**1. Introduction**

In recent years, the AC–DC (alternating current–direct current) matrix converters (MC) have received significant attention in various fields. The AC–DC MC is derived from indirect MCs and inherited several advantages of the MCs, such as bidirectional power flow, sinusoidal input waveforms, controllable input power factor, high power density, and compact design [1–5]. There are various applications of the AC–DC MCs in various fields such as electric vehicles, photovoltaic generation systems, grid-connected converters, microgrids, fuel cell power systems, and battery chargers. [6–9]. Basically, it is a single-stage bidirectional current source AC–DC converter, which rectifies the sinusoidal AC signals to the pure DC signals. Di fferent modulation strategies have been applied for the AC–DC MCs, such as the Alesina–Venturini [5], the pulse-width modulation (PWM) [10,11], the model predictive control (MPC) [12–14], and the space vector modulation (SVM) [15–21]. Predictive control strategies for the AC–DC MCs under unbalanced grid voltage were proposed in [12] and [13]. Literature [14] presented a unity power factor predictive control method for the AC–DC MC. The most wide modulation control strategy for the AC–DC MCs has been considered the SVM method. A unity power factor fuzzy battery charger using the ultra-sparse matrix rectifier was designed and implemented in [15] with only three switches; however, it is a unidirectional converter. The direct power factor control strategy for the three-phase AC–DC MCs was illustrated in [16] based on applying the reduced general direct SVM approach of the AC–AC MC theory. Modulation and control strategies of the AC–DC MC for the battery energy storage system applications were investigated in [17] and [18]. Literature [19] studied the optimal zero-vector configuration to reduce the output inductor current ripple for the space-vector-modulated AC–DC MCs. The optimized modulation strategy to deduce the charging current ripple for vehicle to grid (V2G) applications using the AC–DC MC was presented in [20]. An input power factor control method was studied in [21] based on the concept of the virtual

capacitor. A controlled rectifier was implemented in [22] using the AC–AC MC theory. Literature [23] presented a digitally controlled switch mode power supply based on the MC without any modulation block. Dynamic characteristics of the matrix rectifier researches were studied in [24]. One of the most important issues for the AC–DC MC operation is DC ripples. Generally, increasing the switching frequency or increasing the inductor size of output filter can reduce the DC ripples. However, higher switching frequency leads to higher switching losses and larger size of output inductor results in the increase of size and cost of the converter. There are several approaches based on the SVM algorithm to reduce DC ripples [19,20,25]. In [19], an optimal zero-vector configuration was proposed to reduce DC ripples, however, the e ffectiveness of this approach is mainly maintained at low modulation because a period of zero-vector at high-modulation operation is very short compared with periods of two active vectors. Thus, this optimal configuration does not e ffectively reduce the DC ripples in wide operation ranges. The approach [20] proposed a sectional optimized modulation strategy, which can reduce DC ripples within the whole operation range by dividing many di fferent sectors, and di fferent groups of vectors are selected to synthesize the current vector. However, the zero vector configuration is not optimized at low-modulation operations. A recent work in [25] reduced DC ripples by dividing 12 di fferent sectors and using only active vectors to synthesize the current vector. Since only active vectors are used, the operation of this method is not guaranteed at low-modulation operation.

In this paper, the virtual space vector modulation (VSVM) for the AC–DC MC is proposed to reduce the DC current ripples within the whole modulation range. Previously, the VSVM concept was proposed to suppress the common-mode voltage of a two-level voltage source inverter (VSI) [26], and balance the neutral-point potential of a three-level neutral-point-clamped (NPC) inverter [27,28]. However, none of these approaches have been tried to reduce the DC current ripples using the VSVM, at the best knowledge of the authors. In this proposed VSVM method, each virtual vector is synthesized by two nearest active vectors, and each virtual sector is defined with the area between two virtual vectors. The current reference vector is synthesized by two virtual vectors and one zero vector in every switching period. The main principle of the proposed VSVM is reducing the dwelling period of the largest active current vector in each sector. In addition, the optimized switching patterns are proposed to further reduce the DC current ripples at both high- and low-power operations. Simulation and experimental results are demonstrated to verify validity and e ffectiveness of the proposed VSVM for reducing the DC current ripples of the AC–DC MCs.

### **2. Topology and Modulations of AC–DC Matrix Converter**

### *2.1. The Topology of AC–DC Matrix Converter*

The topology of the AC–DC MC is shown in Figure 1. It is made up of an array of six bidirectional power semiconductor switches, with the ability to conduct current in both directions. Each bidirectional switch is generally constructed by two insulated-gate bipolar transistors (IGBTs) connected in series with a common emitter. An input filter is used to suppress the high-frequency harmonic generated by the operation of converter and the grid. At the output terminal, the LC filter is used to smooth the output current. The AC–DC MC is powered by the AC voltage sources; thus the AC voltages are not allowed to be shorted. In addition, because of the inductive nature of the load, the load terminal must never be opened. Therefore, the AC–DC MC operates by connecting only one bidirectional switch in the upper-arm and only one bidirectional switch in the lower-arm at any instant. This leads to there being nine switching current vectors for the operation of the AC–DC MC.

**Figure 1.** Topology of AC–DC (alternating current–direct current) matrix converter.

### *2.2. Modulations of AC–DC Matrix Converter*
