im
Journal of	JET v°iume 10 (2°1?) p.p. n-2?
Issue 4, December 2017 Type of article 1.01
Technology	www.fe.um.si/en/jet.html
A STRAIGHTFORWARD ANALYTICAL WAY OF EVALUATING THE SINGLE-PHASE INVERTER SPWM FREQUENCY SPECTRUM
ANALITIČEN POSTOPEK OCENITVE FREKVENČNEGA SPEKTRA SPWM IZHODNE NAPETOSTI ENOFAZNEGA RAZMERNIKA
Alenka Hren1R, Franc Mihalič2
Keywords: Fourier analysis, sinusoidal pulse-width modulation (SPWM), over-modulation phenomenon, single-phase inverter, total harmonic distortion (THD)
Abstract
For a DC-AC converter (inverter), a key element in renewable power supply systems, an output voltage of sinusoidal shape is required to assure a high-quality sustainable energy flow. Thus, through the modulation process, this property must be "incorporated" into the output voltage. This operation incurs some harmonic distortion into the inverter output voltage, which can have an undesired influence on the load. In the single- or three-phase systems (grid-connected, uninterrupted power supply systems (UPS) or motor drives), the high quality Total Harmonic Distortion (THD) factor must be considered; the voltage harmonic content also must be limited.
This paper provides a comprehensive spectrum analysis of a three-level output voltage in singlephase inverter. The output voltage is generated by triangular Sinusoidal Pulse-Width Modulation (SPWM) and, by using the Fourier analysis, Bessel functions and trigonometric equality, the high
Corresponding author: dr. Alenka Hren, Tel.: +386 2 220 7332, Mailing address: University of Maribor, Faculty of Electrical Engineering and Computer Science, Koroška cesta 46, 2000 Maribor, E-mail address: alenka.hren@um.si
2 dr. Franc Mihalič, University of Maribor, Faculty of Electrical Engineering and Computer Science, Koroška cesta 46, 2000 Maribor, E-mail address: franc.mihalic@um.si
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harmonic components are extracted in a straightforward analytical way. Finally, the overmodulation phenomenon is also considered, and the procedure is experimentally validated.
Povzetek
Za zagotavljanje visokokakovostnega trajnostnega pretoka energije iz obnovljivih virov, mora razmerniško vezje, ki je ključni sestavni element pretvorniških sistemov, na izhodu zagotavljat sinusno obliko napetosti. To lastnost oz. obliko "vgradimo" v izhodno napetost razsmernika z izbranim modulacijskih postopkom, ki pa ob osnovni harmonski kompenenti vnaša v izhodno napetost tudi višje harmonske komponente. Te negativno vplivajo na breme razsmerniškega vezja in njegov izkoristek delovanja. V enofaznem ali trifaznem sistem (omrežne povezave, sistemi neprekinjenega napajanja ali motorni pogoni), mora razmerniško vezje delovati tudi z ustreznim, dovolj majhnim, faktorjem popačitve (Total Harmonic Distortion - THD), ki pa je odvisen prav od vsebnosti višjih harmonskih komponent v izhodni napetosti.
V članku je opisan postopek celovite analize harmonskega spektra trinivojske izhodne napetosti enofaznega razsmernika, pri čemer izhodno trinivojsko napetost generiramo s pomočjo sinusne trikotne modulacije (Sinusoidal Pulse-Width Modulation - SPWM). Z uporabo Fourierjeve analize, Besselovih funkcijin trigonometričnih enakosti lahko posamezne višje harmonske komponente izračunamo analitično. Opisan je tudi način delovanja razmernika in postopek izračuna harmonskih komponent v področju nadmodulacije. Pravilnost postopka analitičnega izračuna je eksperimentalno verificirana.
