Investigation of Constant SVPWM and Variable RPWM Strategies on Noise Generated by an Induction Motor Powered by VSI Two- or Three-Level

Abstract

A three-phase inverter generates non-sinusoidal voltages, contains high order harmonics, and concentrates on switching frequency multiples. Supplying an induction machine (IM) with a voltage source inverter (VSI) increases the acoustic noise content which becomes unbearable, particularly for systems needing a moderate level of electric traction. The discrete tonal bands produced by the IM stator current spectrum controlled by the fixed pulse width modulation (PWM) technique have damaging effects on the electronic noise source. Moreover, it has been factually proven that the noise content is strongly associated with the harmonics of the source feeding electric machine. Thus, the harmonic content is influenced by the control strategy VSI to produce pulse width modulation (PWM). Currently, the investigation of new advanced control techniques for variable speed drives has developed into a potential investigation file. Two fundamental topologies for a three-phase inverter have been suggested in the literature, namely two- and three-level topologies. Therefore, this paper investigated the effect of variable and fixed PWM strategies, such as random PWM (RPWM) and space vector PWM (SVPWM), on the noise generated by an IM, powered with a two- or three-level inverter. Simulation results showed the validity and efficiency of the proposed variable RPWM strategy in reducing sideband harmonics for both the two and three levels at different switching frequencies and modulation indexes. The proposed PWM strategies were further evaluated by the results of equivalent experiments on an IM fed by a two-level VSI. The experimental measurements of harmonic current and noise spectra demonstrate that the acoustic noise is reduced and dispersed totally for the RPWM strategy.

1. Introduction

Induction machines (IM) are used in industry for electric vehicle applications due to their high and efficient performance. Therefore, IMs are often employed for drive motors at variable speed. Nevertheless, the electric motor is the source of unpleasant acoustic noise. Emerging, stricter standards and rising sensitivity to environmental noise pollution has made noise a primordial research topic, especially for electrical motors [1,2]. The authors in [2] described the various acoustic noise nuisances and their reduction methods. Thus, they classified the acoustic noise origins in four categories: electronic, mechanical, magnetic, and aerodynamic. Our work focuses on the study of electronic noise. When an electrical machine is driven by VSI, then an audible noise level is observed. Hence, the acoustic noise emission is caused by non-sinusoidal voltages. The spectrum frequency obtained from the inverter output produces many voltage and current harmonics, which increase motor vibration and noise and subsequently reduce system performance. As a result, this can have considerable relevance for some applications, particularly for electric vehicles. In addition, the power source used is frequently PWM control. Hence, the electrical motor emits high acoustic noise. The voltage harmonic spectra created by the converter contain high-order harmonics. Thus, the harmonic content is augmented, and the generated acoustic noise content may become intolerable [3,4].

The acoustic noise-impact produced by the combination of IMs and a static converter constitutes a research topic. The high-content noise of the PWM inverter-IM originates from electronic sources [1,5,6]. PWM controls include constant-frequency and variable-frequency controls. Sine PWM (SPWM) and space vector PWM (SVPWM) are well-known fixed-switching frequency PWM methods used in industry. Using the sinusoidal PWM strategy, the concentration of harmonic components of the inverter occurs in particular ranges localized around switching frequency integer multiples [7,8]. Consequently, the authors in [1] examined an experimental process to define the vibration and acoustics of IM driven by a SVPWM strategy. This study concerned experimental tests at different carrier and fundamental frequencies. The acoustic noise spectral components showed that there were also important components around the switching frequency. Motor drives controlled with fixed PWM produced disagreeable tonal noise. The authors in [9] examined a simple variable-frequency VFPWM technique for a 6 kW IM drive. Experimental tests proved the uniform distribution of noise over a wide frequency range, and a significant reduction of the current harmonic compared with fixed PWM.

In recent years, a well-established concept has involved the application of the randomized PWM strategy. Some techniques are applied to diminish the noise and peak of the electromagnetic interference for electrical drives [7,9,10,11,12,13]. The effects of SVPWM, RPWM, and selective harmonic elimination PWM SHEPWM were also established in our study [14]. Consequently, the experimental results show that the noise and current harmonic were significantly reduced with random RPWM. Thus, this topic is of high interest, considering the numerous articles published lately. In [15], the study suggested a chaotic PWM technique to disperse the power spectra of IMs. Experimental studies confirmed that carrier signals, acoustic noise, and current spectra with the proposed PWM would be distributed along chaotic frequencies and without specific frequency concentration. A trapezoidal PWM approach to disperse noise spectra and decrease the current total harmonic distortion (THD) was suggested in [16]. This method was validated experimentally for a permanent magnet synchronous motor. Another method was developed by authors in [17] using a fixed and variable frequency RPWM for a dual inverter-fed IM. The obtained results validated that the variable-frequency VSF-RPWM decreased the THD current and that the harmonic spectra were distributed by randomizing the pulses. The authors in [10] employed a dual randomized PWM (DRPWM) scheme based on a control of a two-level inverter. The results proved the performance of the DRPWM approach in spreading and reducing current harmonic and noise spectra.

