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

Abstract

We present an advanced design methodology for pulse-width-modulated (PWM) DC-AC signal generation. Using design methods based on the Walsh transform, AC sinusoidal signals can be approximated by suitable PWM signals. For different AC amplitudes, the switching instants of the PWM signals can be efficiently computed by using appropriate systems of explicit linear equations. However, the equation systems provided by conventional implementations of this approach are typically only valid for a restricted interval of AC amplitudes and, in general, a supervised implementation of several equation systems is necessary to cover the full AC amplitude range. Additionally, obtaining suitable equation systems for designs with a large number of switching instants requires solving a complex optimization problem. In defining the constitutive pulses of a PWM signal, a suitable partition of the time interval is used as a reference system. In the new methodology, pulses are chosen to be symmetric with respect to the partition points, and the switching times are specified by means of switching ratios with respect to the endpoint subintervals. This approach leads to particularly simple Walsh series representations, introduces a remarkable computational simplification, and achieves excellent results in reducing the harmonic distortion.

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 𝑡_𝑖^(𝑠). 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 𝑡_𝑖^(𝑠), and (ii) harmonic distortion must be selectively reduced to produce a high-quality AC signal.

Figure 1 

Schematic structure of a voltage PWM inverter.

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.

A second line of solution is based on Walsh series representations. The orthogonal systems of Walsh functions [18–20] allow to establish an elegant link between the switching times 𝑡_𝑖^(𝑠) and the harmonic amplitudes of a PWM signal [21, 22]. By taking advantage of this link, it is possible to give satisfactory responses to the design challenges: (i) systems of simple linear equations ℒ=𝑡_𝑖^(𝑠)=𝑝_𝑖𝐴_ac+𝑟_𝑖,𝑖=1,…,2𝑀, (1) can be obtained for fast on-line computation of the switching instants 𝑡_𝑖^(𝑠) for a given AC amplitude 𝐴_ac and, simultaneously, (ii) selective elimination of undesirable harmonics can be carried out.

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.

The system of Walsh functions in [0,𝑇)𝒲_𝑇=wal_𝑇(𝑛,𝑡),𝑛∈ℕ₀ (2) is a sequence of rectangular functions that alternate the values ±1 and take the value 0 at the discontinuities, which are always located at points of the form 𝑡=(𝑘𝑇)/2^𝑝. In this work, the functions wal_𝑇(𝑛,𝑡) are arranged in the increasing sequency-order, where the index 𝑛 indicates the number of changes of sign. The first 16 elements of 𝒲_𝑇 are schematically depicted in Figure 2, where the line with label wal_𝑛 corresponds to the graphic of the function wal_𝑇(𝑛,𝑡), 0≤𝑛≤15. For example, the line with label wal₁ corresponds to the Walsh function wal_𝑇(1,𝑡), which takes the value +1 in [0,𝑇/2), the value −1 in (𝑇/2,𝑇), and has a change of sign at 𝑡=𝑇/2. 𝒲_𝑇 is a complete and orthogonal system in [0,𝑇). Absolutely integrable functions in [0,𝑇) can be represented in the form of Walsh series 𝑓(𝑡)=^∞_𝑛=0𝑤_𝑛wal_𝑇(𝑛,𝑡), (3) with 𝑤_𝑛=1𝑇^𝑇₀𝑓(𝑡)wal_𝑇(𝑛,𝑡)𝑑𝑡. (4) For 𝑇=1, we obtain the system of normalized Walsh functions 𝒲₁=wal₁(𝑛,𝑡),𝑛∈ℕ₀, (5) which is a complete and orthonormal system in [0,1). Arbitrary functions wal_𝑇(𝑛,𝑡) can be expressed in terms of normalized functions in the form wal_𝑇(𝑛,𝑡)=wal₁𝑡𝑛,𝑇. (6) By introducing the normalized time 𝜏=𝑡/𝑇, the coefficients in (4) can be computed as 𝑤_𝑛=¹₀𝑓(𝑇𝜏)wal₁(𝑛,𝜏)𝑑𝜏, (7) and a function 𝑓(𝑡) defined in [0,𝑇) can be represented as a normalized Walsh series in the form 𝑓(𝑡)=^∞_𝑛=0𝑤_𝑛wal₁𝑡𝑛,𝑇. (8) A signal 𝑓(𝑡) has quarter-wave odd (QWO) symmetry in the interval [0,𝑇) if 𝑓(𝑡) presents odd symmetry with respect to 𝑡=𝑇/2 in the interval [0,𝑇), and it also has even symmetry with respect to 𝑡=𝑇/4 in the interval [0,𝑇/2). The system of sinusoidal signals 𝑠_𝑇(2𝑘−1,𝑡)=sin(2𝑘−1)𝜔₀𝑡,𝑘=1,2,…, (9) with 𝜔₀=2𝜋/𝑇, and the sytem of Walsh functions wal_𝑇(4𝑛−3,𝑡),𝑛=1,2,… (10) have QWO symmetry in [0,𝑇). Signals 𝑓(𝑡) that are absolutely integrable in [0,𝑇), and present QWO symmetry in the interval, can be represented by a Fourier series of the form 𝑓(𝑡)=^∞_𝑘=1𝑎_2𝑘−1sin(2𝑘−1)𝜔₀𝑡, (11) where 𝑎_2𝑘−1=2𝑇^𝑇₀𝑓(𝑡)sin(2𝑘−1)𝜔₀𝑡=8𝑑𝑡𝑇₀^𝑇/4𝑓(𝑡)sin(2𝑘−1)𝜔₀𝑡𝑑𝑡. (12) Analogously, 𝑓(𝑡) can be represented by a Walsh series of the form 𝑓(𝑡)=^∞_𝑛=1𝑤_4𝑛−3wal_𝑇(4𝑛−3,𝑡), (13) where 𝑤_4𝑛−3=1𝑇^𝑇₀𝑓(𝑡)wal_𝑇()=44𝑛−3,𝑡𝑑𝑡𝑇₀^𝑇/4𝑓(𝑡)wal_𝑇(4𝑛−3,𝑡)𝑑𝑡. (14)