1 INTRODUCTION
The efficiency and stable operation of switching mode power inverters are of crucial importance for the sustainable production of the renewable energy sources connected to the utility grid, [1], [2], or that are a part of the standalone multifunctional power systems, [3], and are both closely related to the used modulation strategy. The Pulse-Width Modulation (PWM) strategy has been in a focus of research for many decades, [4], and remains an active research topic, [5], [6], due to its widespread usage in many fields of applications. Its usage is the most popular in the control of switching mode power converters for which it continues to represent a state-of-the-art solution.
Although, PWM has been used for many years and is well-described in many textbooks, [7]-[10], the PWM algorithms for switching power converters have been the subject of intensive research, [11]-[15], and some initiations of the analytical approach have been established, especially in [11]. Some PWM research work has been dedicated to the analytical way to understand the ac-ac converter modulation strategies, [12], [16], [17]. The necessity of the analysis of the PWM strategies in single-phase inverters began with the exploitation of the back-up systems (UPS) and with taking advantage of acoustics equipment in home appliances, [18]-[23]. Single-phase inverters in back-up systems are designed to adapt to the changing needs of load and input voltage sources, and the filter function can be solved by using passive filters, [7]. To minimize the filters' weight and size, it is also important to know the harmonic components of the inverter output voltage. Finally, it is generally accepted that the performance of an inverter that operates with arbitrary switching strategy is closely related to the frequency spectrum of its output voltage, [11].
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A Straightforward Analytical way ofEvaluating the Single-phase Inverter SPWM Frequency Spectrum
PWM can be implemented in many different forms. Pulse frequency is the most important parameter related to the PWM method and can be either constant or variable. A constant frequency PWM signal is obtained by comparing the modulation function with the carrier signal that can be in a sawtooth or a triangular shape. The most commonly used PWM form for a single-phase inverter is a naturally sampled PWM with a triangular (double-edge) carrier signal and a sinusoid as a modulation function, known as SPWM, since this kind of PWM improves the harmonic content of the pulse train considerably, [24].
In many textbooks, the authors only describe the "modern" approach for frequency spectrum evaluation based on Fast Fourier Transformation (FFT), probably due to the comfortability of this sophisticated mathematical tool. Understanding the PWM process using FFT was explored as a short-cut due to its spread appearance in many computer software tools, such as Matlab, LabVIEW SPICE, EWB, Simplorer, among others. These algorithms are also available in some electronic measuring instruments.
This paper presents a step-by-step analytical approach to the exact evaluation of single-phase inverter frequency spectrum obtained by naturally sampled SPWM. The proposed analytical way of evaluating SPWM frequency spectrum gives a comprehensive and deep insight into the mechanism of the harmonic components generation as well as a better foundation for understanding or even designing the SPWM devices in inverters. The main goal is to follow the SPWM procedure exactly by using the Fourier analysis, Bessel functions, and trigonometric equality in order to extract the high harmonic components in an analytical way. The switching (existing) function introduced by Wood, [9], is used for a mathematical description of the modulation function. Additionally, the over-modulation phenomenon in a single-phase inverter and its analysis are considered and are presented in Section 3, where the obtained results were also experimentally verified in order to prove the procedure's correctness. Conclusions are summarized in Section 4.
2 SINGLE-PHASE FULL-BRIDGE INVERTER
When the high-efficiency, low-cost, and compact structure are of primary concern, the transformer-less inverter's topologies based on bridge configuration are the primary choice. Fig. 1 shows a single-phase full-bridge inverter circuit with a DC input voltage (vin) and AC output voltage (voet), the semiconductor switches' structure and the load structure. The inverter consists of two legs (half-bridges) with two semiconductor switches (IGBTs or MOSFETs and diode as indicated in Fig. 1), voltage sources indicated by (V/2) and current source (indicated by Load), representing the inverter output filter consisting of inductor L, capacitor C, and load resistance R.
