An Improved Wavelet PWM Technique with Output Voltage Amplitude Control for Single-phase Inverters

Abstract

Unlike existing pulse-width modulation (PWM) techniques, such as sinusoidal PWM and random PWM, the wavelet PWM (WPWM) technique based on a Harr wavelet function can achieve a high fundamental component for the output voltage, low total harmonic distortion, and simple digital implementation. However, the original WPWM method lacks output voltage control. Thus, the practical application of the WPWM technique is limited. This study proposes an improved WPWM technique that can regulate output voltage amplitude with the addition of a parameter. The relationship between the additional parameter and the output voltage amplitude is analyzed in detail. Experimental results verify that the improved WPWM exhibits output voltage control in addition to all the merits of the WPWM technique.

I. INTRODUCTION

Inverters are important components of power electronic technology that are widely used in AC motor speed control, induction heating, uninterruptible power supply, and other fields. As a core aspect of inverters, the modulation strategy exerts a significant influence on the performance of inverters. A good control strategy can reduce harmonic pollution and switching losses and increase DC voltage utilization. At present, the commonly used modulation strategies [1]-[6] include sinusoidal pulse width modulation (SPWM), random pulse width modulation (RPWM), selective harmonic elimination, and a series of other optimization methods. Searching for an optimal control scheme to improve the output performance of inverters remains a critical research area [7]-[9].

In recent years, a wavelet modulation technique has been proposed for inverters [10]-[20]. Such wavelet modulation method is based on the Harr wavelet function and a non-dyadic multi-resolution analysis. The control signal generated by the wavelet modulation is a series of pulses with different widths. Therefore, this wavelet modulation technique is referred to as the wavelet PWM (WPWM) technique in the present work. In comparison with existing PWM techniques, such as the SPWM and RPWM techniques, the WPWM method is able to produce output voltage and current whose fundamental components exhibit higher magnitude; it also features lower total harmonic contents. However, the WPWM technique is not able to regulate the output voltage amplitude when the parameters are fixed.

Thus, we aim to analyze the feasibility of regulating the output voltage amplitude and propose an improved WPWM technique to realize the regulation of output voltage amplitude through the addition of a parameter. The rest of this paper is organized as follows. The basic theory of the WPWM technique is introduced in Section II. The improved WPWM technique is proposed in Section III. The experimental results are provided in Section IV. The conclusion is given in Section V.

 

II. FUNDAMENTALS OF THE WPWM TECHNIQUE

The WPWM technique is based on a sample and reconstruction of a reference-modulating signal in a non-uniform recurrent manner by using set sampling and synthesis basis functions [10]-[11]. These sampling basis functions are generated as dilated and translated versions of the scale-based linearly combined scaling function φ(j,k)(t). Furthermore, these synthesis basis functions are generated as dilated and translated versions of the scale-based linearly combined synthesis scaling function The scale-based linearly combined scaling function is defined at scale j as

where j = 0, 1, 2, 3, … and ϕH(t) is the Harr scaling function, which given by

The synthesis scaling function associated with φ(t) can be defined as

Using these two dual scaling functions, we can expand a continuous time signal xc(t) as

where j, k ∈ Z, and Z represent a set of integer numbers.

Such form of signal processing suggests that a continuous time signal can be recovered from its samples using sets of synthesis functions

The literature [10] proves that the switching pulses for inverters can be generated by using dilated and shifted versions of the synthesis scaling function When each cycle of xc(t) is divided by a finite number of sample groups D, the length of the time interval of the sample group [td1, td2] changes as scale j changes. That is,

Generally, the sampled signal for a single-phase inverter is a sinusoidal one SM(t) = sin(ωmt) with a period of Tm, which exhibits quarter-cycle symmetry. The procedure of the WPWM technique given in the literature [10] shows that the output voltage of an inverter mainly depends on the initial value of scale j (j0) and the number of sample groups (D) in a period.

When turning switching devices on and off at time points (td1 and td2), a voltage pulse waveform can be obtained. Thus, according to the Fourier analysis, the amplitude of the kth harmonic voltage Vk can be defined as

where k = 1,3,5,7,⋯, Vdc is the amplitude of the input DC voltage for inverters, and V1 is the amplitude of fundamental voltage.

The voltage modulation ratio of an inverter is defined as m = V1/Vdc ; from (5), m can be determined by

From (6), the value of m under different initial values of scale can be calculated via MATLAB. The results of m vs. D when j0 = 0, 1, 2, 3 are shown in Fig. 1. As shown in the figure, m increases as j0 increases under the same D, and m increases as D increases under the same j0. However, once D and j0 are fixed, m is determined. Thus, the inverter controlled by the WPWM method lacks output voltage amplitude control, which is a flaw of the WPWM method for many applications.

