## Posts

Showing posts with the label Alternating Current Circuits

### Circuit containing Inductor and Capacitor in Series (L-C Series Circuit )

Mathematical Analysis of L-C Series Circuit : Let us consider, a circuit containing inductor $L$ capacitor $C$ and these are connected in series. If an alternating voltage source is applied across it then the resultant voltage of the L-C circuit $V=V_{L} - V_{C} \qquad(1)$ We know that: $V_{L} = iX_{L}$ $V_{C} = iX_{C}$ So from equation $(1)$ $V= iX_{L} - iX_{C}$ $V=i \left(X_{L} - X_{C} \right)$ $\frac{V}{i}=\left(X_{L} - X_{C} \right)$ $Z=\left(X_{L} - X_{C} \right) \qquad(2)$ Where $Z \rightarrow$ Impedance of L-C circuit. $X_{L} \rightarrow$ Inductive Reactance which has value $\omega L$ $X_{C} \rightarrow$ Capacitive Reactance which has value $\frac{1}{\omega C}$ So from equation $(2)$, we get $Z=\left( \omega L - \frac{1}{\omega C} \right) \qquad(3)$ The phase of resultant voltage: The phase of resultant voltage from current is $90^{\circ}$ as shown in the figure above. The Impedance and Phase at Re

### Alternating Current Circuit containing Capacitance only (C-Circuit)

Alternating Current Circuit Containing Capacitance Only (C-Circuit): Let us consider, A circuit containing a capacitor of capacitance $C$ only which is connected with an alternating EMF i.e electromotive force source i.e. Let us consider, A circuit containing a coil of inductance $L$ only which is connected with an alternating EMF i.e electromotive force source i.e. $E=E_{\circ}sin\omega t\qquad(1)$ When alternating emf is applied across the capacitor plates then the charge on capacitor plates varies continuously and correspondingly current flows in the connecting leads. Let the charge on the capacitor plates is $q$ and the current in the circuit at any instant is $i$. Since there is no resistance in the circuit then the instantaneous potential difference is $\frac{q}{C}$ across the capacitor plates must be equal to the applied emf i.e. $\frac{q}{C} = E_{\circ} sin \omega t$ $q = CE_{\circ} sin \omega t \qquad(2)$ The instantaneous current $i$ in the circuit is,