent phases. This allows the slot leakage flux to circulate through the small teeth, preventing it from linking with adjacent phase windings. Furthermore, thermal insulation panels are installed on both sides of the small teeth, ultimately achieving the design goal of low thermal coupling and no electromagnetic coupling between phase windings. The integration of electromagnetic isolation, thermal isolation, and the open-winding structure significantly enhances the fault-tolerant capability of the motor.

The coil span is equal to the pole pitch, resulting in a winding factor of one for all harmonic orders. The back EMF of the phase winding is a flat-top wave with third-harmonic components. Therefore, injecting the third harmonics into the phase current can not only reduce its peak value and lighten the load on power devices, but also generate a constant electromagnetic torque to improve the load capacity of the motor.

The voltage equation of the motor can be obtained as

$${\mathit{u}}_{\mathrm{s}}=R{\mathit{i}}_{\mathrm{s}}+L{\displaystyle \frac{\mathrm{d}{\mathit{i}}_{\mathrm{s}}}{\mathrm{d}t}}+{\mathit{e}}_{\mathrm{s}}$$

(1)

where R and L are the phase resistance and inductance, respectively. u_s, i_s, and e_s are the matrices of phase voltage, current, and permanent magnet back EMF, respectively:

$$\left\{\begin{array}{l}{\mathit{u}}_{\mathrm{s}}={\left[\begin{array}{ccccc}{u}_{\mathrm{A}}& u_{\mathrm{B}}& u_{\mathrm{C}}& u_{\mathrm{D}}& u_{\mathrm{E}}\end{array}\right]}^{\mathrm{T}}\\ {\mathit{i}}_{\mathrm{s}}={\left[\begin{array}{ccccc}{i}_{\mathrm{A}}& i_{\mathrm{B}}& i_{\mathrm{C}}& i_{\mathrm{D}}& i_{\mathrm{E}}\end{array}\right]}^{\mathrm{T}}\\ {\mathit{e}}_{\mathrm{s}}={\left[\begin{array}{ccccc}{e}_{\mathrm{A}}& e_{\mathrm{B}}& e_{\mathrm{C}}& e_{\mathrm{D}}& e_{\mathrm{E}}\end{array}\right]}^{\mathrm{T}}\end{array}\right.$$

(2)

The back EMF matrix is

$${\mathit{e}}_{\mathrm{s}}={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\mathit{\psi }}_{\mathrm{f}}$$

(3)

where ψ_f is the matrix of permanent magnet flux linkage. In addition to the fundamental and third-harmonic components, it further includes a zero-sequence component dominated by the fifth harmonic:

$${\mathit{\psi }}_{\mathrm{f}}=\left[\begin{array}{l}{\psi }_{\mathrm{fA}}\\ {\psi }_{\mathrm{fB}}\\ {\psi }_{\mathrm{fC}}\\ {\psi }_{\mathrm{fD}}\\ {\psi }_{\mathrm{fE}}\end{array}\right]={\psi }_{\mathrm{f}1}\left[\begin{array}{l}\cos {\omega }_{e}t\\ \cos \left({\omega }_{e}t-\gamma \right)\\ \cos \left({\omega }_{e}t-2\gamma \right)\\ \cos \left({\omega }_{e}t-3\gamma \right)\\ \cos \left({\omega }_{e}t-4\gamma \right)\end{array}\right]+{\psi }_{\mathrm{f}3}\left[\begin{array}{l}\cos 3{\omega }_{e}t\\ \cos 3\left({\omega }_{e}t-\gamma \right)\\ \cos 3\left({\omega }_{e}t-2\gamma \right)\\ \cos 3\left({\omega }_{e}t-3\gamma \right)\\ \cos 3\left({\omega }_{e}t-4\gamma \right)\end{array}\right]+{\psi }_{\mathrm{f}5}\left[\begin{array}{l}\cos 5{\omega }_{e}t\\ \cos 5{\omega }_{e}t\\ \cos 5{\omega }_{e}t\\ \cos 5{\omega }_{e}t\\ \cos 5{\omega }_{e}t\end{array}\right]$$

(4)

Here, γ = 2π/5; ψ_f1, ψ_f3, and ψ_f5 are the amplitudes of the fundamental, third-harmonic, and fifth-harmonic permanent magnet flux linkage components, respectively. ω_e is the fundamental angular frequency.

To simplify the mathematical model and decouple the phase variables, the motor equations are transformed into the synchronous rotating reference frame. Based on the amplitude-invariant principle, the Clarke transformation is first applied to map the five-phase stationary variables into the α-β stationary frame:

$${\mathit{T}}_{\mathrm{Clark}}={\displaystyle \frac{2}{5}}\left[\begin{array}{ccccc}1& \cos \gamma & \cos 2\gamma & \cos 3\gamma & \cos 4\gamma \\ 0& \sin \gamma & \sin 2\gamma & \sin 3\gamma & \sin 4\gamma \\ 1& \cos 3\gamma & \cos 6\gamma & \cos 9\gamma & \cos 12\gamma \\ 0& \sin 3\gamma & \sin 6\gamma & \sin 9\gamma & \sin 12\gamma \\ 1/2& 1/2& 1/2& 1/2& 1/2\end{array}\right]$$

(5)

Subsequently, the components in the α-β stationary frame are transformed into the d-q rotating frame by Park transformation:

$${\mathit{T}}_{\mathrm{Park}}=\left[\begin{array}{ccccc}\cos {\omega }_{e}t& \sin {\omega }_{e}t& 0& 0& 0\\ -\sin {\omega }_{e}t& \cos {\omega }_{e}t& 0& 0& 0\\ 0& 0& \cos 3{\omega }_{e}t& \sin 3{\omega }_{e}t& 0\\ 0& 0& -\sin 3{\omega }_{e}t& \cos 3{\omega }_{e}t& 0\\ 0& 0& 0& 0& 1\end{array}\right]$$

(6)

