# Glossary¶

Technical terms used in the lab assignment material.

Alpha-beta full transform

Alpha-beta (Clarke) transform is a projection of three phase quantities onto two stationary axes. The usual transform, which is in fact a simplification of the full transform is presented in section αβ (Clarke) transform.

The full transormation adds a third axis (γ) to capture imbalances (sum of phases ≠ 0)

$v_{\alpha \beta \gamma}(t)= T_{\alpha \beta \gamma} v_{abc}(t)$

with transformation matrix

$\begin{split}T_{\alpha \beta\gamma} = \frac 23 \begin{bmatrix} 1&-{\frac 12}&-{\frac 12}\\ 0& {\frac {{\sqrt {3}}}{2}}&-{\frac {{\sqrt {3}}}{2}}\\ {\frac 12}&{\frac 12}&{\frac 12} \end{bmatrix}\end{split}$

so that $$v_\gamma$$ is simply the barycenter ($$(v_a + v_b + v_c)/3$$) of the three phase quantities. It is zero in a balanced situation.

Inverse transformation (αβγ to abc) uses the matrix

$\begin{split}T_{\alpha \beta\gamma}^{-1} = \frac 23 \begin{bmatrix} 1& 0 & 1\\ -{\frac 12}& {\frac {{\sqrt {3}}}{2}} & 1\\ -{\frac 12}&-{\frac {{\sqrt {3}}}{2}} &1 \end{bmatrix}\end{split}$

and for the simplified transform (αβ to abc), just use the first two columns of $$T_{\alpha \beta\gamma}^{-1}$$.

Apparent power

Product of the rms voltage and the rms current, expressed in volt-amperes (VA). Often denoted S. Computation: S = V.I.

In sinusoidal regime, it can be computed from active and reactive powers: ($$S^2 = P^2 + Q^2$$) or from the magnitude of the complex power.

Complex power

For a circuit element operated in sinusoidal regime, we define the complex power flowing into the component as:

$\underline{S} = \frac{1}{2}\underline{V}.\underline{I}^*$

where $$\underline{V}$$ is the complex amplitude of the voltage at the element terminals, and $$\underline{I}$$ is the complex amplitude of the current flowing through the component (“receptor convention”). “*” is the complex conjugate operator.

Complex power can be decomposed into a real part and a imaginary part. They are respectively the active and reactive power flowing into the component:

$\underline{S} = P + jQ$

and its magnitude is the apparent power ($$S^2 = P^2 + Q^2$$).

Notice: in the field of power systems, it is usual to use complex vectors which magnitude is the RMS value rather than the amplitude of the underlying sinusoidal voltage or current (such vectors are “shorter” by a factor $$\sqrt{2}$$). Advantage is that the formula for the complex power simplifies to:

$\underline{S} = \underline{V}_{RMS}.\underline{I}_{RMS}^*$

Warning: when working with a single phase equivalent circuit, the formula is multiplied by 3 to account for all three phases!

IGBT

Insulated-Gate Bipolar Transistor: a type of power semiconductor device. There are available with voltage ratings between about 400 V to 6500 V with a current of up to 1000 Amps in a single module. Very popular in applications with more than a few kVA.

You can check Infineon product catalog for an overview of available IGBTs (bare dies, discrete components, packaged modules…).

You can also look at the IGBT page on Wikipedia for a short history and photos of opened IGBT modules.

Inverter

a type of power electronics converter that can transfer energy from a DC source to an AC source.

Most inverter topologies allow the power flow to be reversed (energy transfer from the AC side to the DC side). When this is the usual operating point of the converter, the inverter is often called an active rectifier.

Single phase equivalent circuit

In a balanced three phase circuit in sinusoidal regime, the current and voltages on each phase are the same, only globally shifted by ±120°. Therefore, to simplify the representation, only one phase is depicted, with the neutral line (even if absent the actual circuit). This simplified view is called “single phase equivalent circuit”.

In this circuit, the laws of electricity apply like in a regular single phase circuit (Ohm’s law…), but with one difference: the power that flow in or out of each depicted component should be the power of the underlying three phase component. Therefore, the formula for complex power is multiplied by 3 compared to single phase:

$\underline{S} = \frac{3}{2} \underline{V}.\underline{I}^*$