Research on Space Vector Overmodulation Technology of Two-Level PWM Converters

Abstract

In this paper, the characteristics of the voltage transfer ratio (VTR) and low-order harmonics of the major modern space vector overmodulation strategies for two-level PWM converters are discussed under different VTRs, and the complexity of the modulation algorithm is subsequently elaborated. First, the principles of these overmodulation strategies are summarized. After that, the accuracy and linearization of the VTR are obtained using the Fourier transform analysis, and the relationship between the main low harmonic components and the VTR is obtained in the same way. All the analysis results are verified through experiments. In addition, the merits and demerits of these overmodulation strategies and applications are revealed by comparing the simplicity of their modulation algorithm, digitization, and other characteristics. Finally, this paper reveals the essence of space vector overmodulation strategies. The results facilitate the design, performance prediction, and development of high-performance overmodulation strategies, as well as the transplantation of space vector overmodulation strategies into different converter topologies.

1. Introduction

With the rapid development of power electronic technology and further improvement of control theory, motor speed regulation technology is maturing. Overmodulation is an effective method to achieve the high output voltage magnitude of the converter without changing the topology structure [1]. In addition, it can expand the steady-state operation area of the motor, which has invariable dominance [2]. In the wide range of speed regulation systems of the motors driven by two-level converters, the technology of overmodulation is relatively mature [3,4]. Compared with the method of adding a boost circuit, only the control or modulation strategy of the converters is required to change, which is easy to implement in engineering [5].

To improve the voltage transfer ratio (VTR) of the PWM converters, many scholars began to study the overmodulation technique in the early 1990s. Considering the required harmonic characteristics and control complexity of power converters, various strategies have been continuously proposed.

The overmodulation techniques can be divided into two categories following the degree of their modulation effect: one is the minimum-phase-error PWM (MPEPWM) [6,7], which only changes the magnitude of the reference voltage without changing the phase. In contrast, both the magnitude and phase of the reference voltage are changed in the minimum-magnitude-error PWM (MMEPWM) [8,9]. In addition, overmodulation techniques can be divided into two categories based on their modulation principle: In [10,11], a PWM overmodulation technique was proposed to improve the VTR realized by adjusting the reference output voltage magnitude (i.e., the magnitude of the modulating wave), and the authors also discussed the harmonic components. Theoretically, due to the modulating wave adopting the standard sine function, only when the magnitude of the reference modulating waveform is infinite can the fundamental waveform magnitude be obtained, which corresponds to a square modulated waveform. This result deteriorates the fundamental magnitude of the output voltage. To address this problem, the authors added an inverse gain compensation link to achieve the desired output voltage. In addition, the problems caused by overmodulation severely threaten the system reliability of the hybrid modular multilevel converter (MMC). The authors of [12] introduced a structure of self-balancing branches into the hybrid MMC to eliminate voltage deviations. In another study [13], the authors proposed an approach of optimizing the SHCC reference to realize voltage balancing, which minimizes the power losses caused by the SHCC injection. Another technique is the space vector overmodulation technique, which improves the VTR by adjusting the magnitude and phase of the reference output voltage vector [5,6,8,14,15,16]. In [5], a multimode space vector overmodulation was proposed based on three-degree-of-freedom converters. Bolognani proposed a space vector strategy based on a six-step mode for high dynamic performance and high switching frequency drives [8]. In [14], the actual output voltage of a current controller was selected as the closest voltage vector at the same angle as that required by the current error compensator. In [16], a novel control technology combining discontinuous pulse width modulation (DPWM) and overmodulation technology was proposed to better utilize direct current. Compared with the PWM overmodulation, the space vector overmodulation has one more modulation degree, which makes it more flexible [17]. Therefore, a remarkable number of studies in the field of two-level PWM converter overmodulation have been dedicated to space vector overmodulation in recent years.

Higher voltage utilization is related to the external performance of the overmodulation strategy. However, whether or not it can be consistent with the command voltage is of greater significance. In other words, an excellent overmodulation strategy should have a smaller low-harmonic distortion rate and more accurate VTR, so the components of harmonic can be used for evaluation. Generally, the harmonic components of the output voltage can be calculated using the fast Fourier transform (FFT) method [7,10,18,19], and the double Fourier integral transform method also offers a possibility to analyze the harmonic components [19,20,21,22,23]. However, since the switching frequency is much larger than the fundamental frequency, a single Fourier integral transform was used in this paper to evaluate the low-order harmonic components [24,25,26,27], and the analysis was verified through experiments.

