Space Vectors

Definition

The space vector transformation is implemented in motulator.common.utils.abc2complex() and its inverse in motulator.common.utils.complex2abc(). We use peak-valued complex space vectors, marked with boldface in equations [Hinkkanen et al., 2024]. As an example, the stator current space vector is

(1)\[ \iss = \frac{2}{3}\left(\iA + \iB\e^{\jj 2\pi/3} + \iC\e^{\jj 4\pi/3}\right)\]

where \(\iA\), \(\iB\), and \(\iC\) are the time-varying phase currents. The subscript s denotes stator quantities and the superscript s denotes stationary coordinates. The components are \(\iss = i_\upalpha + \jj i_\upbeta\), represented in code as i_s_ab.

The space vector excludes the zero-sequence component

(2)\[ i_0 = \frac{1}{3}\left(\iA + \iB + \iC\right)\]

Zero-sequence voltage exists at converter outputs, but zero-sequence current cannot flow (\(i_0 = 0\)) in delta-connected systems or when the star point is isolated. Therefore, zero-sequence voltage produces no power or torque.

Coordinate Transformation

Models and controls often use synchronous coordinates aligned with the rotor or control system variables. Consider a general coordinate system at angle \(\thetac\) relative to stationary coordinates, rotating at \(\omegac\):

(3)\[ \frac{\D\thetac}{\D t} = \omegac\]

The transformation to these coordinates is

(4)\[ \is = \e^{-\jj\thetac}\iss\]

In rotor coordinates (\(\thetac = \thetam\)), components are traditionally denoted as \(\is = \id + \jj \iq\), coded as i_s_dq. Controller coordinates use simply i_s. Similar notation applies to other space vectors.