Figure 2 

Schematic representation of the Walsh functions wal𝑇(𝑛,𝑡) for 0≤𝑛≤15.

Let us now consider the 𝑁-term truncated Walsh series 𝑓_𝑁(𝑡)=^𝑁_𝑛=1𝑤_𝑛wal_𝑇(4𝑛−3,𝑡). (15) Clearly, 𝑓_𝑁(𝑡) has QWO symmetry and can be represented by the Fourier series 𝑓_𝑁(𝑡)=^∞_𝑘=1𝑎_𝑘sin(2𝑘−1)𝜔₀𝑡, (16) with 𝑎_𝑘=8𝑇₀^𝑇/4𝑓_𝑁(𝑡)sin(2𝑘−1)𝜔₀𝑡𝑑𝑡. (17) Note that, for simplicity, we have introduced the notations 𝑤_𝑛=𝑤_4𝑛−3 and 𝑎_𝑘=𝑎_2𝑘−1 in (15) and (16), respectively. By substituting (15) in (17), we obtain 𝑎_𝑘=^𝑁_𝑛=1𝑏_𝑘,𝑛𝑤_𝑛, (18) where 𝑏_𝑘,𝑛=8𝑇₀^𝑇/4wal_𝑇(4𝑛−3,𝑡)sin(2𝑘−1)𝜔₀𝑡𝑑𝑡. (19) Using the 𝐾×𝑁 matrix 𝐵_𝐾,𝑁=𝑏_{𝑘,𝑛𝑘=1,…,𝐾;𝑛=1,…,𝑁} (20) the vector of Fourier coefficients 𝐴_𝐾=𝑎₁,…,𝑎_𝐾^𝑇 (21) can be easily computed from the vector of Walsh coefficients 𝑊_𝑁=𝑤₁,…,𝑤_𝑁^𝑇 (22) in the form 𝐴_𝐾=𝐵_𝐾,𝑁𝑊_𝑁. (23) It is worth to be noted that, by considering the property given in (6) and the angles 𝛼=2𝜋𝑇𝑡, (24) a period-independent expression for the coefficients 𝑏_𝑘,𝑛 can be obtained in the form 𝑏_𝑘,𝑛=4𝜋₀^𝜋/2wal_2𝜋(4𝑛−3,𝛼)sin[(2𝑘−1)𝛼]𝑑𝛼. (25) Thus, for example, to compute the element 𝑏_2,4 we only have to consider the nonzero values of the Walsh function wal_2𝜋(13,𝛼) in the quarter-period interval [0,𝜋/2)wal_2𝜋()=𝜋13,𝛼1,0<𝛼<8,𝜋−1,8𝜋<𝛼<4,𝜋1,4<𝛼<3𝜋8,−1,3𝜋8𝜋<𝛼<2, (26) resulting 𝑏_2,4=4𝜋₀^𝜋/2wal_2𝜋()()=413,𝛼sin3𝛼𝑑𝛼𝜋₀^𝜋/8sin(3𝛼)𝑑𝛼−^𝜋/4_𝜋/8()+sin3𝛼𝑑𝛼_{3𝜋/8𝜋/4}sin(3𝛼)𝑑𝛼−^𝜋/2_3𝜋/8()=4sin3𝛼𝑑𝛼3𝜋1−2cos3𝜋8−cos3𝜋4+cos9𝜋8=0.2836. (27) The particular values corresponding to the matrix 𝐵_4,8 are displayed in  (28)𝐵_4,8=1.2732−0.5274−0.1049−0.2533−0.02490.0103−0.0519−0.12540.42441.0246−0.68460.2836−0.0860−0.2077−0.31080.12870.25470.61480.9201−0.3811−0.2037−0.49180.3286−0.13610.1819−0.07530.37880.9144−0.75050.31090.06180.1493. (28)

