Impedance & RC Filters
A capacitor is a resistor whose value depends on frequency. Pair it with a real resistor and you can choose which frequencies survive.
Builds on: 2.3 Capacitors & the RC Time Constant2.2 Voltage Dividers5.1 Alternating Current
Reactance: resistance with a frequency knob
To steady DC, a charged capacitor is a wall (Lesson 2.3). But wiggle the voltage and the capacitor never finishes charging — current flows continuously to chase the changes. The faster the wiggle, the easier the flow. This frequency-dependent “resistance” is called capacitive reactance:
A divider that plays favourites
Now revisit the voltage divider (Lesson 2.2) and replace the bottom resistor with a capacitor. At low frequencies XC is huge → the output gets nearly everything. At high frequencies XC collapses → the output is shorted away. Congratulations: a low-pass filter. Swap R and C and you get the high-pass — blocks the lows, passes the highs.
The boundary between “passed” and “blocked” is the cutoff frequency, where XC = R:
The transition is gentle, not a cliff — an octave above cutoff a low-pass still leaks nearly half the amplitude (0.45×). Engineers describe the rolloff in decibels: this single-RC filter falls 6 dB per octave; sharper filters stack more stages.
Filters are everywhere
- Tone controls & EQ: bass and treble knobs are literally variable RC filters.
- Speaker crossovers: low-pass to the woofer, high-pass to the tweeter.
- Cleanup: low-pass filters smooth noisy sensor lines and — remember this for the capstone — turn fast PWM pulses into a steady average.
- Radio: add an inductor for LC resonance and you can select one station out of the whole spectrum. (A perfect topic for a future unit.)
Nothing new was invented here: XC slots into the divider formula where R₂ used to be. Advanced electronics keeps re-using the same five ideas at higher speed — that’s the secret nobody tells beginners.
⚡ Lab — The Frequency Sieve
An RC filter with a sine generator, plus its full frequency-response curve.
- Low-pass, fc ≈ 1.6 kHz (1 kΩ + 100 nF): sweep the input from 20 Hz to 20 kHz and watch the output die.
- Park the input at fc: exactly 0.71× and 45° of phase lag.
- Flip to high-pass — the response curve mirrors. Bass gone, treble through.