**1. Introduction**

The concept of modular voltage source inverters (VSI) with coupled reactors considered in this paper was first reported in [1] (therein dubbed multipulse voltage source converters (VSC) with coupled reactors). The idea has its roots in the well-known multipulse AC–DC converter topologies widely used in a variety of applications, including adjustable speed drives (ASD), high-voltage direct current transmission (HVDC), aircraft power systems, and renewable energy conversion systems [2–5]. The use of multipulse topologies for the DC–AC power conversion has broad coverage in the literature (see, for example, in [6–11]). The topologies proposed in [8–11] are based on the use of a transformer in an integrating circuit, which significantly increases the size of the device and is an expensive solution. If the application does not require galvanic isolation of the DC side from the AC side, the integrated circuit can be based on coupled reactors, as proposed in [1]—a solution which greatly reduces the size and cost of the circuit.

Solutions presented in [1,7–11] are focused on maximizing the achievable magnitude of output voltage, while the control of lower magnitudes is left out. The possible voltage synthesis using all allowed combinations of switch states has not been seriously addressed so far. In this paper, a control of the output voltage of the modular voltage source inverters proposed in [1,12] is considered. These converters contain standard inverter modules, but they are connected by special coupled reactors. The idea of modular VSI with coupled reactors draws on the properties of multipulse diode rectifiers with similar coupled reactors [13–15] and other converters with integrating magnetic circuits [4,16,17]. Coupled reactors were selected as integrating elements because their rated power is below 20% of that

of a transformer with similar integrating properties [5,13,14]. As the inverter modules are connected in parallel, these circuits can be used for high-current systems. Where increased operating voltages are required, multilevel inverters can be used as modules.

To clarify the terminology used in the sequel, note that the idea of "pulses" contained in the term "multipulse" has a simple interpretation in the case of diode or thyristor AC–DC converters: it denotes a section of AC input voltage transferred to the DC output. The "number of pulses", denoted *M*, is used to mean the number of such sections transferred in one fundamental period of the AC voltage. Although for DC–AC inverters with fully controlled power switches this idea of pulses is much less tangible, the number of pulses can still be used for the sake of discussion—as a constructional parameter of the inverter.

Transistor-based modular VSI with coupled reactors can be controlled in a variety of ways. One of the methods mimics the workings of multipulse rectifiers. This method relies on applying a succession of only *M* inverter voltage vectors in every fundamental period of the synthesized output voltage. This paper demonstrates that by appropriate selection of all available basic voltage vectors, it is possible to achieve coarsely quantized pulse amplitude modulation (CQ-PAM). This control approach, leading to staircase output voltage waveforms, enjoys very low switching frequency of power transistors (equal to the fundamental frequency of the output voltage). A drawback of this modulation method is low amplitude resolution. A workaround might be application of a controlled source of DC voltage, but this option is complex and costly and thus it is left out in this paper.

High-resolution voltage control can be achieved by means of pulse width modulation (PWM). Unlike CQ-PAM, PWM requires relatively high switching frequency (a multiple of the fundamental frequency of output voltage), which can be particularly disadvantageous in high-speed motor drive applications. For example, a two-pole motor operating at 150,000 rpm would call for 5 kHz fundamental frequency of the output voltage, meaning at least several dozens of kilohertz of the transistor switching frequency for PWM controlled inverter. Such high switching frequencies can cause significant mismatches between the transistor on/off commands and the actual turn-on/turn-off timing. What is more, the switching losses and electromagnetic interference resulting from high switching frequency can be prohibitive [18]. The problem of adequate voltage control increases with the increase of motor speed and/or its power. For instance, motor drives in the 1 kW power range are reported to operate at speeds of 500,000 to 1,000,000 rpm [19,20]. Concerning higher power drives, the authors of [21] report on a 60 kW drive operating at 100,000 rpm, while the authors of [22] describe a 300 kW drive operating at 60,000 rpm.