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

Let us consider the switching signal 𝑓(𝑡) displayed in Figure 3, where 𝑉_dc is a constant and the values 𝑡_𝑖^(𝑠) denote the switching time instants. Broadly speaking, the objective in PWM DC-AC signal generation consists in approximating a sinusoidal signal 𝑠(𝑡)=𝐴_acsin𝜔₀𝑡,𝜔₀=2𝜋𝑇, (29) by means of a suitable switching signal 𝑓(𝑡). For a given PWM signal 𝑓(𝑡) with QWO symmetry in [0,𝑇) and Fourier representation 𝑓(𝑡)=^∞_𝑘=1𝑎_2𝑘−1sin(2𝑘−1)𝜔₀𝑡, (30) the goodness of approximation includes satisfying a three-criteria objective: (C1)Matching the main frequency amplitude𝑎₁=𝐴_ac, (31)(C2)canceling the first 𝑀−1 non-fundamental harmonics𝑎₃=0,𝑎₅=0,…,𝑎_2𝑀−1=0, (32)(C3)producing a low distortion factorDF=100𝐴_ac^∞_𝑛=2𝑎_2𝑛−12𝑛−1². (33)

Figure 3 

Bipolar pulse-width-modulated signal with amplitude 𝑉dc and switching time instants 𝑡𝑖(𝑠).

The conventional design method described in Section 3.1 allows to obtain systems of linear equations ℒ_𝑐=𝑡_𝑖^(𝑠)=𝑝_𝑖𝐴_ac+𝑟_𝑖,𝑖=1,…,2𝑀, (34) which, for a given AC amplitude 𝐴_ac, make possible to compute a suitable PWM signal 𝑓(𝑡) satisfying the design conditions (C1)–(C3). However, it must be highlighted that the linear systems ℒ_𝑐 provided by this approach are typically only valid for a restricted range of AC amplitudes 𝐴_ℒmin<𝐴_ac<𝐴_ℒmax (35) and, consequently, a supervised implementation of several linear systems can be necessary to cover the full sinusoidal amplitude range.

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.

Remark 1. Unipolar PWM signals, which take the values 𝑉_dc and 0, are also used in PWM DC-AC signal generation. In this paper, the discussion will be focused on bipolar signals. A detailed account of the unipolar case can be found in [26].

Remark 2. Note that the relative importance of the harmonic amplitude 𝑎_2𝑛−1 in the distortion factor (33) decreases with the harmonic order. This is a convenient property for a wide class of practical applications, where the negative effects of the harmonics decrease with the harmonic order.

Remark 3. Some applications can require fast variations of the AC amplitude; consequently, the real-time computation of the PWM switching instants 𝑡_𝑖^(𝑠) will demand simple and highly effective numerical procedures.