By applying the aforementioned transformations to Equation (1), the stator voltage equations in the synchronous reference frame can be obtained as:

$$\left[\begin{array}{c}{u}_{d1}\\ u_{q1}\\ u_{d3}\\ u_{q3}\\ u_0\end{array}\right]=R\left[\begin{array}{c}{i}_{d1}\\ i_{q1}\\ i_{d3}\\ i_{q3}\\ i_0\end{array}\right]+{\mathit{L}}_{dq}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\begin{array}{c}{i}_{d1}\\ i_{q1}\\ i_{d3}\\ i_{q3}\\ i_0\end{array}\right]+ {\omega }_{\mathrm{e}}\left[\begin{array}{c}-L_{q1}{i}_{q1}\\ L_{d1}{i}_{d1}+{\psi }_{f1}\\ -3L_{q3}{i}_{q3}\\ 3L_{d3}{i}_{d3}+3{\psi }_{f3}\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ e_0\end{array}\right]$$

(7)

where L_dq = diag[L_d1 L_q1 L_d3 L_q3 L₀] is the inductance matrix, while subscripts 1, 3, and 0 denote the fundamental, third-harmonic, and zero-sequence components, respectively. e₀ is the zero-sequence back EMF. Thus, the zero-sequence voltage is obtained as

$$u_0=R{i}_0+L_0{\displaystyle \frac{\mathrm{d}{i}_0}{\mathrm{d}t}}+e_0$$

(8)

Figure 2 shows the topology of the common DC bus OW-FPPMSM drive system, where the winding beginnings and endings are marked by the uppercase letters ABCDE and the lowercase letters abcde, respectively. As the midpoints of the dual inverters are coincident, the ZSV circulates through the shared DC bus, resulting in the generation of ZSC.

Since the inverter output voltages are pulse signals with identical amplitudes but different duty cycles, each inverter generates a CMV, as expressed in Equation (9).

$$\left\{\begin{array}{l}{u}_{\mathrm{cm}1}={\displaystyle \frac{U_{\mathrm{dc}}}{5}}\left(S_{\mathrm{A}1}+S_{\mathrm{B}1}+S_{\mathrm{C}1}+S_{\mathrm{D}1}+S_{\mathrm{E}1}\right)-{\displaystyle \frac{U_{\mathrm{dc}}}{2}}\\ u_{\mathrm{cm}2}={\displaystyle \frac{U_{\mathrm{dc}}}{5}}\left(S_{\mathrm{A}2}+S_{\mathrm{B}2}+S_{\mathrm{C}2}+S_{\mathrm{D}2}+S_{\mathrm{E}2}\right)-{\displaystyle \frac{U_{\mathrm{dc}}}{2}}\end{array}\right.$$

(9)

where U_dc is the DC-bus voltage, and S_X1 and S_X2 represent the switching states of phase X leg in the first inverter (INV1) and the second inverter (INV2), respectively (for X = A, B, C, D, E). Specifically, S_X1 = 1 indicates that the upper switch is on and the lower switch is off, whereas S_X1 = 0 indicates the opposite state.

According to Figure 2, the phase voltages output by the inverters are expressed as:

$$\left[\begin{array}{c}{u}_{\mathrm{A}}\\ u_{\mathrm{B}}\\ u_{\mathrm{C}}\\ u_{\mathrm{D}}\\ u_{\mathrm{E}}\end{array}\right]=U_{\mathrm{dc}}\left[\begin{array}{c}{S}_{\mathrm{A}1}-S_{\mathrm{A}2}\\ S_{\mathrm{B}1}-S_{\mathrm{B}2}\\ S_{\mathrm{C}1}-S_{\mathrm{C}2}\\ S_{\mathrm{D}1}-S_{\mathrm{D}2}\\ S_{\mathrm{E}1}-S_{\mathrm{E}2}\end{array}\right]$$

(10)

When the CMVs of the two inverters are unequal, a ZSV is generated, as expressed in Equation (11).

$$u_0=u_{\mathrm{cm}1}-u_{\mathrm{cm}2}={\displaystyle \frac{1}{5}}\left(u_{\mathrm{A}}+u_{\mathrm{B}}+u_{\mathrm{C}}+u_{\mathrm{D}}+u_{\mathrm{E}}\right)$$

(11)

Based on Equations (8) and (11), the zero-sequence equivalent circuit of the common DC bus OW-FPPMSM drive system is obtained, as shown in Figure 3.

In the common DC bus configuration, the zero-sequence equivalent circuit forms a closed loop. Both the zero-sequence back-EMF and ZSV from the dual inverters excite the ZSC. This current does not contribute to electromagnetic torque. However, it increases electromagnetic losses and leads to undesirable effects, such as shaft voltage, torque ripple, and winding overheating. Therefore, the ZSC must be suppressed.

3. Improved ZSV-Free SVPWM Strategy

3.1. Conventional ZSV-Free SVPWM Strategy

The dual inverters in the OW-FPPMSM control system contain 10 legs, generating 2¹⁰ = 1024 basic voltage vectors. According to Equation (11), to eliminate the ZSV, the switching states of the dual inverters must satisfy Equation (12). A total of 252 basic voltage vectors meet this condition.

$$S_{\mathrm{A}1}+S_{\mathrm{B}1}+S_{\mathrm{C}1}+S_{\mathrm{D}1}+S_{\mathrm{E}1}=S_{\mathrm{A}2}+S_{\mathrm{B}2}+S_{\mathrm{C}2}+S_{\mathrm{D}2}+S_{\mathrm{E}2}$$

(12)

In the selection of basic voltage vectors, those with larger magnitudes are preferred. Meanwhile, the switching between adjacent vectors should change the switching state of only one phase leg. The distribution of the final selected basic voltage vectors in the fundamental and third-harmonic planes is shown in Figure 4.

The subscripts in the figure can be converted into binary form to represent the values of S_A1–S_E1 and S_A2–S_E2, corresponding to the switching states of the two inverters. However, in practical operation, dead time is inserted into the PWM signals to prevent the simultaneous conduction of the two switches in one inverter leg, which leads to a ZSV at five times the fundamental frequency [10]. A longer dead time results in a larger ZSV. In addition, the zero-sequence component of the back EMF also induces ZSC. Therefore, this control method requires further optimization.

