Improved Switching Strategy for Selective Harmonic Elimination in DC-AC Signal Generation via Pulse-Width Modulation

1. Introduction

Pulse-width-modulated (PWM) signals are a class of two-value piecewise functions that change values at a set of controlled switching instants $$. One important field of application of PWM signals is the generation of alternating current (AC) from direct current (DC) sources by means of electronic circuits called inverters [1–3]. More precisely, voltage PWM inverters use suitable switching signals to produce a sinusoidal AC voltage, with the desired amplitude and frequency, from a constant DC voltage source. The structure of a voltage PWM inverter is schematically depicted in Figure 1. In the practical implementation of this approach, two important design challenges arise: (i) fast and efficient algorithms are required for adaptive real-time computation of the switching instants $$, and (ii) harmonic distortion must be selectively reduced to produce a high-quality AC signal.

A first attempt to deal with these design problems is provided by the method of programmed harmonic elimination [4–8]. This method, however, presents the serious drawback of having to solve nonlinear equation systems. Some efforts to simplify these systems have been made in [9–12], and soft computing strategies have been considered in [13–17]. Nevertheless, determining a proper initial estimate for the switching times still remains as an unsolved difficulty.

This line of work is started in [23] and extended in [24–26], where the main elements of what we call conventional design method are presented. In this case, the main drawback is that the linear systems ℒ are only valid for a restricted interval of AC amplitudes and, in general, a supervised implementation of several linear systems ℒ is necessary to cover the full AC amplitude range. Moreover, obtaining suitable systems ℒ for a large number of switching instants requires solving a complex optimization problem.

The objective of this paper is to present an improved version of the conventional design method that produces linear systems ℒ valid for larger AC amplitude ranges. The new method, that we call advanced design method, also introduces a significant computational simplification and, moreover, presents interesting regularity patterns that make it possible to identify efficient design strategies for problems with a large number of switching instants. Some preliminary and partial results related to the improved method have been presented in [27–30].

The paper is organized as follows. In Section 2, some basic notations and fundamental theoretical elements are provided. In Section 3, the conventional design method is formulated in terms of switching ratios, and the main elements of the advanced design method are introduced. In Section 4, the problem of obtaining optimal switching strategies is discussed. Finally, conclusions are presented in Section 5.

2. Theoretical Background

In this section, we provide some notations and theoretical elements that facilitate a clear presentation of the design methodologies discussed in the paper. In particular, we include a detailed discussion of Walsh series representations for signals with quarter-wave odd symmetry and their relation with the corresponding Fourier series representations.

3. Design of PWM Signals for DC-AC Signal Generation

The advanced design method, presented in Section 3.2, introduces a clever modification in the conventional method that facilitates produceing linear systems ℒ_a with larger amplitude ranges. The new approach also introduces a significant computational simplification and, moreover, presents interesting regularity patterns that make it possible to identify efficient design strategies for large dimension problems.

The formulation of the conventional design method presented in this paper is based on the switching ratios with respect of the interval endpoint, which allow an elegant formulation of the conventional method and leads naturally to the new one.

3.1. Conventional Design Method

Let us assume that the interval

$$ ()

defines the operational range of the sinusoidal amplitude A_ac in (29). Let us also consider the signal

$$ ()

where $$ is the DC amplitude, and $$ is an amplitude-normalized PWM signal that takes the values ±1 and has QWO symmetry in [0, T). To satisfy the approximation criteria (C1) and (C2), the conventional design method uses PWM signals $$ with M switching cycles (pulses) in the quarter-wave interval [0, T/4). The ith switching cycle starts at the switching-down instant $$, where the signal $$ shifts from +1 to −1, and ends at the switching-up instant $$, where the signal shifts back from −1 to +1. The switching instants are selected according to the following rules.

• (R1)

  The quarter-wave interval [0, T/4) is divided into N = 2^p intervals

  $$ ()
  where the partition points are
  $$ ()
  and p is the smallest integer that satisfies
  $$ ()

• (R2)

  A system of M indexes is chosen as

  $$ ()
  and a switching-down time is selected in each interval
  $$ ()

• (R3)

  The M switching-up instants, $$, i = 1, …, M, are computed as follows:

  $$ ()

• (R4)

  Each interval I_j, 1 ≤ j ≤ N, can contain at most one switching time $$ or $$.

