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

Authors: Francisco Palacios-Quiñonero; Jesus Vicente-Rodrigo; Maria A. Molina-Hernández; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 ti(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 ti(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 ofprogrammed 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 ti(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
(1)ℒ={ti(s)=piAac+ri,i=1,…,2M},
can be obtained for fast on-line computation of the switching instants ti(s) for a given AC amplitude Aac 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 systemsℒ 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 Section2, 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 ofWalsh functions in [0,T)(2)𝒲T={walT(n,t),n∈ℕ0}
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)/2p. In this work, the functions walT(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 𝒲T are schematically depicted in Figure 2, where the line with label waln corresponds to the graphic of the function walT(n,t), 0≤n≤15. For example, the line with label wal1 corresponds to the Walsh function walT(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. 𝒲T is a complete and orthogonal system in [0,T). Absolutely integrable functions in [0,T) can be represented in the form of Walsh series
(3)f(t)=∑n=0∞wnwalT(n,t),
with
(4)wn=1T∫0Tf(t)walT(n,t)dt.
For T=1, we obtain the system of normalized Walsh functions(5)𝒲1={wal1(n,t),n∈ℕ0},
which is a complete and orthonormal system in [0,1). Arbitrary functions walT(n,t) can be expressed in terms of normalized functions in the form
(6)walT(n,t)=wal1(n,tT).
By introducing the normalized time τ=t/T, the coefficients in (4) can be computed as
(7)wn=∫01f(Tτ)wal1(n,τ)dτ,
and a function f(t) defined in [0,T) can be represented as a normalized Walsh series in the form
(8)f(t)=∑n=0∞wnwal1(n,tT).‍
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
(9)sT(2k-1,t)=sin[(2k-1)ω0t],k=1,2,…,
with ω0=2π/T, and the sytem of Walsh functions
(10)walT(4n-3,t),n=1,2,…
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
(11)f(t)=∑k=1∞‍a2k-1sin[(2k-1)ω0t],
where
(12)a2k-1=2T∫0T‍f(t)sin[(2k-1)ω0t]dt=8T∫0T/4‍f(t)sin[(2k-1)ω0t]dt.
Analogously, f(t) can be represented by a Walsh series of the form
(13)f(t)=∑n=1∞‍w4n-3walT(4n-3,t),
where
(14)w4n-3=1T∫0T‍f(t)walT(4n-3,t)dt=4T∫0T/4f(t)walT(4n-3,t)dt.Figure 2
Schematic representation of the Walsh functionswalT(n,t) for 0≤n≤15.Let us now consider theN-term truncated Walsh series
(15)fN(t)=∑n=1N‍w^nwalT(4n-3,t).
Clearly, fN(t) has QWO symmetry and can be represented by the Fourier series
(16)fN(t)=∑k=1∞‍a^ksin[(2k-1)ω0t],
with
(17)a^k=8T∫0T/4‍fN(t)sin[(2k-1)ω0t]dt.
Note that, for simplicity, we have introduced the notations w^n=w4n-3 and a^k=a2k-1 in (15) and (16), respectively. By substituting (15) in (17), we obtain
(18)a^k=∑n=1N‍bk,nw^n,
where
(19)bk,n=8T∫0T/4‍walT(4n-3,t)sin[(2k-1)ω0t]dt.
Using the K×N matrix
(20)BK,N=[bk,n]k=1,…,K;n=1,…,N
the vector of Fourier coefficients
(21)A^K=[a^1,…,a^K]T
can be easily computed from the vector of Walsh coefficients
(22)W^N=[w^1,…,w^N]T
in the form
(23)A^K=BK,NW^N.
It is worth to be noted that, by considering the property given in (6) and the angles
(24)α=2πTt,
a period-independent expression for the coefficients bk,n can be obtained in the form
(25)bk,n=4π∫0π/2wal2π(4n-3,α)sin[(2k-1)α]dα.
Thus, for example, to compute the element b2,4 we only have to consider the nonzero values of the Walsh function wal2π(13,α) in the quarter-period interval [0,π/2)(26)wal2π(13,α)={-1,0<α<π8,-1,π8<α<π4,-1,π4<α<3π8,-1,3π8<α<π2,
resulting
(27)b2,4=4π∫0π/2wal2π(13,α)sin(3α)dα=4π[∫0π/8sin(3α)dα-∫π/8π/4sin(3α)dα+∫π/43π/8‍sin(3α)dα-∫3π/8π/2‍sin(3α)dα]=43π[1-2(cos3π8-cos3π4+cos9π8)]=0.2836.
