Reactance phase convention ($v(t)=\mathrm{Re}(\hat{V}{e}^{\mathrm{i}\omega t})$)

Use complex amplitudes only for signals at one common positive angular frequency $\omega$. Define the real terminal voltage and electric current byThe complex amplitudes here use peak magnitudes; both can instead use root mean square magnitudes without changing their ratio. Choose positive electric current entering the terminal assigned positive voltage, so $p=vi$ is absorbed power. For an ideal linear inductor and capacitor, their terminal laws $v_L=Ldi/dt$ and $i_C=Cd{v}_C/dt$ giveThe derivative multiplies a complex amplitude by $\mathrm{i}\omega$: this single algebraic step explains both signs. Taking $i(t)=I_0\cos \omega t$ as a reference gives $v_L(t)=-\omega L{I}_0\sin \omega t$ and $v_C(t)=I_0\sin \omega t/(\omega C)$ for the sinusoidal response. Thus inductor voltage leads electric current by $\pi /2$, while capacitor voltage lags electric current by $\pi /2$. A constant offset or a free transient is outside this single-frequency representation. ^[ref].

With the alternative $e^{-\mathrm{i}\omega t}$ convention, the same real signals use conjugated complex amplitudes and $Z_L=-\mathrm{i}\omega L$, $Z_C=+\mathrm{i}/(\omega C)$. Their physical leads and lags stay the same; the sign of $\mathrm{Im}Z$ changes. Also, some textbooks call the positive magnitude $1/(\omega C)$ the capacitive reactance and write the series expression as $X_L-|X_C|$. This article consistently uses signed $X_C=-1/(\omega C)$ and hence $X=X_L+X_C$. ^[ref].