3.2. Improvement of ZSV-Free SVPWM Strategy

In the conventional method, to avoid ZSV generation, Equation (12) must be satisfied at all times. Consequently, the two inverters are always switched on or off simultaneously. Within a switching period, the dwell time of nonzero voltage vectors is determined by the reference voltage vector, whereas the zero voltage vectors U₀ and U₃₁ only affect the CMV and are independent of the reference voltage vector. Based on this, the switching order and time offset of the two inverters can be controlled to generate a ZSV with the required duration and direction, thereby compensating the ZSC in the system. The reference ZSV $u_0^{\ast }$ is obtained through the ZSC feedback loop. Taking the fundamental reference voltage vector ${\mathit{u}}_1^{\ast }$ in Sector I as an example, the detailed implementation is shown in Figure 5.

Since $u_0^{\ast }$ is an AC variable, different adjustments should be taken according to its direction. Figure 5 shows the case for $u_0^{\ast }>0$. Based on the conventional method, INV1 advances the first switching instant of each leg by T_Δ/2 and delays the second by T_Δ/2. In contrast, INV2 delays the first switching instant by T_Δ/2 and advances the second by T_Δ/2. After the adjustment, the dwell time of the nonzero vector in both inverters remains unchanged. However, INV1 increases the dwell time of U₃₁ by T_Δ and decreases U₀ by T_Δ; INV2 shows the opposite change, it decreases the dwell time of U₃₁ by T_Δ and increases U₀ by T_Δ. As a result, 10 ZSV pulses with an amplitude of +U_dc/5 and a duration of T_Δ are generated within one switching period. For $u_0^{\ast }<0$, the adjustments are reversed, resulting in 10 ZSV pulses with an amplitude of −U_dc/5.

According to the principle of volt-second balance, the inverter can generate a ZSV equivalent to the reference when T_Δ satisfies the following equation. Here, T_s denotes the PWM switching period.

$$T_{\Delta }=\left|u_0^{\ast }\right|T_{\mathrm{s}}/\left(10\times {\displaystyle \frac{1}{5}}{U}_{\mathrm{dc}}\right)={\displaystyle \frac{\left|u_0^{\ast }\right|T_{\mathrm{s}}}{2U_{\mathrm{dc}}}}$$

(13)

In the PWM generation method, the triangular carrier waveform is centrally symmetric, so adding or subtracting a value to the modulation signal will symmetrically shift the PWM edges. In this way, the rising and falling edges move either toward or away from the center simultaneously. Therefore, the duty cycle is modified. For this work, by substituting $u_0^{\ast }$ into Equation (13) instead of its absolute value $\left|u_0^{\ast }\right|$, the offset T_Δ/2 inherently becomes a signed variable. Consequently, the corresponding ZSV can be generated by simply adding T_Δ/2 to the modulation signal of INV1 and subtracting T_Δ/2 from that of INV2, regardless of the sign of the offset. Figure 6 shows the offset process of the PWM signal for INV1. When T_Δ is positive, the duty cycle of S_X1 increases, while that of S_X2 decreases, producing a positive ZSV. Conversely, when T_Δ is negative, the duty cycle of S_X1 decreases, and that of S_X2 increases, resulting in a negative ZSV (for X = A, B, C, D, E).

3.3. Third-Harmonic Current Injection

Since the rotor permanent magnet flux contains a certain third-harmonic component, the control system adopts a strategy with the third-harmonic current injection.

First, the reference voltage vector ${\mathit{u}}_3^{\ast }$ is obtained through the third-harmonic current feedback loop. Once the basic voltage vectors are determined based on the sector of ${\mathit{u}}_1^{\ast }$ in the fundamental plane, the corresponding vectors in the third-harmonic plane are located to synthesize ${\mathit{u}}_3^{\ast }$.

As shown in Figure 7, the distribution of the basic voltage vectors in the dual planes follows a specific mapping pattern. When ${\mathit{u}}_1^{\ast }$ lies in the N-th sector, the four nearest voltage vectors are selected, as shown in Figure 7a, where the phase angles of U_i and U_iii are (2N−3)π/10. In the third-harmonic plane, shown in Figure 7b, the phase difference between U_i and U_iii is π, and that between U_ii and U_iv is also π. Furthermore, U_i lags U_iv by 3π/5, and the phase angle of U_i is (6N − 9)π/10. It can be seen that the basic voltage vectors in both planes are uniquely determined by the sector number of ${\mathit{u}}_1^{\ast }$. Consequently, once the reference vectors ${\mathit{u}}_1^{\ast }$ and ${\mathit{u}}_3^{\ast }$ are obtained, the dwell times of these basic voltage vectors can be calculated to enable the third-harmonic current injection.

3.4. Control System Design

To verify the effectiveness of the proposed method in suppressing ZSC and injecting third-harmonic current, a control system is established, as shown in Figure 8. The system adopts the i_d = 0 vector control, and an additional ZSC feedback loop is incorporated.

The injection ratio of the third-harmonic current to the fundamental is set as k, and the value of k is determined to minimize the peak value of the phase current. By aligning the valley of the third-harmonic current with the positive peak of the fundamental current, the phase current can be expressed as

$$i=I_1\sin \left({\theta }_e\right)+k{I}_1\sin \left(3{\theta }_e\right)$$

(14)

where I₁ is the amplitude of the fundamental current, and ${\theta }_e$ is its phase.

Based on the symmetry of the phase current waveform, ${\theta }_e$ can be defined within the range of (0, π/2). In this interval, the derivative of the phase current, with respect to ${\theta }_e$, is given by

$${\displaystyle \frac{\mathrm{d}i}{\mathrm{d}{\theta }_e}}=I_1\cos \left({\theta }_e\right)+3k{I}_1\cos \left(3{\theta }_e\right)$$

(15)

When the derivative is zero, the phase current reaches its peak value, i_max. It follows that:

$$\sin \left({\theta }_e\right)={\displaystyle \frac{\sqrt{3k\left(3k+1\right)}}{6k}}$$

(16)

Substituting it into Equation (14), i_max can be expressed as:

$${\left(i_{\max }\right)}^2=I_1^2\left(k^2+k+{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{27k}}\right)$$

(17)

As the third-harmonic content increases, the peak value of the phase current first decreases and then increases. Taking the derivative of the above expression with respect to k and setting it to zero, the value of k can be obtained as:

$$k={\displaystyle \frac{1}{6}} \mathrm{or}-{\displaystyle \frac{1}{3}}$$

(18)

Clearly, k = 1/6 should be chosen, at which the peak phase current is minimized.