In this paper, the modern space vector overmodulation strategies based on two-level converters are discussed. First, the significance, application, and classification of overmodulation and the method of characteristic analysis are briefly introduced in Section 1. In Section 2, the basic principle of each space vector overmodulation strategy is summarized. The linearity of VTR, low-order harmonic components, and the complexity of the modulation algorithm are evaluated in Section 3 with the single Fourier transform method. In Section 4, a test system based on two-level converters is built, and the correctness of the theoretical analysis is verified through experiments. Section 5 compares the characteristics of the space vector overmodulation technique. Finally, the essence of the space vector overmodulation strategy is revealed in Section 6.

2. Space Vector Overmodulation Technology of Two-Level PWM Converters

To simplify the analysis, the reference VTR m, output fundamental VTR m_{out_1}, space vector V_ref corresponding to the reference output voltage, and the six effective voltage vectors V_l(l = 1, 2, 3, 4, 5, 6) are defined as:

$$m=\frac{V_{\mathit{ref}}}{\frac{2}{3}{V}_{\mathit{dc}}}{,m}_{\mathit{out}\text{\_}1}=\frac{V_{\mathit{out}\text{\_}1}}{\frac{2}{3}{V}_{\mathit{dc}}}$$

(1)

$${\mathit{V}}_{\mathit{ref}}{=V}_{\mathit{ref}}{e}^{{j\theta }_{\mathit{out}}}$$

(2)

$${\mathit{V}}_l=\frac{2}{3}{V}_{\mathit{dc}}{e}^{j\frac{\pi \left(l-1\right)}{3}}$$

(3)

where V_dc is the DC bus voltage, V_{out_1} is the fundamental magnitude of the output voltage, and V_ref is the magnitude of the reference output voltage vector; the output phase θ_out = 2πf_outt + φ₀, f_out is the reference output voltage frequency, and φ₀ is the reference output voltage initial phase angle.

Achieving the desired reference voltage vector by combining eight different voltage vectors of the two-level converter is the basic principle of SVPWM. Figure 1 shows the vector diagram of the two-level converters. Ignoring the effects of nonlinearities such as switching harmonics and the dead zones of the converters, the SVPWM exhibits excellent linear VTR in the linear modulation region. The output voltage of harmonic components is almost zero (the linear modulation region is the inner circle in Figure 1).

When the converter operates outside the linear modulation region (the region between the inner and outer circles in Figure 1), the space vector overmodulation strategy is utilized to obtain the reference voltage magnitude, but the output voltage contains some low-order harmonics.

There are four widely used strategies in the space vector overmodulation technique: strategy A is based on the dual-mode space vector overmodulation of sub-trajectory, strategy B adopts the single-mode space vector overmodulation of sub-trajectory, strategy C uses the dual-mode space vector overmodulation of superimposed limit trajectory, and strategy D is grounded on the single-mode space vector overmodulation of limit trajectory superposition.

2.1. Strategy A: Dual-Mode Space Vector Overmodulation Based on Sub-Trajectory

A dual-mode space vector overmodulation strategy based on sub-trajectory was proposed in the literature [6], as shown in Figure 2. The strategy distinguished the overmodulation region into two intervals of 0.866 < m ≤ 0.909 and 0.909 < m ≤ 0.954, which are called the overmodulation regions I and II, respectively.

The upper bound of the overmodulation is the maximum output voltage vector V_{out_max} with double effective vector synthesis modulation, whereas the lower bound is the maximum linear output voltage vector V_{linear_max} with a double effective vector and zero vector synthesis modulation. The VTR m₀ corresponding to V_{linear_max} is 0.866, while the m₁ corresponding to V_{out_max} is 0.909. The upper-limit vector of the overmodulation region II is the output voltage vector of a two-level inverter with single effective vector modulation, and the corresponding VTR m₂ is 0.954.

