RC circuit
A resistor and a capacitor: the simplest circuit with memory. See how it charges, how long it takes and how it filters out high frequencies.
The time constant τ
When a voltage is applied to an RC circuit, the capacitor does not charge instantly: the charging current has to flow through the resistor, and that current keeps falling as the capacitor voltage approaches the source voltage. The result is an exponential curve:
VC(t) = Vin · (1 − e−t/τ) with τ = R · C
The time constant τ sets the pace. After one τ the capacitor reaches 63.2% of the final voltage; after 3τ, 95%; and after 5τ, 99.3%, which in practice counts as fully charged. Discharging is the mirror-image curve: in τ it loses 63.2% of its voltage. With R = 10 kΩ and C = 1 µF, τ = 10 ms.
The RC circuit as a low-pass filter
For a sine wave, the capacitor’s impedance is ZC = 1/(j·2π·f·C): at low frequency it is almost an open circuit and the whole voltage appears across it; at high frequency it is almost a short circuit and the output tends to zero. The circuit lets low frequencies through and attenuates high ones. The boundary is the cutoff frequency:
fc = 1 / (2π · R · C)
At the cutoff frequency the output is 1/√2 ≈ 0.707 of the input (−3 dB) and lags by 45°. Above fc the attenuation grows by 20 dB per decade: every time the frequency is multiplied by 10, the amplitude is divided by 10. The phase shift tends to −90°.
How to read a Bode plot
A Bode plot shows gain in decibels and phase shift against frequency on a logarithmic scale. On that scale, the RC filter’s response is almost two straight lines (the dashed asymptotes) that meet at fc. Move the frequency slider: the vertical line marks where the signal sits on the Bode plot, and the bottom chart shows how attenuated and delayed the output is compared with the input.
Frequently asked questions
How long does a capacitor take to charge?
It depends on the time constant τ = R · C. After one τ the capacitor reaches 63 % of the final voltage, and after 5τ it reaches 99.3 %, which in practice counts as fully charged. With 10 kΩ and 100 µF, τ = 1 s, so it takes about 5 seconds to charge.
How do you calculate the cutoff frequency of an RC filter?
The cutoff frequency is fc = 1 / (2π · R · C). With R = 1 kΩ and C = 100 nF you get fc ≈ 1.59 kHz. To choose the parts, pick a standard capacitor value first and then solve for the resistor: R = 1 / (2π · fc · C).
What exactly happens at the cutoff frequency?
At the cutoff frequency the output drops by 3 dB, that is to 70.7 % of the input amplitude, and the signal lags by 45°. Above it, a first-order RC filter attenuates 20 dB per decade: at ten times the cutoff frequency the amplitude is roughly one tenth.
What is an RC circuit used for?
It is one of the most common circuits in electronics: it filters noise at the input of an ADC or a sensor, debounces push buttons, sets timing in timers such as the 555, smooths a PWM signal into a DC voltage and couples audio stages while blocking DC.