Nowadays, improved RPWM techniques have been established for three-phase inverters in the literature. A wide distribution in acoustic noise frequencies can be accomplished by random carrier frequency using a random signal [9,10,11,18]. The application of random switching frequency PWM (RSFPWM) is a widely studied method in spreading the noise frequency. When selecting a variable PWM technique, reduced calculation burden is frequently considered as an appropriate characteristic. The aim is also to obtain a spread spectra frequency in a simple manner with no compromise on any aspects of motor controls, such as harmonic content, low computational effort, and converter design (two- or three-level inverter). The assessment of RSFPWM and SVPWM for two-level and multilevel converter topologies in terms of acoustic and harmonic efficiency has not been well examined, with the exclusion of the SVPWM scheme for VSI two-level topology [1,3,4,9,19]. The widespread use and different benefits offered by these techniques are important to examine the characteristics in acoustic noise for both two- and three-level inverters.

Table 1. Different studies reported in the literature on acoustic noise emitted by an induction motor fed by PWM inverter.

Reference	PWM Strategy Used	Inverter	Year
[8]	Sinusoidal PWM (SPWM) and Random PWM (RPWM)	Two-Level	1991
[19]	Advanced Bus-Clamping PWM and Space Vector PWM SVPWM	Two-Level	2013
[1]	SVPWM	Two-Level	2015
[9]	Variable-switching Frequency PWM (VFPWM) and SVPWM	Two-Level	2016
[7]	RPWM	Two-Level	2016
[4]	SVPWM and Modified SVPWM	Two-Level	2017
[3]	Hybrid Random PWM (HRPWM), RPWM and SVPWM	Two-Level	2018
[14]	SHEPWM, SVPWM and RPWM	Two-Level	2019
[16]	VFPWM and SVPWM	Two-Level	2023
[13]	Random Triangular Carrier Wave SVPWM and SVPWM	Three-Level	2024

summarizes the different studies reported in the literature based on the acoustic noise emitted by an electric motor fed by a PWM inverter. It can be seen that the two-level inverter has been widely developed and examined, particularly when using the SVPWM technique. Nowadays, the PWM three-level neutral point clamped (NPC) inverter thus constitutes an interesting research topic area to explore the performance on acoustic noise emitted by an electric motor. Therefore, the existing paper presents a detailed study of fixed SVPWM and variable RPWM techniques in terms of acoustic noise emitted by an IM powered by both three-phase two- and three-level inverters at different conditions.

This document presents a detailed analysis of fixed SVPWM and variable RPWM methods for acoustic noise produced by an IM driven by both two- and three-level VSI. In addition, the comparative simulation results include the dynamic response of drives, harmonic current spectra, dominant peaks, and switching numbers for both methods under various operating conditions. Furthermore, the influence of the proposed PWM methods is evaluated experimentally for a two-level VSI-powered IM in terms of the harmonic current spectra and the noise frequency spectra. This document is structured in the following way: Section 2 presents the acoustic noise for electric machines. In Section 3, the functioning principles of SVPWM and RPWM for two-level and multilevel inverters are presented. In Section 4, two models of both two- and three-level NPC inverters utilizing PWM and applying an IM with Simulink are presented, and the obtained simulation results discussed. An experimental setup of a controlled two-level VSI using an IM is presented in Section 5. Additionally, the experimental tests of the emitted noise for SVPWM and RPWM strategies are given and discussed in Section 5. Some general conclusions are provided in Section 6.

2. Noise Generation in Electrical Machines

Electric motor control is of high interest for industry with their relevant performances of high efficiency, low cost, and almost free maintenance. Nevertheless, feeding such a machine by an inverter has been the root of certain troublesome problems, such as acoustic noise. The noise emitted by an induction machine fed by a voltage inverter is an environmental problem, causing irritation to the human ear. The global noise emission is divided into four main sources in an electrical machine. These sources provide the excitation forces acting on the internal stator core structure. The ambient air pressure varies periodically under the effect of vibrations, and this results in the noise creation and the sound wave is audible to the human ear. Consequently, considerable efforts have been made to identify the noise sources in electrical motors [2]. These may be subdivided further into four main sources: electronic, mechanical, magnetic, and aerodynamic. The integration of miniature motors is intended for almost any application, such as miniature motors for domestic applications and electric vehicles. The electric motor fed by an inverter for an electric vehicle is presented in Figure 1.

Our work focuses on the study of electronic noise and accordingly switching the harmonics. In electric machines, the harmonic components of the air gap generate forces, tending to dynamically deform its structure. Furthermore, the harmonics absorbed by the stator coils produce supplementary stator flux density components, thus generating other magnetic force components. These harmonics are narrow range components located near the switching frequency of integer multiples [1,2,3,4,9,19,20,21]. The expression of the voltage harmonic frequency is given by (1):

$${\mathrm{F}}_{\mathrm{n}}=\left|{\mathrm{n}}_1{\mathrm{F}}_{\mathrm{c}}+{\mathrm{n}}_2\mathrm{F}\right|$$

(1)

where F_c is the switching frequency, (n₁, n₂) the numbers of integers, and F the fundamental frequency.