The first sector is used as an example, when the two-level converter operates in the overmodulation region I, as shown in Figure 2a. The output voltage vector V_out is a segmented combination of the V_{out_max} and the compensation output voltage vector V_ar, i.e.,

$${\mathit{V}}_{\mathit{out}}=\left\{\begin{array}{l}{\mathit{V}}_{\mathit{ar}},\frac{\pi }{3}\left(l-1\right)\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}\left(l-1\right)+a_r‖\frac{\pi }{3}l-a_r\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}l\\ {\mathit{V}}_{\mathit{out}\text{\_}\mathit{max}},\frac{\pi }{3}\left(l-1\right)+a_r\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}l-a_r\end{array}\right.$$

(4)

where V_ar and V_{out_max} are expressed as follows:

$$\left\{\begin{array}{l}{\mathit{V}}_{\mathit{ar}}{=V}_{\mathit{ar}}{e}^{{j\theta }_{\mathit{out}}}=\frac{V_{\mathit{dc}}}{\sqrt{3}\mathit{cos}\left(\pi /6-a_r\right)}{e}^{{j\theta }_{\mathit{out}}}\\ {\mathit{V}}_{\mathit{out}\text{\_}\mathit{max}}{=V}_{\mathit{out}\text{\_}\mathit{max}}{e}^{{j\theta }_{\mathit{out}}}=\frac{V_{\mathit{dc}}}{\sqrt{3}\mathit{cos}\left(\pi /6-{\theta }_{\mathit{out}}\right)}{e}^{{j\theta }_{\mathit{out}}}\end{array}\right.$$

(5)

In the above equations, a_r is the dividing angle of the overmodulation region I. When the angle θ between the reference output voltage vector and the adjacent effective voltage vector is less than a_r, V_out is equal to V_ar; when this is greater than a_r, V_out is equal to V_{out_max}.

When the two-level converter operates in the overmodulation region II, as shown in Figure 2b, V_out is the segmented combination of V_ap and the adjacent effective voltage vector V_l, i.e.,

$${\mathit{V}}_{\mathit{out}}=\left\{\begin{array}{l}{\mathit{V}}_l,0\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}\left(l-1\right)+a_h\\ {\mathit{V}}_{\mathit{ap}},\frac{\pi }{3}\left(l-1\right)+a_h\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}l-a_h\\ {\mathit{V}}_{l+1},\frac{\pi }{3}l-a_h\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}l\end{array}\right.$$

(6)

where V_ap and a_p are expressed as follows:

$${\mathit{V}}_{\mathit{ap}}=\frac{V_{\mathit{dc}}}{\sqrt{3}\mathit{cos}\left(\pi /6-a_p\right)}{e}^{j\left(a_p+\left(l-1\right)\frac{\pi }{3}\right)}$$

(7)

$$a_p=\frac{{\theta }_{\mathit{out}}-\pi /3\left(l-1\right)-a_h}{1-6a_h/\pi }$$

(8)

In these equations, a_h is the dividing angle of the overmodulation region II. When the angle θ between the reference output voltage vector and the adjacent effective voltage vector is less than a_h, V_out is equal to V_l or V_l+1; when θ is greater than or equal to a_h, V_out is equal to V_ap.

2.2. Strategy B: Single-Mode Space Vector Overmodulation Based on Sub-Trajectory

A single-mode space vector overmodulation strategy based on sub-trajectory was proposed in the literature [8]. This space vector overmodulation strategy no longer divides the overmodulation region. When the two-level converter operates in the overmodulation region, as shown in Figure 3, taking the first sector as an example, the V_out can be expressed as follows:

$${\mathit{V}}_{\mathit{out}}=\left\{\begin{array}{l}{\mathit{V}}_{\mathit{ref}},\frac{\pi }{3}\left(l-1\right)\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}\left(l-1\right)+a_g\\ \left|{\mathit{V}}_{\mathit{ref}}\right|e^{j\left(a_g+\frac{\pi }{3}\left(l-1\right)\right)},\frac{\pi }{3}\left(l-1\right)+a_g\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}l-\frac{\pi }{6}\\ \left|{\mathit{V}}_{\mathit{ref}}\right|e^{j\left(\frac{\pi }{3}l-a_g\right)},\frac{\pi }{3}l-\frac{\pi }{6}\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}l-a_g\\ {\mathit{V}}_{\mathit{ref}},\frac{\pi }{3}l-a_g\le {\theta }_{\mathit{out}}\text{<}\frac{\pi }{3}l\end{array}\right.$$

(9)

on the basis of this geometric relationship, V_ref and a_g are expressed as follows:

$$V_{\mathit{ref}}=\frac{V_{\mathit{dc}}}{\sqrt{3}\mathit{cos}\left(\pi /6-a_g\right)}$$

(10)

2.3. Strategy C: Dual-Mode Space Vector Overmodulation Based on Limit Trajectory Superposition

A dual-mode space overmodulation strategy based on limit trajectory superposition was proposed in the literature [7], as shown in Figure 4. This strategy also divides the overmodulation region into two intervals, and the boundary curve is still the hexagonal trajectory corresponding to V_{linear_max}.

