Inductors

Figure E.2: An RLC filter, input $= v_e(t)$ , output $= v_C(t) = v_L(t)$ .

An inductor can be made physically using a coil of wire, and it stores magnetic flux when a current flows through it. Figure E.2 shows a circuit in which a resistor $R$ is in series with the parallel combination of a capacitor $C$ and inductor $L$ .

The defining equation of an inductor $L$ is

$$\phi(t) = Li(t) \protect$$ (E.3)

where $\phi(t)$ denotes the inductor's stored magnetic flux at time $t$ , $L$ is the inductance in Henrys (H), and $i(t)$ is the current through the inductor coil in Amperes (A), where an Ampere is a Coulomb (of electric charge) per second. Differentiating with respect to time gives

$$v(t) = L\frac{di(t)}{dt}, \protect$$ (E.4)

where $v(t)= d \phi(t)/ dt$ is the voltage across the inductor in volts. Again, the current $i(t)$ is taken to be positive when flowing from plus to minus through the inductor.

Taking the Laplace transform of both sides gives

$$V(s) = Ls I(s) - LI(0),$$

by the differentiation theorem for Laplace transforms.

Assuming a zero initial current in the inductor at time 0 , we have

$$R_L(s) \isdef \frac{V(s)}{I(s)} = Ls.$$

Thus, the driving-point impedance of the inductor is $Ls$ . Like the capacitor, it can be analyzed in steady state (initial conditions neglected) as a simple resistor with value $Ls$ Ohms.