3.1. Conventional Design Method

Let us assume that the interval 0≤𝐴_ac≤𝐴^∗_ac (36) defines the operational range of the sinusoidal amplitude 𝐴_ac in (29). Let us also consider the signal 𝑓(𝑡)=𝑉_dc𝑓(𝑡), (37) where 𝑉_dc=𝐴^∗_ac is the DC amplitude, and 𝑓(𝑡) 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 𝑓(𝑡) with 𝑀 switching cycles (pulses) in the quarter-wave interval [0,𝑇/4). The 𝑖th switching cycle starts at the switching-down instant 𝑡_𝑖^(𝑑), where the signal 𝑓(𝑡) shifts from +1 to −1, and ends at the switching-up instant 𝑡_𝑖^(𝑢), 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 𝐼_𝑗=𝑡_𝑗−1,𝑡_𝑗,𝑗=1,…,𝑁, (38) where the partition points are 𝑡_𝑗𝑇=𝑗ℎ,𝑗=0,…,𝑁,ℎ=,4𝑁 (39) and 𝑝 is the smallest integer that satisfies 𝑁=2^𝑝≥4𝑀. (40)(R2) A system of 𝑀 indexes is chosen as 𝐽^(𝑑)=𝑗₁^(𝑑),…,𝑗_𝑀^(𝑑),1≤𝑗₁^(𝑑)<⋯<𝑗_𝑀^(𝑑)≤𝑁, (41) and a switching-down time is selected in each interval 𝑡_𝑖^(𝑑)∈𝐼_𝑗𝑖^(𝑑),𝑖=1,…,𝑀. (42)(R3) The 𝑀 switching-up instants, 𝑡_𝑖^(𝑢), 𝑖=1,…,𝑀, are computed as follows: 𝑡_𝑖^(𝑢)=𝑡_𝑗𝑖^(𝑑)₊₁if𝑗_𝑖^(𝑑)<𝑁2,𝑡_𝑗𝑖^(𝑑)𝑁if2≤𝑗_𝑖^(𝑑). (43)(R4) Each interval 𝐼_𝑗, 1≤𝑗≤𝑁, can contain at most one switching time 𝑡_𝑖^(𝑑) or 𝑡_𝑖^(𝑢). Using the switching ratios with respect to the interval endpoint 𝜙_𝑖=𝑡_𝑗𝑖^(𝑑)−𝑡_𝑖^(𝑑)ℎ𝑇,𝑖=1,…,𝑀,ℎ=,4𝑁 (44) the switching-down instants can be expressed in the form 𝑡_𝑖^(𝑑)=𝑗_𝑖^(𝑑)−𝜙_𝑖ℎ,𝑖=1,…,𝑀. (45) In the conventional method, the switching-up instants 𝑡_𝑖^(𝑢) are independent of 𝜙_𝑖 and can be written in terms of the step ℎ as 𝑡_𝑖^(𝑢)=𝑗_𝑖^(𝑑)+𝜆_𝑖ℎ,𝜆_𝑖=1if𝑗_𝑖^(𝑑)<𝑁2,𝑁0if2≤𝑗_𝑖^(𝑑). (46) A PWM signal 𝑓(𝑡), 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 𝑗₁^(𝑑)=3 and 𝑗₂^(𝑑)=7. The ratio 𝜙₁ defines the switching-down instant 𝑡₁^(𝑑)=(3−𝜙₁)ℎ; as 𝑗₁^(𝑑)<𝑁/2, the corresponding switching-up instant is located at 𝑡₁^(𝑢)=4ℎ. The ratio 𝜙₂ defines 𝑡₂^(𝑑)=(7−𝜙₂)ℎ. Now, 𝑁/2≤𝑗₂^(𝑑) and the associated switching-up instant is 𝑡₂^(𝑢)=7ℎ.

Figure 4 

Switching structure in the conventional design method. In this case, pulses are not symmetric with respect to the partition points.