When the VSI runs in the overmodulated region I, as shown in Figure 4a, V_out is synthesized by V_{linear_max} and V_{out_max} and linearly weighted, i.e.,

$${\mathit{V}}_{\mathit{out}}{=k}_1{\mathit{V}}_{\mathit{out}\text{\_}\mathit{max}}+(1-k_1){\mathit{V}}_{\mathit{linear}\text{\_}\mathit{max}}$$

(11)

the weighted coefficient k₁ is expressed as follows:

$$k_1=\frac{m-m_0}{m_1-m_0}$$

(12)

When the VSI runs in overmodulated region II, as shown in Figure 4b, V_out is synthesized by V_{out_max} and V_l and linearly weighted, i.e.,

$${\mathit{V}}_{\mathrm{out}}{=k}_2{\mathit{V}}_l+(1-k_2){\mathit{V}}_{\mathit{out}\text{\_}\mathit{max}}$$

(13)

the weighted coefficient k₂ is expressed as follows:

$$k_2=\frac{m-m_1}{m_2-m_1}$$

(14)

The dual-mode space overmodulation strategy based on limit trajectory superposition is realized by the linear superimposed principle.

2.4. Strategy D: Single-Mode Space Vector Overmodulation Based on Limit Trajectory Superposition

A single-mode overmodulation strategy based on limit trajectory superposition was proposed in the literature [9], as shown in Figure 5. V_out in the whole overmodulation region is synthesized by V_{linear_max}, and V_l linearly weighted, i.e.,

$${\mathit{V}}_{\mathrm{out}}=k{\mathit{V}}_l+(1-k){\mathit{V}}_{\mathit{linear}\text{\_}\mathit{max}}$$

(15)

the weighted coefficient k is expressed as follows:

$$k=\frac{m-m_0}{m_2-m_0}$$

(16)

3. Characteristics of Space Vector Overmodulation Technology

3.1. VTR

The output voltage magnitude and low-order harmonic components are the key indexes to evaluate the performance of the overmodulation strategy. Therefore, an excellent evaluation of the overmodulation strategy should have an accurate reference output voltage (i.e., the fundamental VTR m_{out_1} is equal to the VTR m), and the output voltage low-order harmonic components should be almost zero (i.e., the minimum THD) throughout the overmodulation region. The VTR and low-order harmonics of the above space vector overmodulation strategies are analyzed in detail, as described below.

The output voltage is affected by both the modulation wave and the carrier wave, so the output voltage spectrum can be accurately obtained by using a double Fourier integral transform [19,20,21,22,23]. However, in general, the carrier frequency is not considered in a low-order harmonic analysis since the carrier frequency is much larger than the modulation wave. Therefore, the fundamental magnitude of the output voltage and the low-order harmonic components are calculated by using a single Fourier integral transform [24,25,26,27].

3.1.1. Strategy A: Dual-Mode Space Vector Overmodulation Based on Sub-Trajectory

According to the principle of this strategy, the output fundamental voltage magnitude V_{out_1} and m_{out_1} in the overmodulation region I can be obtained by using a single Fourier integral transform as follows:

$$V_{\mathit{out}\text{\_}1}=\frac{V_{\mathit{dc}}}{2\pi }\left(2\sqrt{3}\mathit{ln}\left|\frac{1+\mathit{cos}(\pi /3+a_r)}{1-\mathit{cos}(\pi /3+a_r)}\right|+\frac{4\sqrt{3}{a}_r}{\mathit{cos}(\pi /6-a_r)}\right)$$

(17)

$$m_{\mathit{out}\text{\_}1}=\frac{3}{4\pi }\left(2\sqrt{3}\mathit{ln}\left|\frac{1+\mathit{cos}(\pi /3+a_r)}{1-\mathit{cos}(\pi /3+a_r)}\right|+\frac{4\sqrt{3}{a}_r}{\mathit{cos}(\pi /6-a_r)}\right)$$

