A Voltage Source Inverter (VSI) is a type of power electronic inverter that converts a DC voltage source into an AC output voltage with a desired frequency and magnitude. It is widely used in applications like motor drives, renewable energy systems, and uninterruptible power supplies (UPS).

Let's break down what you've presented and add some context:

1. Voltage Space Vector:

The equation $u_s = (2/3)u_d[S_a + aS_b + a^2S_c]$ represents the voltage space vector $u_s$. Here:

• $u_d$ is the DC bus voltage.
• $S_a$, $S_b$, and $S_c$ are the switching states of the three phases (a, b, and c) of the inverter. They can be either 1 (switch ON) or 0 (switch OFF).
• $a = e^(j2π/3)$ is a complex operator that represents a 120-degree phase shift.

This equation transforms the three-phase switching states into a single complex vector, making analysis and control of the VSI much simpler.

2. Six Active Configurations:

You correctly state that there are six active switching configurations. These correspond to the different combinations of $S_a$, $S_b$, and $S_c$ that result in a non-zero voltage space vector. Each of these configurations produces a voltage space vector with a magnitude of $(2/3)u_d$ and a phase angle of $(k-1)π/3$, where $k$ ranges from 1 to 6.

3. Null Vectors:

In addition to the six active vectors, there are two null vectors (where $u_s = 0$). These occur when all switches are either ON or OFF. You've correctly included these in your representation by stating $k=0$ and $k=7$.

4. Characteristic Hexagon:

The six active voltage space vectors, when plotted in the complex plane, form a hexagon. This hexagon is called the "characteristic hexagon" of the VSI. The vertices of the hexagon represent the tips of the six active voltage vectors. This hexagon visually represents the maximum possible voltage space vectors that the VSI can generate.

Key Concepts and Implications:

• Space Vector Modulation (SVM): The concept of the characteristic hexagon is fundamental to Space Vector Modulation (SVM). SVM is a widely used technique for controlling VSIs. It involves approximating a desired voltage vector by switching between the adjacent active vectors and the zero vectors.
• Simplified Analysis: Using space vectors simplifies the analysis of VSI behavior. Instead of dealing with three separate phase voltages, we can work with a single complex vector.
• Visualization: The characteristic hexagon provides a clear visualization of the VSI's capabilities in terms of voltage generation.

🧠Graphical representation of the characteristic hexagon

https://gist.github.com/viadean/68b569778c698a1b281d9a0a01fed351