Fig. 1.m vs. D when j0 = 0, 1, 2, 3.

 

III. IMPROVED WPWM TECHNIQUE

A. Principle of the Improved WPWM Technique

The output of inverters can be controlled in many ways, one of which is the use of pulse-width modulation (PWM). The principle of PWM is to control the output voltage amplitude of inverters by varying the widths of the pulses. The control signals for inverters generated by the WPWM technique are a type of PWM signal, and the width of the control signals when D and j0 are fixed is determined. Hence, the WPWM technique lacks output voltage amplitude control. Varying the widths of the pulses generated by the WPWM technique when D and j0 are given obviously deserves research attention.

According to (4), if a new parameter μ is added to calculate the time interval of a sample group, then the width of the pulses can be adjusted. As d represents the sampling group, the new parameter μ may be added in four ways:

Method 1: td1 = d + 2−(μ*j+1), td2 = d + 1 - 2−(μ*j+1) ;

Method 2: td1 = d + 2−μ*(j+1), td2 = d + 1 - 2−μ*(j+1) ;

Method 3: td1 = d + μ*2-(j+1), td2 = d + 1 - μ*2-(j+1);

Method 4: td1 = d + μ*2-(j+1), td2 = d + μ*[1 - 2-(j+1)].

The original WPWM can be considered as a special case when μ = 1.0.

The comparison of Methods 1–4 is discussed on the basis of the following aspects:

(1) Pulse width, i.e., the value of td2-td1. As td2-td1≥0 must be satisfied, μ*j≥0 is required for Method 1, μ*(j+1)≥1 is required for Method 2, μ≤2j is required for Method 3, and μ*(1-2−j)≥0 is required for Method 4. Given that j ≥ 0, the value of μ when j is given can be easily determined. Further study reveals that 0 ≤ μ ≤ 1 can be satisfied in Methods 1 and 4 when the voltage modulation ratio m is regulated from 0 to its maximum. However, the value of μ in Methods 2 and 3 changes into j, which makes the choice of μ considerably difficult for different output voltage amplitudes.

(2) Complexity of the algorithm. Obviously, the complexity of Method 1 and the difficulty involved in its digital implementation increase because the power of 2 is not a constant when 0 ≤ μ ≤ 1.

Thus, Method 4 is selected to regulate the width of the pulses. The algorithm for implementing the improved WPWM technique with Method 4 is set up according to the literature [20]. Its flowchart is shown in Fig. 2, where Tm = 1/fm is the period of the modulating sinusoidal signal and td3 = td2|μ = 1.0 is used to determine the value of j and thereby ensure that similar to those in the original WPWM, the output waveforms exhibit quarter-cycle symmetry.

Fig. 2.Flowchart of the improved WPWM technique.

B. Analysis of the Improved WPWM Technique

In Method 4, the pulse width changes to td2-td1 = μ*( 1 − 2−j). Then, m is expressed by

According to (7), m decreases as μ decreases when choosing 0 ≤ μ ≤ 1. Thus, the output voltage amplitude can be controlled by changing the parameter μ.

To determine the relationship between the parameter μ and the output voltage amplitude of inverters, fm = 50 Hz, D = 30, and j0 = 0 are selected; μ is changed in step 0.01 from 0.01 to 1; and the voltage modulation ratio m of the inverter is calculated on the basis of (7). The curve of m vs. μ is illustrated in Fig. 3. The curve resembles a straight line. By fitting the curve with a linear function [21]-[22], the fitting result is derived as m = 0.0008＋1.1223*μ, and the fitting errors are shown in Fig. 4. As shown in the figure, the fitting error is less than 8 × 10−4, which is too small to be neglected.

Fig. 3.m vs. μ.

Fig. 4.Fitting errors.

In this work, a single-phase full-bridge (FB) inverter is selected as an example. The input voltage Vdc = 100V , the output frequency fm = 50 Hz, D = 30, and j0 = 0. The simulation results of the output voltage and its spectrum for μ = 0.8 and μ = 0.9 are shown in Fig. 5. The simulated output voltage amplitudes are close to the theoretical values m = 0.0008+1.1223*0.8 = 0.8986 when μ = 0.8 and m = 0.0008+1.1223*0.9 = 1.0109 when μ = 0.9. This result proves that the inverter output voltage amplitude can be regulated by parameter μ.

Fig. 5.Simulation results of the output voltage and its spectrum at different μ.