For the PWM signal 𝑓(𝑡), defined by the system of switching indexes 𝐽^(𝑑)=𝑗₁^(𝑑),…,𝑗_𝑀^(𝑑), (47) and the vector of switching ratios Φ_𝑀=𝜙₁,…,𝜙_𝑀^𝑇, (48) let us consider the truncated Walsh series 𝑓_𝑁(𝑡)=^𝑁_𝑛=1𝑤_𝑛wal_𝑇(4𝑛−3,𝑡), (49) 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 𝐼_𝑗=𝑗−1ℎ,𝑗ℎ,𝑗=1,…,2^𝑟𝑇,ℎ=2^𝑟, (50) and observing that 4𝑁−3=4⋅2^𝑝−3<2^𝑝+2, (51) it follows that the Walsh functions wal_𝑇(4𝑛−3,𝑡),𝑛=1,…,𝑁 (52) are constant in the interior points of the intervals 𝐼_𝑗 defined in the rule (R1). From (14), the coefficient 𝑤_𝑛 in (49) can be computed as 𝑤_𝑛=4𝑇₀^𝑇/4𝑓(𝑡)wal_𝑇()=44𝑛−3,𝑡𝑑𝑡𝑇^𝑁_𝑗=1wal_𝑇4𝑛−3,𝑡̂_𝑗^𝑗ℎ_{(𝑗−1)ℎ}𝑓(𝑡)𝑑𝑡, (53) where ℎ=𝑇/(4𝑁), and 𝑡_𝑗̂∈((𝑗−1)ℎ,𝑗ℎ). Using the normalized time 𝜏=𝑡/𝑇, we can obtain the following period-free expression for 𝑤_𝑛: 𝑤_𝑛=4^𝑁_𝑗=1wal₁4𝑛−3,𝜏_𝑗^𝑗/(4𝑁)_{(𝑗−1)/(4𝑁)}𝑓(𝜏𝑇)𝑑𝜏, (54) with 𝜏_𝑗∈((𝑗−1)/4𝑁,𝑗/4𝑁). By computing the integrals in (54), we get 𝑤_𝑛2=−𝑁^𝑀_𝑖=1𝜙_𝑖wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑)+1𝑁^𝑁_𝑗=1wal₁4𝑛−3,𝜏_𝑗−2^𝑀_𝑖=1𝜆_𝑖wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑). (55) The vector of Walsh coefficients 𝑊_𝑁=𝑤₁,…,𝑤_𝑁^𝑇 (56) admits the matrix expression 𝑊_𝑁=𝐶_𝑁,𝑀Φ_𝑀+𝐷_𝑁, (57) where 𝐶_𝑁,𝑀 is a matrix with elements 𝑐_𝑛,𝑖2=−𝑁wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑),𝑛=1,…,𝑁,𝑖=1,…,𝑀, (58) and 𝐷_𝑁=[𝑑₁,…,𝑑_𝑁]^𝑇 is a vector with elements 𝑑_𝑛=1𝑁^𝑁_𝑗=1wal₁4𝑛−3,𝜏_𝑗−2^𝑀_𝑖=1𝜆_𝑖wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑). (59) According to the discussion in Section 2, the vector 𝐴_𝑀=𝑎₁,…,𝑎_𝑀^𝑇, (60) which contains the first 𝑀 coefficients of the Fourier series representation 𝑓_𝑁(𝑡)=^∞_𝑘=1𝑎_𝑘sin(2𝑘−1)𝜔₀𝑡, (61) can be computed in the form 𝐴_𝑀=𝐵_𝑀,𝑁𝑊_𝑁, (62) and considering (57), we obtain 𝐴_𝑀=𝐸Φ_𝑀+𝐹, (63) with 𝐸=𝐵_𝑀,𝑁𝐶_𝑁,𝑀,𝐹=𝐵_𝑀,𝑁𝐷_𝑁. (64) For the sinusoidal signal (29), let us consider a given AC amplitude 0≤𝐴_ac≤𝐴^∗_ac and the system of switching indexes (41). If there exists a vector of switching ratios Φ_𝑀=𝜙₁,…,𝜙_𝑀^𝑇,𝜙_𝑖∈(0,1),𝑖=1,…,𝑀, (65) that satisfies Φ_𝑀=𝐸⁻¹𝐴_𝑀−𝐹 (66) with 𝐴_𝑀=𝐴₁,0,…,0^𝑇,𝐴₁=𝐴_ac𝐴^∗_ac, (67) then the conditions (C1) and (C2) are satisfied by the signal 𝑓(𝑡)=𝑉_dc𝑓(𝑡), (68) where 𝑉_dc=𝐴^∗_ac and 𝑓(𝑡) 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 𝐽^(𝑑)=𝑗₁^(𝑑)=3,𝑗₂^(𝑑)=7, (69) the matrices (58) and (59) take the values 𝐶_8,21=−4111−1−11−1−1−1−1−111−111,𝐷₈=1431111−1−1−1, (70) 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 ℒ_𝑐=𝜙₁=−1.0155𝐴₁𝜙+0.9555,₂=−1.5931𝐴₁+1.5317. (71) To compute the regulation range of the system (71), we can consider the auxiliary system ℒ^′_𝑐=𝐴₁₁=−0.9847𝜙₁𝐴+0.9409,₁₂=−0.6277𝜙₂+0.9615, (72) obtained by isolating 𝐴₁ in the different equations of the system (71), and compute the values 𝐴_1min=max_1≤𝑖≤2inf_{𝜙𝑖∈(0,1)}𝐴_1𝑖,𝐴_1max=min_1≤𝑖≤2sup_{𝜙𝑖∈(0,1)}𝐴_1𝑖. (73) Clearly, for any normalized amplitude 𝐴₁ satisfying 𝐴_1min<𝐴₁<𝐴_1max, (74) a suitable system of switching ratios 𝜙_𝑖∈[0,1] can be computed. For the current example, we obtain the particular values 𝐴_1min=0.3373,𝐴_1max=0.9409, (75) 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 𝑓(𝑡) is carried out following the same rules (R1), (R2), and (R4) used in the conventional method, but in this case, the switching indexes 𝑗_𝑖^(𝑑) satisfy 1≤𝑗₁^(𝑑)<⋯<𝑗_𝑀^(𝑑)<𝑁, (76) and the switching-down instants 𝑡_𝑖^(𝑑)=𝑗_𝑖^(𝑑)−𝜙_𝑖ℎ,𝑖=1,…,𝑀 (77) are followed by the switching-up instants 𝑡_𝑖^(𝑢)=𝑗_𝑖^(𝑑)+𝜙_𝑖ℎ,𝑖=1,…,𝑀. (78) This new switching scheme is displayed in Figure 5, where the symmetrical arrangement of the switching instants 𝑡_𝑖^(𝑑) and 𝑡_𝑖^(𝑢) with respect to the interval endpoint 𝑡_𝑗𝑖^(𝑑) can be clearly appreciated. The coefficients (54) of the truncated Walsh series (49) have now the form 𝑤_𝑛2=−𝑁^𝑀_𝑖=1𝜙_𝑖wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑)+wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑)₊₁+1𝑁^𝑁_𝑗=1wal₁4𝑛−3,𝜏_𝑗. (79) The vector of Walsh coefficients (56) admits the matrix expression (57), but now the elements of the matrix 𝐶_𝑁,𝑀 take the form 𝑐_𝑛,𝑖2=−𝑁wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑)+wal₁4𝑛−3,𝜏_𝑗𝑖^(𝑑)₊₁, (80) and the elements of the vector 𝐷_𝑁 are 𝑑_𝑛=1𝑁^𝑁_𝑗=1wal₁4𝑛−3,𝜏_𝑗. (81) The coefficients 𝑐_𝑛,𝑖 can be easily computed by considering the 𝑁×𝑁 matrix 𝑆_𝑁, with elements 𝑠_𝑛,𝑘=wal₁4𝑛−3,𝜏_𝑘,𝜏_𝑘∈𝑘−1,𝑘4𝑁4𝑁. (82) 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 𝑐_𝑛,𝑖2=−𝑁𝑠_{𝑛,𝑗𝑖}^(𝑑)+𝑠_{𝑛,𝑗𝑖}^(𝑑)₊₁, (83) and the 𝑖th column of 𝐶_𝑁,𝑀 can be obtained by adding the columns 𝑗_𝑖^(𝑑) and 𝑗_𝑖^(𝑑)+1 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 𝑑_𝑛=1𝑁^𝑁_𝑘=1𝑠_𝑛,𝑘 (84) 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 𝐷_𝑁=[1,0,…,0]^𝑇. (85) 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.

