THE COMPARISON REGARDING THD BETWEEN DIFFERENT MODULATION STRATEGIES IN SINGLE-PHASE FLYING CAPACITOR MULTILEVEL PWM INVERTER ADRIAN ŞCHIOP1 Keywords: Modulation strategies, Flying capacitor, Single-phase multilevel PWM inverter, Simulink. In this paper the modulation strategies for single-phase multilevel PWM inverter with flying capacitors are analyzed. The analyzed strategies are: phase disposition PWM method, phase-shifted PWM method, the saw-tooth rotation PWM method and the carrier redistribution PWM method. The results are obtained through simulations by Simulink TM. 1. INTRODUCTION Multilevel inverters are generally classified as: diode-clamping inverters, flying capacitor inverters and cascade inverters [1]. Several modulation techniques for three phase flying capacitor multilevel PWM inverter are presented in the literature [2 7]. Some of these techniques obtain natural voltage balancing for the flying capacitor like: phase-shifted PWM method, saw-tooth rotation PWM method and carrier redistribution PWM method while for phase disposition PWM method the redundancy states are chosen for voltage balancing. In this paper a comparison between the four methods is made for single phase flying capacitor multilevel PWM inverter regarding Total Harmonic Distortion factor (THD). The influence on the THD of the disposition of the carrier signals is also investigated. 2. DESCRIPTION OF FLYING CAPACITOR MULTILEVEL INVERTER For the three level flying capacitor inverter the switches S a1 and S a1, S a2 and S a2, S b1 and S b1, S b2 and S b2, from Fig. 1 are driven complementary. An inverter leg may be represented as in Fig. 2. We will say that transistors S a2 and S a2 form switching cell a2 and transistors S a1 and S a1 form switching cell a1. Let us define 1University of Oradea, 1 Universităţii, 410087 Oradea, E-mail: aschiop@uoradea.ro Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 55, 3, p. 330 340, Bucarest, 2010
2 Modulation strategies in single-phase flying capacitor inverter 331 the connection functions y a1 and y a2 for switching cell. For example, for a2 cell from Fig. 2, y a2 = 1 when S a2 is in ON state and y a2 = 0 when S a2 is in OFF state. Depending on adjacent switching states of the capacitor, the current through, for example C a1, is i a when S a2 and S a1 are in ON state, i a when S a2 and S a1 are in ON state, or zero when S a1 and S a2 are in ON state, or when S a1 and S a2 are in ON state. Consequently, adequate driving of the adjacent transistors can modulate the current through C a1. In [8] it was shown that natural voltage balancing occurred when y a1 and y a2 have equal durations for each time period T P. Obtaining the equal durations of all connection functions of commutation cells for an inverter leg is possible by using two carrier signals with a T P /2 phase shift between them, or by using carrier redistribution PWM method or the saw tooth rotation PWM method [2 7]. S a2 S b2 a2 cell a1 cell U e C 2 C a1 S a1 S a1 C b1 S b1 S b1 U e 0 C 2 S a2 C a1 S a1 i a A 0 S a2 S b2 S a2 S a1 A B Fig. 1 Scheme of the three level flying capacitor inverter. Fig. 2 Modelling of inverter leg. In [9] the expression of phase voltage for flying capacitor n level inverters depending on connection functions is derived. Customizing this expression for the three level inverter leg with S a1, S a1, S a2 and S a2 transistors, this will be: u A0 = y a2 u C2 + u Ca1 (y a2 + y a1 )[(1 y a2 )+(1 y a1 )]( 1) (1 ya1), (1) where u C2 and u Ca1 are the voltages across C 2 and C a1 capacitors. The interphase voltage u AB will be: u AB = u A0 u B0 ; u AB = y a2 u C2 + u Ca1 (y a2 + y a1 )[(1 y a2 )+(1 y a1 )]( 1) (1 ya1) y b2 u C2 u Cb1 (y b2 + y b1 )[(1 y b2 )+(1 y b1 )]( 1) (1 yb1) (2), where u Cb1 is the voltages across C b1 capacitor; y b1 and y b2 are connection functions for the leg with S b1, S b1, S b2 and S b2 transistors. The Simulink TM model for three level capacitor clamped inverter obtained by (2) is presented in Fig. 3. Simulink TM model of flying capacitor multilevel inverter employs the following blocks of the Simulink TM library: Source, Sum, Relay, Constant, Gain, Product and Math Function. The Triangle block is a mask
