RC Circuit Calculator
Calculate the time constant (τ = RC) and voltage response of an RC circuit during charging and discharging. Determine how long it takes for a capacitor to charge or discharge to a specific voltage. See also our Capacitance Calculator and Capacitor Energy Calculator.
How RC Circuits Work
An RC circuit consists of a resistor and capacitor connected in series. When voltage is applied, the capacitor charges through the resistor following an exponential curve. The time constant τ (tau) = R × C determines how quickly the capacitor charges or discharges. After one time constant, the capacitor reaches 63.2% of its final value. After five time constants (5τ), it is considered fully charged (99.3%).
The exponential behavior occurs because as the capacitor charges, the voltage across it increases, reducing the voltage across the resistor and thus reducing the charging current. This creates a self-limiting process where the rate of change decreases as the capacitor approaches its final voltage.
RC Circuit Formulas
Time Constant:
τ = R × C (seconds)
Charging (capacitor voltage):
V(t) = V₀ × (1 - e^(-t/τ))
Discharging (capacitor voltage):
V(t) = V₀ × e^(-t/τ)
Current:
Charging: I(t) = (V₀/R) × e^(-t/τ)
Discharging: I(t) = -(V₀/R) × e^(-t/τ)
Time to reach specific voltage:
Charging: t = -τ × ln(1 - V/V₀)
Discharging: t = -τ × ln(V/V₀)
Example Calculation
A 10kΩ resistor with a 100µF capacitor charged from a 12V supply:
τ = R × C = 10,000 × 100×10⁻⁶ = 1.0 second
Charging voltages:
At t = 1s (1τ): V = 12 × (1-e⁻¹) = 12 × 0.632 = 7.585V
At t = 2s (2τ): V = 12 × (1-e⁻²) = 12 × 0.865 = 10.375V
At t = 3s (3τ): V = 12 × (1-e⁻³) = 12 × 0.950 = 11.402V
At t = 5s (5τ): V = 12 × (1-e⁻⁵) = 12 × 0.993 = 11.919V
Time to reach 9V:
t = -1.0 × ln(1 - 9/12) = -1.0 × ln(0.25) = 1.386s
Initial charging current: I₀ = 12V / 10kΩ = 1.2mA
RC Time Constant Reference Table
| Time Constants | Charging % | Discharging % | Charging V (12V) | Discharging V (12V) |
|---|---|---|---|---|
| 0.5τ | 39.3% | 60.7% | 4.72V | 7.28V |
| 1τ | 63.2% | 36.8% | 7.59V | 4.41V |
| 2τ | 86.5% | 13.5% | 10.38V | 1.62V |
| 3τ | 95.0% | 5.0% | 11.40V | 0.60V |
| 4τ | 98.2% | 1.8% | 11.78V | 0.22V |
| 5τ | 99.3% | 0.7% | 11.92V | 0.08V |
Frequently Asked Questions
What is a time constant?
The time constant τ = RC is the time it takes for the capacitor voltage to reach 63.2% of its final value during charging (or drop to 36.8% during discharging). It has units of seconds when R is in ohms and C is in farads. The time constant characterizes how "fast" or "slow" the circuit responds to changes.
Why 5 time constants for "fully charged"?
After 5τ, the capacitor has reached 99.3% of its final voltage. Mathematically, the exponential never reaches exactly 100%, but 99.3% is close enough for practical purposes. In most circuits, the remaining 0.7% is within component tolerances and measurement uncertainty. Some precision applications may require 7τ or more.
How do I design an RC delay circuit?
Choose the desired delay time, then select R and C values whose product equals the needed time constant. For a 1-second delay: R=10kΩ, C=100µF (τ=1s). The trigger threshold determines the actual delay — at 63.2% threshold, delay = 1τ; at 90% threshold, delay = 2.3τ. Use a Schmitt trigger or comparator for clean switching at the threshold.
What is the cutoff frequency of an RC filter?
The -3dB cutoff frequency of an RC low-pass or high-pass filter is f = 1/(2πRC) = 1/(2πτ). At this frequency, the output is attenuated to 70.7% (-3dB) of the input. For example, R=10kΩ and C=100nF gives f = 159 Hz. Above this frequency (low-pass) or below it (high-pass), the signal is progressively attenuated at 20dB/decade.
Can I use RC circuits for power supply filtering?
Yes, RC filters are used for decoupling and noise filtering in power supplies. However, the resistor causes voltage drop and power loss under load. For main power filtering, LC filters (inductor + capacitor) are preferred because inductors have lower DC resistance. RC filters are best for low-current signal filtering and local IC decoupling where the current draw is small.
How does ESR affect RC circuit behavior?
Equivalent Series Resistance (ESR) is the internal resistance of a real capacitor. It adds to the external resistance R, making the effective time constant τ = (R + ESR) × C. For electrolytic capacitors, ESR can be 0.1-10Ω, which is significant in low-resistance circuits. ESR also causes power dissipation in the capacitor during rapid charge/discharge cycles, generating heat.
Practical Applications
- Timer circuits: 555 timer uses RC time constant for oscillation frequency
- Debouncing: RC filter removes switch bounce (typical: 10kΩ + 100nF = 1ms)
- Audio filters: Tone controls and crossover networks use RC filtering
- Power-on reset: RC delay ensures proper initialization sequence
- ADC sampling: RC settling time determines maximum sampling rate
- Snubber circuits: RC snubbers suppress voltage spikes across switches