To address the above problems, this paper proposes a hybrid modulation combining CQ-PAM and PWM. This approach is capable of combining advantages of PWM (virtually unlimited resolution of voltage control) and those of CQ-PAM (radically reduced switching frequency). The CQ-PAM is intended for use at steady state, while the PWM ensures smooth passage through the transients. A somewhat similar idea of hybrid modulation was proposed in [23,24], but it relies on a combination of two different PWM methods: space-vector pulse width modulation (SVPWM) is used for smooth transients, while selective harmonic elimination PWM (SHE-PWM) is applied for low switching frequency operation at steady state. The CQ-PAM proposed here is even more effective in the reduction of switching frequency than SHE-PWM and—unlike the latter—can easily be computed in real-time. Both CQ-PAM and PWM can rely on selecting and applying appropriate voltage vectors. At steady state, a succession of only *M* different vectors per fundamental period is used for *M*-pulse inverters, with all vectors being applied for time intervals of the same length. At each interval, the voltage vector closest to the reference vector is selected. Concerning the PWM, vector selection and duty cycle computations are carried out using the barycentric coordinates [25]. This approach speeds up the calculations and allows easy inclusion of the DC link voltage fluctuations in the algorithm. The proposed modulation is exposed and discussed using the simplest case of modular VSI, that is, a 12-pulse inverter with two-level component inverter modules. The application of the proposed approach can easily be extended to inverters with higher pulse numbers and/or with multilevel

inverter modules [1,12]. Section 2 derives the formula linking the output voltage of modular VSI with coupled reactors with the voltages supplied by component inverter modules and finds the required turns ratio of the coupled reactors. Section 3 presents the proposed modulation method, while Section 4 presents and discusses selected simulation results pertaining to modular VSI using two-level and three-level component inverters and the CQ-PAM. A simulation of the passage between two different steady states using the proposed SVPWM is also demonstrated. The laboratory test results of the same two topologies are presented in Section 5.

#### **2. The 12-Pulse Modular Voltage Source Inverter with Coupled Reactors**

Twelve-pulse topology has been known as early as in the 1980s [14,26] and widely used in industry to date [5,27–29]. However, the first attempts to use this topology (with coupled reactors) in the inverter mode of operation were made only several years ago [1,7]. To the best of the authors' knowledge, there have been no other reports on the use of multipulse topologies for DC–AC conversion. This section analyses the output voltage of the considered modular VSI as a result of vector summation of the component inverter voltages and determines the required turns ratio of the coupled reactors. A schematic diagram of the 12-pulse modular VSI with coupled reactors is shown in Figure 1, while Figure 2 establishes vectorial notation of voltages used in the following analysis.

**Figure 1.** Twelve-pulse modular voltage source inverters (VSI) with coupled reactors.

**Figure 2.** Vectorial equivalent circuit of the 12-pulse modular VSI.

According to the diagram in Figure 2, the output voltage vector of the analyzed modular VSI is given by

*Energies* **2020**, *13*, 4450

$$
\hat{\mathcal{V}}\_{\rm o} = \hat{\mathcal{V}}\_{1} - \Delta \hat{\mathcal{V}}\_{1} - \Delta \hat{\mathcal{V}}\_{2} \tag{1}
$$

where

$$
\Delta \hat{\mathcal{V}}\_1 = \left( \hat{\mathcal{V}}\_1 - \hat{\mathcal{V}}\_2 \right) \cdot \frac{N\_\mathcal{A} + N\_\mathcal{B}}{2 \cdot N\_\mathcal{A} + N\_\mathcal{B}} \tag{2}
$$

$$
\Delta \hat{V}\_2 = a^{-1} \cdot \left( \hat{V}\_1 - \hat{V}\_2 \right) \cdot \frac{N\_\text{B}}{2 \cdot N\_\text{A} + N\_\text{B}} \tag{3}
$$

