I have a question regarding impedances. While it is intuitive that \$ R \$-element has impedance \$ R \$, it's not so easy for me to figure out impedances of \$ L \$ and \$ C \$.
Here is my, probably incorrect, reasoning. For element \$ C \$ following differential equation is satisfied: \$ i(t) = C \frac{\mbox{d} u}{\mbox{d} t}\$. Our input is \$u\$, and utput is \$i\$. By transforming this equation we have
$$ \frac{U(s)}{I(s)} = G(s) = \frac{1}{sC} $$
Since we cosider only sinusoidal inputs we know, that output will also be sinusoidal of the same frequency, but different magnitude and phase, so it's fine to use Fourier instead of Laplace, then let \$s:=j \omega\$, then
$$G(s)=\frac{1}{j \omega C} = -j \frac{1}{\omega C} $$, which is indeed equal to impedance of element \$C\$.
Now - is that a coincidence, or the actual reasoning behind impedances and whole method of solving circuits based on them? If so, why take \$u\$ as an input, and \$i\$ as an output? Is it to make analogy to well known \$R=\frac{U}{I} \$?
If I tried to use not necessarily sinusoidal voltage and current, would I use the more general form: \$\frac{1}{sC}\$? Would it work?