2Space vector pulse width modulation technique

One of the most popular modulation approaches for two-level converters is space vector pulse width modulation (SVPWM), which is increasingly being used in the control of multilevel converters. This is an advanced and computation-intensive PWM technique. The SVPWM increases the output capability of sinusoidal PWM without distorting the line-to-line output voltage waveform.

The concept of space voltage vectors corresponding to various switching states has been applied in the study of impact of various switching states on the capacitor charge balancing. An advantage of the SVPWM is the instantaneous control of switching states and the freedom to select vectors in order to balance the NP. Additionally, one can realize output voltages with almost any average value using the nearest three vectors, which is the method that results in the best spectral performance. The SVPWM method is an advanced, computation-intensive PWM method and is possibly the best among all the PWM techniques for variable frequency drive applications. Because of its superior performance characteristics, it has been finding widespread application in recent years. If the switching frequency is high enough, the losses due to the harmonics can be almost neglected, and the SVPWM is a better solution in terms of inverter output voltage, harmonic losses, and number of switching per cycle [13].

2.1Features of SVPWM

The SVPWM technique is more popular than conventional technique because of the following excellent features:

2.2Space vector concept

The space vector concept is derived from the rotating field of ac machine that is used for modulating the inverter output voltage. In this modulation technique, the three-phase quantities can be transformed to their equivalent two-phase quantity either in synchronously rotating frame or in stationary frame. From this two-phase component, the magnitude of the reference vector can be found and is used for modulating the inverter output. The process of obtaining the rotating space vector is explained in the following section, considering the stationary reference frame.

Let the three-phase sinusoidal voltage component be

$\begin{array}{l}{\text{V}}_{\text{a}}={\text{V}}_{\text{m}}\text{sin}\text{ωt}\\ {\text{V}}_{\text{b}}={\text{V}}_{\text{m}}\text{sin}\left(\text{ωt}-\frac{2\text{π}}{3}\right)\\ {\text{V}}_{\text{c}}={\text{V}}_{\text{m}}\text{sin}\left(\text{ωt}+\frac{2\text{π}}{3}\right).\end{array}$

When this three-phase voltage is applied to the ac machine, it produces a rotating flux in the air gap of the ac machine. This rotating flux component can be represented as single rotating voltage vector. The magnitude and angle of the rotating vector can be found by means of Clark’s transformation as explained below in the stationary reference frame. The representation of rotating vector in complex plane is shown in Fig. 2.1.

Fig. 2.1: Representation of the rotating vector in a complex plane space vector representation of the three-phase quantity.

$\overline{{\text{V}}^{\text{*}}}={\text{V}}_{\text{d}}+{\text{jV}}_{\text{q}}=\frac{2}{3}({\text{V}}_{\text{a}}+{\text{aV}}_{\text{b}}+{\text{a}}^2{\text{V}}_{\text{c}}),$

where a = ej2π/3.

$\overline{\text{|V|}}=\sqrt{{\text{V}}_{\text{d}}^2+{\text{V}}_{\text{q}}^2};\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}a}={\text{tan}}^{-1}\left(\frac{{\text{V}}_{\text{q}}}{{\text{V}}_{\text{d}}}\right)$

${\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}V}}_{\text{d}}+{\text{jV}}_{\text{q}}=\frac{2}{3}({\text{V}}_{\text{a}}+{\text{e}}^{\text{j}\frac{2\text{π}}{3}}{\text{V}}_{\text{b}}+{\text{e}}^{-\text{j}\frac{2\text{π}}{3}}{\text{V}}_{\text{c}})$

Equating real and imaginary parts:

$$\begin{gathered} \begin{array}{l}{\text{V}}_{\text{d}}=\frac{2}{3}({\text{V}}_{\text{a}}+\cos \frac{2\text{π}}{3}{\text{V}}_{\text{b}}+\text{cos}\frac{2\text{π}}{3}{\text{V}}_{\text{c}})\\ {\text{V}}_{\text{q}}=\frac{2}{3}(0.{\text{V}}_{\text{a}}+\sin \frac{2\text{π}}{3}{\text{V}}_{\text{b}}-\text{sin}\frac{2\text{π}}{3}{\text{V}}_{\text{c}})\end{array} \\ \begin{array}{l}\left[\begin{array}{c}{\text{V}}_{\text{d}}\\ {\text{V}}_{\text{q}}\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}1& \cos \frac{2\text{π}}{3}& \text{cos}\frac{2\text{π}}{3}\\ 0& \sin \frac{2\text{π}}{3}& -\sin \frac{2\text{π}}{3}\end{array}\right]\left[\begin{array}{c}{\text{V}}_{\text{a}}\\ {\text{V}}_{\text{b}}\\ {\text{V}}_{\text{c}}\end{array}\right]\\ \left[\begin{array}{c}{\text{V}}_{\text{d}}\\ {\text{V}}_{\text{q}}\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}1& -0.5& -0.5\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}{\text{V}}_{\text{a}}\\ {\text{V}}_{\text{b}}\\ {\text{V}}_{\text{c}}\end{array}\right]\end{array} \end{gathered}$$