332 Adrian Şchiop 3 subsystem that allows set up of the amplitude, frequency, phase and offset of carrier signals. The Math Function blocks realizes the ( 1) (1 ya1) and ( 1) (1 yb1) operations. The Gain blocks multiplies the values of connection functions with values of voltages on flying capacitors. In Fig. 3, at the output of the Relay blocks, the connections functions y a1, y a2, y b1 and y b2 for each switching cell are obtained. The time length of connection functions depends on carrier signal given by Triangle blocks and on modulator signal given by Sine Wave block. For PWM modulation strategies the connection functions yield the value 1 when modulating signal is greater than carrier signal. Fig. 3 Simulink model for the three level flying capacitor inverter. 3. MODULATION METHODS In the PWM method, the connection functions are obtained by comparison of the carrier signals with the modulating signals. The case when the carrier signals are overlapping and are covering a continuous range (mode A) and the case when the carrier signals are phase shifted between them (mode B) are analyzed. When the carrier signals are overlapped there are two operation sub-modes: A1 when using 2(n 1) triangular carrier signals overlapping, (n 1) carrier signals for one leg and the other (n 1) for the other leg, Fig. 4. In A2 mode are using (n 1) carrier signals for both legs and two modulating signal phase shifted by T/2, where T is the period of output voltage. The carrier signals can be in phase like in Fig. 5, or phase opposition like in Fig. 6. For mode B there are three sub-modes: B1 when carrier signals are triangular and phase shifted between them like Fig. 7, B2 saw-
4 Modulation strategies in single-phase flying capacitor inverter 333 tooth rotation PWM method that uses, in fact, saw-tooth carriers signal, Fig. 8 and B3 carrier redistribution PWM method that is using as carrier signals a sum of triangular and trapezoidal signals (Fig. 9). Fig. 4 A1 mode modulation in single phase three level inverter. Fig. 5 A2 mode modulation with in phase carrier signals for three level inverter. Fig. 6 A2 mode modulation with phase opposition carrier signals for three level inverter. vt1, vt2, va * vt1, vt2, va * v * a v t1 v t2 * A B C D E F v a A B C D E v t1 v t2 y a1 y a2 y a1 y a2 1 0 1 0 Fig. 7 B1 mode modulation for three level inverter. Fig. 8 B2 mode modulation for three level inverter.
334 Adrian Şchiop 5 vt1, vt2, v * a 1 0 v t1 v * a v t2-1 0 Tp/2 T P y a1 y a2 0 T p/2 T P 0 T p/2 T P Fig. 9 Carrier signal for a switching cell when carrier redistribution PWM method is used for three level inverter. 4. SIMULATION RESULTS The methods previously presented are analyzed by simulation in Simulink TM environment regarding the harmonic content and total harmonic distortion factor (THD). The analyses are performed for single phase three level and four level flying capacitor PWM inverter. In the next figures the harmonics of line voltage are presented for different amplitude and frequency modulation indexes m a and m f. m a = A m /(n 1)A P ; m f = f P /f m, (3) were A m and A p are the peak to peak values of modulator and carrier signals. f P and f m are the frequencies of carrier and modulator signals. It must be noted that the analysis refers to the voltage at idling for 50 Hz. The simulation results are presented for two cases. In the first case m f is 1,250 Hz and m a is 0.9. In the second case m f is chosen as the value from which the harmonics begin to be grouped around the multiply of modulation frequency. In both cases the THD is calculated for the first 200 harmonics. For low values of m f, for graphics clarity, only the first 100 harmonics are displayed. As one can see from Fig. 10 and Fig. 11, for the A1 mode, if m f > 9, the harmonics are grouped around the frequencies multiple of m f. f ha1 = km f f m ; m f > 9, k N*. (4) If m f < 9 the harmonics are not grouped around the m f and the fundamental amplitude of the interphase voltage is dependent on m f. m a =0,6; m f =9 ma=0,6; mf=9
6 Modulation strategies in single-phase flying capacitor inverter 335 Fig. 10 A1 mode for three level inverter. Fig. 11 A1 mode for four level inverter. For the A2 mode, when the carrier signals are in phase, Fig. 12 and Fig. 13, if m f > 6, the harmonics are grouped around twice of frequencies of mode A1. f ha2 = 2km f f m ; m f > 6, k N*. (5) If m f < 6 the harmonics are not grouped around the 2m f and the fundamental amplitude of the interphase voltage is dependent on m f. ma=0.6; mf=6 ma=0.6; mf=6 Fig. 12 A2 mode for three level inverter. Fig. 13 A2 mode for four level inverter. When the carrier signals are in phase opposition (A2PO), Fig. 14 and Fig. 15, if m f > 9, the harmonics are grouped like in A1 mode. In this case the number of line voltage levels decreases in comparison to the case when carrier signals are in phase. If m f < 9 the harmonics are not grouped around the m f and the fundamental amplitude of the interphase voltage is dependent on m f. f ha2po = km f f m ; m f > 9, k N*. (6)
