RC Circuit Calculator
Enter a resistance and capacitance to instantly compute the RC time constant (tau), cutoff frequency, capacitive reactance, impedance, phase shift, and filter gain. Switch between series and parallel configurations, add a supply voltage to see the charging and discharging voltage at any time t, and view the full charge curve in the chart below.
What is the RC time constant?
The RC time constant (tau, in seconds) is the product of resistance R (in Ohms) and capacitance C (in Farads): tau = R x C. It describes how quickly a capacitor charges or discharges through a resistor. After one time constant, a charging capacitor reaches 63.2% of the supply voltage; after five time constants it is considered fully charged at 99.3%. The same fractions apply in reverse during discharging: after one tau the voltage has fallen to 36.8%, and after five tau it is effectively zero. This exponential behavior is the foundation of RC timing circuits, debounce filters, and integrators.
Cutoff frequency and RC filters
Every RC circuit acts as a frequency-selective filter. The cutoff (or corner) frequency fc = 1 / (2 * pi * R * C) is the point at which the output amplitude falls to 1/sqrt(2), approximately 70.7%, of its low-frequency value, which corresponds to -3 dB. In a low-pass filter (output taken across the capacitor), signals below fc pass largely unchanged while higher frequencies are attenuated. In a high-pass filter (output across the resistor), the situation is reversed: signals above fc pass and lower frequencies are attenuated. The attenuation beyond the cutoff rolls off at -20 dB per decade (6 dB per octave) for a single-stage RC filter.
Phase shift and impedance
A capacitor introduces a phase shift between current and voltage. In a series RC circuit, the impedance is Z = sqrt(R^2 + Xc^2), where Xc = 1 / (2 * pi * f * C) is the capacitive reactance at signal frequency f. For a low-pass filter, the output lags the input by an angle theta = -arctan(f / fc); at exactly the cutoff frequency, the lag is -45 degrees. For a high-pass filter, the output leads by theta = +arctan(fc / f), giving +45 degrees at fc. These phase relationships matter in oscillator design, audio crossovers, and signal processing where timing alignment is critical.
Practical design guidance
When designing an RC filter or timing circuit, start with the desired time constant or cutoff frequency and then choose convenient standard component values. If tau = R x C = 1 ms is needed, you could use 10 kOhm with 100 uF, or 100 kOhm with 10 uF. Prefer resistors in the 1 kOhm to 100 kOhm range for audio frequencies to minimize noise and loading effects. For high-frequency RF filters, move to smaller capacitances (pF to nF range) with lower resistances. Remember that real capacitors have tolerances of 5-20% and equivalent series resistance (ESR) that affect high-frequency performance, so include margin in timing-critical designs.
Capacitor charge level vs. time constants
| Time (multiples of tau) | Charging (% of V) | Discharging (% of V) | Note |
|---|---|---|---|
| 0.5 tau | 39.3% | 60.7% | |
| 1 tau | 63.2% | 36.8% | Definition of tau |
| 2 tau | 86.5% | 13.5% | |
| 3 tau | 95.0% | 5.0% | Considered charged for many designs |
| 4 tau | 98.2% | 1.8% | |
| 5 tau | 99.3% | 0.7% | Fully charged/discharged by convention |
Standard reference values for charging (% of supply voltage) and discharging (% remaining).
Frequently asked questions
What does the RC time constant tell you?
The time constant tau = R x C tells you how quickly the capacitor in the circuit charges or discharges. After one tau, a charging capacitor reaches 63.2% of the applied voltage. After five time constants, it is considered fully charged (99.3%). A smaller tau means a faster response, useful for high-frequency filtering. A larger tau means a slower response, useful for long-duration timers or low-frequency smoothing.
How do I calculate the cutoff frequency of an RC filter?
The cutoff frequency fc = 1 / (2 * pi * R * C). At this frequency, the output voltage falls to 70.7% (-3 dB) of the input. For example, with R = 10 kOhm and C = 100 uF, tau = 10,000 x 0.0001 = 1 s and fc = 1 / (2 * pi * 1) = approximately 0.159 Hz. To raise fc, reduce R or C; to lower fc, increase either.
What is the difference between a low-pass and high-pass RC filter?
In a low-pass filter, the output is taken across the capacitor. Low-frequency signals pass through with little attenuation, while high-frequency signals are blocked. In a high-pass filter, the output is taken across the resistor, so high-frequency signals pass and low-frequency signals are attenuated. Both share the same cutoff frequency fc = 1 / (2 * pi * R * C), but the phase shift differs: the low-pass output lags the input by up to 90 degrees, while the high-pass output leads by up to 90 degrees.
What is capacitive reactance?
Capacitive reactance Xc = 1 / (2 * pi * f * C) is the opposition a capacitor presents to an AC signal at frequency f. Unlike resistance, it varies with frequency: at low frequencies Xc is very high (the capacitor blocks DC), and at high frequencies Xc is low (the capacitor passes the signal freely). Reactance is measured in Ohms and combines with resistance to form the total impedance Z = sqrt(R^2 + Xc^2) in a series RC circuit.
How many time constants does it take to fully charge a capacitor?
By convention, a capacitor is considered fully charged after 5 time constants (5 tau), at which point it holds 99.3% of the supply voltage. At 3 tau it holds 95%, which is sufficient for many practical designs. Theoretically, a capacitor never reaches exactly 100% because the charging follows an exponential curve that asymptotically approaches the supply voltage.
What is the phase shift at the cutoff frequency?
At exactly the cutoff frequency, the phase shift is 45 degrees. For a low-pass filter, the output lags the input by 45 degrees. For a high-pass filter, the output leads the input by 45 degrees. Below fc, the lag approaches 0 degrees for a low-pass filter; above fc, it approaches 90 degrees. The reverse applies for a high-pass filter.
How do series and parallel RC circuits differ?
In a series RC circuit, the resistor and capacitor are connected in a chain: the same current flows through both. The total impedance is Z = sqrt(R^2 + Xc^2). In a parallel RC circuit, the resistor and capacitor share the same two terminals, so the same voltage appears across both but the currents differ. The parallel impedance magnitude is Z = R*Xc / sqrt(R^2 + Xc^2), which is always less than either R or Xc alone. The time constant tau = R x C is the same in both configurations because it depends only on the product of R and C.