The particular values corresponding to the matrix B4,8 are displayed in  (28)(28)B4,8=[1.2732-0.5274-0.1049-0.2533-0.0249-0.0103-0.0519-0.12540.4244-1.0246-0.6846-0.2836-0.0860-0.2077-0.3108-0.12870.2547-0.6148-0.9201-0.3811-0.2037-0.4918-0.3286-0.13610.1819-0.0753-0.3788-0.9144-0.7505-0.3109-0.0618-0.1493].


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

Let us consider the switching signalf(t) displayed in Figure 3, where Vdc is a constant and the values ti(s) denote the switching time instants. Broadly speaking, the objective in PWM DC-AC signal generation consists in approximating a sinusoidal signal
(29)s(t)=Aacsin(ω0t),ω0=2πT,
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
(30)f(t)=∑k=1∞‍a2k-1sin[(2k-1)ω0t],
the goodness of approximation includes satisfying a three-criteria objective: (C1)
Matching the main frequency amplitude(31)a1=Aac,(C2)
canceling the firstM-1non-fundamental harmonics(32)a3=0,a5=0,…,a2M-1=0,(C3)
producing a low distortion factor(33)DF=100Aac∑n=2∞‍(a2n-12n-1)2.Figure 3
Bipolar pulse-width-modulated signal with amplitudeVdc and switching time instants ti(s).Theconventional design method described in Section 3.1 allows to obtain systems of linear equations
(34)ℒc={ti(s)=piAac+ri,i=1,…,2M},
which, for a given AC amplitude Aac, 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 ℒc provided by this approach are typically only valid for a restricted range of AC amplitudes
(35)[Aℒ]min<Aac<[Aℒ]max
and, consequently, a supervised implementation of several linear systems can be necessary to cover the full sinusoidal amplitude range.Theadvanced design method, presented in Section 3.2, introduces a clever modification in the conventional method that facilitates produceing linear systems ℒa 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 theswitching 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 Vdc 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 amplitudea2n-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 instantsti(s) will demand simple and highly effective numerical procedures.


### 3.1. Conventional Design Method

Let us assume that the interval(36)0≤Aac≤Aac*
defines the operational range of the sinusoidal amplitude Aac in (29). Let us also consider the signal
(37)f(t)=Vdcf^(t),
where Vdc=Aac* 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 Mswitching cycles (pulses) in the quarter-wave interval [0,T/4). The ith switching cycle starts at the switching-down instant ti(d), where the signal f^(t) shifts from +1 to -1, and ends at the switching-up instant ti(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=2p intervals
(38)Ij=[tj-1,tj),j=1,…,N,
where the partition points are
(39)tj=jh,j=0,…,N,h=T4N,
and p is the smallest integer that satisfies
(40)N=2p≥4M.(R2)
A system ofM indexes is chosen as
(41)J(d)=[j1(d),…,jM(d)],1≤j1(d)<⋯<jM(d)≤N,
and a switching-down time is selected in each interval
(42)ti(d)∈Iji(d),i=1,…,M.(R3)
TheM switching-up instants, ti(u), i=1,…,M, are computed as follows:
(43)ti(u)={tji(d)+1ifji(d)<N2,tji(d)ifN2≤ji(d).(R4)
Each intervalIj, 1≤j≤N, can contain at most one switching time ti(d) or ti(u).Using the switching ratios with respect to the interval endpoint
(44)ϕi=tji(d)-ti(d)h,i=1,…,M,h=T4N,
the switching-down instants can be expressed in the form
(45)ti(d)=(ji(d)-ϕi)h,i=1,…,M.
In the conventional method, the switching-up instants ti(u) are independent of ϕi and can be written in terms of the step h as
(46)ti(u)=(ji(d)+λi)h,λi={1ifji(d)<N2,0ifN2≤ji(d).
