Editing RL circuit (section)

===Frequency domain considerations===
These are [[frequency domain]] expressions. Analysis of them will show which frequencies the circuits (or filters) pass and reject. This analysis rests on a consideration of what happens to these gains as the frequency becomes very large and very small.

As {{math|''ω'' → ∞}}:
:<math>G_L \to 1 \quad \mbox{and} \quad G_R \to 0\,.</math>

As {{math|''ω'' → 0}}:
:<math>G_L \to 0 \quad \mbox{and} \quad G_R \to 1\,.</math>

This shows that, if the output is taken across the inductor, high frequencies are passed and low frequencies are attenuated (rejected). Thus, the circuit behaves as a ''[[high-pass filter]]''. If, though, the output is taken across the resistor, high frequencies are rejected and low frequencies are passed. In this configuration, the circuit behaves as a ''[[low-pass filter]]''. Compare this with the behaviour of the resistor output in an [[RC circuit]], where the reverse is the case.

The range of frequencies that the filter passes is called its [[Bandwidth (signal processing)|bandwidth]]. The point at which the filter attenuates the signal to half its unfiltered power is termed its [[cutoff frequency]]. This requires that the gain of the circuit be reduced to 
:<math>G_L = G_R = \frac{1}{\sqrt 2}\,.</math>

Solving the above equation yields
:<math>\omega_\mathrm{c} = \frac{R}{L} \mbox{ rad/s} \quad \mbox{or} \quad f_\mathrm{c} = \frac{R}{2\pi L} \mbox{ Hz}\,,</math>
which is the frequency that the filter will attenuate to half its original power.

Clearly, the phases also depend on frequency, although this effect is less interesting generally than the gain variations.

As {{math|''ω'' → 0}}:
:<math>\phi_L \to 90^{\circ} = \frac{\pi}{2} \mbox{ radians} \quad \mbox{and} \quad \phi_R \to 0\,.</math>

As {{math|''ω'' → ∞}}:
:<math>\phi_L \to 0 \quad \mbox{and} \quad \phi_R \to -90^{\circ} = -\frac{\pi}{2} \mbox{ radians}\,.</math>

So at [[Direct current|DC]] (0&nbsp;[[Hertz|Hz]]), the resistor voltage is in phase with the signal voltage while the inductor voltage leads it by 90°. As frequency increases, the resistor voltage comes to have a 90° lag relative to the signal and the inductor voltage comes to be in-phase with the signal.