The evolution of electric drive systems, particularly with the adoption of wide-bandgap semiconductors like Silicon Carbide (SiC), has enabled significant strides in power density and efficiency. The superior material properties of SiC, such as a bandgap three times wider and a critical breakdown field strength ten times higher than traditional silicon, facilitate operation at higher switching frequencies and voltages. However, this advancement introduces pronounced challenges, with high-frequency common-mode voltage (CMV) being a primary contributor to electromagnetic interference (EMI) issues within the electric drive system. The rapid voltage transitions generated by the inverter stage couple through parasitic capacitances, leading to bearing currents, increased stress on motor insulation, and conducted emissions that can disrupt sensitive control electronics.
Conventional CMV mitigation approaches are broadly categorized into hardware and software methods. Hardware solutions, such as passive filters or active cancellation circuits, add cost, volume, and complexity to the electric drive system. Software techniques, which modify modulation algorithms, offer a cost-effective alternative. Various Pulse Width Modulation (PWM) strategies, including active zero-state PWM, have been explored but often suffer from limited linear modulation range or increased current distortion. Model Predictive Control (MPC) has emerged as a powerful alternative for controlling electric drive systems due to its intuitive concept, fast dynamic response, and ability to handle multiple constraints and objectives directly. Finite Control Set MPC (FCS-MPC) evaluates a finite set of voltage vectors from the inverter and selects the one that minimizes a predefined cost function, often based on current tracking error. To improve steady-state performance, multi-vector approaches like dual-vector or three-vector MPC have been developed, synthesizing a reference voltage over a sampling period. However, the inclusion of zero vectors in these schemes inherently generates high CMV levels. While weighting factors in the cost function can balance current tracking and CMV suppression, their tuning is non-trivial and performance can be suboptimal. Recent strategies employ voltage vector preselection to inherently avoid zero vectors, thereby suppressing CMV. Yet, a critical limitation persists: to avoid CMV spikes caused by simultaneous switching of multiple phases, these methods restrict the selection to adjacent voltage vectors, severely limiting the available voltage vectors for current control and leading to significant current distortion, especially near current zero-crossings.
This work addresses the dual challenge of effective CMV suppression and low current distortion in high-frequency SiC-based electric drive systems. The core of the problem lies in the restricted vector selection of existing CMV-aware MPC methods. We propose a novel multi-vector modulation model predictive control strategy. The approach begins with an analysis of the impact of different switching states on SiC device switching characteristics, highlighting the detrimental role of zero vectors. Subsequently, we introduce a multi-vector synthesis method that combines forward-adjacent and reverse-adjacent voltage vectors, expanding the effective control space without reintroducing CMV spikes. A precise time allocation method based on the principles of modulated MPC is derived, and a dedicated dead-time compensation strategy is implemented to prevent CMV suppression failure during sector transitions. The effectiveness of the proposed strategy is validated through comprehensive simulation and experimental studies on a permanent magnet synchronous motor (PMSM) electric drive system.
Analysis of Non-Zero Vector Impact on Current Control in an Electric Drive System
In a two-level three-phase voltage source inverter, the common-mode voltage \(U_{cm}\) measured between the motor neutral point (n) and the DC-link midpoint (o) can be expressed as:
$$U_{cm} \approx U_{no} = \frac{U_{ao} + U_{bo} + U_{co}}{3}$$
where \(U_{ao}, U_{bo}, U_{co}\) are the pole voltages. The eight possible switching states produce discrete CMV levels. The two zero vectors (000, 111) generate the highest CMV magnitude of \(\pm U_{dc}/2\), while the six active vectors produce \(\pm U_{dc}/6\) or 0. Therefore, eliminating zero vectors from the modulation scheme is a fundamental step for CMV reduction.
For a surface-mounted PMSM, the discrete-time prediction model in the synchronous rotating (d-q) reference frame is given by:
$$i_d(k+1) = (1 – \frac{R_s T_s}{L_d})i_d(k) + \frac{T_s \omega_e L_q i_q(k)}{L_d} + \frac{T_s u_d(k)}{L_d}$$
$$i_q(k+1) = (1 – \frac{R_s T_s}{L_q})i_q(k) – \frac{T_s \omega_e (L_d i_d(k) + \psi_f)}{L_q} + \frac{T_s u_q(k)}{L_q}$$
where \(i_d, i_q\) are stator currents, \(u_d, u_q\) are stator voltages, \(R_s\) is stator resistance, \(L_d, L_q\) are inductances, \(\psi_f\) is permanent magnet flux linkage, \(\omega_e\) is electrical angular velocity, and \(T_s\) is the sampling period. The standard cost function for current tracking is:
$$g = [i_d^*(k+1) – i_d(k+1)]^2 + [i_q^*(k+1) – i_q(k+1)]^2$$
where \(i_d^*, i_q^*\) are the reference currents.
