Abstract: Matrix converters have attracted much attention due to their numerous excellent characteristics. However, the large number of power switches required and the complexity of topology and control have limited their development to some extent. This paper studies an indirect space vector modulation strategy based on a virtual DC link combined with a four-step commutation control strategy, and verifies the control strategy in simulation. The simulation results show sinusoidal AC voltage and input current, achieving good results. Keywords: Matrix converter; Indirect space vector modulation (SVPWM); Four-step commutation strategy 0 Introduction Matrix converters (MCs), as a novel device that directly realizes AC-AC conversion, have always attracted much attention. They use a controllable bidirectional switch array to modulate and transform the input voltage, generating an output voltage of arbitrary frequency. They do not require large energy storage components, and the input and output currents are controllable and have regenerative capabilities, thus offering significant advantages over traditional converters. However, the MC structure is complex, and determining the required switch combination based on input conditions and output requirements is very difficult. Early direct modulation methods were complex and had limited practical application value. In recent years, the introduction of the indirect space vector modulation concept has greatly simplified the modulation strategy of matrix converters and has been widely used in practice. Another characteristic of matrix converters is the commutation problem of their switching devices. The inherent characteristics of matrix converters dictate that their inputs cannot be short-circuited at any time; and their outputs cannot be open-circuited at any time. Before the introduction of a new commutation strategy based on four-step commutation, reliable switching on and off has always been a challenge. This paper studies the direct MC with an inductive load in simulation, using the indirect space vector method combined with a four-step commutation strategy to achieve MC switch control. Simulations were conducted in both open-loop and closed-loop control with an asynchronous motor. The direct MC topology is shown in Figure 1. [align=center] Figure 1 Direct MC Topology[/align] A switching matrix model was established using MATLAB, and the relevant switching generation program was written in M ​​language. Based on the implementation of open-loop control with load, a dual closed-loop simulation with an asynchronous motor was performed. Simulation results show that the control strategy achieved a good modulation effect. 1. Direct MC Indirect Space Vector Modulation Principle [align=center] Figure 2 Equivalent AC-DC-AC Structure of Matrix Converter [/align] As shown in Figure 2, theoretically, a direct MC formed by nine bidirectional switches arranged in a 3*3 configuration can be equivalent to a virtual connection between a rectifier and an inverter. Applying space vector modulation (SVPWM) technology to both the "virtual rectifier" and the "virtual inverter" modulates the bidirectional switches, and combining the two processes achieves sinusoidal input and output waveforms and a controllable input power factor. The MC input side is a three-phase voltage source, and the switching action must ensure that the input side is not short-circuited; the output side is generally a three-phase inductive load, which can be equivalent to a three-phase current source. Since the MC has no freewheeling path, the switching action must ensure that the output side is not short-circuited. These are the two basic constraints for MC switch selection. 1.1 DC/AC Space Vector Modulation Principle modulates the "virtual inverter" part in Figure 2. A DC voltage is applied between inputs P and N, and the output is a three-phase AC voltage. Due to the constraint that the output cannot be disconnected, one of the two switches connected to the inputs P and N respectively for the three-phase outputs A, B, and C must be turned on, resulting in eight switch combinations, forming the six-sector output line voltage vector diagram in Figure 3. The output line voltage space vector U[sub]OL[/sub] at any given time can be synthesized from two adjacent non-zero vectors (selected from one zero vector). The sector angle in Figure b represents the position of U[sub]OL[/sub] in the current sector. According to the SVPWM modulation principle, the required output voltage U[sub]OL[/sub] is synthesized from the two switch vectors in the sector: (1) Where: is the action time of the corresponding switch vector, and T[sub]S[/sub] is the switching period. The corresponding duty cycle can be expressed as: (2) Where: is the fundamental amplitude of the output phase voltage, and V[sub]dc[/sub] is the average DC side voltage. [align=center] Figure 3 Space Vector Modulation of Virtual Inverter Output Voltage Figure 4 Space Vector Modulation of Virtual Rectifier Input Phase Current 1.2 AC/DC Space Vector Modulation Principle Based on the detected input phase voltage space vector U[sub]iph[/sub] and the set input phase difference φ[sub]i[/sub], the desired input phase current space vector I[sub]i[/sub] can be determined. Vector modulation is performed on the input phase current of the "virtual rectifier" section in Figure 2. According to the constraint of no short circuit between input phases, each DC output terminal is connected to only one phase AC input terminal, resulting in nine current vector switch combination states, which constitute the input phase current space vector diagram shown in Figure 4. It can be seen that I[sub]i[/sub] at any time can be synthesized from two adjacent non-zero vector switch vectors (selected from one zero switch vector). In Figure b, the sector angle θ[sub]sc[/sub] represents the position of I[sub]i[/sub] in the current sector. For the required output voltage I[sub]i[/sub], using the two switching vectors of the sector, we have: (3) Where: T[sub]u[/sub], T[sub]y[/sub], T[sub]0[/sub] are the action time of the corresponding switching vectors, and T[sub]S[/sub] is the switching period. The corresponding duty cycle can be expressed as: (4) Where: is the input current amplitude, and I[sub]dc[/sub] is the average value of the output current. 1.3 AC-AC equivalent transformation of the three-phase matrix converter Connect the DC bus between the "virtual rectifier" and the "virtual inverter", and its function is equivalent to the actual circuit of the three-phase matrix converter, as shown in Figure 5. [align=center] Figure 5 Synthesis of three-phase MC switching states After determining the sector and sector angle of the voltage and current vectors, the output line voltage vector synthesis and the input phase current vector synthesis are combined in each sampling period. Figure 5 shows the combination. Figure 5(a) shows the switching connection state of the corresponding virtual link, and Figure 5(b) shows the switching connection state when switching to the actual three-phase MC. The synthesized vector has a total of 5 switching states, which are determined by the switching vector, , and the zero vector respectively. The duration of these five switching states within the cycle can be obtained by multiplying equation (2) and equation (4): (5) where m is the MC space vector pulse width modulation coefficient and satisfies the input phase voltage amplitude. According to the above method, the switching state and duty cycle information of any sector combination and any vector combination can be obtained. 2 Four-step commutation strategy The matrix converter topology shown in Figure 1 determines that the input cannot be short-circuited and the output cannot be open-circuited. A relatively reliable current commutation method that follows this principle is to use a four-step commutation strategy to control the switching commutation process, which effectively ensures the safety of the input and output circuits. [align=center] Figure 6 Two-phase to single-phase matrix converter[/align] Figure 6 shows a schematic diagram of a two-phase to single-phase matrix converter. The first two switching devices of the converter are shown in Figure 1. In steady state, a pair of devices in the bidirectional switching unit are triggered to conduct, allowing current to flow bidirectionally. The following explanation assumes the load current flows in the direction shown in the diagram, and the upper bidirectional switch (S[sub]Aa[/sub]) is closed. When commutation to S[sub]Ab[/sub] is required, the current direction determines which device in the switch ceases to conduct, and that device is turned off. Under this current direction assumption, device S[sub]Aa2[/sub] is turned off. The device in the switching unit that is about to conduct current is then triggered, in this case, S[sub]Ab1[/sub]. Either at this point or when the switching device (S[sub]Aa1[/sub]) turns off, the load current shifts to the switching device. The other device in the switching unit, S[sub]Ab2[/sub], turns on, allowing current to flow in reverse. This process is illustrated in the timing diagram in Figure 7. The delay between each switching transition is determined by the device characteristics. [align=center]Figure 7 Four-step commutation timing diagram[/align] 3 Simulation Verification Based on the above analysis and controller design, open-loop and asynchronous machine simulations were performed in MATLAB. 