PX157 - E3 - kirchhoff's laws for AC circuits

kirchhoff's rules are valid if:
- the size of the circuit is much smaller than the wavelength of the source
- the transit time of the electromagnetic radiation is much less than the period of the AC source
- note: for $50 H z$ AC supply: $λ f = c ⟹ λ = 6 \times 10^{6} m$

impedance in series:

{\tilde{Z}}_{e q} = {\tilde{Z}}_{1} + {\tilde{Z}}_{2}

impedance in parallel:

\frac{1}{{\tilde{Z}}_{e q}} = \frac{1}{{\tilde{Z}}_{1}} + \frac{1}{{\tilde{Z}}_{2}}

eg:
all elements are in series:

\begin{aligned} {\tilde{Z}}_{e q} & = {\tilde{Z}}_{R} + {\tilde{Z}}_{C} + {\tilde{Z}}_{L} \\ = R + \frac{1}{j ω C} + j ω L \\ = R + j (ω L - \frac{1}{ω C}) \end{aligned}

\begin{aligned} {\tilde{Z}}_{e q} & = {\tilde{Z}}_{R} + {(\frac{1}{{\tilde{Z}}_{C}} + \frac{1}{{\tilde{Z}}_{L}})}^{- 1} \\ = R + {(j ω C + \frac{1}{j ω L})}^{- 1} \\ = R + {(j (ω C - \frac{1}{ω L}))}^{- 1} \\ = R + \frac{j}{(\frac{1}{ω L} - ω C)} \end{aligned}