3. GENERAL SPACE VECTOR PWM METHOD
3.3 Space vector PWM for VSI
Space vector modulation method for VSI used in the simulations of this thesis is presented in this section. It should be noted that the space vector PWM method provided in this section concentrates only in a linear modulation region. An overmodulation region is ne-glected in the space vector PWM implementation in this thesis, however it is discussed in the next chapter in Section 4.1
VSI has two alternatives where the output of an inverter phase can be connected to: pos-itive or negative DC bus voltage. This approach generates eight different switching vec-tors. Table 1 presents all the possible switching vectors and corresponding switch states for each switches. The subscripts in the switching vector names denotes whether the phase outputs u, v and w are connected to the positive or negative DC bus voltages. The positive connected phase output is denoted as ‘p’ when the output voltage is half of the DC bus voltage, i.e. Udc/2. The negative connected phase output is marked ‘n’ and the phase out-put voltage is - Udc/2 in this occasion. Individual IGBT switch states are presented as ‘1’
and ‘0’. ‘1’ means that the switch is in on-state and conducting, while ‘0’ designates that the switch is in off-state and not conducting. Naming of the switches refers to Figure 1.
Table 1. Switching vectors and switch states for 2L VSI
S1u S2u S1v S2v S1w S2w
svpnn 1 0 0 1 0 1
svppn 1 0 1 0 0 1
svnpn 0 1 1 0 0 1
svnpp 0 1 1 0 1 0
svnnp 0 1 0 1 1 0
svpnp 1 0 0 1 1 0
svppp 1 0 1 0 1 0
svnnn 0 1 0 1 0 1
Switches S1 and S2 of each phase leg work as a pair: Switch pairs are not allowed to have the same state in any occasion to avoid short circuiting in DC bus. More about blanking time is discussed in the next chapter in Section 4.2.
The available switching vectors are obtained by replacing the phase output values uu, uv
and uw in the equation 3.1 with corresponding voltages Udc/2 or -Udc/2. In order to clarify the presentation and space vector scheme, the switching vectors are presented in a com-plex coordinates in Figure 8.
svpnn Im
Re svppn
svnpn
svnpp
svpnp svnnp
svppp
svnnn
Figure 8. Switching vectors for VSI in a complex coordinates.
The switching vectors can be divided in two categories: active vectors and zero vectors.
The active vectors are vectors that enables current paths for inverter phase outputs. There are six active vectors and two zero vectors in Figure 8. The lengths of the active vectors are 2/3Udc for all the VSI’s active vectors. Zero vectors are formed by clamping all the phase outputs to either positive or negative DC bus. Therefore, they don’t provide current paths and the resulting switching vector lengths are zero.
3.3.1 Switching times and sequence for VSI
As said, the reference voltage vector is formed by averaging the discrete switching vec-tors. In order to do this, we need to calculate the switching times. Switching time calcu-lations along with modulation sequence generation are explained in this section.
Switching vector presentation can be divided into six sectors (Figure 9 (a)). All the sectors are symmetric. Thus, the reference switching vector formation presented in the Figure 9 (b) is similar for all the sectors. Reference voltage vector is simply formed by using two nearest active switching vectors in VSI’s space vector PWM scheme.
Im
Re Sector 1
Sector 2 Sector 3
Sector 4
Sector 5
Sector 6
sv i
Re sv i+1
svref
Im
θref
svsvnnnppp
svppp
t1·sv i
t2·sv i+1
(a) (b)
Figure 9. Switching vector presentation for VSI divided into six sectors (a). For-mation of the reference voltage vector is presented in (b).
Switching times for reference switching vector located in any of the sectors can be solved using trigonometry by the following equations [15]
𝑡1 = √3 𝑇𝑠|𝑠𝑣𝑈𝑟𝑒𝑓|
𝑑𝑐 sin (𝜃𝑖+1− 𝜃𝑟𝑒𝑓) and (3.3) 𝑡2 = √3 𝑇𝑠
|𝑠𝑣𝑟𝑒𝑓|
𝑈𝑑𝑐 sin (𝜃𝑟𝑒𝑓− 𝜃𝑖), (3.4)
where t1 is the switching time for svi, t2 is the switching time for svi+1, Ts is the cycle time of modulation period, svref is the reference switching vector, Udc is the DC bus voltage, θi+1 is the angle of the switching vector svi+1 respective to the real axis, θref is the angle of the reference switching vector respective to the real axis and θi is the angle of switching vector svi respective to real axis (Figure 9 (b)).
Application time of zero vectors is easy to compute after t1 and t2 have been calculated as
𝑡0 = 𝑇𝑠− 𝑡1− 𝑡2, (3.5)
where t0 is the switching time for zero vectors. Now we how to select the active vectors and calculate the switching times. An example of a whole modulation period is given in Figure 10.
t
Figure 10. Upper IGBT switches’ control signals and switching vectors applied during one modulation period. The reference switching vector is located in the
Sector 1 in this example.
In this example, the reference vector is located in the Sector 1. The modulation period is divided into eight segments to split the applying time of each switching vector evenly.
The modulation period always starts by applying the zero vector svnnn in a space vector scheme for VSI. The svnnn is applied for the time of t0/2. The next applied switching vector is selected so that only one switch state is changed at a time. In this example, it means svpnn is followed by svnnn. It is applied for the time of t1/2. The next active vector is svppn
with the applying time of t2/2. After the first and the second active switching vectors are applied, the other zero switching vector svppp is followed with the switching time of t0/4.
After these steps, the sequence is repeated in a reverse order. The resulting switching vector can be stated as
𝑠𝑣𝑟𝑒𝑓= 𝑡𝑇1
𝑠𝑠𝑣 𝑖 +𝑡𝑇2
𝑠𝑠𝑣 𝑖+1+𝑡0𝑇/2
𝑠 𝑠𝑣𝑝𝑝𝑝+𝑡0𝑇/2
𝑠 𝑠𝑣𝑛𝑛𝑛. (3.6) As said, the modulation period is generated by carefully choosing the order of the switch-ing vectors. The Table 2 presents the order of the switchswitch-ing vectors in which they should be applied to minimize the state changes during a modulation period.
Table 2. Order of the switching vectors for VSI.
t0/2 t1 t2 t0/2 t0/2 t2 t1 t0/2