RL Low-Pass Cutoff⚠ unverified

Understanding the Parameters

• $L$ and $R$ set the time constant $\tau = L/R$ (see Rl Time Constant); the cutoff is $f_c = 1/(2\pi\tau) = R/(2\pi L)$. Note the inverse dependence on $R$ compared with the RC filter.

• $f_c$ — where the inductive reactance equals the resistance, $X_L = R$.

Derivation (Approaching a Proof)

With $L$ in series and the output across $R$, the transfer function is a voltage divider of $Z_L = j\omega L$ (see Impedance Inductor) and $R$:

$$H(j\omega) = \frac{R}{R + j\omega L} = \frac{1}{1 + j\omega L/R}.$$

Its magnitude is $1/\sqrt{1 + (\omega L/R)^2}$, which hits the half‑power value $1/\sqrt2$ when $\omega L/R = 1$, i.e. $X_L = R$:

$$\omega_c = \frac{R}{L} \quad\Rightarrow\quad f_c = \frac{R}{2\pi L}.$$

The response is first‑order and identical in shape to the RC low‑pass — only the element setting the time constant differs.

History

The RL filter is the magnetic dual of the RC filter. RC forms dominate signal electronics (capacitors are small and cheap), while RL forms appear in power electronics and EMI suppression, where inductors/chokes naturally block high‑frequency noise.

Related Concepts: RC Low-Pass Cutoff, Rl Time Constant, Impedance Inductor, Inductive Reactance

Notes: Registry calculator rl-lowpass-cutoff (unverified). Ideal inductor assumed; winding resistance shifts the corner.