$$\mathcal{V}\_1 = \mathcal{U}\_\mathcal{M} \cdot \mathbf{e}^{j\frac{\pi}{3} \cdot \text{ert}\left[\frac{3}{\pi}\omega t\right]},\\\mathcal{V}\_2 = \mathcal{U}\_\mathcal{M} \cdot \mathbf{e}^{j\frac{\pi}{3} \cdot \text{ert}\left[\frac{3}{\pi}(\omega t - \phi)\right]}\tag{4}$$

$$a = e^{j\frac{2\cdot n}{3}} \tag{5}$$

Thus, on the basis of Equations (1)–(3), the output voltage can be described by

$$\mathcal{V}\_{\rm o} = \mathcal{V}\_{\rm 1} - \left(\mathcal{V}\_{\rm 1} - \mathcal{V}\_{\rm 2}\right) \cdot \frac{N\_{\rm A} + N\_{\rm B}}{2 \cdot N\_{\rm A} + N\_{\rm B}} - a^{-1} \cdot \left(\mathcal{V}\_{\rm 1} - \mathcal{V}\_{\rm 2}\right) \cdot \frac{N\_{\rm B}}{2 \cdot N\_{\rm A} + N\_{\rm B}} \tag{6}$$

After simple algebra, Equation (6) can be rewritten as

$$\dot{\mathcal{V}}\_{\rm o} = \dot{\mathcal{V}}\_{1} \cdot \left(\frac{N\_{\rm A}/N\_{\rm B} - a^{-1}}{2 \cdot N\_{\rm A}/N\_{\rm B} + 1}\right) + \dot{\mathcal{V}}\_{2} \cdot \left(\frac{N\_{\rm A}/N\_{\rm B} - a}{2 \cdot N\_{\rm A}/N\_{\rm B} + 1}\right) \tag{7}$$

It is assumed that the control signals of the inverter modules in the considered 12–pulse system are phase-shifted by *φ* = 30◦, meaning the same phase shift between vectors *V*ˆ <sup>1</sup> and *V*ˆ 2. As a result, Equation (7) can be converted to

$$\mathcal{V}\_{\rm o} = \left(\mathcal{V}\_{1} + \mathcal{V}\_{2}\right) \cdot \left[\frac{\left(N\_{\rm A}/N\_{\rm B}\right)}{2 \cdot \left(N\_{\rm A}/N\_{\rm B}\right) + 1}\right] - \left(\mathcal{V}\_{1} \cdot a^{-1} + \mathcal{V}\_{2} \cdot a\right) \cdot \left[\frac{1}{2 \cdot \left(N\_{\rm A}/N\_{\rm B}\right) + 1}\right] \tag{8}$$

In order to determine the appropriate turns ratios of the reactor coils, assume that the magnitude of voltage across a coil is proportional to its number of turns. Then, from Figure 2 it immediately follows that

$$\frac{|\hat{V}\_1 - \hat{V}\_2|}{2 \cdot N\_\text{A} + N\_\text{B}} = \frac{|\Delta \hat{V}\_1|}{N\_\text{A} + N\_\text{B}} = \frac{|\Delta \hat{V}\_2|}{N\_\text{B}} \tag{9}$$

The desirable turns ratio is such that ensures symmetric contribution of the component voltages *V*ˆ <sup>1</sup> and *V*ˆ <sup>2</sup> to the output voltage *V*ˆ o, as illustrated in Figure 3. From the vector diagram in Figure 4, which is an enlargement of the shaded fragment in Figure 3, the following relationship can be found using Thales's theorem,

$$\frac{|\mathcal{V}\_1 - \mathcal{V}\_2|}{2} \cdot \tan(\lambda) = \left| \Delta \mathcal{V}\_2 \right| \cdot \sin \left( 60^\circ \right) \tag{10}$$

where *λ* = *<sup>φ</sup>* <sup>2</sup> = 15◦. Now, using this value of *λ* and substituting Equation (10) to Equation (9), one arrives at

$$\frac{2 \cdot N\_{\text{A}} + N\_{\text{B}}}{2} \cdot \tan(15^{\circ}) = N\_{\text{B}} \cdot \sin\left(60^{\circ}\right) \tag{11}$$

whereupon, using basic trigonometric relationships, an explicit formula for the required turns ratio of the reactor coils is found:

$$\frac{N\_{\rm A}}{N\_{\rm B}} = \frac{\sin\left(60^{\circ} - 15^{\circ}\right)}{\sin\left(15^{\circ}\right)} = 2.732\tag{12}$$