The ZSC caused by the dead-time effect and zero-sequence back-EMF is dominated by an AC component at five times the fundamental frequency. The proportional-resonant (PR) controller can effectively track AC signals. It achieves the maximum gain at the resonant frequency. Therefore, a PR controller is employed in the ZSC loop to obtain the reference ZSV. The transfer function of the quasi-PR controller is:

$$G\left(s\right)=k_p+{\displaystyle \frac{2k_r{\omega }_{c}s}{s^2+2{\omega }_{c}s+{\omega }_0^2}}$$

(19)

where ω₀ is the resonant frequency, which should be set to five times the fundamental frequency; ω_c is the cut-off frequency used to broaden the controller bandwidth; and k_p and k_r are the proportional and resonant gains, respectively.

Figure 9 shows the frequency responses of the PR controller at a resonant frequency of 400 Hz. Increasing k_r enhances the gain at the resonant frequency, thereby improving the tracking performance for sinusoidal signals at that frequency. However, an excessively large k_r may amplify noise and induce oscillations. On the other hand, increasing ω_c broadens the resonant bandwidth while maintaining nearly the same peak gain, which improves robustness to frequency deviations. But an overly large ω_c may weaken the frequency selectivity and degrade the tracking accuracy. To provide sufficient system bandwidth and ensure stability, the final parameters selected for this study are k_p = 70, k_r = 10, and ω_c = 2π rad/s.

The main design parameters of the OW-FPPMSM studied in this work are listed in

Table 1. Main parameters of OW-FPPMSM.

Parameter	Value	Parameter	Value
Rated power (kW)	15	d-axis inductance (mH)	1.595
DC bus (V)	270	q-axis inductance (mH)	1.827
Rated speed (r/min)	1350	Fundamental PM flux linkage (Wb)	0.0945
Coil turns	18	Third-harmonic PM flux linkage (Wb)	0.0034

.

3.5. Simulation Results

This motor works in a cryogenic environment filled with liquid oxygen and operates under short-time intermittent duty. To ensure safe testing at room temperature, derated operation is adopted. The effectiveness of the proposed strategy is not affected by temperature variations as it does not rely on motor parameters.

The simulation is conducted at a speed of 800 r/min with a load of 20 N·m. Initially, only the third-harmonic current is injected, and the ZSC suppression strategy is implemented at 0.5 s. Figure 10a shows the phase current waveforms. Due to the injected third-harmonic current, the phase currents exhibit flat-top waveforms. However, without ZSC control, the waveforms show certain distortions. After 0.5 s, the current quality is significantly improved. A Fast Fourier Transform (FFT) analysis is performed on the steady-state currents before and after 0.5 s, with the results shown in Figure 10b. The third-harmonic contents are 16.68% and 16.73% of the fundamental component, respectively, which satisfy the control requirement. The fifth-harmonic content decreases from 5.87% to 0.29% of the fundamental component.

Since the third harmonic is intentionally injected, a modified THD excluding the third-harmonic component is defined as:

$${\mathrm{THD}}_{\mathrm{ex}3}=\sqrt{{\mathrm{THD}}^2-{\left(I_3/I_1\right)}^2}$$

(20)

where I₃ is the amplitude of the third-harmonic current. Under this definition, the THD_ex3 of the phase current is reduced from 7.06% to 2.66%, showing a significant improvement.

Figure 11a shows the ZSC waveform, whose peak value decreases from 1.2 A to approximately 0.2 A. FFT results are shown in Figure 11b. The fifth-harmonic current decreases from 0.82 A to 0.06 A, corresponding to a reduction of approximately 92.7%, indicating the effective suppression by the proposed strategy. Due to the asymmetry of the phase currents, certain fundamental and second-order harmonic components appear in the ZSC. These components are also eliminated after applying the improved control strategy.

To verify the effect of the third-harmonic injection, comparative simulations are carried out by setting the injection ratio to zero. Figure 12 shows the phase-A current waveforms before and after the injection. When k = 0, the phase current exhibits a standard sinusoidal waveform with a peak value of approximately 14 A. When k = 1/6, the waveform becomes flat-topped, and the peak value is reduced to approximately 12 A, indicating that third-harmonic injection effectively reduces the peak phase current.

Figure 13 shows the torque and speed responses during the process. It can be observed that, in the acceleration stage, the injection of third-harmonic current slightly increases the output torque, thereby accelerating the speed response and enabling the motor to reach the reference speed more quickly.

As shown in Figure 12 and Figure 13, the injection of third-harmonic current significantly reduces the peak value of the phase current and slightly enhances the electromagnetic torque. Under this condition, no noticeable increase in torque ripple is observed. In addition, the RMS current after the injection becomes:

$$I_{\mathrm{RMS}}=\sqrt{I_1^2+I_3^2}=\sqrt{I_1^2+{\left(I_1/6\right)}^2}\approx 1.014I_1$$

(21)

It can be seen that, although harmonic injection may introduce extra losses, the variation in RMS current remains relatively small. Therefore, the overall impact on efficiency and thermal performance is not expected to be significant.

4. Asymmetric-CMV SVPWM Strategy

4.1. Reselect Basic Voltage Vector

In the previous section, the SVPWM method does not use the basic voltage vectors with the largest magnitude. Among 1024 available basic voltage vectors, the maximum magnitude is 1.294U_dc. There are 10 such vectors, and they are distributed at 0°, 36°, 72°, …, 324°. Using these vectors can improve the utilization of the DC bus voltage. In addition, a nonzero ZSV is generated because the CMVs of the two inverters are unequal. The reference ZSV can, therefore, be synthesized directly. Considering the injection of the third-harmonic current, the basic voltage vectors shown in Figure 14 are finally selected.

