Capacitors and inductors

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Capacitors

A capacitor is a device that stores energy in an electric field which is generated by equal and opposite electric charge on opposing electrodes or plates within the capacitor. In turn, the charge Q is proportional to the voltage across the capacitor,

$ Q = CV\frac{}{} $

where C is the capacitance (units of farads (F) ≡ coulombs/volt). The current through the capacitor is the time rate of change of the charge and so the terminal relation of voltage and current for a capacitor is

$ i_C(t) = \frac{dQ}{dt} = C \frac{dV_C}{dt} $
R-C circuit driven by a step voltage V0u(t).

In an R-C circuit where a capacitor is charged via a current-limiting resistor from a step voltage source V0u(t) (the voltage is zero for t<0 and V0 for t>0), the current through the resistor is identical to the current through the capacitor. The resistor current is found simply by taking the voltage drop across the resistor divided by R,

$ i = \frac{V_0-V_C}{R} = C \frac{dV_C}{dt} $

for t>0. This equation may be rewritten in terms of voltages as

$ V_0-V_C = RC \frac{dV_C}{dt} $

This first-order differential equation has the well-known solution

$ V_C(t) = A e^{-t/\tau} + B\frac{}{} $
Transient step voltage response of R-C circuit.

where τ=RC is the "RC" time constant. At time t=0 there is no charge or voltage on the capacitor so VC(0)=0 and A=−B. Also as t→∞ then VC→B a constant and so i→0 and the voltage across the resistor also goes to zero. We conclude that B=V0 and, therefore,

$ V_C(t) = V_0 (1-e^{-t/\tau})\frac{}{} $


Inductors

In contrast to a capacitor, an inductor is a device that generates and stores energy in a magnetic field internally. In turn, the time rate of change of this magnetic field generates a voltage (Faraday's law of induction),

$ V_L(t) = L \frac{di_L}{dt} $

where L is the inductance (units of henrys (H) ≡ volt⋅second/ampere). For a constant current, the voltage across the inductor is identically zero.

L-R circuit driven by a step voltage V0u(t).

In an L-R circuit where an inductor is driven in series with a current-limiting resistor from a step voltage source V0u(t),

$ V_L = L \frac{di_L}{dt} = V_0 - i_L R $

for t>0, or

$ \frac{L}{R} \frac{di_L}{dt} = \frac{V_0}{R} - i_L $

This is almost identical to the case of the R-C circuit above, except here we are solving for the current through the inductor instead of voltage. Here we make the same arguments about the inductor current as we did for the capacitor voltage, that is, at time t=0 we must have iL=0 and as t→∞ then iL→V0/R. Therefore,

$ i_L(t) = \frac{V_0}{R} (1-e^{-t/\tau}) $
Transient step response of L-R circuit.

where now τ=L/R is the "L over R" time constant. The inductor voltage may be found simply by taking the derivative of the current,

$ v_L(t) = L \frac{V_0}{R} \frac{d}{dt}(1-e^{-t/\tau}) = V_0 \frac{L}{R}\frac{1}{\tau}e^{-t/\tau} = V_0 e^{-t/\tau} $