Simplify Complex Operations

As compared to real numbers, a new dimension represented by the imaginary number $j$ is added in the complex representation. A complex number $V$ is simply a combination of two real numbers, one on the real axis and one on the imaginary axis, as shown in the figure below. We call the x-component $V_I$ and the y-component $V_Q$.

$$\begin{equation}\begin{aligned} V_I &= A\cos \theta \\ V_Q &= A\sin \theta \end{aligned} \end{equation}$$

where $A$ is the distance from the origin (magnitude) and $\theta$ is the angle from the x-axis.