RC time constant and RLC circuit calculator

RC and RL time constant and the time to charge or discharge to any percent; series RLC resonant frequency, impedance, phase, Q factor and bandwidth.

업데이트 2026년 9월 28일검증한 예제: 7

Time constant τ

유효 자릿수: 6; 가장 가까운 값, 중간값은 0에서 먼 쪽으로

Time to reach the level: 2.30259s
Final value: 5V
Energy stored when fully charged: 0.00125J

τ = 1 s: after one τ the capacitor voltage has risen to 63.2% of 5 V, and it takes 2.303 s to reach 90%. After 5τ it is within 1% of the final value.

Capacitor voltage over five time constants

계산 방법 S

Time constant
$\tau = RC = (10{,}000)(0.0001) = 1\ \mathrm{s}$
Rise towards the final value
$x(t) = X_\infty\left(1 - e^{-t/\tau}\right),\quad X_\infty = 5\ \mathrm{V}$
Time to reach 90%
$t = -\tau\ln(1 - 0.9) = 2.30259\ \mathrm{s}$
Stored energy when fully charged
$E = \tfrac12 CV^2 = 0.00125\ \mathrm{J}$

RC time constant and RLC circuit calculator 소개

An RC circuit responds with time constant τ = RC and an RL circuit with τ = L/R: after one τ the capacitor voltage or inductor current has covered 63.2% of the way to its final value, and reaching a fraction p takes t = −τ ln(1 − p). For a series RLC circuit the calculator finds the resonant frequency f₀ = 1/(2π√(LC)), the impedance and phase at your signal frequency from the reactances 2πfL and 1/(2πfC), and Q = (1/R)√(L/C).

The default, 10 kΩ with 100 µF on a 5 V supply, gives τ = 1 s and 2.30 s to reach 90%. RC time constants set timer delays, switch debounce periods and filter cut-off frequencies, 1/(2πRC). With 10 Ω, 10 mH and 1 µF in series, f₀ = 1,591.55 Hz and Q = 10.

Components are ideal and the input is a step: no capacitor leakage and no inductor winding resistance beyond R. After 5τ the response is within 0.67% of its final value (e⁻⁵), the usual rule for "fully charged".

계산 예제

10 kΩ and 100 µF charging to 90%

Circuit: RC
Resistance R: 10
Resistance unit: kΩ
Capacitance C: 100
Capacitance unit: µF
Process: Charging / rising
Source voltage: 5 V
Time to reach this level: 90%

Time constant τ: 1 s
Time to reach the level: 2.30259 s
Energy stored when fully charged: 0.00125 J

검증 출처: Python 3.8 math: τ = 10⁴ × 10⁻⁴ = 1 s; t = τ ln 10; E = ½CV² = ½·10⁻⁴·25

Capacitor voltage after one τ

Circuit: RC
Resistance R: 10
Resistance unit: kΩ
Capacitance C: 100
Capacitance unit: µF
Process: Charging / rising
Source voltage: 5 V
Time to reach this level: 90%
Value at time: 1 s

Value at the chosen time: 3.1606 V

검증 출처: Python 3.8 math: 5(1 − e⁻¹) = 3.1606028

Discharging to half: t = τ ln 2

Circuit: RC
Resistance R: 10
Resistance unit: kΩ
Capacitance C: 100
Capacitance unit: µF
Process: Discharging / decaying
Source voltage: 5 V
Time to reach this level: 50%

Time to reach the level: 0.693147 s

검증 출처: Python 3.8 math: ln 2 = 0.6931472 (τ = 1 s)

RL: 100 Ω and 10 mH

Circuit: RL
Resistance R: 100
Resistance unit: Ω
Inductance L: 10
Inductance unit: mH
Process: Charging / rising
Source voltage: 5 V
Time to reach this level: 90%
Show times in: s
Value at time: 50 µs

Time constant τ: 0.0001 s
Final value: 0.05 A
Value at the chosen time: 0.019673 A

검증 출처: Python 3.8 math: τ = L/R = 10⁻⁴ s, I = 5/100, i(50 µs) = 0.05(1 − e^−0.5) = 0.0196735

자주 묻는 질문

What is the RC time constant?

τ = R × C, in seconds when R is in ohms and C in farads. It is the time a charging capacitor takes to reach 63.2% (1 − e⁻¹) of the supply voltage, or a discharging one to fall to 36.8%. 10 kΩ with 100 µF gives τ = 1 s; 1 kΩ with 1 µF gives 1 ms.

How long does it take a capacitor to fully charge?

About five time constants. After 5τ the voltage is 99.33% of the supply (1 − e⁻⁵), which engineers treat as fully charged; in theory it never quite gets there. Reaching 90% takes τ ln 10 ≈ 2.303τ and 99% takes τ ln 100 ≈ 4.605τ, so a 10 kΩ, 100 µF circuit reaches 99% in 4.6 s.

How do you calculate the resonant frequency of an RLC circuit?

f₀ = 1/(2π√(LC)), with L in henries and C in farads; in a series circuit R does not shift it. 10 mH with 1 µF resonates at 1,591.55 Hz, and 100 µH with 100 pF at 1.59 MHz. At f₀ the inductive and capacitive reactances are equal and cancel, so the impedance falls to R alone and the current peaks.

What is the Q factor of a series RLC circuit?

Q = (1/R)√(L/C), the ratio of either reactance at resonance to the resistance. It sets the −3 dB bandwidth, Δf = f₀/Q: 10 Ω, 10 mH and 1 µF give Q = 10 and a 159 Hz band around 1,592 Hz. Above Q = 0.5 the circuit rings (underdamped); exactly 0.5 is critically damped, reached here at R = 2√(L/C) = 200 Ω.

What is the time constant of an RL circuit?

τ = L/R. It is the time for the current to rise to 63.2% of its final value V/R after the switch closes, or to fall to 36.8% once the source is removed and the coil discharges through R. 10 mH with 100 Ω gives τ = 100 µs, so on 5 V the current reaches 99% of 50 mA in about 460 µs.

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정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 7. 예를 들어 “10 kΩ and 100 µF charging to 90%”은 Python 3.8 math: τ = 10⁴ × 10⁻⁴ = 1 s; t = τ ln 10; E = ½CV² = ½·10⁻⁴·25와 대조해 확인합니다.

이 계산 방법의 출처는 무엇인가요?

OpenStax University Physics Volume 2, §10.5 RC circuits; §14.4 RL circuits; §15.3 RLC series circuits with AC; HyperPhysics — Series RLC resonance and Q.

이 계산기 소개

\tau_{RC} = RC,\ \tau_{RL} = \frac{L}{R},\quad v(t) = V(1 - e^{-t/\tau});\qquad f_0 = \frac{1}{2\pi\sqrt{LC}},\ |Z| = \sqrt{R^2 + (X_L - X_C)^2},\ Q = \frac1R\sqrt{\frac{L}{C}}

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Capacitor voltage over five time constants

Time constant

Rise towards the final value

Time to reach 90%

Stored energy when fully charged