PWM methods carrier based

Carrier based methods are simplest for the realization, yet not always straightforward to understand. The switching state of the inverter leg is determined by comparison of the modulating signal providing information on the voltage reference, and carrier signal providing information of the switching period. As dual inverter stands between the simplest two-level and multilevel inverters that are more complex, the carrier-based modulation methods can be adopted both by extending two-level methods or applying true multilevel methods, such are level-shifted and phase shifted modulations.

3.4.1. Independent modulation

This modulation is a direct extension from single two-level method. Using (2.27) from Section 2.2, the approach as two three-phase inverters can be used to define total voltage reference v* divided in two equal halves for inverters H and L:

2

*/

* v

v_H = , v_L^* =−v^*/2 (45)

The chosen method for single inverters is the popular sinusoidal modulation with third harmonic injection, with modified references va^*, vb^*, vc^*:

due to its excellent performances for single inverter. However, the result is improper multilevel waveform shown in Fig. 16(a), showing large voltage excursion from maximal to minimal voltage levels. The vectors v_oH and v_oL overlap in at the beginning and the end of the switching period, as already seen in Fig. 10(c), thus forming the total zero vector v_O even for a relatively large modulating index value. For this reason, the space vector plot in Fig. 16(b) shows only

0 Fig. 16. Independent modulation m = 0.75 (a) output voltages and (b) space vector plot.

maximal vectors. Similarly, for the applied simplest proportional modulation the output would comprise all available vectors in the plot analogous as in Fig. 16(b) since there is no such distinct overlap between two switching patterns of Fig. 10(a). Both theoretical and experimental switching pattern and waveform are shown in Fig. 17.

Benefits of the method are simple modulation and large degree of freedom. The voltage references can be given independently regarding both amplitude and phase (within available dc voltage limits) [12]. This approach has been proposed for automotive applications [13], [14], setting one inverter to supply only active power whereas the other provides necessary reactive power for the drive. As a benefit, only one supply can be used since inverter providing reactive power does not need dc source.

3.4.2. Level-shifted modulation

Level-shifted modulation is a general principle applied to all multilevel converters in general, where for l-levels there are (l-1) carriers shifted by l/(l-1) of the Vdc, so each carrier occupies its own “trace”. Common modulation signals passes gradually to every carrier and in this way modulates corresponding part of the multilevel inverter, while all “lower” switches are on and

“higher” are off.

0

Fig. 18. Phase shifted modulation m = 0.75 (a) Modulation signals for the first phase and (b) waveform.

S1L Fig. 17. Independent modulation voltage v1 for in outer triangle ACE (a) switching pattern, (b) experimental results

with zoomed detail in the outer triangle ACE, (m = 0.75, fs = 2 kHz).

In practical implementation (e.g. DSP) it is more convenient to produce level-shifted modulating signals and use common carrier. The second approach described in Section 2.2. to achieve multilevel modulation is to use treatment as three single-phase choppers. However, in order to obtain proper multilevel waveform modulation signals need to be additionally adjusted.

With the reference to (2.23), modulating signal v1S for A- and D-leg switches will be determined, taking into account that leg voltage vAN (Fig. 2.10) is always positive

}

Therefore, for given voltage reference the obtained leg voltages need to be:

⎩⎨

The modulation can be achieved by using common triangular carrier and references 2

as illustrated in Fig. 18(a), taking in account that for the inverter L modulation logic has to be opposite to provide corresponding negative values. A detailed switching period for v^* inside OCD triangle is depicted in Fig. 18(b). Due to the symmetry, analogous conclusions will be valid for the remaining modulating signals v2S and v3S of B-E and C-F-leg switches respectively.

The resulting proper multilevel output voltage is depicted in Fig. 19(a), and the space vector plot in Fig. 19(b) shows all applied levels. Note that one of the inverters (H) applies all three NTVs (v_aH, v_bH, v_oH), while the other uses just two of them (v , _bL v ). Yields that period-_oL averaged vectors v_H and v cannot be collinear. In Chapter IV will be shown that the omission _L of the one of the active vector is actually stipulated by the demand of proper multilevel waveform. There are more solutions for this modulation, depending on the chosen modulating signals, carriers, symmetry of carriers for inverters H and L like in Fig. 20 and Fig. 21.

0 Fig. 19. Level-shifted modulation m = 0.75 (a) output voltages and (b) space vector plot.

3.4.3. Phase-shifted modulation

Phase-shifted modulation is based on the harmonic analysis of PWM signals given in more details in the appendix. The principle of the method is to phase shift carriers by 2π/(l-1), where l is the number of the voltage levels by each leg. Using harmonic result for single two-level inverter

and output voltage expressions from Section 2.2, it is possible to set phase shift in order to obtain output voltage free of some harmonics that existed in single inverter cases. For example phase shifting by 180° (phase opposition [15]) cancels sideband harmonics around even c. However, the harmonic analysis is behind the scope of this work, while the major concentration is on the proper multilevel waveform so this method will be not discussed further.

2 Fig. 20 Level-shifted modulation (a) modulator (b) switching pattern, m = 0.8.

0 0.005 0.01 0.015 0.02

-1.5

Fig. 21. Phase-shifted modulation (m = 0.75, fs = 5 kHz) with: (a) modulation signals and carrier, (b) zoomed switching period with the reference vector in the outer triangle ACE.