A CMV-suppressing dual-vector MPC preselects only the six active vectors for evaluation. To prevent large CMV spikes during commutation, it further restricts vector transitions to only adjacent vectors (e.g., from vector 100, only 110 and 101 are allowed as the next vector). This drastic reduction in the candidate vector set is the root cause of current distortion. In certain sectors of the complex plane, the available adjacent vectors may all exert the same directional influence on the d-axis or q-axis current error. For instance, if the control requirement is to decrease \(i_d\), but all allowed vectors only act to increase it, the controller cannot correct the error, leading to sustained current distortion and increased Total Harmonic Distortion (THD).
To systematically analyze this, we partition the plane into 12 sectors based on the combined position of the voltage and current vectors. The influence of each of the six active voltage vectors on the derivatives of the d and q-axis currents (\(di_d/dt\), \(di_q/dt\)) is evaluated for each sector. The table below summarizes whether a given vector increases (+) or decreases (-) the current component in a specific sector.
| Sector | Vectors with \(di_d/dt > 0\) | Vectors with \(di_q/dt > 0\) |
|---|---|---|
| 1 | 101, 100, 110 | 110, 010, 011 |
| 2 | 100, 110, 010 | 110, 010, 011 |
| 3 | 100, 110, 010 | 010, 011, 001 |
| 4 | 110, 010, 011 | 010, 011, 001 |
| 5 | 110, 010, 011 | 011, 001, 101 |
| 6 | 010, 011, 001 | 011, 001, 101 |
| 7 | 010, 011, 001 | 001, 101, 100 |
| 8 | 011, 001, 101 | 001, 101, 100 |
| 9 | 011, 001, 101 | 101, 100, 110 |
| 10 | 001, 101, 100 | 101, 100, 110 |
| 11 | 001, 101, 100 | 100, 110, 010 |
| 12 | 101, 100, 110 | 100, 110, 010 |
This analysis confirms that within a given sector, the subset of adjacent vectors available under the strict CMV suppression rule may lack the necessary vector to correct a specific current error direction, leading to the distortion observed in the electric drive system’s current waveform.
Proposed Multi-Vector Modulation Model Predictive Control
The proposed strategy aims to synthesize a reference voltage that provides full control over both d and q current components while strictly avoiding zero vectors and non-adjacent vector transitions that cause CMV spikes. The core idea is to employ a combination of four active vectors within one sampling period: two primary optimal vectors and two auxiliary reverse-adjacent vectors.
The control sequence for each sampling period \(T_s\) is as follows:
- Primary Vector Selection: Evaluate the six active vectors using the standard cost function \(g\) and select the optimal vector \(V_{opt1}\).
- Secondary Vector Selection: From the subset containing \(V_{opt1}\) and its two adjacent vectors, evaluate the cost function again to select the second optimal vector \(V_{opt2}\). This ensures \(V_{opt1}\) and \(V_{opt2}\) are always adjacent.
- Auxiliary Vector Addition: To expand the control capability, two additional vectors \((V_{adj1}, V_{adj2})\) are introduced. These are defined as the non-zero vectors that are 180 degrees apart from \(V_{opt1}\) and \(V_{opt2}\), respectively. For example, if \(V_{opt1}=100\) (phase angle 0°), then \(V_{adj1}\) is 011 (phase angle 180°). Crucially, \(V_{adj1}\) is adjacent to \(V_{adj2}\), and the transition sequence \(V_{adj1} \rightarrow V_{opt1} \rightarrow V_{opt2} \rightarrow V_{adj2}\) involves only adjacent vector changes, thus preventing CMV spikes.
- Time Allocation: The four vectors are applied within \(T_s\) to synthesize the desired reference voltage. Their duty cycles \(T_1, T_2, T_3, T_4\) (for \(V_{adj1}, V_{opt1}, V_{opt2}, V_{adj2}\) respectively) are calculated based on the principle of deadbeat current control. The goal is to nullify the predicted current error at the end of the period. For a reference located in Sector I (between vectors 100 and 110), using vectors \(V_3(010)\), \(V_1(100)\), \(V_2(110)\), and \(V_6(101)\) as \(V_{adj1}, V_{opt1}, V_{opt2}, V_{adj2}\), the time allocation is derived by solving:
$$T_1 e_{d,3} + T_2 e_{d,1} + T_3 e_{d,2} + T_4 e_{d,6} = 0$$
$$T_1 e_{q,3} + T_2 e_{q,1} + T_3 e_{q,2} + T_4 e_{q,6} = 0$$
$$T_1 + T_2 + T_3 + T_4 = T_s$$
where \(e_{d,x}, e_{q,x}\) are the current errors predicted when applying vector \(V_x\) alone for \(T_s\). The solution provides the optimal dwell times, ensuring the combined effect of the four vectors achieves the desired current tracking.
This multi-vector approach effectively enlarges the synthesized voltage vector space. By including a pair of opposite-sector vectors (\(V_{adj1}, V_{adj2}\)), it guarantees that for any required adjustment in \(i_d\) or \(i_q\), at least one vector in the applied set provides the necessary corrective action, thereby eliminating the inherent current distortion of the strict dual-vector method in the electric drive system.