The SPWM processes can generally be divided into two groups with respect to the inverter output voltage that can be either in two-level (+ Vd and -Vd) or three-level shape (with +Vd, 0 and -Vd). Since it is well known that the three-level output voltage has better spectrum properties, this kind of SPWM process will be considered in detail in this paper. Moreover, the first harmonic magnitude can be increased over Vd when over-modulation is applied, which means that the modulation index must exceed 1. With over-modulation, an increased magnitude of the first harmonic component is welcome in those situations in which the input voltage is decreased, but as a consequence of this phenomenon, [10], additional spectrum lines appear, which also increases the Total Harmonic Distortion (THD) of the output signal.
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Figure 1: Single-phase inverter structure; the semiconductor switch structure; the (current
source) load structure.
Figure 2: (e) Output volteges: vA0(t), vB0(t) eed vAB(t) with appropriate modulation functions eed (b) Spectrel lines for three-level output voltege.
2.1 Generation of Three-Level Output Voltage
The whole inverter shown in Fig. 1 is divided into two half-bridge structures (legs). By using the first leg (switches Su and S21) the voltage vA0(t) ("first leg" voltage) and by using the second one (switches S12 and S22) the voltage vB0(t) ("second leg" voltage) are generated at the inverter output, both with respect to the neutral point (shown in Fig. 1 and Fig. 2(a), respectively). If voltage vA0(t) precedes vB0(t) for an appropriate phase angle the inverter output voltage vAB(t) that equals the difference between voltages vA0(t) and vB0(t) will have the desired magnitude and desired three-level waveform, as indicated in Fig. 2(a). Voltages vA0(t) and vB0(t) are described as two switching events:
VA0(t) = dA(t)(Vd/2) + de(t)(-Vd/2),	(2.1)
vBo(t) = ds(t)(Vd/2) + db(t)(-Vd/2),	(2.2)
where the switching functions (see Fig. 3(a) and 3(b)) are:
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A Straightforward Analytical way ofEvaluating the Single-phase Inverter SPWM Frequency Spectrum
Figure 3: Thraa-laval switching functions generation (a,b) and, oetpet voltages on tha interval TS
(c): VAo(t), VBo(t) and VAB(t).
dA =
dB =
1, S11 = ON,	(1, S21 = ON,
0, S11 = OFF, a = [0, S21 = OFF,
1, S12 = ON, 0, S12 = OFF,
db =
1, S22 = ON, 0, S22 = OFF.
(2.3)
(2.4)
In order to avoid the short circuit between the battery	terminals P and N, the following conditions must be fulfilled:
dA(t) + da(t) = 1,	(2.5)
dB(t) + db(t) = 1.	(2.6)
Applying the above conditions in (2.1) and (2.2) follows to:
VAo(t) = (2dA(t) - 1)(Vd/2),	(2.7)
VBo(t) = (2dB(t) - 1)(Vd/2).	(2.8) Referring to Fig. 3(c) (up and in the middle), the average value of the voltages vA0(t) and vB0(t) over the interval [0,Ts] can be evaluated as:
VA0
= -I1vA0(Wt = (2Da(t)- 1)Vd /2,
' c
(2.9)
B0
=1 It6VB0(t)dt = (2Db(t)- 1)Vd /2,
(2.10)
where DA(t)=tonA/Ts and DB(t)=tonB/Ts represent the corresponding duty cycle functions. If Ts << T = 2n/u>o holds, the following approximation can be introduced:
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VA0
ev
(t) T = VoutA (t),	(2.11)
's
VB0
(t) T_ = VoutB(t).	(2.12)
Functions voutA[t) and voutB(t) represent the desired inverter output voltages for each half-bridge that can be expressed as:
V
VoutA (t) = + -cos(®ot),	(2.13)
VoutB (t) = - i2cos(®ot).	(2.14)
Now, the duty cycle functions Da(0 and Db(0 can be evaluated from (2.9) to (2.14), respectively:
1	1 V/2	1 1
Da (t) = - + -... ,cos(®ot) = - + - mi cos(®ot),	(2.15)
2	2 (id / 2) 22
1 1 V/2 1 1
Da (t) =----cos(®0t) =---m, cos(®ot),	(2.16)
A 2 2(id/2) o 2 2 1 ' 0 " ' 1
where m, =V/ id is modulation index.
