Capacitor

A capacitor is a passive electrical component used to store energy electrostatically in an electric field. A resistor dissipates energy, whereas a capacitor stores energy. Typical capacitance values range from about 1 pF (10−12 F) to about 1 mF (10−3 F). Capacitors are widely used in electronic circuits for blocking direct current while allowing alternating current to pass.

 An ideal capacitor is wholly characterized by a constant capacitance C, defined as the ratio of charge ±Q on each conductor to the voltage V between them:
C= \frac{Q}{V}
Sometimes charge build-up affects the capacitor mechanically, causing its capacitance to vary. In this case, capacitance is defined in terms of incremental changes:
C= \frac{\mathrm{d}Q}{\mathrm{d}V}

DC circuits

If the capacitor is initially uncharged while the switch is open, and the switch is closed at t0, it follows from Kirchhoff's voltage law that:
V_0 = v_\text{resistor}(t) + v_\text{capacitor}(t) = i(t)R + \frac{1}{C}\int_{t_0}^t i(\tau) \mathrm{d}\tau
Taking the derivative and multiplying by C, gives a first-order differential equation:
RC\frac{\mathrm{d}i(t)}{\mathrm{d}t} + i(t) = 0
At t = 0, the voltage across the capacitor is zero and the voltage across the resistor is V0. The initial current is then I(0) =V0/R. With this assumption, solving the differential equation yields
\begin{align} I(t) &= \frac{V_0}{R} e^{-\frac{t}{\tau_0}} \\ V(t) &= V_0 \left( 1 - e^{-\frac{t}{\tau_0}}\right) \end{align}
where τ0 = RC is the time constant of the system. The case of discharging a charged capacitor likewise demonstrates exponential decay, but with the initial capacitor voltage replacing V0 and the final voltage being zero.

AC circuits

Impedance, the vector sum of reactance and resistance, describes the phase difference and the ratio of amplitudes between sinusoidally varying voltage and sinusoidally varying current at a given frequency.

 The reactance and impedance of a capacitor are respectively
\begin{align} X &= -\frac{1}{\omega C} = -\frac{1}{2\pi f C} \\ Z &= \frac{1}{j\omega C} = -\frac{j}{\omega C} = -\frac{j}{2\pi f C} \end{align}
where ω is the angular frequency of the sinusoidal signal.
The −j phase indicates that the AC voltage V = ZI lags the AC current by 90°.

 A capacitor connected to a sinusoidal voltage source will cause a displacement current to flow through it. In the case that the voltage source is V0cos(ωt), the displacement current can be expressed as:
I = C \frac{dV}{dt} = -\omega {C}{V_\text{0}}\sin(\omega t)
At sin(ωt) = -1, the capacitor has a maximum (or peak) current whereby I0 = ωCV0. The ratio of peak voltage to peak current is due to capacitive reactance (denoted XC).
X_C = \frac{V_\text{0}}{I_\text{0}} = \frac{V_\text{0}}{\omega C V_\text{0}} = \frac{1}{\omega C}
XC approaches zero as ω approaches infinity. If XC approaches 0, the capacitor resembles a short wire that strongly passes current at high frequencies. XC approaches infinity as ω approaches zero. If XC approaches infinity, the capacitor resembles an open circuit that poorly passes low frequencies.

 The current of the capacitor may be expressed in the form of cosines to better compare with the voltage of the source:
I = - {I_\text{0}}{\sin({\omega t}}) = {I_\text{0}}{\cos({\omega t} + {90^\circ})}
In this situation, the current will lead the voltage by 90°.


Parallel and series connection








 

 

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