Similarly, the relationship between μ and m can be derived by a series of fitting equations for different D. The fitting function m = f(μ) at fm = 50 Hz and j0=0 for different D when the fitting error is controlled to be less than 8 × 10−4 is presented in Table I. The calculation results of the total harmonic distortion (THD, total of the former 101th harmonics) vs. m for different D are shown in Fig. 6. The figure clearly shows that THD decreases as m increases.

TABLE IFITTING FUNCTION M = F(Μ) FOR DIFFERENT D

Fig. 6.THD vs. m at different D.

 

IV. EXPERIMENTAL RESULTS

To verify the improved WPWM technique, we implement the algorithm of the improved WPWM technique by using a digital signal processor (TMS320F28335). The input voltage (Uin) of the prototype is Uin = 50 V. MOSFET IRF450 is selected as a switch, TLP250 is used as a driver, and resistor R = 50 Ω is used as the load. The experiments are carried out by choosing m = 0.8 and μ = 1 (the values of m and μ are calculated with the expression in Table I) when j0 = 0, fm = 50 Hz, and D = 26, 30, 36, 40. The experimental results of the output voltage Uab and its spectrum are shown in Figs. 7–10, which show that the spectrum of Uab is the rms (root mean square) value of different orders of harmonic voltage. As V1(V1 = m*Vdc) is the amplitude of fundamental voltage, the theoretical rms value of the fundamental voltage is defined by For example, the theoretical rms value of the fundamental component of the inverter output voltage at m = 0.8 is close to The experimental results and the corresponding theoretical rms values for different D are listed in Table II.

Fig. 7.Output voltage and its spectrum for a single-phase FB inverter with the improved WPWM at j0 = 0, D = 26, and f = 50 Hz.

Fig. 8.Output voltage and its spectrum for a single-phase FB inverter with the improved WPWM at j0 = 0, D = 30, and f = 50 Hz.

Fig. 9.Output voltage and its spectrum for a single-phase FB inverter with the improved WPWM at j0 = 0, D = 36, and f = 50 Hz.

Fig. 10.Output voltage and its spectrum for a single-phase FB inverter with the improved WPWM at j0 = 0, D = 40, and f = 50 Hz.

TABLE IIEXPERIMENTAL RESULTS FOR DIFFERENT D

Table II shows that the derived fitting equation m = f(μ) for different D is feasible because the error between the theoretical rms value and the experimental value is very small and the improved WPWM technique can regulate the output voltage amplitude of inverters with an extra parameter μ, in addition to all the merits of the WPWM technique.

To test the regulation ability of the improved WPWM technique, we design a digital control scheme with a voltage feedback loop for a single-phase FB inverter (Fig. 11). The experimental setup is shown in Fig. 12. The experiments are carried out by choosing j0 = 0, fm = 50 Hz, and D = 30; the reference voltage (Vref) for the output voltage (Uo) is set to 28 Vrms (40 Vmax) and 35 Vrms (50 Vmax). The experimental waveforms when Vref = 28 Vrms and Vref = 35 Vrms after changing the input voltage from 40 V to 50 V or from 50 V to 40 V are shown in Figs. 13 and 14, respectively. The experimental waveforms when Vref = 28 Vrms and Vref = 35 Vrms after changing the load from 50 Ω to 27 Ω or from 27 Ω to 50 Ω are shown in Figs. 15 and 16, respectively. Fig. 13 shows that the output voltage can be stable when m is changed between 0.8 and 1. Fig. 14 shows that the output voltage can be stable when m is changed between 1 and 1.11. Figs. 15–16 show that the output voltage can be regulated to a stable value quickly.

Fig. 11.Digital control scheme with voltage feedback loop.

Fig. 12.Experimental platform.

Fig. 13.Input and output voltages when the reference voltage is 28 Vrms.

Fig. 14.Input and output voltages when the reference voltage is 35 Vrms.

Fig. 15.Load current and output voltage when the reference voltage is 28 Vrms.

Fig. 16.Load current and output voltage when the reference voltage is 35 Vrms.

 

V. CONCLUSIONS

This work analyzes the feasibility of modifying the WPWM technique based on the Harr wavelet function and proposes an improved WPWM technique to realize output voltage regulation through an extra parameter. The relationship between the additional parameter and the output voltage amplitude for different numbers of sample groups can be represented by a group of linear functions. The experimental results for a single-phase FB inverter prove that the improved WPWM technique can regulate the output voltage amplitude linearly through the additional parameter with simple digital implementation, which is beneficial for inverters that require constant output voltage in a constant frequency.