The distribution of the basic voltage vectors in the fundamental and third-harmonic planes also follows a specific mapping pattern. When ${\mathit{u}}_1^{\ast }$ lies in the N-th sector (0 to π/5 defined as Sector I), the four nearest voltage vectors are shown in Figure 15a, where the phase angles of U_i and U_iii are (N − 1)π/5. In the third-harmonic plane shown in Figure 15b, the phase difference between U_i and U_iii is π, and that between U_ii and U_iv is also π. Moreover, U_i lags U_iv by 3π/5, and the phase angle of U_i is (3N − 3)π/5. It can be seen that the basic voltage vectors in both planes are uniquely determined by the sector number of ${\mathit{u}}_1^{\ast }$. Consequently, once the reference vectors ${\mathit{u}}_1^{\ast }$ and ${\mathit{u}}_3^{\ast }$ are obtained, the dwell times of U_i-U_iv can be calculated accordingly.

On the zero-sequence axis shown in Figure 15c, the directions of the vectors depend on the parity of the sector number, N, as expressed in Equation (22). Once the reference ZSV is obtained, the dwell times of U_0-31 and U_31-0 can be determined with the constraint of the switching period T_s.

$$\left\{\begin{array}{l}{\mathit{U}}_{\mathrm{i}}=-0.6U_{\mathrm{dc}}\\ {\mathit{U}}_{\mathrm{ii}}=-0.2U_{\mathrm{dc}}\\ {\mathit{U}}_{\mathrm{iii}}=0.2U_{\mathrm{dc}}\\ {\mathit{U}}_{iv}=0.6U_{\mathrm{dc}}\end{array}\right.\left(\mathrm{odd}\right)\hspace{1em}\left\{\begin{array}{l}{\mathit{U}}_{\mathrm{i}}=0.6U_{\mathrm{dc}}\\ {\mathit{U}}_{\mathrm{ii}}=0.2U_{\mathrm{dc}}\\ {\mathit{U}}_{\mathrm{iii}}=-0.2U_{\mathrm{dc}}\\ {\mathit{U}}_{iv}=-0.6U_{\mathrm{dc}}\end{array}\right.\left(\mathrm{even}\right)$$

(22)

As observed, the proposed strategy not only employs basic voltage vectors with larger magnitudes but also allows for direct generation of the ZSV. Therefore, it inherently possesses the capability to suppress ZSC.

4.2. Adjustment of INV2 Switching Sequence

Generally, the ZSC is dominated by the fifth-harmonic component. The above asymmetric-CMV SVPWM strategy can effectively suppress this harmonic. However, it introduces undesired high-frequency disturbances related to the switching period. Taking ${\mathit{u}}_1^{\ast }$ in Sector I as an example, the following section provides a detailed analysis and proposes a corresponding solution.

When ${\mathit{u}}_1^{\ast }$ lies in Sector I, the selected basic voltage vectors are applied in the following sequence: U_0-31, U_16-15, U_24-7, U_25-6, U_29-2, and U_31-0. Taking phase A as an example, the switching sequences of phase-A legs in the two inverters, together with the phase-A voltage, are shown in Figure 16.

It can be seen that, within one switching period, the switching states of INV1 and INV2 are always complementary. As a result, the phase-A voltage changes from −U_dc to +U_dc and then returns to −U_dc. The phase voltage changes only twice in each period, and the variation is large. Consequently, the phase current rises or falls significantly in different intervals, leading to the sawtooth waveform shown in Figure 17.

As shown in Figure 16, the +U_dc interval is centrally distributed within each switching period. The sawtooth component in the phase current varies synchronously with the phase voltage, thereby exhibiting an approximate center symmetry. This symmetry indicates, on the one hand, that the frequency of the sawtooth components corresponds to the PWM switching period. On the other hand, it causes the phases of these harmonics in different currents to be close to each other. When these harmonics are superimposed, they result in a harmonic component of the same frequency in the ZSC.

To suppress the ZSC, the switching timing of the inverters needs to be reorganized. Since the voltage vector generated by the dual-inverter system is the difference between the voltage vectors of the two individual inverters, the relationship between the reference vector ${\mathit{u}}^{\ast }$ and the basic vectors can be expressed as

$$\begin{array}{cc}\hfill {\mathit{u}}^{*}{T}_{\mathrm{s}}=& {\mathit{U}}_{0\text{-}31}{T}_0+{\mathit{U}}_{16\text{-}15}{T}_1+{\mathit{U}}_{24\text{-}7}{T}_2+{\mathit{U}}_{25\text{-}6}{T}_3+{\mathit{U}}_{29\text{-}2}{T}_4+{\mathit{U}}_{31\text{-}0}{T}_5\hfill \\ \hfill =& \left({\mathit{U}}_0{T}_0+{\mathit{U}}_{16}{T}_1+{\mathit{U}}_{24}{T}_2+{\mathit{U}}_{25}{T}_3+{\mathit{U}}_{29}{T}_4+{\mathit{U}}_{31}{T}_5\right)-\hfill \\ \hfill & \left({\mathit{U}}_0{T}_5+{\mathit{U}}_2{T}_4+{\mathit{U}}_6{T}_3+{\mathit{U}}_7{T}_2+{\mathit{U}}_{15}{T}_1+{\mathit{U}}_{31}{T}_0\right)\hfill \end{array}$$

(23)

From the above equation, the synthesis of the reference voltage vector depends on the dwell times of the basic vectors, regardless of their application sequence. This is also consistent with the volt-second balance principle. Therefore, without changing the PWM duty cycle, the high level in the switching sequence of INV2 can be shifted to the center of the switching period, making the switching states follow the same 0-1-0 pattern as INV1. The resulting switching sequences of the phase-A legs in both inverters, along with the corresponding phase-A voltage, are shown in Figure 18.

It can be seen that the adjusted phase-A voltage switches only between 0 and +U_dc, instead of between −U_dc and +U_dc, as before. This reduces the voltage step by 50%. The phase voltage changes four times in one period, leading to a shorter duration for each individual voltage state.

After the adjustment, the two inverters no longer switch their states synchronously. Since the switching order and time offset of the two inverters are not fixed, ZSV pulses with varying polarities and durations are generated, as shown in Figure 19.

