RC Filters: Reading the Bode Plot Behind Low-Pass, High-Pass and Band-Pass

One resistor, one capacitor, one corner frequency

A resistor and a capacitor in series form the simplest possible filter. A capacitor's impedance Z_C = 1/(jωC) falls as frequency rises, so where you take the output — across the capacitor or across the resistor — decides whether the circuit favours low or high frequencies. Taking the output across the capacitor gives a low-pass filter: at low ω the capacitor's impedance dominates and nearly all the input voltage appears across it; at high ω its impedance collapses toward zero and the output is shorted out. Swap the two components and you get a high-pass filter with the opposite behaviour.

Low-pass:   H(jω) = 1 / (1 + jωRC)
High-pass:  H(jω) = jωRC / (1 + jωRC)

cutoff (−3 dB) frequency, both cases:  f_c = 1 / (2πRC)

The Bode plot: magnitude in decibels, phase in degrees

A Bode plot shows a filter's transfer function magnitude in decibels, |H|_dB = 20·log₁₀|H(jω)|, against frequency on a logarithmic axis, alongside a phase plot. For the RC low-pass, far below f_c the magnitude sits flat at 0 dB (all the signal passes); far above f_c it falls in a straight line at −20 dB per decade (equivalently 6 dB per octave) — a direct consequence of |H| going as 1/ω once ωRC ≫ 1. At exactly f_c the magnitude has dropped by 3 dB, which is where the output power is half the input power, and the phase has shifted by exactly −45°, swinging from 0° far below f_c to −90° far above it. This is the standard first-order rolloff: every additional RC stage cascaded on adds another −20 dB/decade to the slope far past its own cutoff.

Band-pass: cascading a high-pass and a low-pass

A band-pass filter passes a range of frequencies while rejecting both very low and very high ones. The simplest RC realisation cascades a high-pass stage (rejects low frequencies) into a low-pass stage (rejects high frequencies), with the high-pass cutoff set well below the low-pass cutoff so there is a passband between them. Its quality factor Q — how sharply it selects its band relative to its centre frequency — is comparatively poor for a passive RC cascade, because each stage loads the other; LC or active (op-amp) filters are used when a sharp, high-Q band-pass response is needed, but the RC cascade is enough to demonstrate the principle and is genuinely used for coarse tone shaping in audio circuits.

Why the corner isn't a hard wall

A first-order RC filter's rolloff is gentle by engineering standards — −20 dB/decade means a frequency 10× past the cutoff is only attenuated by a factor of 10 in amplitude, not eliminated. Sharper filters (Butterworth, Chebyshev, elliptic) stack multiple reactive stages, each contributing another pole to the transfer function and another −20 dB/decade of eventual rolloff, at the cost of more components and, for anything beyond a maximally-flat Butterworth response, ripple in the passband or stopband. The single RC stage modelled here is the building block every one of those more elaborate filters is assembled from.

Time domain: the same RC, a different picture

Everything above describes steady-state sinusoidal response, but the same RC pair also has a clean time-domain story: driven by a step input, the low-pass output charges toward the new voltage as V(t) = V_final(1 − e^(−t/RC)), an exponential approach whose time constant τ = RC is the same RC that set the cutoff frequency (f_c = 1/(2πτ)). A filter with a low cutoff frequency (large RC) is, equivalently, a circuit that responds sluggishly to a sudden step — frequency response and transient response are two views of the same underlying pole.

Frequently asked questions

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