Using the switching ratios with respect to the interval endpoint

$$ ()

the switching-down instants can be expressed in the form

$$ ()

In the conventional method, the switching-up instants $$ are independent of ϕ_i and can be written in terms of the step h as

$$ ()

A PWM signal $$, designed according to the rules (R1)–(R4), is schematically displayed in Figure 4. In this case, we have M = 2 switching cycles. The interval [0, T/4) is divided into N = 2³ = 4M subintervals, with step h = T/32. The switching-down indexes are $$ and $$. The ratio ϕ₁ defines the switching-down instant $$; as $$, the corresponding switching-up instant is located at $$. The ratio ϕ₂ defines $$. Now, $$ and the associated switching-up instant is $$.

For the PWM signal $$, defined by the system of switching indexes

$$ ()

and the vector of switching ratios

$$ ()

let us consider the truncated Walsh series

$$ ()

where N = 2^p ≥ 4M is the number of subintervals in [0, T/4). Taking into account that the Walsh functions wal_T(j, t) with j < 2^r are constant in the intervals

$$ ()

and observing that

$$ ()

it follows that the Walsh functions

$$ ()

are constant in the interior points of the intervals I_j defined in the rule (R1). From (14), the coefficient $$ in (49) can be computed as

$$ ()

where h = T/(4N), and $$. Using the normalized time τ = t/T, we can obtain the following period-free expression for $$:

$$ ()

with $$. By computing the integrals in (54), we get

$$ ()

The vector of Walsh coefficients

$$ ()

admits the matrix expression

$$ ()

where C_N,M is a matrix with elements

$$ ()

and D_N = [d₁, …, d_N] ^T is a vector with elements

$$ ()

According to the discussion in Section 2, the vector

$$ ()

which contains the first M coefficients of the Fourier series representation

$$ ()

can be computed in the form

$$ ()

and considering (57), we obtain

$$ ()

with

$$ ()

For the sinusoidal signal (29), let us consider a given AC amplitude $$ and the system of switching indexes (41). If there exists a vector of switching ratios

$$ ()

that satisfies

$$ ()

with

$$ ()

then the conditions (C1) and (C2) are satisfied by the signal

$$ ()

where $$ and $$ is the amplitude-normalized PWM signal defined by the switching instants (45) and (46), corresponding to the switching ratios (65).

To illustrate the implementation of the conventional method, let us consider the problem of regulating the fundamental amplitude and canceling the first nonfundamental harmonic. In this case, M = 2 and the number of subintervals in [0, T/4) is N = 8. For the particular system of switching indexes

$$ ()

the matrices (58) and (59) take the values

$$ ()

and the matrix B_2,8 is formed by the first two rows of the matrix B_4,8 displayed in (28). Equation (66) produces the linear system

$$ ()

To compute the regulation range of the system (71), we can consider the auxiliary system

$$ ()

obtained by isolating $$ in the different equations of the system (71), and compute the values

$$ ()

Clearly, for any normalized amplitude $$ satisfying

$$ ()

a suitable system of switching ratios ϕ_i ∈ [0,1] can be computed. For the current example, we obtain the particular values

$$ ()

which mean that the linear system (71) allows computing a suitable PWM signal to regulate the amplitude of the fundamental frequency in the range 33.7%–94.1% of the maximum AC amplitude.

3.2. Advanced Design Method

In the new approach, the design of the normalized PWM signal $$ is carried out following the same rules (R1), (R2), and (R4) used in the conventional method, but in this case, the switching indexes $$ satisfy

$$ ()

and the switching-down instants

$$ ()

are followed by the switching-up instants

$$ ()

This new switching scheme is displayed in Figure 5, where the symmetrical arrangement of the switching instants $$ and $$ with respect to the interval endpoint $$ can be clearly appreciated. The coefficients (54) of the truncated Walsh series (49) have now the form

$$ ()

The vector of Walsh coefficients (56) admits the matrix expression (57), but now the elements of the matrix C_N,M take the form

$$ ()

and the elements of the vector D_N are

$$ ()

The coefficients c_n,i can be easily computed by considering the N × N matrix S_N, with elements

$$ ()

The nth row of S_N contains the sign sequence of the Walsh function wal₁(4n − 3, τ) for a homogeneous partition of the interval [0,1/4) with step h = 1/(4N). Now, we have

$$ ()

and the ith column of C_N,M can be obtained by adding the columns $$ and $$ of S_N, and multiplying by −2/N. The coefficients d_n can also be expressed in terms of the elements s_n,k as

$$ ()

and could be computed by adding the columns of S_N. However, it is not necessary to perform this computation because (81) always produces the vector

$$ ()

Clearly, the rest of the design procedure described in Section 3.1 and the discussion about the regulation range remain valid for the new approach.