In the figure, each ZSV pulse has an amplitude of either +U_dc/5 or −U_dc/5, where t₁, t₂, and t₃ are:

$$\left\{\begin{array}{l}{t}_1=\left(T_5-T_0\right)/2\\ t_2=\left(T_0+T_1-T_4-T_5\right)/2\\ t_3=\left(T_3+T_4+T_5-T_0-T_1-T_2\right)/2\end{array}\right.$$

(24)

Therefore, the equivalent ZSV generated by the inverters within one switching period is:

$$\begin{array}{cc}\hfill u_0=& {\displaystyle \frac{U_{\mathrm{dc}}}{5}}\left(4t_1-4t_2+2t_3\right)/T_{\mathrm{s}}\hfill \\ \hfill =& \left(-U_{\mathrm{dc}}{T}_0-0.6U_{\mathrm{dc}}{T}_1-0.2U_{\mathrm{dc}}{T}_2+\right.\left.0.2U_{\mathrm{dc}}{T}_3+0.6U_{\mathrm{dc}}{T}_4+U_{\mathrm{dc}}{T}_5\right)/T_{\mathrm{s}}\hfill \end{array}$$

(25)

Compared with Equation (22), the ZSV remains unchanged after adjusting the switching sequence of INV2. Thus, the capability of the system to suppress the ZSC is not affected. Figure 20 shows the phase currents and ZSC after adjustment. Since the high-frequency harmonics with the same frequency and similar phase are eliminated from the phase currents, the corresponding sawtooth ripples in the ZSC also disappear.

FFT analysis is performed on both the phase currents and ZSC before and after the adjustment, as shown in Figure 21. The fundamental frequency is 80 Hz. The PWM period is 10⁻⁴ s, corresponding to a carrier frequency of 10 kHz, which is equivalent to 125 times the fundamental frequency. Before adjustment, both the phase currents and ZSC contain a 125th harmonic component (i.e., 10 kHz), with a content of 8.83% of the fundamental current. After adjustment, this component is effectively eliminated. The THD_ex3 of the phase current decreases from 10.22% to 2.55%. In addition, the fifth-harmonic component in the ZSC remains essentially unchanged, indicating that the proposed modification does not affect the suppression performance on the fifth harmonic of ZSC.

5. Analysis of Two Modulation Strategies

5.1. Modulation Index Comparison

When the reference voltage vector is synthesized using the nearest four vectors, the maximum linear modulation region is bounded by the inscribed circle of the equivalent decagon. Taking 0.5U_dc as the base value, the modulation indices in the fundamental and third-harmonic planes are defined as

$$\left\{\begin{array}{l}{m}_1=2\left|{\mathit{u}}_1^{\ast }\right|/U_{\mathrm{dc}}\\ m_3=2\left|{\mathit{u}}_3^{\ast }\right|/U_{\mathrm{dc}}\end{array}\right.$$

(26)

Assume that, in the fundamental plane, the dwell time of large vector U_L1 is λ times that of medium vector U_M1. Consequently, the maximum linear modulation indices for the two planes are obtained as Equation (27). Here, U_L3 and U_M3 are the corresponding voltage vectors mapped into the third-harmonic plane from U_L1 and U_M1, respectively.

$$\left\{\begin{array}{l}{m}_{1\max }=\left|\lambda {\mathit{U}}_{L1}+{\mathit{U}}_{M1}\right|{\displaystyle \frac{2\cos \left(\mathrm{\pi }/10\right)}{\left(\lambda +1\right)U_{\mathrm{dc}}}}\\ m_{3\max }=\left|\lambda {\mathit{U}}_{L3}-{\mathit{U}}_{M3}\right|{\displaystyle \frac{2\cos \left(3\mathrm{\pi }/10\right)}{\left(\lambda +1\right)U_{\mathrm{dc}}}}\end{array}\right.$$

(27)

For the improved ZSV-free SVPWM strategy (S1) and the asymmetric-CMV SVPWM strategy (S2), the magnitudes of the large and medium vectors in both planes are listed in

Table 2. Magnitudes of large and medium vectors in two planes.

Strategy	|UL1|	|UM1|	|UL3|	|UM3|
S1	1.231Udc	0.761Udc	0.291Udc	0.471Udc
S2	1.294Udc	0.8Udc	0.494Udc	0.8Udc

.

Based on the data in the table, within the same plane, the magnitude ratios of the corresponding vectors between S1 and S2 are consistent, as shown in Equation (28).

$$\left\{\begin{array}{l}{\displaystyle \frac{\left|{\mathit{U}}_{L1}\left(\mathrm{S}2\right)\right|}{\left|{\mathit{U}}_{L1}\left(\mathrm{S}1\right)\right|}}={\displaystyle \frac{\left|{\mathit{U}}_{M1}\left(\mathrm{S}2\right)\right|}{\left|{\mathit{U}}_{M1}\left(\mathrm{S}1\right)\right|}}=1.05\\ {\displaystyle \frac{\left|{\mathit{U}}_{L3}\left(\mathrm{S}2\right)\right|}{\left|{\mathit{U}}_{L3}\left(\mathrm{S}1\right)\right|}}={\displaystyle \frac{\left|{\mathit{U}}_{M3}\left(\mathrm{S}2\right)\right|}{\left|{\mathit{U}}_{M3}\left(\mathrm{S}1\right)\right|}}=1.70\end{array}\right.$$

(28)

Therefore, in the fundamental and third-harmonic plane, the ratios of the maximum linear modulation indices of S2 to S1 are:

$$\left\{\begin{array}{l}{\displaystyle \frac{m_{1\max }\left(\mathrm{S}2\right)}{m_{1\max }\left(\mathrm{S}1\right)}}=1.05\\ {\displaystyle \frac{m_{3\max }\left(\mathrm{S}2\right)}{m_{3\max }\left(\mathrm{S}1\right)}}=1.70\end{array}\right.$$

(29)

It can be seen that the maximum linear modulation indices of S2 are greater than those of S1 in both planes. Theoretically, S2 can provide higher phase voltage, thereby extending the maximum output torque and speed of the motor.

