Capacitor and Inductor
Let us apply the findings from the last chapter to capacitors and inductors.
Capacitor
For the capacitor, the following applies:
\[
\begin{gathered}
u_C(t) = U_{C0} + \frac{1}{C} \int i_C(t)\,dt \\[6pt]
\text{Assumed capacitor current: } i_C(t) = \hat{\imath}_C \cdot \sin(\omega t) \\[6pt]
u_C(t) = U_{C0} + \frac{1}{C} \cdot \frac{1}{\omega} \cdot \hat{\imath}_C \cdot \left[-\cos(\omega t)\right] \\[6pt]
\text{Empty capacitor: } U_{C0} = 0\,\mathrm{V} \\[6pt]
u_C(t) = \frac{1}{\omega C} \cdot \hat{\imath}_C \cdot \left[-\cos(\omega t)\right] \\[6pt]
\text{Peak values: } \hat{u}_C = \frac{1}{\omega C} \cdot \hat{\imath}_C
\end{gathered}
\]
The peak value of the voltage depends on the peak value of the current, the angular frequency omega and the capacitance of the capacitor. Let us compare this with Ohm's law.
\[
\begin{gathered}
\text{Ohm's law: } U = R \cdot I \\[6pt]
R = \frac{U}{I} \\[6pt]
\text{Capacitor: } \hat{u}_C = \frac{1}{\omega C} \cdot \hat{\imath}_C \\[6pt]
\text{Reactance: } X_C = \frac{\hat{u}_C}{\hat{\imath}_C} = \frac{1}{\omega C} \\[6pt]
\hat{u}_C = X_C \cdot \hat{\imath}_C
\end{gathered}
\]
The relationship between the peak values of voltage and current at the capacitor is represented by the “reactance” XC.
Inductor
For the inductor, very similar relationships apply as for the capacitor:
\[
\begin{gathered}
i_L(t) = I_{L0} + \frac{1}{L} \int u_L(t)\,dt \\[6pt]
\text{Assumed inductor voltage: } u_L(t) = \hat{u}_L \cdot \sin(\omega t) \\[6pt]
i_L(t) = I_{L0} + \frac{1}{L} \cdot \frac{1}{\omega} \cdot \hat{u}_L \cdot \left[-\cos(\omega t)\right] \\[6pt]
\text{Inductor without current: } I_{L0} = 0\,\mathrm{A} \\[6pt]
i_L(t) = \frac{1}{\omega L} \cdot \hat{u}_L \cdot \left[-\cos(\omega t)\right] \\[6pt]
\text{Peak values: } \hat{\imath}_L = \frac{1}{\omega L} \cdot \hat{u}_L \rightarrow \hat{u}_L = \omega L \cdot \hat{\imath}_L \\[6pt]
\text{Ohm's law: } \hat{u}_L = \omega L \cdot \hat{\imath}_L = X_L \cdot \hat{\imath}_L \\[6pt]
\text{Reactance: } X_L = \frac{\hat{u}_L}{\hat{\imath}_L} = \omega L
\end{gathered}
\]
Summary of reactances
If only resistors are installed in a network with an AC voltage source, the “normal” Ohm's law applies to the resistors. For the inductor and the capacitor, Ohm's law also applies, but with the reactance X instead of the resistance R.
\[
\begin{gathered}
\text{Resistor: } \hat{u}_R = R \cdot \hat{\imath}_R \\[6pt]
\text{Capacitor: } \hat{u}_C = X_C \cdot \hat{\imath}_C \text{ with } X_C = \frac{1}{\omega C} \\[6pt]
\text{Inductor: } \hat{u}_L = X_L \cdot \hat{\imath}_L \text{ with } X_L = \omega L
\end{gathered}
\]