(18)

From Equation (18), it can be seen that m_{out_1} has a nonlinear relationship with the boundary angle a_r in the overmodulation region I, as shown in Figure 6. To obtain the linearized output between the fundamental output voltage and the reference output voltage in the overmodulation region I, the approximate linear function of m and a_r is expressed as follows:

$$a_r=\left\{\begin{array}{l}-38.3911\times m + 33.7703;0.866\le m\text{<}0.8683\\ -9.4720\times m + 8.6598 ;0.8683\le m\text{<}0.9065\\ -29.3911\times m + 26.7165;0.9065\le m\text{<}0.909\end{array}\right.$$

(19)

V_{out_1} and m_{out_1} of the overmodulation region II can be expressed as follows:

$$V_{\mathit{out}\text{\_}1}=\frac{V_{\mathit{dc}}}{2\pi }\left(8{\mathit{sin}(a}_h)+4\sqrt{3}\mathit{cos}(\frac{\pi }{6}-a_h{)+D}_1\right)$$

(20)

$$m_{\mathit{out}\text{\_}1}=\frac{3}{4\pi }\left(8{\mathit{sin}(a}_h)+4\sqrt{3}\mathit{cos}(\frac{\pi }{6}-a_h)+D_1\right)$$

(21)

where

$$D_1={\displaystyle {\int }_{\frac{\pi }{3}+a_h}^{\frac{2\pi }{3}-a_h}\frac{2\sqrt{3}\mathit{cos}(\frac{\pi }{3}+\frac{{\theta }_{\mathit{out}}-\pi /3-a_h}{1-6a_h/\pi })}{\mathit{sin}(\frac{\pi }{3}+\frac{{\theta }_{\mathit{out}}-\pi /3-a_h}{1-6a_h/\pi })}{e}^{-{jp\theta }_{\mathit{out}}}{d\theta }_{\mathit{out}}}$$

(22)

It can be seen from Equation (21) that m_{out_1} is nonlinearly related to a_h in the overmodulation region II, as shown in Figure 7.

To obtain the linearized output of fundamental output voltage and the reference output voltage of the overmodulated region II, the approximate linear function of m with respect to the division angle a_h is expressed as follows:

$$a_h=\left\{\begin{array}{l}6.667\times m-6.0567;0.9089\le m<0.9360\\ 11.7171\times m-10.7844;0.9360\le m<0.9527\\ 44.8611\times m-42.3142;0.9513\le m<0.9549\end{array}\right.$$

(23)

The overmodulation strategy obtains the reference output voltage accurately by using the segmented processing of the nonlinear numerical relationship between the VTR and a_h, so as to realize the linearization of the fundamental component and the output VTR in the overmodulation region [18].

3.1.2. Strategy B: Single-Mode Space Vector Overmodulation Based on Sub-Trajectory

Following the principle of this strategy, the output fundamental voltage magnitude in the overmodulation region can be obtained by using the single Fourier integral transform as follows:

$$V_{\mathit{out}\text{\_}1}=\frac{2\sqrt{3}{V}_{\mathit{dc}}}{\pi }\left[\frac{a_g}{\mathit{cos}\left(\frac{\pi }{6}{-a}_g\right)}+\mathit{cos}\left(\frac{\pi }{3}{+a}_g\right)+\mathit{tan}(\frac{\pi }{6}-a_g)\left(1-\mathit{cos}\left(\frac{\pi }{6}-a_g\right)\right)\right]$$

(24)

Substituting Equation (10) into Equation (24), the relationship between m_{out_1} and m can be obtained as follows:

$$m_{\mathit{out}\text{\_}1}=\frac{3}{\pi }\left[2m\left(\pi /6-\mathit{arccos}\frac{\sqrt{3}}{2m}\right)+\sqrt{4m^2-3}\right]$$

(25)

It can be seen from Equation (25) that m_{out_1} has a nonlinear relationship with m.