5.2. ZSV Modulation Range

For the S1 modulation strategy, the equivalent ZSV generated by the dual inverters can be obtained from Equation (13) as

$$u_{0\text{-}\mathrm{S}1}={\displaystyle \frac{\left(T_5-T_0\right)}{T_{\mathrm{s}}}}{U}_{\mathrm{dc}}$$

(30)

where T₀ and T₅ are the dwell times of the zero vectors U₀ and U₃₁ for INV1, respectively. u_0-S1 reaches its maximum when T₀ is minimized and T₅ is maximized. In this case, T₀ and T₅ are given by:

$$\left\{\begin{array}{l}{T}_0=0\\ T_5=T_{\mathrm{s}}-T_1-T_2-T_3-T_4\end{array}\right.$$

(31)

Substituting these into Equation (30), the maximum ZSV can be obtained as:

$$u_{0\text{-}\mathrm{S}1}\left(\max \right)=U_{\mathrm{dc}}-\left|{\mathit{u}}_1^{\ast }\right|\cos \left[{\theta }_1-{\displaystyle \frac{\left(\mathrm{N}-1\right)\mathrm{\pi }}{5}}\right]-\left|{\mathit{u}}_3^{\ast }\right|\cos \left[{\theta }_3-{\displaystyle \frac{\left(3\mathrm{N}-3\right)\mathrm{\pi }}{5}}\right]$$

(32)

where θ₁ and θ₃ are the phase angles of ${\mathit{u}}_1^{\ast }$ and ${\mathit{u}}_3^{\ast }$, respectively. Conversely, u_0-S1 reaches its minimum when T₀ is maximized and T₅ is minimized:

$$u_{0\text{-}\mathrm{S}1}\left(\min \right)=\left|{\mathit{u}}_1^{\ast }\right|\cos \left[{\theta }_1-{\displaystyle \frac{\left(\mathrm{N}-1\right)\mathrm{\pi }}{5}}\right]+\left|{\mathit{u}}_3^{\ast }\right|\cos \left[{\theta }_3-{\displaystyle \frac{\left(3\mathrm{N}-3\right)\mathrm{\pi }}{5}}\right]-U_{\mathrm{dc}}$$

(33)

For the S2 modulation strategy, the equivalent ZSV generated by the dual inverters in the odd and even ${\mathit{u}}_1^{\ast }$ sectors are, respectively, expressed as:

$$u_{0\text{-}\mathrm{S}2}=\left\{\begin{array}{l}{\displaystyle \frac{-T_0-0.6T_1-0.2T_2+0.2T_3+0.6T_4+T_5}{T_{\mathrm{s}}}}{U}_{\mathrm{dc}} \left(\mathrm{odd}\right)\\ {\displaystyle \frac{-T_0+0.6T_1+0.2T_2-0.2T_3-0.6T_4+T_5}{T_{\mathrm{s}}}}{U}_{\mathrm{dc}} \left(\mathrm{even}\right)\end{array}\right.$$

(34)

Similarly, the maximum and minimum values can be obtained as:

$$u_{0\text{-}\mathrm{S}2}\left(\max \right)=\left\{\begin{array}{ll}{U}_{\mathrm{dc}}-\left|{\mathit{u}}_1^{\ast }\right|\cos \left[{\theta }_1-{\displaystyle \frac{\left(\mathrm{N}-1\right)\mathrm{\pi }}{5}}\right]-\left|{\mathit{u}}_3^{\ast }\right|\cos \left[{\theta }_3-{\displaystyle \frac{\left(3\mathrm{N}-3\right)\mathrm{\pi }}{5}}\right]& \left(\mathrm{odd}\right)\\ U_{\mathrm{dc}}-\left|{\mathit{u}}_1^{\ast }\right|\cos \left({\theta }_1-{\displaystyle \frac{\mathrm{N}\mathrm{\pi }}{5}}\right)-\left|{\mathit{u}}_3^{\ast }\right|\cos \left({\theta }_3-{\displaystyle \frac{3\mathrm{N}\mathrm{\pi }}{5}}\right)& \left(\mathrm{even}\right)\end{array}\right.$$

(35)

$$u_{0\text{-}\mathrm{S}2}\left(\min \right)=\left\{\begin{array}{ll}\left|{\mathit{u}}_1^{\ast }\right|\cos \left({\theta }_1-{\displaystyle \frac{\mathrm{N}\mathrm{\pi }}{5}}\right)+\left|{\mathit{u}}_3^{\ast }\right|\cos \left({\theta }_3-{\displaystyle \frac{3\mathrm{N}\mathrm{\pi }}{5}}\right)-U_{\mathrm{dc}}& \left(\mathrm{odd}\right)\\ \left|{\mathit{u}}_1^{\ast }\right|\cos \left[{\theta }_1-{\displaystyle \frac{\left(\mathrm{N}-1\right)\mathrm{\pi }}{5}}\right]+\left|{\mathit{u}}_3^{\ast }\right|\cos \left[{\theta }_3-{\displaystyle \frac{\left(3\mathrm{N}-3\right)\mathrm{\pi }}{5}}\right]-U_{\mathrm{dc}}& \left(\mathrm{even}\right)\end{array}\right.$$

(36)

Comparing Equations (32), (33), (35) and (36), it can be seen that, for both control strategies, the maximum and minimum ZSVs depend on the magnitudes and phase angles of the fundamental and third-harmonic reference voltage vectors. According to the previous simulation results, under the proposed strategy, the phase currents exhibit flat-top waveforms, indicating a constant phase difference between the fundamental and third-harmonic currents. However, as shown in Equation (1), the phase voltage is related to the current, back EMF, and winding impedance. This implies that, at different speeds and load torques, the phase difference between the voltage and current in the dual planes is not fixed, leading to the absence of a constant phase relationship between the fundamental and third-harmonic voltages. Therefore, the maximum and minimum ZSVs are affected by four independent variables—$\left|{\mathit{u}}_1^{\ast }\right|$, θ₁, $\left|{\mathit{u}}_3^{\ast }\right|$, and θ₃—making it challenging to perform a quantitative analysis.

Qualitatively, as the motor speed increases or the load becomes heavier, the reference voltage vectors ${\mathit{u}}_1^{\ast }$ and ${\mathit{u}}_3^{\ast }$ gradually increase, leading to a decrease in u_0-S1(max) and u_0-S2(max) and an increase in u_0-S1(min) and u_0-S2(min). This indicates that the range of ZSV generated by the control system gradually narrows, approaching zero. As a result, the capability to suppress ZSC correspondingly decreases.

