I'm going through the book "System Dynamics" by Katsuhiko Ogata. Specifically I'm reading about frequency response. According to the book to get the magnitude of the sinusoidal transfer function $G(j\omega)$:
$$
|G(j\omega)| = \lvert{\frac{X(j\omega)}{P(j\omega)}}\rvert =\frac{|X(j\omega|}{|P(j\omega)|}
$$
I know that the magnitude of a complex number is $\sqrt{(RealPart)^2+(ImPart)^2}$
So, there's an example in the book where:
$$
G(j\omega) = \frac{1}{(k-m\omega^2)+jb\omega}
$$
I tried to solve it following the formula $|G(j\omega)|=\frac{|X(j\omega|}{|P(j\omega)|}$:
$$
\frac{|1|}{|(k-m\omega^2)+jb\omega|} = \frac{1}{\sqrt{(k-m\omega^2)^2+(jb\omega)^2}}=\frac{1}{\sqrt{(k-m\omega^2)^2-b^2\omega^2}}
$$
The last result is from $j=\sqrt{-1}$ , so $j^2 = -1$. The problem is that in the book the magnitude is:
$$
\frac{1}{\sqrt{(k-m\omega^2)^2+b^2\omega^2}}
$$
I can't figure out why $+$ instead of $-$ in denominator. Any help will be appreciated. Thank you
Best Answer
By definition, the imaginary part of the complex number $a+ib$ (where $a$ and $b$ are real numbers) is $b$, not $ib$.