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Inductors

Figure E.2: An RLC filter, input $ = v_e(t)$, output $ = v_C(t) = v_L(t)$.
\begin{figure}\input fig/rlc.pstex_t \end{figure}

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.

Mechanical Equivalent of an Inductor is a Mass

The mechanical analog of an inductor is a mass. The voltage $ v(t)$ across an inductor $ L$ corresponds to the force $ f(t)$ used to accelerate a mass $ m$. The current $ i(t)$ through in the inductor corresponds to the velocity $ {\dot x}(t)$ of the mass. Thus, Eq.$ \,$(E.4) corresponds to Newton's second law for an ideal mass:

$\displaystyle f(t) = m a(t), $

where $ a(t)$ denotes the acceleration of the mass $ m$.

From the defining equation $ \phi=Li$ for an inductor [Eq.$ \,$(E.3)], we see that the stored magnetic flux in an inductor is analogous to mass times velocity, or momentum. In other words, magnetic flux may be regarded as electric-charge momentum.

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Capacitors