RC Low-Pass Cutoff⚠ unverified

Understanding the Parameters

• $R$ and $C$ set the time constant $\tau = RC$ (see RC Time Constant); the cutoff is just $f_c = 1/(2\pi\tau)$. Larger $R$ or $C$ lowers the cutoff (the filter is "slower").

• $f_c$ — the frequency where output power is halved (voltage down to $1/\sqrt2$).

Derivation (Approaching a Proof)

Treat the RC pair as a voltage divider of the resistor's impedance $R$ and the capacitor's impedance $Z_C = 1/(j\omega C)$ (see Impedance Capacitor). The output across $C$ is

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

Its magnitude is $|H| = \dfrac{1}{\sqrt{1 + (\omega R C)^2}}$. The cutoff is defined where the power drops by half, $|H|^2 = \tfrac12$, i.e. $\omega R C = 1$:

$$\omega_c = \frac{1}{RC} \quad\Rightarrow\quad f_c = \frac{\omega_c}{2\pi} = \frac{1}{2\pi R C}.$$

At $f_c$ the output lags by $45^\circ$; far above it $|H| \approx 1/(\omega RC)$ falls as $1/f$, i.e. $-20\,$dB/decade (see Filter Order Rolloff).

History

The RC low‑pass is the simplest frequency‑selective network, a direct consequence of Ohm's law applied to a frequency‑dependent capacitive impedance (Steinmetz's phasor method, 1890s). It is ubiquitous as a smoothing/anti‑alias/de‑noising stage and as the prototype "first‑order system" throughout engineering.

Related Concepts: RC High-Pass Cutoff, Lowpass Passive Cutoff, RC Time Constant, Impedance Capacitor, Filter Order Rolloff

Notes: Registry calculator rc-lowpass-cutoff (unverified). Ideal first‑order response; loading by a following stage shifts the corner.