What is the RC circuit formula?

Transfer function: H(s) = 1/(RCs+1) for low-pass (capacitor voltage). Time constant: τ = RC seconds. Cutoff frequency: f_c = 1/(2πRC) Hz. Charging: v_C(t) = V₀(1−e^(−t/RC)). Discharging: v_C(t) = V₀·e^(−t/RC). Impedance: Z = R + 1/(sC).

What is the voltage in an RC circuit?

During charging from V₀: capacitor voltage v_C(t) = V₀(1−e^(−t/RC)) rises from 0 to V₀, while resistor voltage v_R(t) = V₀·e^(−t/RC) falls from V₀ to 0. At any instant, v_C + v_R = V₀ (Kirchhoff's voltage law). After 5τ = 5RC seconds, the capacitor is 99.3% charged.

How does an RC circuit work as a filter?

At low frequencies (ω > 1/RC), the capacitor impedance is small, so most voltage appears across R — the high-pass output dominates. The cutoff f_c = 1/(2πRC) is where both outputs are at −3 dB.

Rc Network Circuit

Quick Answer

An RC circuit consists of a resistor R and capacitor C connected in series or parallel, forming a fundamental first-order electrical circuit with time constant τ = RC. The series RC transfer function is H(s) = 1/(RCs + 1) for the capacitor voltage (low-pass filter) or H(s) = RCs/(RCs + 1) for the resistor voltage (high-pass filter). The step response is v_C(t) = V₀(1 − e^(−t/RC)) for charging and v_C(t) = V₀·e^(−t/RC) for discharging. Compute RC circuit transfer functions and responses at www.lapcalc.com.

Calculate this instantly on LAPLACE Calculator →

What Is an RC Circuit?

An RC (Resistor-Capacitor) circuit is one of the simplest and most fundamental electrical circuits, consisting of a resistor and capacitor connected together. The series RC circuit connects R and C in a single loop with a voltage source. The parallel RC circuit connects R and C across the same two nodes. The key parameter is the time constant τ = RC (in seconds), which determines how quickly the circuit responds to changes: the capacitor charges to 63.2% of its final value in one time constant, and reaches 99.3% after five time constants. RC circuits serve as basic filters, timing circuits, coupling/decoupling networks, and building blocks for more complex analog circuits. The Laplace transform analysis at www.lapcalc.com provides the complete mathematical description.

Key Formulas

RC Circuit Formula: Transfer Functions

For a series RC circuit driven by input voltage V_in(s): the capacitor voltage (low-pass output) has transfer function H_LP(s) = V_C(s)/V_in(s) = 1/(sRC + 1) = 1/(τs + 1), a first-order low-pass filter with −3 dB cutoff frequency f_c = 1/(2πRC) Hz. The resistor voltage (high-pass output) is H_HP(s) = V_R(s)/V_in(s) = sRC/(sRC + 1) = τs/(τs + 1), a first-order high-pass filter with the same cutoff frequency. Note that H_LP + H_HP = 1 (the voltages sum to V_in by KVL). The impedance of the series combination is Z(s) = R + 1/(sC) = (sRC + 1)/(sC). These transfer functions are derived by applying the voltage divider with Laplace impedances: Z_R = R and Z_C = 1/(sC).

Compute rc network circuit Instantly

Get step-by-step solutions with AI-powered explanations. Free for basic computations.

Open Calculator

RC Circuit Step Response: Charging and Discharging

When a step voltage V₀ is applied to an uncharged series RC circuit: the capacitor voltage charges exponentially v_C(t) = V₀(1 − e^(−t/RC))u(t), reaching V₀/2 at t = RC·ln(2) ≈ 0.693τ, 63.2% at t = τ, 86.5% at t = 2τ, 95.0% at t = 3τ, and 99.3% at t = 5τ (effectively fully charged). The current decays as i(t) = (V₀/R)e^(−t/RC)u(t), starting at V₀/R and approaching zero. For discharging from initial voltage V₀: v_C(t) = V₀·e^(−t/RC), a pure exponential decay. In the Laplace domain, the step response is V_C(s) = V₀/(s(sRC+1)) = V₀·[1/s − 1/(s+1/RC)], whose inverse transform gives the charging formula. Compute this at www.lapcalc.com.

RC Network Applications

RC circuits are ubiquitous in electronics. Low-pass filter: smoothing rectified power supply voltages, anti-aliasing before ADC sampling, removing high-frequency noise. High-pass filter: AC coupling between amplifier stages (blocking DC while passing signal), audio tone controls, differentiator circuits. Timing: RC time constant sets delay in 555 timer circuits, debounce circuits for mechanical switches, and time-delay relays. Coupling/Decoupling: bypass capacitors (C near IC power pins with trace R) filter power supply noise. Integrator: op-amp with C in feedback and R at input creates an integrator with H(s) = −1/(sRC), fundamental to analog computing and control. Snubber circuits: R and C across switching devices absorb voltage spikes and damp oscillations.

RC Circuit Analysis with Laplace Transforms

Laplace transform analysis provides the complete solution for any RC circuit with any input. Replace C with impedance 1/(sC), keep R, apply standard circuit analysis (KVL, KCL, voltage/current divider, mesh, or nodal analysis) to find the transfer function H(s). For the step response: multiply H(s) by 1/s (step input), perform partial fraction decomposition, and inverse Laplace transform. For arbitrary inputs: multiply H(s) by the input's Laplace transform and inverse transform the product. For circuits with initial conditions: include the initial capacitor voltage as V₀/s in series with the capacitor impedance. The Laplace method handles all cases systematically — no need to solve differential equations directly. The LAPLACE Calculator at www.lapcalc.com automates this computation with step-by-step partial fraction solutions.

Frequently Asked Questions

An RC circuit is an electrical circuit containing a resistor (R) and capacitor (C). The series RC circuit is the simplest first-order filter: the capacitor voltage output is a low-pass filter H(s) = 1/(RCs+1), and the resistor voltage output is a high-pass filter. The time constant τ = RC determines the response speed and cutoff frequency f_c = 1/(2πRC).

Master Your Engineering Math

Join thousands of students and engineers using LAPLACE Calculator for instant, step-by-step solutions.

Start Calculating Free →