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 theconventional design method. In this case, pulses are not symmetric with respect to the partition points.For the PWM signalf^(t), defined by the system of switching indexes
(47)J(d)=[j1(d),…,jM(d)],
and the vector of switching ratios
(48)ΦM=[ϕ1,…,ϕM]T,
let us consider the truncated Walsh series
(49)f^N(t)=∑n=1N‍w^nwalT(4n-3,t),
where N=2p≥4M is the number of subintervals in [0,T/4). Taking into account that the Walsh functions walT(j,t) with j<2r are constant in the intervals
(50)Ij=((j-1)h,jh),j=1,…,2r,h=T2r,
and observing that
(51)4N-3=4·2p-3<2p+2,
it follows that the Walsh functions
(52)walT(4n-3,t),n=1,…,N
are constant in the interior points of the intervals Ij defined in the rule (R1). From (14), the coefficient w^n in (49) can be computed as
(53)w^n=4T∫0T/4‍f^(t)walT(4n-3,t)dt=4T∑j=1N‍walT(4n-3,t^j)∫(j-1)hjh‍f^(t)dt,
where h=T/(4N), and t^j∈((j-1)h,jh). Using the normalized time τ=t/T, we can obtain the following period-free expression for w^n:
(54)w^n=4∑j=1N‍wal1(4n-3,τ^j)∫(j-1)/(4N)j/(4N)‍f^(τT)dτ,
with τ^j∈((j-1)/4N,j/4N). By computing the integrals in (54), we get
(55)w^n=-2N∑i=1M‍ϕiwal1(4n-3,τ^ji(d))+1N[∑j=1N‍wal1(4n-3,τ^j)-2∑i=1M‍λiwal1(4n-3,τ^ji(d))].
The vector of Walsh coefficients
(56)W^N=[w^1,…,w^N]T
admits the matrix expression
(57)W^N=CN,MΦM+DN,
where CN,M is a matrix with elements
(58)cn,i=-2Nwal1(4n-3,τ^ji(d)),n=1,…,N,i=1,…,M,
and DN=[d1,…,dN]T is a vector with elements
(59)dn=1N[∑j=1N‍wal1(4n-3,τ^j)-2∑i=1M‍λiwal1(4n-3,τ^ji(d))].
According to the discussion in Section 2, the vector
(60)A^M=[a^1,…,a^M]T,
which contains the first M coefficients of the Fourier series representation
(61)f^N(t)=∑k=1∞‍a^ksin[(2k-1)ω0t],
can be computed in the form
(62)A^M=BM,NW^N,
and considering (57), we obtain
(63)A^M=EΦM+F,
with
(64)E=BM,NCN,M,F=BM,NDN.
For the sinusoidal signal (29), let us consider a given AC amplitude 0≤Aac≤Aac* and the system of switching indexes (41). If there exists a vector of switching ratios
(65)ΦM=[ϕ1,…,ϕM]T,ϕi∈(0,1),i=1,…,M,
that satisfies
(66)ΦM=E-1(A^M-F)
with
(67)A^M=[A^1,0,…,0]T,A^1=AacAac*,
then the conditions (C1) and (C2) are satisfied by the signal
(68)f(t)=Vdcf^(t),
where Vdc=Aac* 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
(69)J(d)=[j1(d)=3,j2(d)=7],
the matrices (58) and (59) take the values
(70)C8,2=-14[-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1],D8=14[-3-1-1-1-1-1-1-1],
and the matrix B2,8 is formed by the first two rows of the matrix B4,8 displayed in (28). Equation (66) produces the linear system
(71)ℒc={ϕ1=-1.0155A^1+0.9555,ϕ2=-1.5931A^1+1.5317.
To compute the regulation range of the system (71), we can consider the auxiliary system
(72)ℒc′={[A^1]1=-0.9847ϕ1+0.9409,[A^1]2=-0.6277ϕ2+0.9615,
obtained by isolating A^1 in the different equations of the system (71), and compute the values
(73)[A^1]min=max1≤i≤2(infϕi∈(0,1)[A^1]i),[A^1]max=min1≤i≤2(supϕi∈(0,1)[A^1]i).
Clearly, for any normalized amplitude A^1 satisfying
(74)[A^1]min<A^1<[A^1]max,
a suitable system of switching ratios ϕi∈[0,1] can be computed. For the current example, we obtain the particular values
(75)[A^1]min=0.3373,[A^1]max=0.9409,
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 signalf^(t) is carried out following the same rules (R1), (R2), and (R4) used in the conventional method, but in this case, the switching indexes ji(d) satisfy
(76)1≤j1(d)<⋯<jM(d)<N,
and the switching-down instants
(77)ti(d)=(ji(d)-ϕi)h,i=1,…,M
are followed by the switching-up instants
(78)ti(u)=(ji(d)+ϕi)h,i=1,…,M.
