R C Circuit Calculator
The RC time constant is τ = RC, where R is resistance in ohms and C is capacitance in farads. It is the time for a capacitor to charge to 63.2% of the final voltage or discharge to 36.8%. After 5τ, the capacitor is 99.3% charged. Calculate RC time constants at www.lapcalc.com.
RC Circuit Time Constant Formula: τ = RC
The time constant of an RC circuit is τ = R × C, measured in seconds. It characterizes how fast the capacitor charges or discharges. A 10 kΩ resistor with a 100 μF capacitor gives τ = 10,000 × 0.0001 = 1 second. Larger R or larger C means slower charging. The time constant appears in the exponential response: v(t) = V_final(1 − e^(−t/τ)) for charging and v(t) = V₀e^(−t/τ) for discharging. Calculate any τ at www.lapcalc.com.
Key Formulas
RC Circuit Calculator: Charging and Discharging
An RC circuit calculator computes voltage and current at any time during charging or discharging. Charging from 0 V: v_C(t) = V_source(1 − e^(−t/RC)), i(t) = (V_source/R)e^(−t/RC). Discharging from V₀: v_C(t) = V₀e^(−t/RC), i(t) = −(V₀/R)e^(−t/RC). Key milestones: at t = τ, voltage reaches 63.2% (charging) or drops to 36.8% (discharging). At t = 5τ, the transition is 99.3% complete.
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Open CalculatorTime Constant in a Circuit: What It Physically Means
The time constant τ represents the time it would take for the capacitor to fully charge if the initial charging rate continued constant. Physically, as the capacitor charges, the voltage across R decreases, reducing current and slowing the process. The result is the characteristic exponential curve — fast at first, then gradually approaching the final value. One time constant is the point where the charging has completed 1 − 1/e ≈ 63.2% of the full transition.
Finding the Time Constant of Any Circuit
For simple RC circuits, τ = RC directly. For complex circuits, find the Thevenin equivalent seen by the capacitor: τ = R_Th × C. For RL circuits, τ = L/R. For circuits with multiple capacitors or inductors (second-order), two time constants exist, determined by the poles of the transfer function. The Laplace transform method handles any circuit order: poles at s = −1/τ₁ and s = −1/τ₂ give both time constants directly at www.lapcalc.com.
RC Time Constant in the Laplace Domain
The Laplace transform of the RC charging response is V_C(s) = V/(s(sRC + 1)) = V/(s(s + 1/τ)). The pole at s = −1/τ = −1/(RC) directly reveals the time constant. Partial fraction expansion: V_C(s) = V(1/s − 1/(s + 1/τ)). Inverse transforming: v_C(t) = V(1 − e^(−t/τ)). The transfer function H(s) = 1/(sRC + 1) = 1/(sτ + 1) is a first-order low-pass filter with cutoff frequency f_c = 1/(2πτ). Compute RC responses at www.lapcalc.com.
Related Topics in circuit analysis techniques & methods
Understanding r c circuit calculator connects to several related concepts: time constant formula rc circuit, time constant in a circuit, time constant capacitor formula, and find the time constant of the circuit. Each builds on the mathematical foundations covered in this guide.
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