To illustrate the implementation of the advanced method, let us consider the design problem presented in the previous section, with M = 2, N = 8 and the switching indexes $$ and $$. For a homogeneous partition of the interval [0,1/4] with step h = 1/32, the system of Walsh functions

$$ ()

produces the matrix of sign sequences

$$ ()

According to (83), the matrix C_8,2 is

$$ ()

and from (84), the value of D₈ is

$$ ()

which coincides with the value indicated in (85). Finally, using (64) and (66) with the matrix B_2,8 introduced in the previous subsection, and the matrices C_8,2 and D₈ given in (88) and (89), we obtain the linear system

$$ ()

and from the associated auxiliary system

$$ ()

we get the modulation range

$$ ()

This example clearly shows that the advanced design method introduces an important computational simplification. Moreover, by comparing the values in (75) and (92), it can be appreciated that the advanced method also produces significantly larger modulation ranges.

Remark 4. The sign sequences generated by the Walsh functions wal₁(4n − 3, τ), n = 1, …, N, for a homogeneous partition of [0,1/4] with step h = 1/(4N) are coincident with the sign sequences generated by the Walsh functions wal₁(n − 1, τ), n = 1, …, N, for a homogeneous partition of [0,1] with step h = 1/N. Considering the property (6), the matrix S_N can be computed by using the sign sequences of the functions wal_T(n − 1, t), n = 1, …, N, for a homogeneous partition of [0, T] with step h = T/N. For example, the matrix S₈ presented in (87) can be obtained from the sign sequences of the Walsh functions wal_T(n − 1, t), n = 1, …, 8, for a homogeneous partition of [0, T] with step h = T/8 (see Figure 2).

Remark 5. From the previous remark and (84), we have

$$ ()

In consequence, d₁ = 1, and d_n = 0 for n = 2, …, N, as indicated in (85).

Remark 6. Offline computations are carried out to design the linear systems ℒ_c and ℒ_a. Then, these linear systems can be used for fast real-time computation of suitable PWM signals.

4. Optimal Switching Strategies

The values of the optimal amplitude intervals achieved by the conventional and advanced design methods for the optimal switching index systems (103) and (105), respectively, are presented in Table 2. Also in this case, the data show the superiority of the advanced method, which can practically cover the whole modulation interval with a single linear system ℒ_a. In contrast, the best solution provided by the conventional method only covers the normalized amplitude interval 54.7%–98.5%.

Regarding the harmonic distortion, the values of the distortion factor (33) obtained for the optimal advanced design (see DF values in Figures 8 and 9) are inferior to the corresponding ones obtained for the optimal conventional design (see DF values in Figures 6 and 7). Moreover, the harmonic distributions of the optimal advanced design present a very regular pattern, which contrasts sharply with the uneven harmonic distributions of the optimal conventional design.

5. Conclusions

In this paper, an advanced design method for DC-AC signal generation using pulse-width-modulated (PWM) signals has been presented. The new method is based on the Walsh transform and introduces significant improvements with respect to the current design methodologies. In particular, the new approach presents the following features. (i) Efficient real-time operation. The switching instants of the PWM signal can be easily computed by means of explicit linear systems. Moreover, these linear systems are valid for a wide AC-amplitude range. (ii) Improved harmonic distortion attenuation. A prescribed number of initial harmonics can be fully eliminated, and the residual harmonics present a particularly well-shaped distribution. (iii) Large-dimension extensibility. Optimal switching strategies for PWM signals with a large number of pulses can be directly obtained, without solving costly optimization problems.

In this work, the discussion has been focused on the problem of DC-AC signal generation. However, the proposed approach can also be of interest in other fields as, for example, power measurement [31].