An auxiliary triangular carrier signal vcarr(t) needs to be introduced in order to transform the duty cycle functions DA(t) and DB(t) into a switching function dA(t) and dB(t). Fig. 3(a) and 3(b) show the triangular carrier signal, and both switching functions signal, respectively. The switching functions dA(t) and dB(t) were obtained by a comparison of duty cycle functions DA(t) and DB(t) with vcarr(t) as follows:
d [1, Da (t) > Vcarr (t),	d = |1, Db (t) > V^ (t), [0, Da (t) < V
carr
(t),	B [0, Da (t) < v carr(t).
The above-described procedure enables the generation of the triggering pulses in electrical circuits for all the semiconductor switches in the inverter. When referring to Fig. 4, the comparators (comp) compare the duty cycle functions DA(t) and DB(t) with triangular carrier signal (vcarr) and the signals dA(t), da(t), dB(t) and db(t) are obtained according to (2.5), (2.6) and (2.17).
Figure 4: Moduletor block-scheme.
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A Straightforward Analytical way ofEvaluating the Single-phase Inverter SPWM Frequency Spectrum
The switching signals generated according to (2.17) can be considered to be periodic signals on the time interval [0,7s]. When they are provided to the inverter switches, the three-level voltage (as shown in Fig. 2(a) and Fig. 3, respectively) appears at the inverter output:
vAB(t) =
+Vd, dA (t) = 1, dB (t) = 0, Da (t) > Db (t),
0, dA (t) = 0, dB (t) = 0 v dA (t) = 1,dB(t) = 1,	(2.18)
-Vd, dA (t) = 0, dB (t) = 0,Da (t) < Db (t).
It is well known that any periodic signal of period Ts can be expanded into a trigonometric Fourier series form:
a m
dA (t) = -° + ^ (an cos(naTt) + bn sin(n®Tt)),	(2.19)
2 n=1
where wT is the frequency of the triangular carrier signal (wT = 2n/Ts), and the coefficients a0, an and bn form a set of real numbers associated uniquely with the function dA(t):
2 t0+Ts
ao = — J dA (t)dt,
Ts to 2 t0+Ts
an = — J dA (t)cos(n®Tt)dt,	(2.20)
Ts f0 2 t0+Ts
bn =— J dA (t)sin(na>Tt)dt. Ts t0
Each term an cos(nuiTt) + bn sin(nuiTt) defines one harmonic function that occurs at integer multiples of the triangular carrier signal frequency nuiT. According to the signal waveform of the pulse train shown in Fig. 3(a), the Fourier coefficients can now be evaluated as:
dA (t) =
1, t0 - MEl < t < Da (t)Ts
0 2 t0 + s , (2.21)
' elsewhere.
In order to simplify the coefficient's calculation, the initial time to = 0 is chosen, so the coefficient ao is:
+DA (t)Ts
2 2
°0 = — J 1dt = 2Da (t)	(2.22)
Ts -Da (t)Ts 2
Coefficients an are also calculated from (2.20):
+da (t)Ts
2 2 , > , 2 sin(n*DA (t)) a0 = — J 1cos(n®Tt)dt =----^	(2.23)
Ts da (t)Ts	*	n
2
and all coefficient bn are equal to 0. According to (2.19) the Fourier series of dA(t) is:
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, , 2 ® sin(exDA (t)) dA (t) = Da (t) + -X—-—	(2.24)
" e=1
and also for the switching function dB(t):
,, 2 ® sin(exDA (t)) dB (t) = Db (t) + -£ —-^	(2.25)
X e=1 e
2.2 The Output Voltage Spectrum Calculation
The output voltage vAB(t) can be constructed simply by subtracting outputs vA0(t) and vB0(t). In practice, when the load is connected between terminals A and B, the voltage difference vAB(t) appears on it. When (2.8) is subtracted from (2.7) it follows to:
Vab (t) = (dA (t) - dB (t))Dd	(2.26)