Figure 5 

Switching structure in the advanced design method. In this case, each pulse is chosen to be symmetric with respect to a certain partition point.

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 𝑗₁^(𝑑)=3 and 𝑗₂^(𝑑)=7. For a homogeneous partition of the interval [0,1/4] with step ℎ=1/32, the system of Walsh functions wal₁(1,𝜏),wal₁(5,𝜏),wal₁(9,𝜏),…,wal₁(29,𝜏) (86) produces the matrix of sign sequences 𝑆₈=111111111111−1−1−1−111−1−1−1−11111−1−111−1−11−1−111−1−111−1−11−111−11−11−1−11−111−11−11−11−1. (87) According to (83), the matrix 𝐶_8,2 is 𝐶_8,21=−41222−2−22−2−200000000=−2111−1−11−1−100000000, (88) and from (84), the value of 𝐷₈ is 𝐷₈=1880000000=10000000, (89) 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 ℒ_𝑎=𝜙₁=−0.5877𝐴₁𝜙+1.0583,₂=−0.6933𝐴₁+0.7071, (90) and from the associated auxiliary system ℒ^′_𝑎=𝐴₁₁=−1.7015𝜙₁𝐴+1.8007,₁₂=−1.4424𝜙₂+1.0200, (91) we get the modulation range 𝐴_1min=0.0992,𝐴_1max=1.0200. (92) 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 𝑑_𝑛=^𝑁_𝑘=1wal₁𝑛−1,𝜏_𝑘𝑁,𝜏_𝑘∈𝑘−1𝑁,𝑘𝑁=^𝑁_𝑘=1^𝑘/𝑁_{(𝑘−1)/𝑁}wal₁(𝑛−1,𝜏)wal₁()=0,𝜏𝑑𝜏₁₀wal₁(𝑛−1,𝜏)wal₁(0,𝜏)𝑑𝜏. (93) 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

