Time Constant Calculator
RC and RL transient constant
Required Parameters
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Quick Answer
The time constant (τ) determines how fast an RC or RL circuit responds to a step change. RC: τ = R×C. RL: τ = L/R. After 5τ, the circuit has settled to 99.3% of its final value.
Time Constant Calculator
Calculate the RC or RL time constant (tau) for charging and discharging circuits. The time constant determines how quickly a capacitor charges or an inductor builds up current.
RC Time Constant
tau = R x C
After one time constant, the capacitor reaches 63.2% of its final voltage. After 5 tau, it is considered fully charged (99.3%).
| Time | % Charged | % Remaining |
|---|---|---|
| 1 tau | 63.2% | 36.8% |
| 2 tau | 86.5% | 13.5% |
| 3 tau | 95.0% | 5.0% |
| 4 tau | 98.2% | 1.8% |
| 5 tau | 99.3% | 0.7% |
RL Time Constant
tau = L / R
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Design Notes
The time constant connects two fundamental domains: time response and frequency response. A circuit with τ = 1ms has a bandwidth of fc = 1/(2πτ) = 159 Hz. This means slower circuits (larger τ) filter more noise but respond more sluggishly to input changes. In digital design, minimizing parasitic RC (stray capacitance × trace resistance) is critical for achieving fast rise times: t_rise ≈ 2.2τ.
Common Mistakes
- 1
Forgetting that RL and RC time constants have opposite dependencies on R — increasing R makes RC slower but RL faster.
- 2
Assuming the circuit reaches final value after one time constant. One τ only reaches 63.2% — you need 5τ for practical completion.
- 3
Not accounting for source impedance when calculating the time constant of an RC network driven by a non-ideal source.
Engineering Handbox
1. τ = R × C = 10,000 × 47 × 10⁻⁶ = 0.47 seconds 2. Time to 95%: 3τ = 1.41 seconds 3. Time to 99.3%: 5τ = 2.35 seconds 4. Cutoff frequency: fc = 1/(2π × 0.47) = 0.339 Hz
Knowledge Base
What is one time constant (tau)?
One time constant (τ) is the time for an RC or RL circuit to reach 63.2% of its final value during charging (or decay to 36.8% during discharging). For RC circuits: τ = R × C. For RL circuits: τ = L / R. Example: 10kΩ × 100µF = 1 second.
How many time constants for full charge?
1τ = 63.2%, 2τ = 86.5%, 3τ = 95.0%, 4τ = 98.2%, 5τ = 99.3%. The circuit is generally considered 'fully settled' after 5 time constants. Technically, an exponential curve never reaches 100%, but 5τ is close enough for all practical purposes.
What is the RC time constant formula?
τ = R × C, where τ is in seconds, R in ohms, and C in farads. Example: R = 1MΩ, C = 1µF gives τ = 1 second. R = 10kΩ, C = 100nF gives τ = 1 millisecond. The time constant is independent of voltage — it only depends on component values.
What is the RL time constant formula?
τ = L / R, where τ is in seconds, L in henries, and R in ohms. Note this is L DIVIDED by R, opposite to RC. Higher resistance means FASTER response in RL circuits (shorter τ), while higher resistance means SLOWER response in RC circuits (longer τ).
What is the voltage equation during charging?
V(t) = V_final × (1 - e^(-t/τ)) for charging, and V(t) = V_initial × e^(-t/τ) for discharging. These exponential equations describe the exact voltage at any time t. At t = τ: charging reaches 63.2%, discharging falls to 36.8%.
How is the time constant used in filter design?
The -3dB cutoff frequency of a first-order RC filter is fc = 1/(2πτ) = 1/(2πRC). A longer time constant means a lower cutoff frequency. Example: τ = 1ms gives fc = 159 Hz. This directly connects transient response to frequency response — they are two views of the same behavior.
What determines rise time in digital circuits?
Rise time (10% to 90%) ≈ 2.2 × τ. For a digital signal passing through an RC network with τ = 1ns, rise time ≈ 2.2ns. This sets the bandwidth limit: BW ≈ 0.35/rise_time. Slower rise times mean less bandwidth, which is why minimizing parasitic RC is critical in high-speed design.
How do I measure the time constant of a real circuit?
Apply a step voltage and measure the output with an oscilloscope. The time constant is the time to reach 63.2% of the final value. Alternatively, measure the time to fall to 36.8% during discharge. For RC circuits, you can also measure R and C separately and multiply.
What happens with multiple RC stages?
Two cascaded RC stages do NOT simply give τ_total = τ1 + τ2. The interaction between stages creates a second-order response with different settling behavior. The dominant time constant determines the slowest response. For accurate analysis of cascaded stages, use the full transfer function or simulation.
Why is the time constant important for debouncing?
Mechanical switches bounce for 1-10ms. An RC filter with τ = 5-10ms (e.g., 10kΩ + 1µF) smooths these bounces, providing a clean digital signal. The RC network charges slowly enough to ignore the rapid bouncing but fast enough to respond to intentional presses (human minimum press ≈ 50ms).
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