2.2.1Principle of SVPWM

The SVPWM treats the sinusoidal voltage as a constant amplitude vector rotating at a constant frequency. This PWM technique approximates the reference voltage Vref by a combination of the eight switching patterns. A three-phase voltage vector is transformed into a vector in the stationary d-q coordinate frame, which represents the spatial vector sum of the three-phase voltage.

2.2.2Definition of space vector

The space vector Vsr constituted by the pole voltages V ao, V bo, and Vco is defined as

${\text{V}}_{\text{sr}}={\text{V}}_{\text{ao}}+{\text{V}}_{\text{bo}}\cdot {\text{e}}^{\text{j}\frac{\text{2π}}{\text{3}}}+{\text{V}}_{\text{co}}\cdot {\text{e}}^{\text{j}\frac{\text{4π}}{\text{3}}}.$

The relationship between the phase voltages Van, Vbn, and Vcn and the pole voltages Vao, Vbo, and Vco is given by

${\text{V}}_{\text{ao}}={\text{V}}_{\text{an}}+{\text{V}}_{\text{no}};{\text{V}}_{\text{bo}}={\text{V}}_{\text{bn}}+{\text{V}}_{\text{no}};{\text{V}}_{\text{co}}={\text{V}}_{\text{cn}}+{\text{V}}_{\text{no}}.$

Since Van + Vbn + Vcn = 0,

${\text{V}}_{\text{no}}=\frac{{\text{V}}_{\text{ao}}+{\text{V}}_{\text{bo}}+{\text{V}}_{\text{co}}}{3},$

where Vno is the common mode voltage. From Eqs. (2.9) and (2.10), it is evident that the phase voltages Van, Vbn, and Vcn also result in the same space vector Vsr. The space vector Vsr can also be resolved into two rectangular components, namely Vd and Vq. It is customary to place the d-axis along the A-phase axis of the induction motor.

Hence,

${\text{V}}_{\text{sr}}={\text{V}}_{\text{d}}+{\text{jV}}_{\text{q}}.$

2.2.3Advantages of SVPWM

SVPWM is considered a better technique of PWM implementation owing to its associated advantages mentioned below:

2.3SVPWM for the two-level inverter

2.3.1Three-phase voltage source inverter

A three-phase two-level inverter with a star connected load can be represented, as shown in Fig. 2.2, where Vao, Vbo, and Vco are the inverter output voltages with respect to their return terminal of the dc source marked as ‘O’. These voltages are called pole voltages and Van, Vbn, and Vcn are the load phase voltages with respect to neutral (n). Each switching circuit configuration generates three independent pole voltages Vao, Vbo, and Vco. There are eight possible switching configurations, called the operating states or inverter states.

2.3.2Determination of switching states

The possible pole voltages that can be produced at any time are +0.5 Vdc and −0.5 Vdc. For example, when switches S1, S6, and S5 are closed then a-phase and c-phase are connected to the positive dc bus and b-phase is connected to negative dc bus, the corresponding pole voltages are:

Using this procedure, the inverter state in the above equation is represented by the notation (+ − +) or 1 0 1 and the corresponding switching state is denoted by V6. Thus, every phase of a three-phase voltage source inverter (VSI) can be connected either to the positive or the negative dc bus. The switching states are shown in Fig. 2.3, which are designated using the code numbers 0 to 7.

Fig. 2.2: Three-phase two-level inverter.

In case of switching states V0 and V7, all the three poles are connected to the same dc bus, effectively shorting the load and there will be no power transfer between source and load. These two states are called ‘null states’ or ‘zero states’. In case of other switching states, power transfers between source and load. Hence, these states (V1, V2,…V6) are called ‘active voltage vectors’ or ‘active states’.

In terms of phase voltages of inverter, the voltage space vector can be written as shown below:

${\text{V}}_{\text{sr}}={\text{V}}_{\text{an}}+{\text{V}}_{\text{bn}}{\text{e}}^{\text{j}\frac{\text{2π}}{\text{3}}}+{\text{V}}_{\text{cn}}{\text{e}}^{\text{j}\frac{\text{4π}}{\text{3}}}.$

In terms of the pole voltages of the inverter, the voltage space vector can be written as

${\text{V}}_{\text{sr}}={\text{V}}_{\text{ao}}+{\text{V}}_{\text{bo}}\cdot {\text{e}}^{\text{j}\frac{\text{2π}}{\text{3}}}+{\text{V}}_{\text{co}}\cdot {\text{e}}^{\text{j}\frac{\text{4π}}{\text{3}}}.$

Fig. 2.3: Possible switching states of the two-level inverter.