3. Three-Phase Two- or Three-Level PWM Inverters

3.1. IM Model

An IM is composed of a rotor and a stator. The three stator coils are presented by axes Sa, Sb, and Sc, while the other axes are supported by rotor windings ra, rb, and rc. The electrical angle θ indicates the rotor position relative to the stator, and angles ${\mathsf{\theta }}_{\mathrm{s}}$ and ${\mathsf{\theta }}_{\mathrm{r}}$ represent rotor and stator positions, respectively with reference (d, q). The expression of these angles is determined by the following:

$${\mathsf{\theta }}_{\mathrm{s}}=\mathsf{\theta }+{\mathsf{\theta }}_{\mathrm{r}}$$

(2)

The magnetic and electrical magnitudes are given by the system in Equation (3):

$$\left\{\begin{array}{l}{\mathrm{V}}_{\mathrm{s}}={\mathrm{R}}_{\mathrm{ss}}{\mathrm{I}}_{\mathrm{s}}+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{\mathsf{\psi }}_{\mathrm{s}}\\ 0={\mathrm{R}}_{\mathrm{rr}}{\mathrm{I}}_{\mathrm{r}}+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{\mathsf{\psi }}_{\mathrm{r}}\end{array}\right.$$

(3)

with: ${\mathrm{I}}_{\mathrm{s}}={\left[{\mathrm{i}}_{\mathrm{s}\mathrm{a}}\hspace{0.33em}{\mathrm{i}}_{\mathrm{s}\mathrm{b}}\hspace{0.33em}{\mathrm{i}}_{\mathrm{s}\mathrm{c}}\right]}^{\mathrm{T}}$, ${\mathrm{V}}_{\mathrm{s}}={\left[{\mathrm{V}}_{\mathrm{s}\mathrm{a}}\hspace{0.33em}{\mathrm{V}}_{\mathrm{s}\mathrm{b}}\hspace{0.33em}{\mathrm{V}}_{\mathrm{s}\mathrm{c}}\right]}^{\mathrm{T}}$, ${\mathrm{I}}_{\mathrm{r}}={\left[{\mathrm{i}}_{\mathrm{ra}}\hspace{0.33em}{\mathrm{i}}_{\mathrm{rb}}\hspace{0.33em}{\mathrm{i}}_{\mathrm{rc}}\right]}^{\mathrm{T}}$, ${\mathsf{\psi }}_{\mathrm{s}}={\left[{\mathsf{\psi }}_{\mathrm{s}\mathrm{a}}\hspace{0.33em}{\mathsf{\psi }}_{\mathrm{s}\mathrm{b}}\hspace{0.33em}{\mathsf{\psi }}_{\mathrm{s}\mathrm{c}}\right]}^{\mathrm{T}}$, ${\mathsf{\psi }}_{\mathrm{r}}={\left[{\mathsf{\psi }}_{\mathrm{ra}}\hspace{0.33em}{\mathsf{\psi }}_{\mathrm{rb}}\hspace{0.33em}{\mathsf{\psi }}_{\mathrm{rc}}\right]}^{\mathrm{T}}$, ${\mathrm{R}}_{\mathrm{s}\mathrm{s}}={\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_3,$${\mathrm{R}}_{\mathrm{rr}}={\mathrm{R}}_{\mathrm{r}}{\mathrm{I}}_3$.

• where I₃ is the third-order unit matrix, R_s the stator phase resistance, and R_r the rotor phase resistance.

Thus, the stator and rotor fluxes are expressed as follows:

$$\left\{\begin{array}{l}{\mathsf{\psi }}_{\mathrm{s}}={\mathrm{L}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{s}}+{\left[{\mathrm{L}}_{\mathrm{m}}\left(\mathsf{\theta }\right)\right]}^{\mathrm{T}}{\mathrm{I}}_{\mathrm{r}}\\ {\mathsf{\psi }}_{\mathrm{r}}={\mathrm{L}}_{\mathrm{r}}{\mathrm{I}}_{\mathrm{r}}+{\left[{\mathrm{L}}_{\mathrm{m}}\left(\mathsf{\theta }\right)\right]}^{\mathrm{T}}{\mathrm{I}}_{\mathrm{s}}\end{array}\right.$$

(4)

The electromagnetic and mechanical torque Equations are given by the following:

$${\mathrm{T}}_{\mathrm{e}}={\left[{\mathrm{I}}_{\mathrm{s}}\right]}^{\mathrm{T}}+{\left[\left({\displaystyle \frac{\mathrm{d}\left[{\mathrm{L}}_{\mathrm{m}}\left(\mathsf{\theta }\right)\right]}{\mathrm{d}\mathsf{\theta }}}\right)\right]}^{\mathrm{T}}\left[{\mathrm{I}}_{\mathrm{r}}\right]$$

(5)

$$\mathrm{J}{\displaystyle \frac{\mathrm{d}{\mathsf{\omega }}_{\mathrm{m}}}{\mathrm{d}\mathrm{t}}}={\mathrm{T}}_{\mathrm{e}}-\mathrm{f}{\mathsf{\omega }}_{\mathrm{m}}-{\mathrm{C}}_{\mathrm{r}}$$

(6)

3.2. Two-Level PWM Inverter

The control circuit structure of a two-level inverter driving an IM is also illustrated by Figure 2a. The harmonic content at the inverter output is generally chosen as a performance criterion that should be minimized in most applications. To avoid harmonic problems, the inverter’s power switch control strategy must be designed using an appropriate PWM algorithm. Many PWM methods have been developed and proposed [7,13,20,21,22,23,24,25,26,27,28,29,30,31]. SVPWM is a digital technique founded on the transformation of three reference voltages by a single vector [1,19,28]. For the two-level inverter, there are eight vectors whose two vectors are zero. The non-zero vectors in (α, β) are presented in the standard hexagon whose origin is aligned to the zero vectors, as shown in Figure 2b.