The switching functions dA(t) and dB(t) can be expanded by the Fourier series in (2.24) and (2.25), respectively, and after applying (2.26) the inverter output voltage vAB(t) can be expressed by using the Bessel function as described in [7], [8], [10]:
4Vd ® 1,
vab(t) = mVd cos(®0t) + —!L 2 -[
X e=1e
P7T
cos(—)J1(a)[cos((emT +mo)t) + cos((emT -mo)t)]
flTT
- cos(—)J3(a)[cos((emT + 3mo )t) + cos((emT - 3m o)t)]
flTT
cos(—)J5(a)[cos((emT + 5mo)t) + cos((emT - 5mo)t)]... ]
where a = emn/2. The structure of the spectral line's appearance is evident from (2.27). The SPWM signal vAB(t) has a fundamental component that appears at the frequency uio and, in addition to at the triangular carrier signal frequency 2ewT, also contains the sideband harmonics at frequencies ewT + kwo, e = 1,2,3,...k = ±1,±2,±3,...~. Since the value of cos(en/2) is zero for every odd e, the spectral lines only appear around even multiples of the carrier frequency fT = wT/(2n), which is indicated in Tables 1 and 2, respectively. Fig. 2(b) shows the spectrum lines for the single-phase inverter's three-level output voltage for the Vd = 330 V and mi = 1, line frequency fo = 50 Hz and fT= 2 kHz. From all the analyses above and the obtained results, the following conclusions can be made:
•	The spectrum lines appear only for every even multiplier of fr,
•	The triangular carrier signal frequency fr = 2 kHz is present in the half-bridge voltages (Vyio(t) and VBo(t) not considered separately), but the synthesized inverter's output voltage VAB(t) switching frequency is doubled, so the first higher harmonic component appears next to the 4 kHz, and
•	Filter components are needed at the inverter output to extract the first harmonic component at the fundamental frequency and reject the high switching frequency components of the output voltage vAB(t). The doubled switching frequency allows reduction of the size and weight of the filter components.
(2.27)
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A Straightforward Analytical way ofEvaluating the Single-phase Inverter SPWM Frequency Spectrum
Table 1: Spattral lines aroend tha satond seltipliar of triangle carrier signal frequency 2fT.
a2i	4Vd 1 2T 2T --cos(—)Ji(—s) T 2 2 2	-63.4 V
a23	4Vd 1 2T. , 2T ----cos(—)J3(—sI) T 2 2 2	74.3 V
a25	4Vd 1 ,2t. , 2T . --cos(—)J5(— s,) t 2 2 5 2	-11.6 V
Table 2: Spattral lines aroend tha satond seltipliar of triangle carrier signal frequency 4fT.
a41	4Vd 1 4T . ,4T cos(—)J1(—s,) T 4 2 1 2	-23.7 V
a43	4Vd 1 4T . M --7 cos^)J3^ s,) T 4 2 2	-3.2 V
a45	4Vd 1 4t . ,4T . --cos(—)J5(— sI) t 4 2 2	41.4 V
3 OVER-MODULATION PHENOMENON IN A SINGLE-PHASE INVERTER
The first harmonic magnitude for three-level output voltage (see (2.27)) is defined by af = sIVd and has a position at the angular frequency uio (or frequency fo). Obviously, the maximum magnitude of the first harmonic equals Vd due to the range of sI £ (0,1). The first harmonic magnitude can be increased over Vd when over-modulation is applied, which means that the modulation index must exceed sI £ (0,1). With over-modulation, an increased magnitude of the first harmonic component is welcome in some applications but, as a consequence of this phenomenon, the additional low-frequency spectrum lines appear.
(a)	(b)
Figure 5: Ovar-sodelation protadera (sI = 1.2): (a) Triangle tarriar signal and dety tytla fenttion DA(t), switthing fenttions dA(t) and dB(t) wavaforss. (b) Ovar-sodelatad dasirad half-
bridga voltages VA0os(t) and VB0os(t).