This new switching scheme is displayed in Figure 5, where the symmetrical arrangement of the switching instants ti(d) and ti(u) with respect to the interval endpoint tji(d) can be clearly appreciated. The coefficients (54) of the truncated Walsh series (49) have now the form
(79)w^n=-2N∑i=1M‍ϕi[wal1(4n-3,τ^ji(d))+wal1(4n-3,τ^ji(d)+1)]+1N∑j=1N‍wal1(4n-3,τ^j).
The vector of Walsh coefficients (56) admits the matrix expression (57), but now the elements of the matrix CN,M take the form
(80)cn,i=-2N[wal1(4n-3,τ^ji(d))+wal1(4n-3,τ^ji(d)+1)],
and the elements of the vector DN are
(81)dn=1N[∑j=1N‍wal1(4n-3,τ^j)].
The coefficients cn,i can be easily computed by considering the N×N matrix SN, with elements
(82)sn,k=wal1(4n-3,τ^k),τ^k∈(k-14N,k4N).
The nth row of SN 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
(83)cn,i=-2N(sn,ji(d)+sn,ji(d)+1),
and the ith column of CN,M can be obtained by adding the columns ji(d) and ji(d)+1 of SN, and multiplying by -2/N. The coefficients dn can also be expressed in terms of the elements sn,k as
(84)dn=1N(∑k=1N‍sn,k)
and could be computed by adding the columns of SN. However, it is not necessary to perform this computation because (81) always produces the vector
(85)DN=[1,0,…,0]T.
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 theadvanced 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, withM=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
(86)wal1(1,τ),wal1(5,τ),wal1(9,τ),…,wal1(29,τ)
produces the matrix of sign sequences
(87)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].
According to (83), the matrix C8,2 is
(88)C8,2=-14[-2-2-2-2-2-2-2-2-0-0-0-0-0-0-0-0]=-12[-1-1-1-1-1-1-1-1-0-0-0-0-0-0-0-0],
and from (84), the value of D8 is
(89)D8=18[80000000]=[10000000],
which coincides with the value indicated in (85). Finally, using (64) and (66) with the matrix B2,8 introduced in the previous subsection, and the matrices C8,2 and D8 given in (88) and (89), we obtain the linear system
(90)ℒa={ϕ1=-0.5877A^1+1.0583,ϕ2=-0.6933A^1+0.7071,
and from the associated auxiliary system
(91)ℒa′={[A^1]1=-1.7015ϕ1+1.8007,[A^1]2=-1.4424ϕ2+1.0200,
we get the modulation range
(92)[A^1]min=0.0992,[A^1]max=1.0200.
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 functionswal1(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 SN can be computed by using the sign sequences of the functions walT(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 walT(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
(93)dn=∑k=1N‍wal1(n-1,τ~k)N,τ~k∈(k-1N,kN)=∑k=1N‍∫(k-1)/Nk/N‍wal1(n-1,τ)wal1(0,τ)dτ=∫01‍wal1(n-1,τ)wal1(0,τ)dτ.
In consequence, d1=1, and dn=0 for n=2,…,N, as indicated in (85).Remark 6.
Offline computations are carried out to design the linear systemsℒc and ℒa. Then, these linear systems can be used for fast real-time computation of suitable PWM signals.


## 3.1. Conventional Design Method

Let us assume that the interval(36)0≤Aac≤Aac*
defines the operational range of the sinusoidal amplitude Aac in (29). Let us also consider the signal
(37)f(t)=Vdcf^(t),
where Vdc=Aac* 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 Mswitching cycles (pulses) in the quarter-wave interval [0,T/4). The ith switching cycle starts at the switching-down instant ti(d), where the signal f^(t) shifts from +1 to -1, and ends at the switching-up instant ti(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=2p intervals
(38)Ij=[tj-1,tj),j=1,…,N,
where the partition points are
(39)tj=jh,j=0,…,N,h=T4N,
and p is the smallest integer that satisfies
(40)N=2p≥4M.(R2)
A system ofM indexes is chosen as
(41)J(d)=[j1(d),…,jM(d)],1≤j1(d)<⋯<jM(d)≤N,
and a switching-down time is selected in each interval
(42)ti(d)∈Iji(d),i=1,…,M.(R3)
TheM switching-up instants, ti(u), i=1,…,M, are computed as follows:
(43)ti(u)={tji(d)+1ifji(d)<N2,tji(d)ifN2≤ji(d).(R4)
Each intervalIj, 1≤j≤N, can contain at most one switching time ti(d) or ti(u).Using the switching ratios with respect to the interval endpoint
(44)ϕi=tji(d)-ti(d)h,i=1,…,M,h=T4N,
the switching-down instants can be expressed in the form
(45)ti(d)=(ji(d)-ϕi)h,i=1,…,M.