The general expressions for the reference voltages of amplitude ${\mathrm{V}}_{\mathrm{m}}$ are given by the system in Equation (7).

$$\left\{\begin{array}{l}{\mathrm{V}}_{\mathrm{Aref}}={\mathrm{V}}_{\mathrm{m}}\sin \left(2\mathsf{\pi }\mathrm{Ft}\right)\\ {\mathrm{V}}_{\mathrm{Bref}}={\mathrm{V}}_{\mathrm{m}}\sin \left(2\mathsf{\pi }\mathrm{Ft}-{\displaystyle \frac{2\mathsf{\pi }}{3}}\right)\\ {\mathrm{V}}_{\mathrm{Cref}}={\mathrm{V}}_{\mathrm{m}}\sin \left(2\mathsf{\pi }\mathrm{Ft}-{\displaystyle \frac{4\mathsf{\pi }}{3}}\right)\end{array}\right.$$

(7)

where V_m is the reference voltage amplitude.

The SVPWM strategy is applied by turning the vector V_ref. Thus, for SVPWM control, the command signals are generated by the sector number determination, the duty cycle calculation, and the switching sequence generation. Figure 3 depicts the switching state distribution at sector 1.

The general expressions for the duty cycles for three adjacent vectors (V0, V1, and V2) to a reference vector for a two-level inverter are given as follows:

$$\left\{\begin{array}{l}{\mathrm{d}}_{\mathrm{1}}={\displaystyle \frac{\sqrt{3}{\mathrm{V}}_{\mathrm{ref}}}{\mathrm{E}}}\sin \left(\mathrm{s}{\displaystyle \frac{\mathsf{\pi }}{3}}-\mathsf{\theta }\right)\\ {\mathrm{d}}_2={\displaystyle \frac{\sqrt{3}{\mathrm{V}}_{\mathrm{ref}}}{\mathrm{E}}}\sin \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)\\ {\mathrm{d}}_0=1-\left({\mathrm{d}}_1+{\mathrm{d}}_2\right)\end{array}\right.$$

(8)

where $\mathsf{\theta }$ is the instantaneous angle, and s is the sector.

Various strategies of the random PWM technique have been established for two-level VSI in the literature [7,9,10,12,15,16,17,18]. The three main strategies are as follows: random switching frequency, random pulse width, and random pulse position. Figure 4 displays the usual switching signal q, that can be defined by three variables: the duty cycle d, the switching period T, and the delay report δ. Random switching frequency (RSF) is suggested and implemented into two- or three-level inverters in this work. The random RSF applied to the random PWM consists of leading with the duty d, the delay δ constant, and the randomized period T.

Therefore, the RSF provides the random possibility of adjusting the period $\mathrm{T}(\dots \ne {\mathrm{T}}_{\mathrm{n}-1}\ne {\mathrm{T}}_{\mathrm{n}}\ne {\mathrm{T}}_{\mathrm{n}+1}\ne \dots )$. The limits of the instantaneous period are given by the following:

$$\mathrm{T}\in \left[\overline{\mathrm{T}}\left(1-{\displaystyle \frac{{\mathrm{R}}_{\mathrm{T}}}{2}}\right),\overline{\mathrm{T}}\left(1+{\displaystyle \frac{{\mathrm{R}}_{\mathrm{T}}}{2}}\right)\right]$$

(9)

where $\overline{\mathrm{T}}$ is the statistical mean of the randomized period T, and R_T is the randomness level; this determines the interval in which T is randomized.

The randomness level can be expressed as follows:

$${\mathrm{R}}_{\mathrm{T}}={\displaystyle \frac{{\mathrm{T}}_{\max }-{\mathrm{T}}_{\min }}{\overline{\mathrm{T}}}}$$

Theoretically, the minimum value of the randomness level is (R_T)_min = 0 and the maximum value is (R_T)_max = 2, which gives ${\mathrm{T}}_{\max }=2\overline{\mathrm{T}}$ and T_min = 0.

The random parameters T and R_T may take any probability law. In our work, we used the uniform law because it is the simplest to implement. Thus, we express the instantaneous switching period as follows:

$$\mathrm{T}={\mathrm{T}}_{\min }+{\mathrm{T}}_{\max }-{\mathrm{T}}_{\min }\times \mathrm{R}$$

(10)

R: is a series of uniformly distributed random numbers in the interval [0, 1].

Replacing (T_max − T_min) by their values, we obtain the following:

$$\mathrm{T}={\mathrm{T}}_{\min }+\overline{\mathrm{T}}\times {\mathrm{R}}_{\mathrm{T}}\times \mathrm{R}$$

(11)

The switching angles are generated by comparing the random carrier and the modulation signal. Therefore, Figure 5 depicts the waveforms of the random frequency carrier V_T, the modulation waves signal V_m, and the switching sequence of switch S₁₁ for SVPWM and RPWM control.