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The three-level output voltage signal at the inverter output and over-modulation's influence on the frequency spectrum will be considered in the following subsections. Over-modulation appears when the duty cycle function DA(t) exceeds the magnitude of the high-frequency triangular carrier signal (mI > 1). Fig. 5(a) shows a relationship between the triangular carrier signal and duty cycle function DA(t) and its influences on the switching functions dA(t) and dB(t) when a 20% over-modulation is applied, respectively.
3.1 The Duty Cycle Function Evaluation
Duty cycle functions can be evaluated in over-modulation as follows from (2.15) and (2.16) by applying vA0om(t) and VB0om(t) instead of vAo(t) exceeds VBo(t):
„ _ tA 11 y/2
DA (t) = ^ = - +	VAOom
Ts 2 2Vd/2
(t)
Db (t) = -B =
tB 1 1 V/2
vB0om
(t)
Ts 2 2Vd/2
where of vA0om(t) and vB0om(t) are shown in Fig. 5(b) and are defined as follows:
V
and
vA0om
(t) =
2
-sinwot,
.Vl 2 '
0 < wot < ß, ß< wjt < (x —ß),
V
sin wot, (x — ß) < mot < (x + ß)
.Vi 2
(x + ß) < Wot < {2k —ß)
V
—sin wot, (2x — ß) < wot < 2x,
vB0om(t) =
V
—sinwot,
2o
Vd
0 < wot < ß, ß< Wot < (x —ß),
(3.1)
(3.2)
(3.3)
--sinwot, (x —ß) < wot < (x + ß)	(3.4)
,	(x + ß) < wot < (2x — ß)
V
--sinwot, (2x —ß) < wot < 2x.
The voltages described by (3.3) and (3.4) can be expressed using Fourier series. The functions are odd, and due to this, the coefficients cn = 0, so the voltages vA0om(t) and vB0om(t) are only
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A Straightforward Analytical way ofEvaluating the Single-phase Inverter SPWM Frequency Spectrum
expressed by coefficients bn, yielding:
œ
vA0om (t) = Zbk sin(k®0i),	(3.5)
k
œ
vB0om (t) = "Zbk sin(k®ot),	(3.6)
k
where k = 1,3,5,... and the Fourier coefficients bk can be evaluated from (3.3) by taking the symmetry of the signal during the half of the period:
2 T/2
bk = t J [/KO " f(-®ot)]sin(k®ot)dt,	(3.7)
T 0
where T = 2n/u>o, so it follows:
b = 2Vd
bk =-
K
sin[(k -1)^] sin[(k +1)^]
(k -1) (k +1) where 6 is computed as follows from Fig. 5(b):
1
—cos(kfi)dt k
(3.8)
P = arcsinl = arcsinl — I	(3.9)
I V J { si
Substituting (3.5) and (3.6) into (3.1) and (3.2), respectively, the duty cycle functions become:
1 1 ® Da (t) = - + —2 bk sin(k®0t),
2 Vd k
d k	(3.10)
1 1 ® Db (t) = --—2 bk sin(k®0i), 2 Vd k
3.2 The Over-Modulated Frequency Spectrum Calculation
The over-modulated output voltage can be evaluated from (2.26), (3.3) and (3.4):
VABos (t) = VaOos (t) - VbOos (t) = (¿A (t) - ^ (t))Vd	(3.11)
where the switching functions dA(t) and dB(t) can be evaluated by Fourier series as in (2.24) and (2.25), respectively and when combined with (3.10), the line to line voltage vABos(t) can be expressed as:
Vabos (t) = 2^|bk cos(k^t) J + HFSC,	(3.12)
LFSC
where LFSC means "Low-Frequency Spectral Components" and HFSC "High-Frequency Spectral Components", which can be calculated as follows:
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HFSC = -
4V
f
£ 1 en . 2 — cos—sin e=1e 2
f
Vn\ 2bk cosM
Vd I k
sin emTt
(3.13)
Eq. (3.12) consists of two parts: The first describes the low-frequency spectrum lines (next to the fundamental frequency wo), and the second describes the position and magnitudes of the high-frequency spectrum lines (next to the multipliers of triangle frequency wT). The overmodulation phenomenon is used in order to increase the first harmonic magnitude over the supply voltage when VAB1 > Vd.