336 Adrian Şchiop 7 ma=0.6; mf=9 ma=0.6; mf=9 Fig. 14 A2 mode phase opposition for three level inverter. Fig. 15 A2 mode phase opposition for four level inverter. For B1 mode, Fig. 16 and Fig. 17, when the inverter has an odd number of levels, the number of levels of the output voltage decreases in comparison to the case when the group of carrier signals for inverter legs are 90 o phase shifted. For B1 mode, if m f > 4 for n odd, or m f > 2 for n even, the harmonics of interphase voltage appear as side band of frequency f hb1 : f hb1 = km f (n 1)f m ; m f > 4, k N*; for n odd, (7) f hb1 = 2km f (n 1)f m ; m f > 2, k N*; for n even. For B1 mode, when the carrier groups for the two legs are 90 o phase shifted (B1 90 o PS), Fig. 18 and Fig. 19, if m f > 4 for n odd, or m f > 6 for n even, the harmonics of voltage interphase appear as side bands of frequency f hb1 90PS : f hb1 90PS = km f (n 1)f m ; m f >6, k N* with k 2(2i+1), i N; for n even, f hb1 90PS = 2km f (n 1)f m ; m f > 4, k N*; for n odd, where n is the number of inverter levels. (8)
8 Modulation strategies in single-phase flying capacitor inverter 337 ma=0.6; mf=4 ma=0.6; mf=2 Fig. 16 B1 mode for three level inverter. Fig. 17 B1 mode for four level inverter. ma=0.6; mf=4 ma=0.6; mf=6 Fig. 18 B1 mode with 90 o phase shifted between carrier groups for three level inverter. Fig. 19 B1 mode with 90 o phase shifted between carrier groups for four level inverter.
338 Adrian Şchiop 9 ma=0.6; mf=9 ma=0.6; mf=9 Fig. 20 B2 mode for three level inverter. Fig. 21 B2 mode for four level inverter. In B2 mode Fig. 20 and Fig. 21, if m f > 9, the harmonics appears as side bands of frequency f hb2 : f hb2 = km f (n 1)f m ; m f > 9, k N*. (9) If m f < 9 the harmonics are not grouped around the m f and the fundamental amplitude of the interphase voltage is dependent on m f. For B3 mode, Fig. 22 and Fig. 23, if m f > 4 for n odd, or m f > 2 for n even, the harmonics of voltage interphase appear as side bands of frequency f hb3 : f hb3 = 2km f (n 1)f m ; m f > 4, k N*; for n odd, f hb3 = km f (n 1)f m ; m f > 2, k N*; for n even. (10) Fig. 22 B3 mode for three level inverter. Fig. 23 B3 mode for four level inverter. The THD for analyzed cases are presented in Table 1 and Table 2. From Table 1 we can observe that B3 mode leads to obtaining of lowest values of THD.
10 Modulation strategies in single-phase flying capacitor inverter 339 Table 1 The THD values of line voltage for m f = 25 and m a = 0.9 A1 A2 A2 PO B1 B1 90 o PS B2 B3 3 level 0.3225 0.3106 0.6196 0.5915 0.2802 0.3106 0.2802 4 level 0.2137 0.2068 0.3852 0.1805 0.2412 0.1998 0.1732 Table 2 The THD values of line voltage for m a = 0.6 and those values of m f from which the harmonics begin to be grouped around the multiply of modulation frequency A1 A2 A2PO B1 B1 90 o PS B2 B3 m f =9 m f =6 m f =9 m f =4 m f =4 m f =9 m f =4 3 level 0.3902 0.4176 1.0378 0.9942 0.4014 0.4268 0.4014 m f =9 m f =6 m f =9 m f =2 m f =6 m f =9 m f =2 4 level 0.3125 0.3249 0.5614 0.3204 0.4172 0.3193 0.3025 By increasing the number of levels in the inverter the THD becomes lower, for the same m f, but at the expense of increasing complexity in the control circuit. However, as can be seen from Table 1, even for the three level inverter the modulation techniques B1 90 o PS and B3 lead to a THD below 0.3, while the harmonics are clustered around the four multiples of m f, which facilitates their filtering. From Table 2 we can see that m f values from which the harmonics begin grouped around the multiply of modulation frequency is dependent of modulation techniques and for B1, B1 90 o PS and B3 modes it depends on the number of the inverter levels 5. CONCLUSIONS Several modulation techniques for flying capacitor multilevel PWM inverter are presented in the literature. In this paper a comparison between these modulation techniques for single phase flying capacitors three/four level PWM inverter are analyzed regarding THD and harmonics distribution. The characteristics of discussed PWM methods, obtained by simulations using developed model presented in [9], are as follows: A1 and A2 with phase opposition modes give the same harmonic distribution. The harmonics are grouped around the frequencies multiple of frequency modulation index. In A2 mode the harmonics are grouped around the twice frequencies multiple of frequency modulation index. In B1, B2 and B3 modes the frequencies around which the side bands are grouped depend on the number of inverter levels. B3 mode allows one to obtain the lowest values of THD.
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