According to the results in Section 3, PWM signals with 𝑀=2^𝑝 switching cycles in the quarter-period interval can be conveniently designed to modulate the normalized fundamental-frequency amplitude 𝐴₁ and cancel the first 𝑀−1 nonfundamental odd harmonics. In the design procedure, the quarter-period interval is divided into 𝑁=2^𝑝+2 subintervals, and a suitable system of switching indexes 𝐽^(𝑑)=𝑗₁^(𝑑),…,𝑗_𝑀^(𝑑) (94) is selected. Each system 𝐽^(𝑑) determines a PWM signal 𝑓(𝑡) that satisfies the design conditions for a certain normalized-amplitude interval 𝐴_1min<𝐴₁<𝐴_1max. (95) Obviously, the interval length Δ=𝐴_1max−𝐴_1min (96) is a meaningful design parameter. Values of Δ close to 1 indicate designs of practical interest. Additionally, unfeasible designs are obtained for negative values of Δ.

The main objective of this section is to design PWM signals 𝑓(𝑡) with optimal amplitude ranges (95). For clarity and simplicity, we will initially focus the discussion on signals 𝑓(𝑡) with 𝑀=8 switching cycles in the quarter-period interval. In this case, the interval [0,𝑇/4) is divided into 𝑁=32 subintervals as follows: 𝐼_𝑗=𝑇𝑗−1ℎ,𝑗ℎ,𝑗=1,…,32,ℎ=.128 (97) The design of a specific PWM signal starts by selecting a system of switching indexes 𝐽^(𝑑)=𝑗₁^(𝑑),…,𝑗₈^(𝑑), (98) which determine the starting subintervals for the switching cycles. The number of different index systems 𝐽^(𝑑) is 832≈1.05×10⁷. (99) This huge number clearly shows that strategies based on exhaustive exploration are not viable, even for moderate values of 𝑀.

Exploratory studies, conducted with small values of 𝑀, indicate that constraining the starting point of the switching cycles within blocks of four consecutive subintervals produces no negative effect in the optimization problem. Consequently, the switching indexes 𝑗_𝑖^(𝑑) can be selected satisfying 4𝑖−3≤𝑗_𝑖^(𝑑)<4𝑖+1,𝑖=1,…,8. (100) Under this constrained selection scheme, the number of different systems 𝐽^(𝑑) is reduced to a more tractable value 4⁸=65536. (101) It should be highlighted, however, that these index systems can be infeasible for a variety of reasons, for example, violating the design rule (R4) or condition (76), or producing negative values for Δ. The numbers of feasible switching index systems obtained with the conventional and advanced design methods are presented in Table 1, which also includes the number of feasible index systems that produce amplitude intervals (95) with length Δ in the ranges Δ<20%,20%≤Δ≤40%,Δ>40%. (102) The data in Table 1 clearly indicate the superiority of the advanced method in providing feasible index systems 𝐽^(𝑑). This superiority is particularly remarkable for large values of Δ.

Table 1 

Number of feasible switching index systems 𝐽^(𝑑) for conventional and advanced designs with 𝑀=8 switching cycles.

The optimal index system for the conventional design method is 𝐽_𝑐^(𝑑)=[2,6,9,14,20,22,27,30], (103) which produces the following system of linear equations to compute the switching ratios: ℒ_𝑐=𝜙₁=−0.1418𝐴₁𝜙+1.0034,₂=−0.3350𝐴₁𝜙+0.9945,₃=−0.5201𝐴₁𝜙+1.0077,₄=−0.6621𝐴₁𝜙+0.9891,₅=−0.8191𝐴₁𝜙+1.0164,₆=−0.8771𝐴₁𝜙+0.9723,₇=−1.0310𝐴₁𝜙+1.0607,₈=−0.7059𝐴₁+0.7071. (104) The PWM signal and the harmonic distribution obtained for a DC amplitude 𝑉_dc=100 V, a fundamental frequency of 50 Hz, and the AC amplitude 𝐴_ac=98 V are displayed in Figure 6. The corresponding switching instants can be computed by setting the normalized amplitude 𝐴₁=0.98 in (104), and by considering (45) and (46) with a period 𝑇=20 ms. The PWM signal and the harmonic distribution for the same DC amplitude and frequency, and 𝐴_ac=54 V are displayed in Figure 7.

Figure 6 

Optimal conventional design with 𝑀=8 switching cycles. PWM signal and harmonic distribution for the AC amplitude 𝐴ac=98 V with a fundamental frequency of 50 Hz and 𝑉dc=100 V.

