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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

$\displaystyle \phi(t) = Li(t) \protect$ (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

$\displaystyle v(t) = L\frac{di(t)}{dt}, \protect$ (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

$\displaystyle 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

$\displaystyle 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.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-04-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University