Electronics
Low-pass filter explained
A low-pass filter passes low frequencies and weakens high ones. The simplest is a resistor in series with a capacitor to ground, with the output taken across the capacitor. Its cutoff frequency is fc = 1 ÷ (2π R C), where the output has fallen to 70.7% of the input.
A divider that hears pitch
Think of it as a voltage divider whose bottom part is a capacitor. A capacitor's opposition to AC, its reactance, shrinks as the frequency rises:
Low notes see a huge reactance, so the capacitor keeps nearly all their voltage. High notes see a tiny one and drain away to ground. It's like bass thumping through a wall while the voices don't.
Live experiment
Sweep a note through a filter
A note passes through an RC filter, 1.6 kΩ and 100 nF, with its corner at about 995 Hz. Slide the note from 20 Hz to 20 kHz and watch what reaches Pip shrink past the corner. Then flip it into a high-pass.
“Low- and high-pass filters” is lesson 4 of 6 in Oscillation Coast, Electronics World 4. Free with an account. Lessons unlock in order, so you’ll do the 24 before it first.
The cutoff frequency
The cutoff, or corner, is where the reactance equals R:
There the output is 1 ÷ √2 ≈ 70.7% of the input, which is −3 dB, and it lags the input by 45°.
Say you want a 1 kHz low-pass with a 10 nF capacitor. Turn the formula round: R = 1 ÷ (2π × 1000 × 10⁻⁸) ≈ 15.9 kΩ. The nearest E12 value, 15 kΩ, puts the corner at 1.06 kHz.
Above the corner
Well above the cutoff, the output is about fc ÷ f of the input. Ten times the cutoff frequency leaves about a tenth, −20 dB; a hundred times leaves a hundredth. That steady slope, 20 dB per decade, is what a single resistor and capacitor can do.
The full response is Vout ÷ Vin = 1 ÷ √(1 + (f ÷ fc)²), which gives 70.7% at the corner and the fc ÷ f rule far above it.
Flip it into a high-pass
Swap the two parts, with the capacitor in series and the output across the resistor, and you get a high-pass filter with the same cutoff. Most audio inputs have one built in: a coupling capacitor that passes the signal and blocks any steady DC.
Filters are everywhere. Traditional phone calls keep only about 300 Hz to 3.4 kHz, which is why voices sound thin, and tone knobs are adjustable filters.
Free lesson
Now build it yourself
Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.
“Low- and high-pass filters” is lesson 4 of 6 in Oscillation Coast, Electronics World 4. Free with an account. Lessons unlock in order, so you’ll do the 24 before it first.
Or start from the very first lesson, no account neededWork it out
- RC time constant calculatorτ, charge time, cutoff frequency and the voltage at any moment.
- RC filter cutoff frequency calculatorThe cutoff from R and C, the part for a cutoff you want, and the gain at any frequency.
- RC filter practice problemsCutoff frequency, choosing R or C, and the output either side of the cutoff, with the working.
- Decibel practice problemsRatios to dB and back, for power and voltage, gains that add and the −3 dB point.
Practice questions
Three quick questions from the lessons. Pick an answer to check it.
Show the answer
Answer: the capacitor. At high frequencies the capacitor's reactance gets small and drains the signal to ground, so the voltage across it falls. Across the resistor you'd get a high-pass.
- Why not “the resistor”? Taking the output across the resistor makes a high-pass filter: the capacitor blocks low frequencies, so the resistor's voltage grows with frequency.
- Why not “the source”? The source's voltage is the input itself. Taken there, nothing would be filtered at all.
Show the answer
Answer: 70.7% of the input. At fc, XC = R and the output is 1 over √2 ≈ 70.7% of the input. That's half the power, since 0.707² = 0.5, which is where the 50% idea comes from.
- Why not “50% of the input”? 50% is the power at the cutoff, not the voltage. Half the power means the voltage is 1 ÷ √2 of the input, about 70.7%.
- Why not “100% of the input”? The full input only comes through well below the cutoff. At the cutoff the capacitor's reactance equals R, and the output is already 3 dB down.
Show the answer
Answer: 90° out of phase. The resistor's voltage is in phase with the current, and a capacitor's voltage lags its current by a quarter cycle. So they're 90° out of phase, at the same frequency.
- Why not “in phase”? Only the resistor's voltage is in step with the current. A capacitor's voltage lags its current by a quarter cycle, so the two can't be in step.
- Why not “180° out of phase”? 180° apart means exactly opposite, as a coil's and a capacitor's voltages are. A resistor and a capacitor are only a quarter cycle, 90°, apart.
“Low- and high-pass filters” is lesson 4 of 6 in Oscillation Coast, Electronics World 4. Free with an account. Lessons unlock in order, so you’ll do the 24 before it first.
Questions people ask
What does a low-pass filter do?
It passes low frequencies and weakens high ones. It never cuts a frequency off completely: above the cutoff, each step higher just leaves a smaller share. Uses include tone controls, removing hiss and smoothing a PWM signal into a steady voltage.
How do you calculate the cutoff frequency of an RC filter?
f_c = 1 ÷ (2π × R × C). With 10 kΩ and 10 nF, that's 1 ÷ (2π × 10,000 × 0.00000001) ≈ 1.6 kHz. To design one, pick C and solve for R = 1 ÷ (2π × f_c × C).
Why is the cutoff called the −3 dB point?
At the cutoff the output voltage is 1 ÷ √2, about 70.7% of the input, which is −3 dB. The voltage hasn't halved: the power has, because 0.707² = 0.5.
What is the difference between a low-pass and a high-pass filter?
Swap the parts. A low-pass has the resistor in series and the capacitor to ground, with the output across the capacitor. A high-pass has the capacitor in series and the output across the resistor, so it passes high frequencies and blocks DC. Both have the same cutoff formula.
Related topics
- RC time constant explainedτ = R × C: how long a capacitor takes to charge through a resistor, and the 63% rule.
- Decibels explaineddB as a ratio on a log scale: 10 log for power, 20 log for voltage, and why gains just add.
- AC voltage and RMS explainedPeak, period, frequency and RMS: the numbers that describe an AC sine wave.