3.1.3. Strategy C: Dual-Mode Space Vector Overmodulation Based on Limit Trajectory Superposition

The magnitude of the output fundamental voltage in overmodulation regions I and II is:

$$V_{\mathit{out}\text{\_}1}=\frac{V_{\mathit{dc}}}{2\pi }\left[2\sqrt{3}\mathit{ln}3k_1+\frac{2\sqrt{3}}{3}\pi (1-k_1)\right]$$

(26)

$$V_{\mathit{out}\text{\_}1}=\frac{V_{\mathit{dc}}}{2\pi }\left[4k_2+2\sqrt{3}\mathit{ln}3(1-k_2)\right]$$

(27)

Substituting Equation (12) and Equation (14) into Equation (26) and Equation (27), respectively, m_{out_1} can be expressed as follows:

$$m_{\mathit{out}\text{\_}1}=m$$

(28)

It can be seen from Equation (28) that overmodulation strategy C can accurately obtain the reference output voltage.

3.1.4. Strategy D: Single-Mode Space Vector Overmodulation Based on Limit Trajectory Superposition

The magnitude of the output fundamental voltage and the output fundamental VTR are expressed as follows:

$$V_{\mathit{out}\text{\_}1}=\frac{V_{\mathit{dc}}}{2\pi }[4k+(1-k)\frac{2\sqrt{3}\pi }{3}]$$

(29)

Substituting Equation (16) into Equation (29), m_{out_1} can be expressed as Equation (28), which shows that overmodulation strategy D can accurately obtain the reference output voltage.

Using the data from Figure 6 and Figure 7, and Equations (18), (21), (25), (26) and (28), the relationship between the VTR and the fundamental VTR of each overmodulation strategy can be obtained as shown in Figure 8.

It can be seen from the diagram that overmodulation strategy A can obtain the approximate linearity of the fundamental VTR and the reference VTR after the approximate linearization of Equations (19) and (23). The fundamental VTR of strategy B is nonlinear with the voltage modulation ratio, and the maximum VTR cannot be obtained. Overmodulation strategies C and D can accurately obtain the fundamental magnitude of output voltage in the overmodulation region.

3.2. Low-Order Harmonic Components

Following the principle of each overmodulation strategy, and using the single Fourier integral transform, the nth low-order harmonic components h_n (=V_{out_n}/V_{out_1}) about the output voltage can be obtained, as shown in Figure 9.

Figure 9 shows the main low-order harmonic components of the output voltage, and it can be seen from this figure that when 0.866 < m ≤ 0.909, the fifth and seventh harmonic components of strategies A and C do not exceed 3%, and the fifth and seventh harmonic components of strategies B and D are much larger.

4. Experiment

In order to verify the above theory and analyze the practicality of each overmodulation strategy, a two-level converter test system was built, as shown in Figure 10. The experimental parameters are shown in

Table 1. Parameters of experiment setup.

Parameters	Value
DC bus voltage (Vdc)	100 V
Carrier frequency	10 kHz
Output fundamental frequency	60 Hz

.

4.1. Verification and Analysis of VTR

As mentioned previously, after the linearization of overmodulation strategies A, C, and D, the fundamental VTR was basically consistent with the expected value, while strategy B was lower than the rest. The deviation increased with the increase in the modulation ratio. Compared with Figure 8, the results were basically consistent. Due to the existence of nonlinear factors such as switch tube voltage drop and dead band, there was a percentile deviation between the actual and the theoretical results, as shown in Figure 11 and

Table 2. Experimental results of fundamental modulation ratio output by each strategy.

Modulation Ratio	Strategy A	Strategy B	Strategy C	Strategy D
0.8660	0.8612	0.8581	0.8570	0.8591
0.9000	0.8929	0.8826	0.8951	0.8940
0.9090	0.9012	0.8926	0.8956	0.9024
0.9300	0.9207	0.9064	0.9165	0.9248
0.9548	0.9465	0.9199	0.9427	0.9419
0.9548	0.9465	0.9199	0.9427	0.9419

. Obviously, the experimental data verified the correctness of the theoretical analysis.

4.2. Verification and Analysis of Low-Order Harmonic Components

The experimental data were decomposed via fast Fourier transform, and the modulation ratio and content of harmonics could be obtained. When the fundamental voltage modulation ratio m_{out_1} = 0.909, the output phase voltage of each strategy is shown in Figure 12, and the Fourier transform results are shown in Figure 13. The results were consistent with those of the previous analysis. The harmonic components of strategies A and C were the same under this condition, the harmonic components of strategies B and D were relatively large, and the harmonic components of strategy D were the largest among the four strategies.