6. Experimental Results

6.1. Experimental Platform

To verify the effectiveness of the two proposed ZSC suppression strategies and compare their performances, an experimental platform for the OW-FPPMSM control system was established, as shown in Figure 22. The PE-Expert4 digital control system from Myway (Yokohama, Japan) was employed as the main controller. It enables the simultaneous acquisition of phase currents from five Hall current sensors and the rotor position from a resolver, as well as the generation of PWM signals. Four IPMs (PM450CLA060, Mitsubishi Electric, Tokyo, Japan) are supplied by one DC power cabinet. These IPMs are connected to both ends of the five phase windings of the OW-FPPMSM, respectively. The load is applied by a magnetic powder brake.

6.2. Results for Improved ZSV-Free SVPWM Strategy

The following results are acquired using the host computer software (PE-ViewX, version 3.01) with a 10 kHz sampling rate.

Consistent with the simulation, the reference speed is set to 800 r/min, and the load torque is 20 N·m. Under the conventional strategy, the steady-state phase currents are shown in Figure 23a. Although the currents exhibit a flat-top waveform, significant distortion occurs due to the presence of ZSC. With the improved ZSC-free SVPWM strategy, the waveforms become smoother and more symmetrical, as shown in Figure 23b.

FFT analysis is performed on the two sets of currents, and the results are shown in Figure 24. The third-harmonic contents for the two strategies represent 16.22% and 16.95% of the fundamental component, respectively, both consistent with the control requirements. Meanwhile, the fifth-harmonic content is significantly suppressed, decreasing from 5.56% to 1.09%of the fundamental component. The THD_ex3 of the phase current decreases from 8.21% to 4.38%.

Figure 25 shows the ZSC waveforms under the two strategies and their FFT results. The peak value is reduced from 2.1 A to 0.2 A, and the fifth-harmonic current decreases from 0.87 A to 0.13 A, corresponding to a reduction of approximately 85.1%, validating the effectiveness of the proposed strategy in ZSC suppression. Theoretically, the ZSC is primarily composed of the fifth-harmonic component. However, due to the inherent asymmetry of the phase currents under the conventional strategy, a certain amount of the fundamental and third-harmonic components also exists in the ZSC.

To further evaluate the effectiveness and reliability of the improved strategy, a quantitative comparison of ZSC components is summarized in

Table 3. Statistical analysis of ZSC components.

Harmonic Order	Conventional (A)	Improved (A)	Reduction (%)
1st	0.250 ± 0.058	0.015 ± 0.006	94.0
3rd	0.273 ± 0.062	0.050 ± 0.012	81.7
5th	0.853 ± 0.025	0.128 ± 0.005	85.0

. The results are derived from four sets of data obtained under identical operating conditions, presented as mean ± standard deviation (SD). As indicated in the table, the proposed method achieves a significant reduction across the dominant ZSC components. In particular, the fifth harmonic is suppressed by 85.0%. The low SD values demonstrate the stability and robustness of the improved strategy.

6.3. Results for Asymmetric-CMV SVPWM Strategy

The PWM switching period is set to 10⁻⁴ s, consistent with the simulation settings. Under the asymmetric-CMV SVPWM strategy, the current sampling frequency is increased to 100 kHz to capture the sawtooth components at the carrier frequency.

Under the same speed and load, Figure 26 shows the current waveforms before and after adjusting the switching sequence of INV2. Before the adjustment, distinct high-frequency sawtooth ripples appear in the phase currents. Since these components are nearly in phase, corresponding ripples also appear in the ZSC. After adjustment, the sawtooth components in the phase currents are significantly reduced, and consequently, the ZSC is effectively suppressed.

FFT analysis is performed on the two sets of phase currents and the ZSCs, as shown in Figure 27. The low-frequency components within the 10th order remain relatively unchanged. However, the 10 kHz component corresponding to the carrier frequency, i.e., the 125th high-frequency harmonic, is clearly suppressed. As a result, the ZSC is significantly reduced. The THD_ex3 of the phase current decreases from 13.19% to 6.43%.

6.4. Comparison of Two Proposed Strategies

To ensure experimental safety in an overload condition, the DC bus voltage is set to 50 V. The two modulation strategies are tested at a reference speed of 450 r/min with a load torque of 5 N·m. Figure 28 shows the steady-state speed and output torque, where the torque is calculated from the phase currents and the permanent magnet flux linkage. Under the improved ZSC-free SVPWM strategy, the speed reaches a steady state at 430 r/min. In contrast, the asymmetric-CMV SVPWM strategy enables the motor to reach the reference speed, verifying its superior DC bus voltage utilization.

Figure 29 shows the ZSC waveforms under the two modulation strategies. The peak values of the ZSCs are 0.3 A and 0.4 A, respectively, indicating that the control strategies remain effective in suppressing the ZSC, even under overload conditions.

7. Conclusions

For the OW-FPPMSM control system applied in an aerospace starter–generator system with a common DC bus, this paper proposes two ZSC suppression strategies. The first strategy is derived from the conventional ZSV-free control method. By offsetting the switching instants of the two inverters, ZSV pulses are intentionally generated. The second strategy uses the basic voltage vectors with larger magnitudes, achieving a higher modulation index. By adjusting the switching sequence of INV2, the high-frequency sawtooth components in the phase currents and ZSC are further eliminated. Both strategies also achieve the injection of the third-harmonic current. Finally, the modulation indices and ZSV modulation ranges of these two strategies are analyzed.

Compared with conventional methods that focus on eliminating or passively compensating for ZSV, the proposed strategies do not require a clamped inverter or virtual voltage vectors. Both strategies enable a precise and flexible injection of ZSV. While satisfying the requirements for ZSC suppression, they feature a clear control structure, low computational burden, and strong robustness, making them well-suited for engineering applications that require high real-time performance and reliability.

Considering the strong fault-tolerant capability of the motor, future work will focus on fault-tolerant control methods under short-circuit or open-circuit faults in one or multiple phase windings, where the suppression of ZSC will also be addressed.