Dead-Time Compensation for Robust CMV Suppression
In a practical electric drive system, a dead-time \(t_d\) is inserted between complementary switch commands to prevent shoot-through. This dead-time distorts the output voltage, and its effect is particularly detrimental to CMV suppression at sector boundaries. When the dwell time of a primary vector (\(T_2\) or \(T_3\)) is very short (comparable to or less than \(2t_d\)), the intended adjacent vector transition can be disrupted.
Consider the transition from Sector I to Sector II, with the sequence \(110 \rightarrow 100 \rightarrow 101\). If the dwell time \(T_2\) for vector 100 is less than \(2t_d\), the B-phase upper switch may not have time to turn off before the command for the next vector (101) arrives. Meanwhile, the A-phase upper switch remains on, and the C-phase current freewheels through its lower diode. This results in an effective output state of 111 (all high), which is a zero vector. Consequently, a CMV spike of \(+U_{dc}/2\) appears, causing the CMV suppression algorithm to fail. A similar failure occurs with an effective 000 vector at other boundaries.
To solve this, a dead-time compensation strategy adjusts the calculated dwell times. The compensation ensures that the dwell times for the primary vectors \(V_{opt1}\) and \(V_{opt2}\) are always greater than \(2t_d\), without altering the overall synthesized reference voltage vector. The adjustment is made by borrowing time from the auxiliary vectors. If \(T_2 < 2t_d\), the times are modified as follows:
$$T’_1 = T_1 + 2t_d, \quad T’_2 = T_2 – 2t_d, \quad T’_3 = T_3 + t_d, \quad T’_4 = T_4 – t_d$$
A similar symmetrical adjustment is made if \(T_3 < 2t_d\). This compensation guarantees that the physical switching sequence always follows the intended adjacent-vector pattern, preventing the emergence of effective zero states and ensuring robust CMV suppression throughout the entire electric drive system operation, including sector transitions.
System Performance and Comparative Analysis
The performance of the proposed multi-vector MPC strategy was evaluated in a SiC-based PMSM electric drive system and compared against conventional Space Vector PWM (SVPWM) and a standard CMV-suppressing dual-vector MPC. Key metrics include CMV amplitude/spectrum and current waveform quality (THD).
Common-Mode Voltage Suppression: The SVPWM strategy, which utilizes zero vectors, generates the highest CMV amplitude with significant high-frequency ringing, especially during zero-state intervals. The dual-vector MPC eliminates the zero states, confining the CMV to \(\pm U_{dc}/6\). However, without dead-time compensation, it suffers from \(\pm U_{dc}/2\) spikes at sector boundaries due to the effect described earlier. The proposed multi-vector strategy, incorporating the dead-time compensation, successfully confines the CMV within \(\pm U_{dc}/6\) without any high-voltage spikes. Spectral analysis shows that the proposed method attenuates CMV magnitude across a wide frequency band. Compared to SVPWM, the attenuation can be as high as 60 dB at higher frequencies, significantly reducing the EMI generation potential of the electric drive system.
Current Quality: The current THD is a critical indicator of control performance and losses in the electric drive system. As expected, SVPWM provides the lowest current THD due to its continuous modulation nature. The standard dual-vector MPC, suffering from limited vector selection, exhibits pronounced periodic current distortion, resulting in a THD several times higher than SVPWM. The proposed multi-vector strategy effectively bridges this gap. By expanding the synthesized voltage space through the four-vector combination, it significantly reduces current distortion. Experimental results demonstrate that the current THD under the proposed method is much closer to that of SVPWM and is typically reduced to about one-third of the value produced by the standard dual-vector MPC across various operating points. This represents a major improvement in the torque quality and efficiency of the electric drive system.
Dynamic Response: The inherent deadbeat control characteristic of MPC is retained in the proposed method. The dynamic response to changes in torque or speed reference remains fast and comparable to other FCS-MPC schemes, as the core predictive model and cost function minimization process are preserved. The added computation for time allocation and dead-time compensation is manageable for modern digital signal processors.
Conclusion
This work presents a comprehensive multi-vector modulation model predictive control strategy designed to address the critical challenge of common-mode voltage in high-frequency SiC-based electric drive systems. The analysis first identified the root cause of current distortion in existing CMV-suppressing MPC methods: the severe restriction of candidate voltage vectors necessary to avoid CMV spikes. The proposed solution innovatively combines four active vectors—two optimal adjacent vectors and their corresponding reverse-adjacent counterparts—within one sampling period. This synthesis method maintains the strict adjacent-vector switching sequence to eliminate CMV spikes while vastly improving the controllability of the stator currents. A precise dead-time compensation mechanism was developed to counteract the detrimental formation of effective zero vectors at sector boundaries, ensuring robust CMV suppression under all operating conditions.
The effectiveness of the strategy has been rigorously validated. The results confirm that the proposed method successfully suppresses the CMV amplitude across a wide frequency spectrum, with a maximum attenuation of approximately 60 dB compared to standard modulation, thereby mitigating a major source of EMI in the electric drive system. Simultaneously, it dramatically improves the current quality, reducing the current THD to nearly one-third of that produced by the conventional dual-vector CMV suppression approach. This represents a significant advancement towards high-performance, high-reliability, and low-EMI electric drive systems essential for next-generation applications in automotive, aerospace, and industrial automation.