In the conventional method, the switching-up instants ti(u) are independent of ϕi and can be written in terms of the step h as
(46)ti(u)=(ji(d)+λi)h,λi={1ifji(d)<N2,0ifN2≤ji(d).
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 theconventional design method. In this case, pulses are not symmetric with respect to the partition points.For the PWM signalf^(t), defined by the system of switching indexes
(47)J(d)=[j1(d),…,jM(d)],
and the vector of switching ratios
(48)ΦM=[ϕ1,…,ϕM]T,
let us consider the truncated Walsh series
(49)f^N(t)=∑n=1N‍w^nwalT(4n-3,t),
where N=2p≥4M is the number of subintervals in [0,T/4). Taking into account that the Walsh functions walT(j,t) with j<2r are constant in the intervals
(50)Ij=((j-1)h,jh),j=1,…,2r,h=T2r,
and observing that
(51)4N-3=4·2p-3<2p+2,
it follows that the Walsh functions
(52)walT(4n-3,t),n=1,…,N
are constant in the interior points of the intervals Ij defined in the rule (R1). From (14), the coefficient w^n in (49) can be computed as
(53)w^n=4T∫0T/4‍f^(t)walT(4n-3,t)dt=4T∑j=1N‍walT(4n-3,t^j)∫(j-1)hjh‍f^(t)dt,
where h=T/(4N), and t^j∈((j-1)h,jh). Using the normalized time τ=t/T, we can obtain the following period-free expression for w^n:
(54)w^n=4∑j=1N‍wal1(4n-3,τ^j)∫(j-1)/(4N)j/(4N)‍f^(τT)dτ,
with τ^j∈((j-1)/4N,j/4N). By computing the integrals in (54), we get
(55)w^n=-2N∑i=1M‍ϕiwal1(4n-3,τ^ji(d))+1N[∑j=1N‍wal1(4n-3,τ^j)-2∑i=1M‍λiwal1(4n-3,τ^ji(d))].
The vector of Walsh coefficients
(56)W^N=[w^1,…,w^N]T
admits the matrix expression
(57)W^N=CN,MΦM+DN,
where CN,M is a matrix with elements
(58)cn,i=-2Nwal1(4n-3,τ^ji(d)),n=1,…,N,i=1,…,M,
and DN=[d1,…,dN]T is a vector with elements
(59)dn=1N[∑j=1N‍wal1(4n-3,τ^j)-2∑i=1M‍λiwal1(4n-3,τ^ji(d))].
According to the discussion in Section 2, the vector
(60)A^M=[a^1,…,a^M]T,
which contains the first M coefficients of the Fourier series representation
(61)f^N(t)=∑k=1∞‍a^ksin[(2k-1)ω0t],
can be computed in the form
(62)A^M=BM,NW^N,
and considering (57), we obtain
(63)A^M=EΦM+F,
with
(64)E=BM,NCN,M,F=BM,NDN.
For the sinusoidal signal (29), let us consider a given AC amplitude 0≤Aac≤Aac* and the system of switching indexes (41). If there exists a vector of switching ratios
(65)ΦM=[ϕ1,…,ϕM]T,ϕi∈(0,1),i=1,…,M,
that satisfies
(66)ΦM=E-1(A^M-F)
with
(67)A^M=[A^1,0,…,0]T,A^1=AacAac*,
then the conditions (C1) and (C2) are satisfied by the signal
(68)f(t)=Vdcf^(t),
where Vdc=Aac* 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
(69)J(d)=[j1(d)=3,j2(d)=7],
the matrices (58) and (59) take the values
(70)C8,2=-14[-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1],D8=14[-3-1-1-1-1-1-1-1],
and the matrix B2,8 is formed by the first two rows of the matrix B4,8 displayed in (28). Equation (66) produces the linear system
(71)ℒc={ϕ1=-1.0155A^1+0.9555,ϕ2=-1.5931A^1+1.5317.
