RC & RL Time Constant Calculator

About the RC & RL Time Constant Calculator

A resistor and a capacitor (or an inductor) change in the same way every time: quickly at first, then more and more slowly, with a time constant τ that sets the pace. τ = R × C for a capacitor and τ = L ÷ R for an inductor; after one τ the change is 63.2% done, after five it is 99.3% done. Enter R and C (or L), the source voltage and the starting value, and the calculator gives τ, the voltage and current at any moment you choose, the time to reach a threshold, the 1τ to 5τ table, the stored energy and a chart of the whole curve.

Two presets cover jobs engineers do often. Supercap hold-up works out how long a charged supercapacitor keeps a load above its minimum voltage, for a constant current, a constant power (a DC-DC converter) or a resistor, including the drop across the capacitor’s ESR. I²C pull-up gives the range of pull-up resistors from the bus capacitance, the supply and the speed mode, with the limits of the NXP I²C-bus specification, and rates the standard values in that range.

Without a pass, the free preview works on your own values and shows the names of each table’s first rows and the chart under a watermark, without its numbers; the time constant, the hold-up time, the pull-up range, every other figure, the working, Copy result and the CSV come with a Pro pass.

The first-order step response

Every voltage and current in a circuit with one resistor and one capacitor or inductor follows x(t) = x∞ + (x0 − x∞) × e^(−t/τ), where x0 is the value at the start and x∞ the value it heads for. Solving it for t gives the time to reach a value x: t = τ × ln((x0 − x∞) ÷ (x − x∞)).

• After 1τ, 2τ, 3τ, 4τ and 5τ the change is 63.21%, 86.47%, 95.02%, 98.17% and 99.33% done.
• Half of the change takes τ × ln 2 = 0.693τ; 10% to 90% takes 2.197τ.
• The final value is approached, never reached: that is why 5τ is the usual “fully charged”.
• The same R and C used as a filter have their −3 dB point at fc = 1 ÷ (2πτ).

Capacitors and inductors

• Capacitor charging from V0 toward the source Vs: v(t) = Vs + (V0 − Vs) × e^(−t/τ), and the current is (Vs − v) ÷ R. Discharging through R: v(t) = V0 × e^(−t/τ).
• Inductor energising from a source Vs through R: the current heads for Vs ÷ R, and the voltage across the inductor is (I∞ − i) × R, Vs at the start. De-energising through R: i(t) = I0 × e^(−t/τ).
• Energy: a capacitor stores ½CV², an inductor ½LI². Charging a capacitor from 0 V through any resistor turns as much energy into heat in the resistor as ends up in the capacitor.
• Inductive kick: an inductor’s current cannot stop at once. When the source is removed the voltage across it jumps to I0 × R of the path it discharges through — without such a path (a flyback diode) the voltage rises until something breaks down.

Supercapacitor hold-up time

From i = C × dv/dt, with the capacitor’s ESR in series with the load:

• Constant current I: the terminal voltage drops by I × ESR at once, then falls in a straight line: t = C × (V1 − V2 − I × ESR) ÷ I.
• Constant power P (a DC-DC converter takes more current as the voltage falls): without ESR, t = C × (V1² − V2²) ÷ 2P. With ESR the calculator integrates the exact current, and warns when the load would pass the maximum-power point (the voltage at the load falling to half the capacitor’s) before the minimum voltage.
• Resistor R: the voltage decays with τ = (R + ESR) × C: t = (R + ESR) × C × ln(V1 ÷ Vc,end), with Vc,end = V2 × (R + ESR) ÷ R.

Only the energy between the two voltages is usable: going from 5 V down to 2.5 V uses 75% of what is stored.

I²C pull-up resistors

The NXP I²C-bus specification (UM10204, section 7.1) sets both limits. The resistor must be large enough for every device to pull the line down to VOL: Rp(min) = (VDD − VOL(max)) ÷ IOL, with VOL = 0.4 V at 3 mA in Standard-mode and Fast-mode and at 20 mA in Fast-mode Plus (0.2 × VDD at 2 mA when VDD is 2 V or less). It must be small enough for the line to rise from 30% to 70% of VDD within the rise time: Rp(max) = tr ÷ (0.8473 × Cb), where 0.8473 = ln(0.7 ÷ 0.3).

• Standard-mode (100 kbit/s): tr ≤ 1000 ns, Cb ≤ 400 pF
• Fast-mode (400 kbit/s): tr ≤ 300 ns, Cb ≤ 400 pF
• Fast-mode Plus (1 Mbit/s): tr ≤ 120 ns, Cb ≤ 550 pF

Sources

• NXP UM10204, I²C-bus specification and user manual: pull-up resistor sizing and the bus-line characteristics
• Texas Instruments SLVA689, I2C Bus Pullup Resistor Calculation
• RC time constant: τ = RC and the 63.2% point

Limitations

• Parts are ideal: a real capacitor’s leakage, dielectric absorption and ESR, and an inductor’s winding resistance and core saturation, are not included (except the ESR you enter for the hold-up time).
• R is the whole resistance in the loop: add the source’s output resistance and anything else in series.
• Supercapacitor capacitance and ESR change with temperature and age; use the end-of-life figures from the datasheet for a hold-up time you can rely on.
• The I²C limits assume a resistor pull-up; buses with a current-source pull-up, bus buffers or series resistors on each line need the extra terms of the specification.

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