Cutoff Frequency and the First-Order Low-Pass
Lows pass, highs are blocked
Sweep the frequency from low to high. At first the signal comes out almost untouched, then it shrinks. Find the cutoff frequency ω_c where the output drops to 1/√2 of the input, that is −3 dB.
The capacitor makes a frequency-dependent divider
Put a resistor and a capacitor in series and take the output across the capacitor, and the two form a voltage divider. But the capacitor’s impedance 1/ωC falls as frequency rises. At low frequency the capacitor takes almost all the voltage, so the output nearly equals the input; at high frequency the capacitor’s share shrinks and the output collapses.
The cutoff frequency ω_c = 1/RC
The boundary is the frequency where the resistor’s and capacitor’s impedances are equal. That is ω_c = 1/RC, and because the two equal impedances are at a right angle, the output drops to 1/√2 ≈ 0.707 of the input. In power that is one half, so it is called the −3 dB point. In phase, the output lags the input by exactly 45 degrees.
Above cutoff it rolls off at a fixed slope
Above the cutoff, each tenfold rise in frequency cuts the gain to one tenth — that is −20 dB per decade. This fixed slope is the signature of a first-order filter. On log axes the flat pass-band line and the −20 dB/decade falling line meet at a corner, and that corner is the cutoff frequency.
Back to the first screen
As you raised the frequency, the pass-through gain began flat, then bent down near the cutoff, and at exactly ω_c = 1/RC the output became 0.707 of the input, −3 dB. Above that it fell a steady −20 dB per tenfold. That the same circuit passes or blocks depending on which frequency you send is summed up by a single cutoff frequency.