Basics
An RC filter is a frequency-dependent voltage divider
A capacitor's reactance, Xc = 1/(2πfC), falls as frequency rises. Put R in series and C to ground (low-pass) and higher frequencies are shunted to ground; swap them — C in series, R to ground (high-pass) — and DC and low frequencies are blocked. Either way it's a divider made of R and Xc, and the cutoff frequency fc is where Xc equals R. At that point the output is 1/√2 = 0.707 of the input, −3.01 dB in voltage and exactly half the power — hence 'the −3 dB point' or 'half-power frequency'. The phase at fc is −45° for a low-pass and +45° for a high-pass, and beyond fc a first-order filter rolls off at 20 dB per decade (6 dB per octave).
Cutoff: fc = 1 / (2π × R × C) Low-pass gain: |H| = 1 / √(1 + (f/fc)²) phase φ = −atan(f/fc) High-pass gain: |H| = (f/fc) / √(1 + (f/fc)²) phase φ = 90° − atan(f/fc) Time constant: τ = R × C, Vc(t) = Vs × (1 − e^(−t/τ)) Time to p %: t = −τ × ln(1 − p/100), 10–90 % rise time = ln9 × τ ≈ 2.2τ
Worked examples
- 10 kΩ with 100 nF: fc = 1 ÷ (2π × 10,000 × 100×10⁻⁹) = 159 Hz. A decade above, at 1.59 kHz, the response is −20 dB (one-tenth the voltage); two decades above, −40 dB.
- A 1 kHz low-pass with a 10 nF capacitor: R = 1 ÷ (2π × 1,000 × 10×10⁻⁹) = 15.9 kΩ. The nearest E24 value, 16 kΩ, gives fc = 995 Hz (−0.5 %).
- τ = 10 kΩ × 10 µF = 0.1 s: the capacitor reaches 63.2 % at 1τ, 95 % at 3τ (0.3 s) and 99.3 % at 5τ (0.5 s) — the basis of power-on reset and soft-start delays.
| Elapsed time | Charging (Vc/Vs) | Discharging (remaining) |
|---|---|---|
| 1 τ | 63.2 % | 36.8 % |
| 2 τ | 86.5 % | 13.5 % |
| 3 τ | 95.0 % | 5.0 % |
| 4 τ | 98.2 % | 1.8 % |
| 5 τ | 99.3 % | 0.7 % |
555 timer
An oscillator bouncing between 1/3 and 2/3 of VCC
Inside an NE555 are three resistors dividing VCC into thirds, two comparators, a flip-flop and a discharge transistor. In astable mode the capacitor charges through R1 + R2 up to 2/3 VCC while the output is high, then discharges through R2 alone down to 1/3 VCC while it's low. Exponential charging from 1/3 to 2/3 takes exactly ln2 × RC ≈ 0.693RC, which is where the famous 1.44 (= 1/0.693) comes from. In monostable mode a trigger starts one pulse that lasts while C charges from 0 to 2/3 VCC: ln3 × RC ≈ 1.1RC. Because the thresholds are ratios of VCC, the timing barely moves with supply voltage.
Astable: tH = 0.693 × (R1 + R2) × C, tL = 0.693 × R2 × C
f = 1 / (tH + tL) ≈ 1.44 / ((R1 + 2·R2) × C)
duty = (R1 + R2) / (R1 + 2·R2) → always above 50 %
Design: R1 + 2R2 = 1 / (0.693 × f × C), R1 = (2D − 1)(R1+2R2), R2 = (1 − D)(R1+2R2)
Monostable: t = ln3 × R × C ≈ 1.1 × R × C- R1 = 1 kΩ, R2 = 10 kΩ, C = 10 nF: f = 1.44 ÷ (21 kΩ × 10 nF) = 6.86 kHz at 52.4 % duty. Making R1 much smaller than R2 pushes duty toward 50 %.
- 1 kHz at 60 % duty with C = 100 nF: R1 + 2R2 = 14.4 kΩ, so R1 = 2.89 kΩ and R2 = 5.77 kΩ. With E24 parts (3.0 kΩ and 5.6 kΩ) you get 1,016 Hz at 60.6 %.
- A 1 s one-shot with 10 µF: R = 1 ÷ (1.0986 × 10 µF) = 91 kΩ — conveniently an E24 value.
Standard values
From calculated values to real parts — E24 resistors, E6/E12 capacitors
You can't buy a 15.9 kΩ or 2.89 kΩ resistor off the shelf. Resistors come from the E24 series at ±5 % or E96 at ±1 %; capacitors, with ±10–20 % tolerance, are commonly stocked only in E6 (1.0, 1.5, 2.2, 3.3, 4.7, 6.8) or E12 values. The standard workflow is therefore: fix the capacitor at a stocked value first, then trim with the resistor. The calculator shows the nearest standard values next to each result, along with the frequency and error you'd actually get with them.
Pitfalls
Common mistakes
- Dropping the 2π: 1/(RC) is the angular frequency ω in rad/s. Forget to divide by 2π and your cutoff is off by a factor of 6.28.
- Ignoring source and load impedance: the driving stage's output impedance adds to R, and the next stage's input impedance sits in parallel with C. If the load isn't at least 10× R, buffer it.
- Cascading two RC sections and expecting −40 dB/decade: they load each other and the knee goes soft. For a steep response, buffer between stages or use an active second-order (Sallen-Key) filter.
- Timing with X7R or Y5V ceramics: high-k ceramics lose tens of percent of capacitance with DC bias and temperature. Use C0G/NP0 or film capacitors for timing and filters.
- Skipping 555 decoupling: a bipolar NE555 draws a big supply-current spike on every output transition. Put 0.1 µF (plus 10 µF) right at VCC and 10 nF on the CV pin.
In practice
Tips from KEMI's bench
On real boards RC networks mostly show up as ADC anti-aliasing filters, button debouncing and PWM-to-DC smoothing. In front of a SAR ADC, put fc well below half the sample rate, but also make C much larger than the ADC's internal sampling capacitor (a few nF or more) and keep R below the datasheet's maximum source impedance so the sample settles. For a pushbutton, 10 kΩ + 100 nF (τ = 1 ms) into a Schmitt-trigger input is plenty. The 555 is still handy for single-chip LED flashers and buzzer drivers, but if there's already an MCU on the board, a hardware timer is more accurate and saves parts. If you send us a schematic for review, we check the reasoning behind these RC constants too.