**Figure 3.** Desirable positions of basic output vectors of component inverters in the 12-pulse modular VSI.

**Figure 4.** Details of geometric relationship between the output vector *V*ˆ o of the 12-pulse modular VSI and component vectors *V*ˆ <sup>1</sup> and *V*ˆ 2.

#### **3. Proposed Hybrid Modulation Method**

In one of the proposed operation modes of the considered modular VSI, transistors of inverter modules commute with the fundamental output frequency, which means significant reduction of switching losses and electromagnetic interference (EMI) distortion, and allows high frequencies of output voltages. The problem with this type of control, referred to as the CQ-PAM, is the limited resolution of the output voltage. The proposed solution is the use of PWM as a complementary control method. Both modulation techniques are based on appropriate selection and timing of the available basic vectors, that is, voltage vectors corresponding directly to the on/off states of inverter switches. These vectors can be considered points on a two dimensional *αβ* plane. The basic vector diagram for the considered 12-pulse system is shown in Figure 5a. The basic vectors in Figure 5 are obtained in two steps. First, the leg voltages of component inverter modules (marked in Figure 1) are transformed to phase voltages of the modular VSI by

$$
\begin{bmatrix} u\_{\mathsf{a}} \\ u\_{\mathsf{b}} \\ u\_{\mathsf{c}} \end{bmatrix} = \begin{bmatrix} u\_{1\mathsf{b}} & u\_{1\mathsf{b}} - u\_{2\mathsf{b}} & u\_{1\mathsf{a}} - u\_{2\mathsf{a}} \\ u\_{1\mathsf{c}} & u\_{1\mathsf{c}} - u\_{2\mathsf{c}} & u\_{1\mathsf{b}} - u\_{2\mathsf{b}} \\ u\_{1\mathsf{a}} & u\_{1\mathsf{a}} - u\_{2\mathsf{a}} & u\_{1\mathsf{c}} - u\_{2\mathsf{c}} \end{bmatrix} \cdot \begin{bmatrix} 1 \\ -k\_1 \\ -k\_2 \end{bmatrix} \tag{13}
$$

where

$$k\_1 = \frac{N\_\text{A} + N\_\text{B}}{2 \cdot N\_\text{A} + N\_\text{B}}, \; k\_2 = \frac{N\_\text{B}}{2 \cdot N\_\text{A} + N\_\text{B}} \tag{14}$$

Then, the *αβ* coordinates of the corresponding basic vectors (*V*ˆ <sup>o</sup>) are determined by the Clarke transformation:

$$
\begin{bmatrix} v\_{\mathsf{a}} \\ v\_{\mathsf{f}} \end{bmatrix} = \begin{bmatrix} \frac{2}{3} & -\frac{1}{3} & -\frac{1}{3} \\ 0 & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} \end{bmatrix} \cdot \begin{bmatrix} u\_{\mathsf{a}} \\ u\_{\mathsf{b}} \\ u\_{\mathsf{c}} \end{bmatrix} \tag{15}
$$

In general, the number of different basic vectors for an *M*-pulse inverter with *l*-level inverter modules is *l M* <sup>2</sup> . Therefore, the considered 12-pulse converter exhibits 64 different basic vectors.

**Figure 5.** Space-vector diagrams of three modular VSI topologies (top graphs) and example voltage waveforms (bottom graphs) corresponding to the sequences of vectors indicated by connecting lines: (**a**) 12-pulse 2-level modules; (**b**) 12-pulse 3-level modules; (**c**) 18-pulse 2-level modules.