3.3. Three-Level PWM Inverter

The voltage source inverters are generally classified as two-level (conventional) and multilevel inverters. The multilevel inverters are characterized by superior features compared with the two-level ones, with better output waveform quality and lower switching losses. The control system of the proposed NPC inverter is illustrated in Figure 6. The three branches (A, B, and C) are composed of four switches (Sx1, Sx2, Sx3, and Sx4 for x = 1, 2, 3) and two holding diodes linked to the DC line midpoint [13,29,30,31,32].

The three states (P, O, and N) and voltage values are $\left(\mathrm{E}/2,0,-\mathrm{E}/2\right)$, respectively. The V_a0, V_b0, and V_c0 voltage expressions can be represented as follows:

$$\left\{\begin{array}{l}{\mathrm{V}}_{\mathrm{a}0}={\displaystyle \frac{\mathrm{E}}{2}}\left({\mathrm{S}}_{11}+{\mathrm{S}}_{12}-1\right)\\ {\mathrm{V}}_{\mathrm{b}0}={\displaystyle \frac{\mathrm{E}}{2}}\left({\mathrm{S}}_{21}+{\mathrm{S}}_{22}-1\right)\\ {\mathrm{V}}_{\mathrm{c}0}={\displaystyle \frac{\mathrm{E}}{2}}\left({\mathrm{S}}_{31}+{\mathrm{S}}_{32}-1\right)\end{array}\right.$$

(12)

The SVPWM control is one of the popular PWM strategies for both two-level and multilevel inverter controls. Thus, 3³ = 27 states the switching states of a three-level inverter resulting in 27 space vectors that form a hexagon, as shown in Figure 7a [31,32,33]. Generally, the space diagram is divided into six main sectors (1, 2, …, 6), which can be identified as the two-level. Each sector includes (N − 1)2 triangles. For the NPC three-level, just four triangles are identified in any sector [33]. After defining the reference vector coordinates V_ref in the reference frame (α, β), the triangle in which this vector is situated must be identified. This vector can be located in one of the four triangles, as indicated in Figure 7b.

The three-level SVPWM algorithm can be easily implemented by considering three essential steps: identification of the triangle, the duty cycle calculation, and the control signal generation. After the coordinates of the reference vector V_ref have been defined in reference (α, β), the triangle in which this vector is located must be identified. This vector can be located in one of the four triangles as shown in Figure 7a. Since the four triangles are identical for all six sectors, new coordinates for the reference vector can be defined in a two-phase moving reference (α′, β′) with a rotation of pi/3 to identify the appropriate triangle in the six sectors. These new coordinates are given as follows:

$$\left\{\begin{array}{l}{\mathrm{V}}_{\alpha }^{\prime }=\left|{\mathrm{V}}_{\mathrm{ref}}\right|\cos \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)\\ {\mathrm{V}}_{\beta }^{\prime }=\left|{\mathrm{V}}_{\mathrm{ref}}\right|\sin \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)\end{array}\right.$$

(13)

Hence, the four triangles Tri are identified on the basis of the following system:

$$\left\{\begin{array}{l}\mathrm{when}\hspace{0.33em}\hspace{0.33em}{\mathrm{V}}_{\mathsf{\alpha }}^{\prime }<{\displaystyle \frac{\mathrm{E}}{3}}-{\displaystyle \frac{\sqrt{3}}{3}}{\mathrm{V}}_{\mathsf{\beta }}^{\prime }\hspace{0.33em}\mathrm{Also\; Tri}=\hspace{0.33em}1\hspace{0.33em}\mathrm{else}\\ \mathrm{when}\hspace{0.33em}\hspace{0.33em}{\mathrm{V}}_{\mathsf{\alpha }}^{\prime }>{\displaystyle \frac{\mathrm{E}}{3}}+{\displaystyle \frac{\sqrt{3}}{3}}{\mathrm{V}}_{\mathsf{\beta }}^{\prime }\hspace{0.33em}\mathrm{Also\; Tri}=\hspace{0.33em}4\hspace{0.33em}\mathrm{else}\\ \mathrm{when}\hspace{0.33em}{\mathrm{V}}_{\mathsf{\beta }}^{\prime }<{\displaystyle \frac{\sqrt{3}}{6}}\mathrm{E}\hspace{0.33em}\mathrm{Also\; Tri}=\hspace{0.33em}2\hspace{0.33em}\mathrm{else}\\ \mathrm{Tri}=\hspace{0.33em}3\end{array}\right.$$

(14)

Consider the example shown in Figure 7b, where the reference vector V_ref is located in triangle 4 of the first sector. For this position, V_x = V₁, V_y = V_7, and V_z = V₈. The application time for each vector is determined from the duty cycles that depend on the sector numbers and the region surrounding the reference vector. The general expressions for the duty cycles dx, dy, and dz corresponding to the four triangles possible are indicated in

Table 2. Duty cycles for the three-level NPC inverter.

