RC Time Constant Calculator

Calculate time constant, resistance, capacitance, voltage at any time, or time to reach a target voltage in RC circuits.

RC Circuit Formulas: Time Constant: τ = R × C Where: • τ (tau) = time constant (seconds) • R = resistance (ohms, Ω) • C = capacitance (farads, F) Charging (capacitor initially at 0V): Vc(t) = Vs × (1 - e^(-t/τ)) Discharging (capacitor initially at V0): Vc(t) = V0 × e^(-t/τ) Time to reach voltage: • Charging: t = -τ × ln(1 - Vc/Vs) • Discharging: t = -τ × ln(Vc/V0) Current: • Charging: I(t) = (Vs/R) × e^(-t/τ) • Discharging: I(t) = (V0/R) × e^(-t/τ) Cutoff Frequency (RC filter): fc = 1/(2πτ) = 1/(2πRC) Key Time Points: • 1τ: 63.2% charged / 36.8% remaining • 2τ: 86.5% charged / 13.5% remaining • 3τ: 95.0% charged / 5.0% remaining • 4τ: 98.2% charged / 1.8% remaining • 5τ: 99.3% charged / 0.7% remaining
Example 1 (Time Constant): R = 10 kΩ, C = 100 µF τ = 10,000 × 0.0001 = 1 second fc = 1/(2π × 1) = 0.159 Hz Time to 99% = 5τ = 5 seconds Example 2 (Voltage at Time - Charging): R = 1 kΩ, C = 47 µF, Vs = 12V, t = 50 ms τ = 1000 × 0.000047 = 0.047 s = 47 ms Vc = 12 × (1 - e^(-50/47)) Vc = 12 × (1 - e^(-1.064)) Vc = 12 × 0.655 = 7.86V Example 3 (Time to Voltage - Charging): R = 4.7 kΩ, C = 22 µF, Vs = 5V, Target = 3.3V τ = 4700 × 0.000022 = 0.1034 s = 103.4 ms t = -103.4 × ln(1 - 3.3/5) t = -103.4 × ln(0.34) t = -103.4 × (-1.078) = 111.5 ms Example 4 (Camera Flash - Discharging): R = 100 Ω, C = 1000 µF, V0 = 330V, t = 10 ms τ = 100 × 0.001 = 0.1 s = 100 ms Vc = 330 × e^(-10/100) Vc = 330 × e^(-0.1) Vc = 330 × 0.905 = 298.6V Example 5 (Low-Pass Filter): Cutoff frequency needed: 1 kHz, C = 100 nF R = 1/(2π × fc × C) R = 1/(2π × 1000 × 0.0000001) R = 1591.5 Ω ≈ 1.6 kΩ (use standard value) τ = 1/(2π × 1000) = 0.159 ms Example 6 (555 Timer): For 1 second LED flash: C = 100 µF R = τ/C = 1/0.0001 = 10 kΩ Actually τ ≈ 1.1RC for 555, so use R ≈ 9.1 kΩ

What is the RC time constant?

The RC time constant (τ or tau) is the time required for a capacitor to charge to 63.2% of the applied voltage or discharge to 36.8% of its initial voltage. It's calculated as τ = R × C, where R is resistance in ohms and C is capacitance in farads.

Why is 63.2% significant in RC circuits?

This comes from the exponential function e^(-1) ≈ 0.368. After one time constant, voltage reaches (1 - e^(-1)) = 0.632 or 63.2% during charging. It's the mathematical result of exponential charging/discharging behavior in RC circuits.

How long does it take for a capacitor to fully charge?

Theoretically, infinite time for 100% charge. Practically, after 5 time constants (5τ), the capacitor reaches 99.3% charge, which is considered "fully charged." For τ = 1 ms, full charge takes about 5 ms.

What is the charging formula for RC circuits?

Vc(t) = Vs × (1 - e^(-t/τ)), where Vc is capacitor voltage at time t, Vs is source voltage, τ is time constant, and e is Euler's number (2.718). This describes the exponential rise from 0V to Vs.

What is the discharging formula for RC circuits?

Vc(t) = V0 × e^(-t/τ), where Vc is capacitor voltage at time t, V0 is initial voltage, and τ is time constant. This describes exponential decay from V0 to 0V as the capacitor discharges through the resistor.

How do I calculate the time to reach a specific voltage?

For charging: t = -τ × ln(1 - Vc/Vs). For discharging: t = -τ × ln(Vc/V0). For example, to charge to 50% of source voltage, t = -τ × ln(0.5) ≈ 0.693τ.

What are practical applications of RC circuits?

RC circuits are used in filters (low-pass, high-pass), timing circuits, smoothing power supplies, camera flashes, audio tone controls, coupling/decoupling in amplifiers, and debouncing switches. They're fundamental building blocks in electronics.

How does resistance affect charging time?

Higher resistance increases charging time (larger τ). For example, with C = 100 µF: R = 1 kΩ gives τ = 0.1 s, but R = 10 kΩ gives τ = 1 s. The resistor limits current flow, slowing the charge rate.

How does capacitance affect the time constant?

Larger capacitance increases the time constant. A bigger capacitor stores more charge, taking longer to charge/discharge. For R = 1 kΩ: C = 10 µF gives τ = 10 ms, while C = 100 µF gives τ = 100 ms.

What is the cutoff frequency of an RC filter?

The cutoff frequency (3 dB point) is fc = 1/(2πRC) = 1/(2πτ). This is where the filter attenuates signals by 3 dB (about 70.7% of input). For τ = 1 ms, fc = 159 Hz. It defines the transition between pass and stop bands.

Can I use electrolytic capacitors in RC timing circuits?

Use polarized electrolytics carefully - they must be connected with correct polarity and work only for DC or pulsed DC. For AC coupling or timing circuits that reverse polarity, use non-polarized capacitors (ceramic, film, or tantalum).

How accurate are RC timing circuits?

Component tolerances affect accuracy. Standard resistors (±5%) and capacitors (±10-20%) can give ±15-25% timing error. Use precision components (±1% resistors, ±5% capacitors) for accurate timing, or use crystal oscillators for precision applications.

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