Complex Vector
A pair of real vectors - one as a real part and the second as an imaginary part defines the Complex Vector. \[ \overline{ \mathbf B} = \overline{ \mathbf B}_{Re} + i· \overline{ \mathbf B}_{Im} \]
The canonical form for the complex vector is:
\[ \overline{ \mathbf B} =
\left (\begin{matrix}
1 && i·k\\
-i·k && 1
\end{matrix} \right )
\left (\begin{matrix}
B_x\\
B_y
\end{matrix} \right )
· e^{i·φ}
\]
where
φ - phase,
Bx and By are real components of the magnitude vector, and
k is a coefficient of polarization (-1≤k≤1).
Coefficient of polarization is k = Bmin / Bmax.
k=0 corresponds to linear polarization along the direction defined by the vector (Bx, By). The k=1 corresponds to the counterclockwise circular polarization, k=-1 corresponds to the clockwise circular polarization.
\( B_{max} = \sqrt{B_x^2 + B_y^2}\) is a real magnitude of the complex vector, it corresponds to the maximum momentary value of time-dependent \( \overline {\mathbf B} \).
RMS value of complex vector is: \( B_{RMS} = \sqrt{ \left( B_{min}^2 + B_{max}^2 \right) / 2 } \)
Related Topics
Complex Value.