Fig. 2.4: Three-phase (a, b, c) to two-phase (d, q) transformation.

In the implementation of SVPWM, the (a, b, c) reference frame voltage equations are transformed into the d-q reference frame, which is a stationary reference frame, as depicted in Fig. 2.4. As described in Fig. 2.4, this transformation is equivalent to an orthogonal projection of [a, b, c] onto the two dimensions perpendicular to the vector [1, 1, 1] (the equivalent d-q plane) in a three-dimensional coordinate system. The desired reference voltage vector Vref is obtained in the d-q plane by applying the similar transformation to the desired output voltage. The approximation of the reference vector from the eight switching states is the primary objective of SVPWM technique. The space vector diagram of two-level inverter is as shown in Fig. 2.5.

Fig. 2.5: Space vector diagram of the two-level inverter.

2.3.3Calculation of switching times

The switching times of the SVPWM-based inverter can be calculated using the volt-second relation. Figure 2.6 represents the calculation of switching times based on the voltage-second relation of the reference vector Vsr. The volt-second produced by vectors V1, V2, and V7 or V0 along the d and q axes are the same as those produced by the reference vector Vref.

Fig. 2.6: Representation of the reference vector in terms of the volt-second relation.

$\begin{array}{l}{\text{V}}_{\text{dc}}\cdot {\text{T}}_{\text{1}}+{\text{V}}_{\text{dc}}\cos {60}^{\circ }\cdot {\text{T}}_2=\left|{\text{V}}_{\text{sr}}^{\text{*}}\right|\cos \alpha \cdot {\text{T}}_{\text{s}}\\ {\text{V}}_{\text{dc}}\sin {60}^{\circ }\cdot {\text{T}}_{\text{2}}=\left|{\text{V}}_{\text{sr}}^{\text{*}}\right|\sin \alpha \cdot {\text{T}}_{\text{s}}\end{array}$

After simplifying, we will get the expressions as

$\begin{array}{l}{\text{T}}_{\text{1}}=\frac{(\text{m}\times \sin ({60}^{\circ }-\alpha ))}{\sin ({60}^{\circ })}\\ T_2=\frac{(\text{m}\times \sin (\alpha ))}{\sin ({60}^{\circ })}\\ {\text{T}}_{\text{0}}={\text{T}}_{\text{S}}-{\text{T}}_1-{\text{T}}_2,\end{array}$

where

$\text{m}=\frac{{\text{V}}_{\text{sr}}}{\left(\frac{2}{3}\right){\text{V}}_{\text{dc}}}.$

2.3.4Optimized switching sequence

The aim of SVPWM is to approximate the reference voltage vector (Vref) in a sampling period by time averaging the three voltage vectors. In the SVPWM strategy, the total zero voltage vector time is equally distributed between V0 and V7. Further, the zero voltage vector time is equally distributed symmetrically at the start and at the end of the subcycle in a symmetrical manner. Moreover, to minimize the switching frequency and reduce the number of commutations, it is desirable that the switching sequence between the three voltage vectors involves only one commutation when there is a transfer from one state to the other. This requires the use of both zero vectors (V7 and V0) in a given sector and a reversal of the switching sequence every subcycle.

Thus, SVPWM uses V0 - V1 - V2 - V7–V7 - V2 - V1 - V0 in sector I, V0- V3- V2- V7–V7- V2- V3- V0 in sector II, and so on. Table 2.1 depicts the switching sequence for all sectors. The switching times of the two-level inverter at each sector are shown in Tab. 2.2.

The two-level inverters have certain drawbacks:

Tab. 2.2: Switching times of the two-level inverter.

Multilevel topology has the following advantages over traditional two-level topology:

2.4Conclusions

In this chapter, the most recommended PWM technique applied to power electronic inverters has been discussed in detail as well as the features of the space vector modulation technique, concept of space vector with principle of operation, theoretical analysis of SVPWM, pros and cons, and finally, analysis of space vector modulation to two-level inverter. The analysis of VPWM, including the determination of switching states and calculation of switching times and optimized switching sequence when applied to a two-level VSI, has been presented with operation state diagrams. The main objective of this chapter is to help the reader understand how space vector modulation is implemented in case of a two-level VSI. To enrich the analysis, the simulation results of space vector pulse width-modulated two-level inverter are furnished at the end.