Consider a complex number \(z\): \[ z = a + bj \] where \(a, b\in\mathbb{R}\) correspond to the real and imaginary part of \(z\), respectively; \(j\) denotes the imaginary number \(j=\sqrt{-1}\). \(a\) and \(b\) are often denoted as \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\), where \(\operatorname{Re}(\cdot)\) and \(\operatorname{Im}(\cdot)\) denote the real and imaginary part of the complex number, respectively.
Using a real and imaginary axes, complex number \(z\) can be visually represented as a pair of numbers \((a,b)\) forming a vector on the complex plane. At the same time, complex number \(z\) can be represented in polar coordinate by a simple coordinate transformation: \[ z = a + bj \;\;\; \Longrightarrow \;\;\; z = R(\cos\phi + j\sin\phi ) = Re^{j\phi} \] where \(R=\sqrt{a^2+b^2}\) and \(\phi = \tan^{-1}(b/a)\) (Fig. \(\ref{fig:Cartesian_Polar_Form}\)). \(R\) is called the magnitude of \(z\), denoted as \(|z|\), while \(\phi\) is called the ``phase’’ of complex number \(z\), denotes as \(\angle z\).
FILL_FIGURE
The complex conjugate of complex number \(z\), often denoted as \(\bar{z}\), is a complex number with an
equal real part and an imaginary part equal in magnitude but opposite in
sign: \[
\bar{z} = a - bj
\]
Using \(\bar{z}\), the magnitude and
phase of complex number \(z\) (Eq.
\(\ref{eq:cartesian_polar}\)) can be
represented as: \[
R = \sqrt{z\bar{z}}, \;\;\; \phi =
\frac{1}{2j}\ln\big({\frac{z}{\bar{z}}}\big)
\] It is easy to check that for two complex numbers \(z_1\) and \(z_2\), then the phase of \(z_3 = z_1 z_2\) is a simple sum of \(\angle z_1\) and \(\angle z_2\): \[
z_1 = R_1 e^{j\phi_1}, z_2 = R_2 e^{j\phi_2} \;\;\; \Longrightarrow
\;\;\; z_3 = R_1 R_2 e^{j(\phi_1+\phi_2)},\;\; \angle z_3 = \phi_1 +
\phi_2
\] Moreover, in log scale, magnitude of \(z_3\) is: \[
\log|z_3| = \log|R_1 R_2 | = \log|R_1| + \log|R_2| = \log|z_1| +
\log|z_2|
\] which is simple summation. Property (Eq. \(\ref{eq:phase}\)) and (Eq. \(\ref{eq:magnitude_simp}\)) are useful when
we consider Bode plots.