According to (3.8) and (3.12), the output voltage's low frequency harmonic components can be evaluated as follows:
VA
ABk
= 2bk =-
4Vd
mI
sin[(k -1)P] sin[(k +1)P]
(k -1)
(k +1)
-cos(kp)
(3.14)
for the first, third, fifth and all odd spectral components, the (3.14) can be rewritten as (where in case of the first harmonic the division by zero can be avoided by replacing the function sinx = x for mi >>, or p <<):
VAB1 ='
4Vd
P-
sin(2P)
cos(P)
(3.15)
VAB3 = "
4V,
4Vd
mI
sin(2P) sin(4P)
sin(4P) sin(6P)
-cos(3P)
-—cos(5P)
(3.16) (3.16)
Figure 6: (e) Over-moduleted output voltege vABom(t) eed its first hermoeic compoeeet vAB1(t). (b) The low eed high hermoeic compoeeets spectrum liees.
K
Fig. 6(a) shows the three-level output voltage vAB(t) for Vd = 350 V, mI = 1.2, line frequency fo = 50 Hz and fT= 2 kHz, while its spectrum lines are presented in Fig. 6. From all the analyses above and the obtained results for spectrum evaluation the following conclusions can be made:
• The HFSC spectrum lines calculated by (3.13) appear only for every even multiplier of
fT,
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A Straightforward Analytical way ofEvaluating the Single-phase Inverter SPWM Frequency Spectrum
• The LFSC spectrum lines calculated by (3.12) are zoom-out in Fig. 6, while the calculated values for the magnitude as well as the normalized values VABkn for the first, third and fifth harmonic components are given in Table 3 for the case of 20% overmodulation. The magnitude of the first harmonic component VaS1 that exceeds the DC-voltage Vd by 10.4% was calculated by (3.15).
Table 3: Spattral lines aroend tha satond seltipliar of triangle carrier signal frequency 4fT.
VAB1	4Vd K	sI _ 2		_ sin(2B) 1 B--— + cos(B) 2				386.6 V	VABln	1.1046
VAB3	_ 4VL K		"m, \sin(2B) sin(4B)], 1_„(3B)" —---+— cos(3B) _ 2 L 2 4 J 3 _					-25.1 V	VAB3n	0.0717
VAB5	4Vd K	mL _ 2		rsin(4B) _ sin(6B) 1+ 1cos(5B)l _ 4 6 J 5 _				-12.9 V	VAB5n	0.0369
Fig. 7(a) shows the calculated first, second and third harmonic components versus modulation index changed from 0 to 2. When sI exceeds 1, the magnitudes of harmonic components start to increase as follows from the presented over-modulation analysis.
(a)	(b)
Figure 7: (a) First, third and fifth harsonit tosponants varses sodelation index sI E(0,2). (b) THD varses sodelation index sI E(1,2).
To validate the presented procedure's correctness, a single-phase low-voltage (Vd = 20 V) inverter experimental set up based on the DRV8870DDAR integrated circuit was built in the laboratory. The described SPWM strategy was implemented using a Texas Instruments TMDSCNCD28335 control card that can be programmed in MATLAB SIMULINK and is ideal to use for initial evaluation and system prototyping. Output voltage over ohmic load was measured with a RIGOL DS2102A Digital storage oscilloscope. Fig. 8(a) presents the measured output voltage when the inverter operates in over-modulation with sI = 1.2 while Fig. 8(b) shows its normalized frequency spectrum calculated by FFT in MATLAB. The obtained results show clearly that the normalized values for the first, third, and fifth harmonic components
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match almost perfectly the predicted values calculated with the described procedure (see Table
3).