	dx	dy	dz
Triangle 1	$1-\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-2\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$	$\mathsf{\rho }\sin \left(\mathrm{s}{\displaystyle \frac{\mathsf{\pi }}{3}}-\mathsf{\theta }\right)$	$\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$
Triangle 2	$1-\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$	$1-\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$	$1-\mathsf{\rho }\sin \left(\mathrm{s}{\displaystyle \frac{\mathsf{\pi }}{3}}-\mathsf{\theta }\right)$
Triangle 3	$2-\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-2\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$	$-1+\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$	$\mathsf{\rho }\sin \left(\mathrm{s}{\displaystyle \frac{\mathsf{\pi }}{3}}-\mathsf{\theta }\right)$
Triangle 4	$2-\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-2\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$	$-1+\mathsf{\rho }\sin \left(\mathrm{s}{\displaystyle \frac{\mathsf{\pi }}{3}}-\mathsf{\theta }\right)$	$\mathsf{\rho }\sin \left(\mathsf{\theta }-\left(\mathrm{s}-1\right){\displaystyle \frac{\mathsf{\pi }}{3}}\right)$

.

The random carrier period is also applied for the three-level SVPWM control. After the identification of the small hexagon and sub-sector and the duty cycle calculation, the random frequency carrier Vt is used to generate the switching sequences. Figure 8 shows the waveform random carrier, the modulation wave signal, and the switching sequence of switch S11 for three-level VSI, respectively, for the SVPWM and RPWM controls.

4. Simulation Results

Two numerical simulation models using the proposed PWM strategies of two- or three-level VSI were designed and realized on MATLAB R2016a/Simulink. In addition, the results show the harmonic current spectra and the dynamic response of drives at different switching frequencies and modulation indexes. The machine parameters are displayed in

Table 3. Induction machine parameters.

Parameter	Value	Parameter	Value
Rated power	0.5 kW	Rotor inductance	0.66 H
Motor speed	3000 tr/min	Mutual inductance M	0.63 H
Stator resistance	24 Ω	Friction coefficient	0.00159
Stator inductance	0.66 H	Pair pole number	1
Rotor Resistance	10.88 Ω	Inertia Moment	0.004 kg·m2

.

Figure 9, Figure 10, and Figure 11 demonstrate the stator current of phase A i_sa and the harmonic spectra waveforms respectively, for (m = 0.6, Fc = 5 kHz), (m = 1, Fc = 5 kHz), and (m = 1, Fc = 2.5 kHz), corresponding to an SVPWM and RPWM two-level inverter. Thus, the corresponding THD values are (4.7% and 2.33%), (4.11% and 1.17%), and (8.22% and 2.95%), for the SVPWM and RPWM, respectively. For SVPWM, the dominant magnitudes of current harmonics are concentrated on the switching frequency and multiples, as depicted in Figure 9a, Figure 10a and Figure 11a. It can be also seen that the amplitude of the harmonic components is influenced by the values of the modulation index and the switching frequency. In addition, it is evident that the high harmonic is generated, especially for a lower switching frequency Fc = 2.5 kHz and a lower modulation index m = 0.6. However, for the RPWM method, the harmonic spectra are totally dispersed, as represented by the obtained results. In addition, it is important to note that we can clearly observe that for m = 1 and Fc = 5 kHz the harmonic components are decreased, as compared to both other cases. As a consequence, for a two-level VSI and with the SVPWM method, the harmonic spectra provide high harmonics around the switching frequency. However, with the proposed RPWM method, the harmonic spectrum is spread over the entire frequency band at different conditions.

The current stator and harmonic spectra waveforms pertaining to three level NPC inverters using the SVPWM and RPWM methods, respectively, for (m = 0.6, Fc = 5 kHz), (m = 1, Fc = 5 kHz), and (m = 1, Fc = 2.5 kHz) are illustrated by Figure 12, Figure 13 and Figure 14. Thus, the corresponding calculated THD values are, respectively, (2.55% and 1.56%), (1.71% and 0.48%), and (3.47% and 1.73%). Depending on the obtained results, we noted that the current harmonic content of the three-level NPC inverter is significantly reduced and is lower than that of the two-level inverter. For SVPWM, as shown by the results, lower-order harmonics are denoted as clearly visible near the integer multiples of Fc. Moreover, the highest magnitudes are produced for m = 0.6 and Fc = 2.5 kHz. However, for the RPWM method, a scattered frequency spectrum is noticed. Furthermore, for m = 1 and Fc = 5 kHz, the harmonic spectra are significantly decreased. In conclusion, the three-level NPC using the proposed RPWM has better performance in terms of the harmonic spectra as a good choice for the switching frequency and the modulation index.

Next, the two proposed PWM methods of a two- or three level VSI are examined in terms of THD current, peak magnitude, switching losses, and electromagnetic torque. Figure 15 represents, at a switching frequency Fc = 5 kHz, the variation according to the modulation index of the THD current, the peak magnitude, and the switching losses of two- and three-levels, respectively. As illustrated in Figure 15a, it is of note that when the modulation index value rises, the THD level falls for both SVPWM and RPWM. The current harmonic content of the NPC converter is significantly reduced and lower than the two-level. Hence, current THD with the proposed RPWM for a two- or three-level inverter provides the best performance compared to that of SVPWM. Added to that, the high current harmonic is decreased significantly using RPWM over the total modulation index range for both two- and three-level VSIs, as shown in Figure 15b. Furthermore, the proposed strategies are also investigated as a function of switching losses of the switch S11. The comparison results confirm that the PWM three-level offers a lower number of commutations versus the two-level, as demonstrated in Figure 15c. Consequently, the proposed RPWM method provides a lower number of commutations over the whole modulation range, implying that it generates fewer switching losses. The temporal waveform evolution of the electromagnetic torque for the two topologies of the inverter with (Fc = 5 kHz and m = 1) is presented in Figure 15d and Figure 15e, respectively. The steady state is established after 0.6 s for both inverters. Thus, a high oscillation is generated with SVPWM, but the RPWM method also ensures a better response for both inverter types.