To compute the regulation range of the system (71), we can consider the auxiliary system
(72)ℒc′={[A^1]1=-0.9847ϕ1+0.9409,[A^1]2=-0.6277ϕ2+0.9615,
obtained by isolating A^1 in the different equations of the system (71), and compute the values
(73)[A^1]min=max1≤i≤2(infϕi∈(0,1)[A^1]i),[A^1]max=min1≤i≤2(supϕi∈(0,1)[A^1]i).
Clearly, for any normalized amplitude A^1 satisfying
(74)[A^1]min<A^1<[A^1]max,
a suitable system of switching ratios ϕi∈[0,1] can be computed. For the current example, we obtain the particular values
(75)[A^1]min=0.3373,[A^1]max=0.9409,
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 signalf^(t) is carried out following the same rules (R1), (R2), and (R4) used in the conventional method, but in this case, the switching indexes ji(d) satisfy
(76)1≤j1(d)<⋯<jM(d)<N,
and the switching-down instants
(77)ti(d)=(ji(d)-ϕi)h,i=1,…,M
are followed by the switching-up instants
(78)ti(u)=(ji(d)+ϕi)h,i=1,…,M.
This new switching scheme is displayed in Figure 5, where the symmetrical arrangement of the switching instants ti(d) and ti(u) with respect to the interval endpoint tji(d) can be clearly appreciated. The coefficients (54) of the truncated Walsh series (49) have now the form
(79)w^n=-2N∑i=1M‍ϕi[wal1(4n-3,τ^ji(d))+wal1(4n-3,τ^ji(d)+1)]+1N∑j=1N‍wal1(4n-3,τ^j).
The vector of Walsh coefficients (56) admits the matrix expression (57), but now the elements of the matrix CN,M take the form
(80)cn,i=-2N[wal1(4n-3,τ^ji(d))+wal1(4n-3,τ^ji(d)+1)],
and the elements of the vector DN are
(81)dn=1N[∑j=1N‍wal1(4n-3,τ^j)].
The coefficients cn,i can be easily computed by considering the N×N matrix SN, with elements
(82)sn,k=wal1(4n-3,τ^k),τ^k∈(k-14N,k4N).
The nth row of SN 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
(83)cn,i=-2N(sn,ji(d)+sn,ji(d)+1),
and the ith column of CN,M can be obtained by adding the columns ji(d) and ji(d)+1 of SN, and multiplying by -2/N. The coefficients dn can also be expressed in terms of the elements sn,k as
(84)dn=1N(∑k=1N‍sn,k)
and could be computed by adding the columns of SN. However, it is not necessary to perform this computation because (81) always produces the vector
(85)DN=[1,0,…,0]T.
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 theadvanced 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, withM=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
(86)wal1(1,τ),wal1(5,τ),wal1(9,τ),…,wal1(29,τ)
produces the matrix of sign sequences
(87)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].
According to (83), the matrix C8,2 is
(88)C8,2=-14[-2-2-2-2-2-2-2-2-0-0-0-0-0-0-0-0]=-12[-1-1-1-1-1-1-1-1-0-0-0-0-0-0-0-0],
and from (84), the value of D8 is
(89)D8=18[80000000]=[10000000],
which coincides with the value indicated in (85). Finally, using (64) and (66) with the matrix B2,8 introduced in the previous subsection, and the matrices C8,2 and D8 given in (88) and (89), we obtain the linear system
(90)ℒa={ϕ1=-0.5877A^1+1.0583,ϕ2=-0.6933A^1+0.7071,
and from the associated auxiliary system
(91)ℒa′={[A^1]1=-1.7015ϕ1+1.8007,[A^1]2=-1.4424ϕ2+1.0200,
we get the modulation range
(92)[A^1]min=0.0992,[A^1]max=1.0200.
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 functionswal1(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 SN can be computed by using the sign sequences of the functions walT(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 walT(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
(93)dn=∑k=1N‍wal1(n-1,τ~k)N,τ~k∈(k-1N,kN)=∑k=1N‍∫(k-1)/Nk/N‍wal1(n-1,τ)wal1(0,τ)dτ=∫01‍wal1(n-1,τ)wal1(0,τ)dτ.
In consequence, d1=1, and dn=0 for n=2,…,N, as indicated in (85).Remark 6.
