RC Time Constant Calculator
Enter a resistor R and capacitor C to get the time constant τ = RC and the cutoff frequency of a first-order RC filter. A chart shows how far the capacitor has charged or discharged at the time you choose.
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Results
- Time constant τ = RC
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- Cutoff frequency 1/(2πRC)
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- 5τ (essentially fully charged)
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- Charged at time t
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- Remaining after discharge at time t
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The blue line is charging (as a percentage of the supply voltage) and the green line is discharging (as a percentage of the starting voltage). The horizontal axis is time in units of τ.
See it as a graph
Other inputs stay at their current values while the chosen input varies from 1/4 to 4 times its current value. The red dot is the current value.
What the time constant τ = RC means
When a capacitor C charges through a resistor R, its voltage approaches the supply voltage V as v(t) = V(1 − e^(−t/τ)). Here τ = RC is the time constant; ohms times farads gives seconds. At t = τ the voltage reaches 1 − e^(−1) ≈ 63.2% of its final value. With the defaults of R = 10 kΩ and C = 100 nF, τ = 10 kΩ × 100 nF = 1 ms, so at t = 1 ms the capacitor is 63.21% charged, or, when discharging, 36.79% of the voltage remains. Discharging follows v(t) = V0·e^(−t/τ) with the same τ.
How long it takes to charge fully
Mathematically the voltage only keeps approaching its final value and never reaches exactly 100%, so the question is how close is close enough. It reaches about 86.5% at 2τ, 95.0% at 3τ, 98.2% at 4τ and 99.3% at 5τ, so 5τ is the usual rule of thumb for “essentially fully charged”. With the defaults, 5τ = 5 ms. The time to reach 50% is ln2·τ ≈ 0.69τ, and the 10–90% rise time is ln9·τ ≈ 2.2τ. When a voltage must settle to high accuracy, as at an ADC input, it needs longer than 5τ.
Cutoff frequency and RC filters
For a first-order RC low-pass filter, the cutoff frequency is fc = 1/(2πRC). At this frequency the output amplitude is 1/√2 ≈ 0.707 of the input, or −3 dB, and the phase lags by 45°. Well above fc, the amplitude falls by about 20 dB for every tenfold increase in frequency. With the defaults, fc = 1/(2π × 1 ms) ≈ 159 Hz. Swapping R and C to make a high-pass filter gives the same cutoff formula. Both values assume a source with zero resistance and no load on the output.
Uses, and RC delay in interconnects
RC circuits are used to debounce switches (contacts that bounce and switch on and off rapidly), to filter noise, and to create delays for resets and timers. For debouncing, τ is usually made well longer than the bounce lasts, and the signal is read with a Schmitt-trigger input. Inside a chip, the resistance and capacitance of the wiring cause RC delay. Since both R and C of a wire grow with its length, the delay grows with the square of the length, and treated as a distributed line the 50% delay is about 0.38RC. Note also that capacitors have tolerances, and some types of ceramic capacitor change capacitance considerably with voltage and temperature.
Practice analog circuit questions
FAQ
What is the time constant?
In an RC circuit it is the product of resistance and capacitance, τ = RC, and it describes how fast the circuit responds to a step input. After a time τ, a charging capacitor reaches about 63.2% of its final voltage, and a discharging one falls to about 36.8%. For 10 kΩ and 100 nF, τ = 1 ms.
How long does a capacitor take to fully charge?
In theory it never reaches 100%, but it is at about 99.3% after 5τ, so 5τ is the usual rule of thumb for fully charged. For 10 kΩ and 100 nF that is 5 ms. If you need more accuracy, choose the time from the accuracy required; at 7τ, for example, it is about 99.9%.
What is the formula for the cutoff frequency of an RC filter?
For a first-order RC filter, fc = 1/(2πRC), for both low-pass and high-pass. In terms of the time constant, fc = 1/(2πτ). For 10 kΩ and 100 nF it is about 159 Hz, and at that frequency the amplitude is about 0.707 times the input (−3 dB).