(a)	(b)
Figure 8: (a) Inverter output voltage at mI = 1.2 and (b) Its frequency spectrum.
3.3 The Upper Limit Calculation
The upper limit is obtained when the modulation index mI goes to infinity and, according to Fig. 5(b), the angle p goes to 0. The upper limit can be evaluated as follows:
"	_ 4Vd
VABk,max - lim -
mI
sin[(k - 1)ß] sin[(k + 1)ß]
(k -1)
(k +1)
-cos(kß)
(3.18)
V,
and after calculation through a limit process:
T k
It is evident from (3.19) that the upper limit for the first (fundamental, where k = 1) voltage harmonic component is:
ABk ,max
(3.19)
V
4V
ABk ,max
d
(3.20)
and is indicated as the upper limit line (i.e. at the square-wave output signal) in Fig. 7(a). The over-modulation operation of the inverter also has significant influence on the THD factor that is defined as:
THD -
Jn-
V/2
2 ABk
V
(3.21)
AB1
THD factor dependence on modulation index mI is presented in Fig. 7(b). The maximum permissible THD factor is defined for different load types, so it is possible to set the necessary modulation index mI which defines the first harmonic component magnitude VAbi and, consequently from (3.15), the appropriate DC voltage Vd can be determined. In some special cases in which the available DC voltage supply Vd does not meet the requirements regarding the
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magnitude of the output voltage V/abi (Vd < Vabi ^ V4B1max), this over-modulation property allows us to omit the DC-DC boost converter in the power supply system.
4 CONCLUSION
In this paper, a straightforward, step-by-step SPWM frequency spectrum analysis has been presented for a single-phase inverter as an indispensable part of renewable energy source systems. The precise, analytical approach to SPWM signal analysis is rather unpopular among many engineers due to the relatively long and complicated procedure but, to understand these processes, it is necessary to take advantage of it. The additional reason that the analytical approach is not widely used in practice is due to the widespread usage of the MATLAB Software package. This program enables FFT numerical analysis of the PWM processes but, by using FFT, engineers do not gain an in-depth insight into the connections between the different quantities appearing in the formulas.
For three-level inverter output voltage, and over-modulation, principles of modulation algorithms have been developed here in a traditional analytical way. The electronic circuit of an SPWM modulator can be realized from the described algorithms by using new microcomputer technology and in the paper presented knowledge is also a good base for further investigation into SPWM switching strategies. The over-modulation phenomenon can help engineers to speculate using the inverter's parameters; for example, it can decrease the DC input voltage Vd and the output voltage will still have the necessary magnitude of the first harmonic component Vb . The obtained results were experimentally verified in order to prove the procedure's correctness.
This analysis could be easily extended to three-phase inverters intended for electrical motor drive applications or grid-connected. And finally, the results of this analysis are also appropriate for further investigation of SPWM processes with respect to higher harmonic components' influence on losses for the different types of loads, as well as for filter design in the single-phase inverters used in UPS or grid-connected back-up systems.
References
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[5]	M. Odavic, M. Summer, P. Zanchetta, and J.C. Clare: A theoretical eeelysis of the harmonic content of PWM waveforms for multiplefrequency modulators, IEEE Trees. Power Electronics, vol. 25, no. 1, pp. 131-141, Jan. 2010
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Nomenclature
(Symbbls)	(Symbol meaninn)
bk	Fourier series coefficieets
P	cross-section aagle in the over-moUulatioo
dA(t)	switching funstion
DDt)	duty-cycle funstion
J (a)	BBssel funstions
mi	moUulation indux
m	triaagular carrier signnl frequency
Ok	fundumentalourput voeage ^queeny
FFT	fast Fourier transfosmation
HFSC	eige-freuuency sppctralcomponents
LTSC	low-freuuency sppctralcompondnts
CCWM	sinnrs)¡Uul fapuse-v^iuthi moUulation
THD	tc3tali^ermonic U¡stostion
UCC	uuinterruetiOle ppwer suuply
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