The experimental results of the fifth and seventh harmonic components are shown in Figure 14. When 0.866 < m ≤ 0.909, the fifth and seventh harmonic components of strategies A and C were less than 3%, and the fifth and seventh harmonic components of strategies B and D were far greater than those of strategies A and C. Compared with Figure 9, the results were basically consistent. In addition, the phase current waveform of the motor is shown in Figure 15. It is clear that the current harmonic components of strategies B and D also exceed those of strategies A and C. Therefore, the correctness of the theoretical analysis was verified through these experiments.

5. Characteristic Comparison of Space Vector Overmodulation Techniques

The space vector overmodulation strategies can be divided into single-mode and dual-mode space vector overmodulation strategies based on the analysis and comparison of their characteristics, as shown in Figure 16.

The single-mode overmodulation strategy directly regulates the magnitude and phase of the reference voltage vector, which is a kind of MMEPWM. In contrast, the dual-mode overmodulation strategy divides the overmodulation region into regions I and II, and the boundary VTR of the two regions is the maximum fundamental VTR obtained by only adjusting the magnitude of the reference voltage vector. In detail, the overmodulation region I only adjusts the magnitude of the reference voltage vector, which is a kind of MPEPWM; and the overmodulation region II simultaneously adjusts the magnitude and phase of the reference voltage vector to obtain a higher VTR, which is a kind of MMEPWM. Compared with the single-mode overmodulation, the output waveform of the dual-mode overmodulation strategy reduces the harmonic distortion.

The specific implementation of the space vector overmodulation strategy is divided into two methods: sub-trajectory and limit trajectory superposition.

In the sub-trajectory strategies, the output voltage vector adopts the lower bound trajectory vector of the overmodulation region when the angle θ between the reference output voltage vector and the adjacent effective voltage vector does not exceed the division angle. By contrast, when θ is equal to or larger than the division angle, the output voltage vector adopts the upper bound trajectory vector of the overmodulation region. The realization method of the sub-trajectory needs to establish a nonlinear mathematical relationship between the boundary angle and the fundamental magnitude of the output voltage by using the single Fourier integral transform. After that, approximate linearization is performed to obtain the specific value of the boundary angle α. Therefore, extensive data should be stored in the controller, offline and in advance, resulting in a considerable amount of system storage, which deteriorates algorithm accuracy and is adverse to engineering applications and portability. Moreover, this realization method also needs the symmetry of each sector.

The limit trajectory superposition is realized by the linear superimposed principle; this implementation method only needs to calculate the fundamental magnitude and harmonic components corresponding to the upper and lower limit trajectories. The result not only avoids a large number of Fourier integral or table-searching calculations but also obtains the linearized output voltage. Compared with the sub-trajectory method, the latter is more general and portable.

6. Conclusions

By analyzing and comparing the major modern space vector overmodulation technologies of two-level PWM converters, we divided the modern space vector overmodulation technologies into single-mode and dual-mode modulation strategies following their degree of regulation. On the grounds of the specific implementation method, the modern space vector overmodulation technology was divided into sub-trajectory and limit trajectory superposition strategies.

The dual-mode space vector overmodulation technique fully utilizes the two degrees of the two-level converter. Firstly, the output voltage with a small distortion rate is obtained by adjusting the magnitude of the reference voltage vector on the basis of improving the VTR. After that, the phase and magnitude of the reference voltage vector are simultaneously adjusted to obtain a higher VTR. Compared with the single-mode overmodulation technique, the dual-mode overmodulation technique can further reduce the output voltage THD under the same VTR.

The space vector overmodulation technique can be divided into several modes following the adjustable degree based on the converter topology. For example, the two-level converter can be adjusted by two degrees in the overmodulation region: the magnitude and phase of the reference output voltage. As a result, the two-level converter can be divided into single-mode and dual-mode overmodulation techniques. In contrast, the matrix converter can be adjusted by three degrees in the overmodulation region: the magnitude and phase of the output voltage vector and the phase of the input current. Therefore, the matrix converter can be divided into single-mode, dual-mode, and multi-mode overmodulation techniques.

The implementation method based on the limit trajectory superposition does not require symmetry in each sector and only needs to calculate the fundamental magnitude and harmonic components corresponding to the upper and lower limit trajectories to achieve its goal. Compared with the overmodulation strategy based on the sub-trajectory principle, it has the advantages of generality and portability, thus it can be transferred to converters of different topologies.