3.1 Open-loop simulation To initially verify the SVPWM control strategy, an MC model was built using an ideal switch. Its input uses LC filtering, while the output has a resistive-inductive load. The main simulation parameters are shown in Table 1, and the simulation results are shown in Figures 8-10. Table 1 Main simulation parameters [align=center]Figure 8 Input three-phase phase current Figure 9 Output three-phase line voltage Figure 10 Output three-phase phase current[/align] Figure 8 shows the grid-side three-phase input phase current, Figure 9 shows the output line voltage U[sub]AB[/sub] and the filtered three-phase output line voltage waveforms, and Figure 10 shows the three-phase output phase current. It can be seen from the figures that the output current, voltage, and input current modulated by the ideal simulation model have good sinusoidal characteristics, verifying the correctness of the control strategy. 3.2 Closed-Loop Speed ​​Control of Asynchronous Motor Based on the preliminary verification of the modulation strategy, a matrix converter is used to drive the asynchronous motor. Its structure is shown in Figure 11, simultaneously realizing space vector pulse width modulation of the matrix converter and vector control of the asynchronous motor. The control strategy adopts a rotor field-oriented vector control method based on a current model flux linkage observer. In the MT synchronous rotating coordinate system with the rotor flux linkage direction as the T-axis, the asynchronous motor control system is decomposed into two subsystems: a speed loop and a flux linkage loop, which respectively adjust the motor speed and rotor flux linkage. Through coordinate transformation, the motor stator current is decomposed into M-axis and T-axis components. A T-axis current loop is set within the speed loop to adjust the electromagnetic torque. Given a flux linkage, the required M-axis stator current is calculated and compared with the feedback M-axis current to form a flux linkage loop for flux linkage stability. Stator voltage decoupling is also added to improve system performance. The required output voltage vector can be obtained according to the rotor flux linkage orientation described above, while the current vector measurement detects the input voltage phase and obtains the phase of the input phase current vector based on the required input power factor. This allows for the calculation of switching combinations at any given time, similar to open-loop simulation, to control the required frequency voltage of the MC output and achieve the goal of controlling the motor. The simulation flux linkage was set to 0.96, and the speed was set to 30 rad/s. To test the system's anti-interference performance, a step-incremental load torque was applied, initially 10 N·m, then abruptly increasing to 90 N·m at 0.4 s, before returning to 10 N·m at 1.0 s. The speed regulator, torque regulator, and flux linkage regulator all used PI controllers. The speed regulator parameters were Kp=35, Ki=100, with an output torque command limit of 300 N·m; the torque regulator parameters were Kp=0.008, Ki=0.13; and the flux linkage regulator parameters were Kp=0.021, Ki=0.045. The maximum limit for the final output was 1. The T-axis current, calculated from the torque based on the speed, would be very large during startup due to the small flux linkage; therefore, a limit of 400 A was applied. The motor used is a MATLAB built-in motor model with a rated power of 50*746VA, a rated line voltage of 460V, and a rated frequency of 60Hz. The simulation results are shown in Figures 12-13. [align=center] Figure 11 Simulation structure diagram of motor vector control[/align] [align=center] Figure 12 Output phase current, speed, and electromagnetic torque waveforms Figure 13 Input phase current[/align] The simulation shows that the flux linkage gradually approaches 0.96; while the motor's electromagnetic torque rapidly rises to the maximum limit of 300 N·m at startup, achieving maximum constant torque startup. The speed rises rapidly, reaching the given speed of 30 rad/s for the first time at 0.175s. Afterward, the electromagnetic torque rapidly decreases, and the speed gradually approaches the given speed after a period of overshoot (reaching a maximum speed of 36 rad/s). For load disturbances of 0.4s and 1.0s, the speed does not fluctuate significantly, indicating that the control loop has good disturbance rejection performance. 4. Conclusion This paper analyzes the indirect spatial modulation strategy based on a virtual DC link and, combined with a four-step commutation control strategy, studies the MC (modulation control) in open-loop and closed-loop tractor motor simulation models. Simulation results verify the correctness of the control strategy.