5. Experimental Results Obtained

The photographic view of the experimental system for acoustic noise measurements is described in Figure 16. It contains mainly an IM, a F28335 DSP, and a two-level VSI controlled by the two proposed PWM modulation techniques, namely RPWM and SVPWM. A sound level meter (SL-4033SD, Lutron Electronic Enterprise Co., Ltd., Taipei, Taiwan) was used to measure the instantaneous sound pressure. The sound level meter is attached via a coaxial cable to an oscilloscope. The instantaneous sound pressure is stored as a data format file in the oscilloscope memory. After every test measurement, the data file is transmitted to the PC. Spectral noise analysis is performed in MATLAB R2016a FFT function. The IM used in this study is classified as a small machine. Therefore, all measurement points must be located at a 250 mm distance or more from the main machine surface. In addition, the measurement points should be made when the motor reaches the steady state of the operating mode [34,35].

The experimental stator current waveforms, the harmonic spectrum, and the noise spectra produced by the IM supply by a two-level VSI and controlled by the RPWM and SVPWM techniques pertaining to (m = 0.6 and Fc = 5 kHz), (m = 1 and Fc = 5 kHz), (m = 1 and Fc = 2.5 kHz), respectively, are shown in Figure 17, Figure 18, and Figure 19. The measured THD values are (5.21% and 2.8%), (4.52% and 1.4%), and (3.31% and 9.12%) for the SVPWM and RPWM, respectively. As can be seen from the results of the measured harmonic spectrum, for the SVPWM strategy, the important harmonic is also located at the switching frequency and its multiples. Furthermore, for m = 0.6, the dominant harmonics generated by the sidebands around 2Fc are significant compared to that of Fc. However, for m = 1, the harmonic components around Fc are more important. In addition, for m = 1 and Fc = 2.5 kHz, the level harmonic is high compared to Fc = 5 kHz. Hence, referring to the results using the RPWM, the harmonic current is totally distributed above the whole frequency at different conditions. Therefore, these results confirm that the proposed RPWM has better performance on current harmonics compared to the SVPWM.

The weighted spectra of noise using the RPWM and SVPWM techniques are performed for (m = 0.6 and Fc = 5 kHz), (m = 1 and Fc = 5 kHz), and (m = 1 and Fc = 2.5 kHz), respectively, shown in Figure 20, Figure 21, and Figure 22. Referring to these experimental results, it can be observed that the high-level noise is noticed for the lower modulation index and lower switching frequency (m = 0.6 and Fc = 2.5 kHz), as illustrated by Figure 20. With RPWM, the noise level is −38 dBA, −40 dBA, and −38 dBA, respectively, at frequencies 5100 Hz, 8650 Hz, and 14,950 Hz. For SVPWM, the noise amplitude is −41 dBA, −20 dBA, and −41 dBA, respectively, at frequencies 5050 Hz, 10,000 Hz, and 14,950 Hz. Also, we noted that the lowest noise components are marked for modulation index m = 1 and Fc = 5 kHz, as shown in Figure 21. In addition, as should be expected, the SVPWM control generates noise components concentrated around the integer multiples of the switching frequency. Then, for m = 1 and Fc = 5 kHz, the noise amplitude is −32.6 dBA, −35.9 dBA, and −32.6 dBA, respectively, at frequencies 5050 Hz, 10,000 Hz, and 14,950 Hz. However, it can be distinguished in Figure 21 that the RPWM technique produces a well-spread noise spectrum with lower noise levels compared to SVPWM. The noise level is −58.3 dBA, −60 dBA, and −58.3 dBA, respectively, at frequencies 5100 Hz, 8650 Hz, and 14,950 Hz. Whereas, for m = 1 and Fc = 2.5 kHz, as depicted by Figure 22, the noise level is high compared to others cases. In conclusion, the experimental results demonstrate that RPWM exhibits an enhanced better performance regarding the acoustic noise and the harmonic content.

The current harmonic and frequency components of the motor acoustic noise is influenced by the modulation index. Figure 23 presents the experimental variation in the current THD and in the noise amplitude according to the modulation index for the proposed RPWM and SVPWM at Fc = 5 kHz. It can be observed from Figure 23a that the modulation index rises and the THD level diminishes. In addition, the SVPWM technique offers the maximum THD. However, the total distortion level is considerably reduced for RPWM. In fact, the sensitivity of the human ear can be demonstrated through the appropriate weighting curves. The A-weighting curve is widely used in electrical machine acoustics and is also used in this work. Moreover, the experimental acoustic noise measurement procedure is repeated over the whole modulation range at Fc = 5 kHz. In this case also, the acoustic noise amplitudes are reduced with RPWM over the entire modulation range, as shown in Figure 23b. Consequently, the RPWM method has superior performance on the current harmonic and acoustic noise.

