RC Low-Pass Filter

Frequency Response

So far we've analyzed DC circuits (constant voltages). Real signals have frequency content — they oscillate. Different frequencies pass through a circuit differently.

The RC Low-Pass Filter

Vin ---[R]---+--- Vout
             |
            [C]
             |
            GND

The capacitor's impedance is Xc = 1/(jωC), where ω = 2πf. At low frequencies, Xc is large → Vout ≈ Vin. At high frequencies, Xc is small → Vout ≈ 0.

Voltage Gain (Magnitude)

|H(f)| = 1 / sqrt(1 + (2πfRC)²)

Or equivalently using ω = 2πf:

|H| = 1 / sqrt(1 + (ωRC)²)

The Cutoff Frequency

At f_c = 1/(2πRC), gain = 1/√2 ≈ 0.7071 — the −3 dB point. This is where power is halved.

| f | |H| | Meaning | |---|-----|---------| | 0 | 1.0000 | DC passes through | | f_c | 0.7071 | −3 dB (half power) | | 10·f_c | 0.0995 | mostly blocked | | 100·f_c | 0.0100 | almost completely blocked |

Applications

• Audio: remove high-frequency hiss
• ADC anti-aliasing: remove frequencies above Nyquist
• Power supply: smooth rectified AC into near-DC
• Signal smoothing / averaging

Your Task

Implement double rc_lowpass(double f, double r, double c) that returns the voltage gain magnitude |H(f)|.

Use #include <math.h> for sqrt and define PI.