Figure 7 

Optimal conventional design with 𝑀=8 switching cycles. PWM signal and harmonic distribution for the AC amplitude 𝐴ac=54 V with a fundamental frequency of 50 Hz and 𝑉dc=100 V.

For the advanced method, we obtain the optimal index system 𝐽_𝑎^(𝑑)=[3,7,11,15,19,23,27,31], (105) with an associated linear system ℒ_𝑎=𝜙₁=−0.7935𝐴₁𝜙+0.1434,₂=−1.2594𝐴₁𝜙+1.4115,₃=−0.1541𝐴₁𝜙+0.2644,₄=−2.1312𝐴₁𝜙+2.1001,₅=−1.3492𝐴₁𝜙+1.6192,₆=−1.7046𝐴₁𝜙+1.9318,₇=−1.7874𝐴₁𝜙+1.8833,₈=−2.0562𝐴₁+2.0576. (106) The PWM signals and the harmonic distributions obtained for the DC amplitude 𝑉_dc=100 V, a fundamental frequency of 50 Hz, and the AC amplitudes 𝐴_ac=98 V and 𝐴_ac=54 V are displayed in Figures 8 and 9, respectively. In this case, the switching instants are obtained from the linear system ℒ_𝑎 in (106), (77), and (78).

Figure 8 

Optimal advanced design with 𝑀=8 switching cycles. PWM signal and harmonic distribution for the AC amplitude 𝐴ac=98 V with a fundamental frequency of 50 Hz and 𝑉dc=100 V.

Figure 9 

Optimal advanced design with 𝑀=8 switching cycles. PWM signal and harmonic distribution for the AC amplitude 𝐴ac=54 V with a fundamental frequency of 50 Hz and 𝑉dc=100 V.

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%.

Table 2 

Optimal switching index systems and modulation ranges for conventional and advanced designs with 𝑀=8 switching cycles.

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.

Finally, one of the most outstanding features of the new approach is that the design methodology discussed for 𝑀=8 can be extended to larger values 𝑀=2^𝑝 by selecting the system of switching indexes 𝐽_𝑀^(𝑑)=𝑗₁^(𝑑),…,𝑗_𝑀^(𝑑),𝑗_𝑖^(𝑑)=4𝑖−1,𝑖=1,…𝑀, (107) which, using a common computational notation, can be represented in the form 𝐽_𝑀^(𝑑)=𝑗₁^(𝑑)∶4∶𝑗_𝑀^(𝑑)=[3∶4∶4𝑀−1], (108) where 𝑗₁^(𝑑)=3 and 𝑗_𝑀^(𝑑)=4𝑀−1 are the first and last elements of the index sequence, respectively, and 4 is the step. The PWM signal and the harmonic distribution obtained in the case 𝑀=32 with the index system 𝐽^(𝑑)₃₂=[3∶4∶127] are displayed in Figure 10 (for a fundamental frequency of 50 Hz and 𝑉_dc=𝐴_ac=100 V). This design strategy has been successfully applied to the values 𝑀=2^𝑝, 3≤𝑝≤6, obtaining in all the cases a normalized-amplitude regulation interval of 5%–100% and the well-shaped harmonic distribution observed in Figures 8, 9, and 10.

Figure 10 

Advanced design with 𝑀=32 switching cycles. PWM signal and harmonic distribution corresponding to the switching index system 𝐽(𝑑)32=[3∶4∶127], for a fundamental frequency of 50 Hz and 𝑉dc=𝐴ac=100 V.

Remark 7. The strategy for selective harmonic elimination discussed in Section 3 uses an 𝑁-term truncated Walsh series (49) to represent the PWM signal. Consequently, condition (32) is only approximately satisfied, and the first 𝑀−1 non-fundamental harmonics are not fully eliminated by small values of 𝑀. This fact can be appreciated in Figures 7–10. For 𝑀=2^𝑝, the number of terms in the truncated Walsh series is 𝑁=2^𝑝+2. Hence, an accurate representation of the PWM signal is obtained by moderate values of 𝑀, and this residual harmonic distortion is practically removed (see Figure 10).

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].

Acknowledgments

This work was partially supported by the Spanish Ministry of Economy and Competitiveness through Grant DPI2012-32375/FEDER and by the Norwegian Center of Offshore Wind Energy (NORCOWE) under Grant 193821/S60 from the Research Council of Norway (RCN). NORCOWE is a consortium with partners from industry and science, hosted by the Christian Michelsen Research.