# Capacitor **A passive electrical component that stores energy in an electric field when a voltage is applied across it.** ![[Capacitor.svg|200]] Most capacitors consist of a pair of electrical conductors, commonly known as *plates*, separated by a *dielectric medium*. Capacitors are characterised by their [[capacitance]]. The *voltage* across a capacitor *can not* change *instantaneously*, as this would theoretically destroy and create an electric field instantly. ## Capacitors in series and parallel > [!Series] The total capacitance of a *series* of capacitors is the reciprocal of the sum of the reciprocals of their capacitances. $C_{\text{eq}}=\frac{1}{\frac{1}{C_{1}}+\frac{1}{C_{2}}+\cdots+\frac{1}{C_n}}$ > ![[CapacitorsSeries.svg|600]] >[!Parallel] The total capacitance of *parallel* capacitors is the sum of the individual capacitances. $C_{\text{eq}}=C_{1}+C_{2}+\cdots +C_{n}$ > ![[CapacitorsParallel.svg|600]] ## Impedance > [!Example] Impedance > The [[impedance]] of a capacitor is > $\mathbf{Z}=\frac{1}{j\omega C}=-\frac{j}{\omega C}$ > - $\omega$ - [[angular frequency]], radians per second > - $C$ - [[capacitance]], $\text{F}$ The current *leads* the voltage by $90^{\circ}$ in a capacitor. At DC, a capacitor acts like an *open circuit* as the charge simply accumulates on the plates and the electric field exactly balances the voltage supplied to the capacitor. At high frequencies, the charge on the plates accumulate less at each peak, generating a weak opposing electric field and thus acting like a *short circuit*. ![[CapacitorFrequencyResponse.svg|550]] ## Voltage > [!Voltage] > The *voltage* across a capacitor is > $v(t)=\frac{1}{C}\int_{t_0}^{t}i(t)\;dt+v(t_{0})$ ## Current > [!Current] > The *current* through a capacitor is > $i=C\frac{dV}{dt}$ Note that this implies that the voltage across a capacitor *can not* change *instantaneously*, as this would require an infinitely large current. ## Stored energy > [!Stored energy] > The *energy* stored in the electric field of a capacitor is > $w(t)=\frac{1}{2}C\;v(t)^2=\frac{1}{2}CV^2$