6. Field Oriented Control FOC Using RPWM Technique for Two- and Three-Level Inverter

The filed oriented control (FOC) control block diagram is illustrated in Figure 24. It consists principally of four regulation strategy loops for flux, speed, and two regulation loops for direct and quadrature stator current. Thus, the system resolution is as a multi-variable system with two inputs (stator voltages V_sd and V_sq) and two outputs (stator currents i_sd and i_sq).

In our control system, the four regulators are the proportional–integral (PI) controller, as in the following expression:

$$R(p)=k\left(1+{\displaystyle \frac{1}{\tau p}}\right)$$

(15)

The k and τ constants allow action for the adjustment of the system dynamics and static error.

-

For the flux controller, the $k_f$ and ${\tau }_f$ parameters are given by Equation (16):

$$\left\{\begin{array}{c}{T}_f=T_r\\ k_f={\displaystyle \frac{4}{M}}\end{array}\right.$$

(16)

where T_r = l_r/R_r is the rotor time constant and M is the mutual inductance.

-

For speed regulation, the $k_v$ and ${\tau }_v$ parameters are determined by the following:

$$\left\{\begin{array}{c}{T}_v={\displaystyle \frac{J}{K_l}}\\ k_v=4K_l\end{array}\right.$$

(17)

where $K_l=f+k_l$, f: is the friction coefficient, and $k_l$: is the nominal torque proportionality constant.

-

For speed regulation, the same design as for the d-axis and q-axis current loops, the regulators parameters are expressed by the following:

$$\left\{\begin{array}{c}{T}_d=T_q={\displaystyle \frac{\sigma l_s}{R_s+{\left(\frac{M}{l_r}\right)}^2{R}_r}}\\ K_d=K_q=4\left(R_s+{\left({\displaystyle \frac{M}{l_r}}\right)}^2{R}_r\right)\end{array}\right.$$

(18)

where l_s: is the stator inductance, R_s is the stator resistance, R_r is the rotor resistance, l_r: is the rotor inductance, and $\sigma =1-{\displaystyle \frac{M^2}{l_r{l}_s}}$ is the Blondel dispersion coefficient.

Simulation analysis is realized in order to evaluate the performance of the proposed RPWM technique for two- and three-level inverters on a rotor flux vector-controlled induction machine. The parameters of the IM are shown in Appendix A

Table A1. 0.5-kW IM parameters.

Parameter	Value
Rated power P	0.5 kW
Motor speed ω	3000 tr/min
Stator resistance Rs	24 Ω
Stator inductance ls	0.66 H
Rotor resistance Rr	10.88 Ω
Rotor inductance lr	0.66 H
Mutual inductance M	0.63 H
Friction coefficient f	0.00159
Inertia moment J	0.004 kg·m2

. The load torque is assumed to be proportional to the speed. The proportionality constant at the rated load is 0.067. The motor is operated at no-load at switching frequency Fc = 5 kHz and DC bus voltage VDC = 560 V using the proposed two- and three-level RPWM technique. The simulation results of the FOC control are related to the temporal evolution of the direct and quadrature components of the stator current, real and estimated rotor flux, real and reference speed, and electromagnetic torque using, respectively, the proposed RPWM technique for two- and three-level inverters.

Based on the results in Figure 25a and Figure 26a, the quadrature stator current iq peaks during the start-up phase, then falls to a constant value. Thus, it can be seen that the iq current is proportional to the variation in the load torque, while the direct stator current i_sd remains constant. Figure 25b and Figure 26b show the real and estimated flux responses, which are well established and first-order. We noted that the flux remains substantially constant, which clearly shows the existence of decoupling between the “d” and “q” signals. This is the principle of oriented rotor flux. For the two types of inverter, the real speed perfectly follows the reference speed of the machine, as shown in Figure 25c and Figure 26c. In addition, Figure 25d and Figure 26d demonstrate low torque oscillations, which are higher for the two-level inverter compared to the three-level configuration. In conclusion, the three-level inverter-driven rotor flux vector control of the induction machine with the proposed RPWM technique offers high performance.

7. Conclusions

The noise electronic level is caused by the voltage and current harmonic of the VSI feeding the IM. Thus, the harmonic level depends on the control technique used to produce the control signals. In this study, the impact of the fixed SVPWM strategy and the variable RPWM technique applied both to two- or three-level inverters on the noise emitted by an induction driven motor was analyzed. Two Simulink models were developed, and some simulation results were presented and discussed. The performance results in terms of current THD, harmonic spectra, dominant peaks, and switching number at different switching frequencies and modulation indexes were shown and discussed. The simulation results show that with SVPWM, harmonics were produced near the switching frequency. However, with RPWM, the harmonic spectrum was totally dispersed for both topology inverters, whereas a high level harmonic was noticed for the lower modulation index and lower switching frequency. In addition, the same study for two-level VSI using the proposed PWM strategies was also validated experimentally with a laboratory prototype on stator harmonic spectra and noise spectra. The practical results demonstrate that SVPWM generates noise near the switching frequency and its integer multiples. However, RPWM exhibits an enhanced performance regarding the acoustic noise and the harmonic content over the entire frequency band at different conditions especially for the high modulation index.