Offline computations are carried out to design the linear systemsℒc and ℒa. 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 Section3, PWM signals with M=2p switching cycles in the quarter-period interval can be conveniently designed to modulate the normalized fundamental-frequency amplitude A^1 and cancel the first M-1 nonfundamental odd harmonics. In the design procedure, the quarter-period interval is divided into N=2p+2 subintervals, and a suitable system of switching indexes
(94)J(d)=[j1(d),…,jM(d)]
is selected. Each system J(d) determines a PWM signal f^(t) that satisfies the design conditions for a certain normalized-amplitude interval
(95)[A^1]min<A^1<[A^1]max.
Obviously, the interval length
(96)Δ^=[A^1]max-[A^1]min
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 signalsf^(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:
(97)Ij=[(j-1)h,jh),j=1,…,32,h=T128.
The design of a specific PWM signal starts by selecting a system of switching indexes
(98)J(d)=[j1(d),…,j8(d)],
which determine the starting subintervals for the switching cycles. The number of different index systems J(d) is
(99)(328)≈1.05×107.
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 ofM, 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 ji(d) can be selected satisfying
(100)4i-3≤ji(d)<4i+1,i=1,…,8.
Under this constrained selection scheme, the number of different systems J(d) is reduced to a more tractable value
(101)48=65536.
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
(102)Δ^<20%,20%≤Δ^≤40%,Δ^>40%.
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 systemsJ(d) for conventional and advanced designs with M=8 switching cycles.
Method
Conventional
Advanced
Δ
^
<
20
%
2919
4700
20
%
≤
Δ
^
≤
40
%
1008
2896
Δ
^
>
40
%
11
3843
Total
3938
11439The optimal index system for the conventional design method is(103)Jc(d)=[2,6,9,14,20,22,27,30],
which produces the following system of linear equations to compute the switching ratios:
(104)ℒc={ϕ1=-0.1418A^1+1.0034,ϕ2=-0.3350A^1+0.9945,ϕ3=-0.5201A^1+1.0077,ϕ4=-0.6621A^1+0.9891,ϕ5=-0.8191A^1+1.0164,ϕ6=-0.8771A^1+0.9723,ϕ7=-1.0310A^1+1.0607,ϕ8=-0.7059A^1+0.7071.
The PWM signal and the harmonic distribution obtained for a DC amplitude Vdc=100 V, a fundamental frequency of 50 Hz, and the AC amplitude Aac=98 V are displayed in Figure 6. The corresponding switching instants can be computed by setting the normalized amplitude A^1=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 Aac=54 V are displayed in Figure 7.Figure 6
Optimal conventional design withM=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 withM=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(105)Ja(d)=[3,7,11,15,19,23,27,31],
with an associated linear system
(106)ℒa={ϕ1=-0.7935A^1+0.1434,ϕ2=-1.2594A^1+1.4115,ϕ3=-0.1541A^1+0.2644,ϕ4=-2.1312A^1+2.1001,ϕ5=-1.3492A^1+1.6192,ϕ6=-1.7046A^1+1.9318,ϕ7=-1.7874A^1+1.8833,ϕ8=-2.0562A^1+2.0576.
The PWM signals and the harmonic distributions obtained for the DC amplitude Vdc=100 V, a fundamental frequency of 50 Hz, and the AC amplitudes Aac=98 V and Aac=54 V are displayed in Figures 8 and 9, respectively. In this case, the switching instants are obtained from the linear system ℒa in (106), (77), and (78).Figure 8
Optimal advanced design withM=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 withM=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 ℒa. 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 withM=8 switching cycles.
Method
Conventional
Advanced
J
(
d
)
[
2,6
,
9,14,20,22,27,30
]
[
3,7
,
11,15,19,23,27,31
]
[
A
^
1
]
max
0.985
1.002
[
A
^
1
]
min
0.547
0.059
Δ
^
0.448
0.943Regarding 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 forM=8 can be extended to larger values M=2p by selecting the system of switching indexes
(107)JM(d)=[j1(d),…,jM(d)],ji(d)=4i-1,i=1,…M,
which, using a common computational notation, can be represented in the form
(108)JM(d)=[j1(d):4:jM(d)]=[3:4:4M-1],
where j1(d)=3 and jM(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 J32(d)=[3:4:127] are displayed in Figure 10 (for a fundamental frequency of 50 Hz and Vdc=Aac=100 V). This design strategy has been successfully applied to the values M=2p, 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 withM=32 switching cycles. PWM signal and harmonic distribution corresponding to the switching index system J32(d)=[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 Section3 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=2p, the number of terms in the truncated Walsh series is N=2p+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].



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