How do you find the RC transfer function?

Replace R with impedance R and C with impedance 1/(sC). Apply voltage divider: H(s) = Z_out/(Z_R + Z_C). For capacitor output: H = [1/(sC)]/[R+1/(sC)] = 1/(sRC+1). For resistor output: H = R/[R+1/(sC)] = sRC/(sRC+1). This is the standard Laplace impedance method.

What is the cutoff frequency of an RC circuit?

f_c = 1/(2πRC) Hz, or equivalently ω_c = 1/RC rad/s. At this frequency, the output magnitude is −3 dB (70.7% of the DC value) and the phase is −45°. Below f_c the signal passes; above f_c it's attenuated at −20 dB/decade for the low-pass configuration.

What is the step response of an RC circuit?

Capacitor voltage: v_C(t) = V₀(1−e^(−t/RC)) — exponential charging from 0 to V₀. Resistor voltage: v_R(t) = V₀·e^(−t/RC) — exponential decay from V₀ to 0. At t = RC: v_C = 63.2% of V₀. At t = 5RC: v_C = 99.3% of V₀ (effectively fully charged).

Transfer Function of Rc Circuit

Quick Answer

The transfer function of a series RC circuit depends on the output taken. For the capacitor voltage (low-pass filter): H(s) = V_C(s)/V_in(s) = 1/(RCs + 1), with cutoff frequency f_c = 1/(2πRC) and time constant τ = RC. For the resistor voltage (high-pass filter): H(s) = V_R(s)/V_in(s) = RCs/(RCs + 1). The RC transfer function is derived by applying voltage divider with Laplace impedances: Z_R = R and Z_C = 1/(sC). Compute RC circuit transfer functions and step responses at www.lapcalc.com.

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Transfer Function of RC Circuit: Derivation

The transfer function of a series RC circuit is derived using Laplace-domain impedances. Replace the resistor with impedance Z_R = R and the capacitor with Z_C = 1/(sC). For the capacitor voltage output (low-pass): H_LP(s) = Z_C/(Z_R + Z_C) = [1/(sC)]/[R + 1/(sC)] = 1/(sRC + 1). For the resistor voltage output (high-pass): H_HP(s) = Z_R/(Z_R + Z_C) = R/[R + 1/(sC)] = sRC/(sRC + 1). Note that H_LP + H_HP = 1 (by Kirchhoff's voltage law, V_C + V_R = V_in). The transfer function H_LP(s) = 1/(τs + 1) is the standard first-order low-pass form with time constant τ = RC seconds and DC gain of 1 (unity). The pole is at s = −1/(RC) = −1/τ. This derivation using Laplace impedances is the standard method taught in circuit analysis, with computations supported at www.lapcalc.com.

Key Formulas

RC Transfer Function: Frequency Response

Substituting s = jω into H_LP(s) = 1/(jωRC + 1) gives the frequency response. Magnitude: |H(jω)| = 1/√(1 + (ωRC)²). At ω = 0: |H| = 1 (0 dB) — DC passes fully. At ω = 1/RC = ωc: |H| = 1/√2 (−3 dB) — the cutoff frequency. At ω >> 1/RC: |H| ≈ 1/(ωRC) — rolls off at −20 dB/decade. Phase: ∠H(jω) = −arctan(ωRC). At ω = 0: phase = 0°. At ω = ωc: phase = −45°. At ω → ∞: phase → −90°. The Bode plot shows a flat magnitude below ωc, then −20 dB/dec roll-off, with phase transitioning from 0° to −90° centered at ωc. The cutoff frequency in Hz is f_c = 1/(2πRC) = ωc/(2π). For R = 1 kΩ, C = 1 μF: f_c = 159 Hz, τ = 1 ms.

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RC Circuit Transfer Function: Step and Impulse Response

The step response is found by inverse Laplace transforming Y(s) = H(s)/s = 1/[s(RCs+1)]. Partial fractions: 1/[s(τs+1)] = 1/s − 1/(s+1/τ). Inverse transform: v_C(t) = (1 − e^(−t/τ))u(t). The capacitor charges exponentially toward the input voltage with time constant τ = RC. At t = τ: 63.2%. At t = 5τ: 99.3%. The impulse response is h(t) = (1/τ)e^(−t/τ)u(t) — an exponential decay starting at 1/τ and decaying with time constant τ. This is the inverse Laplace transform of H(s) = 1/(τs+1), representing the system's fundamental response to an instantaneous input. For the high-pass output: step response is v_R(t) = e^(−t/τ)u(t) — an immediate jump followed by exponential decay. All computed at www.lapcalc.com.

RC Transfer Function Applications

The RC transfer function H(s) = 1/(τs+1) appears throughout electronics. Anti-aliasing filter: placed before an ADC to remove frequencies above the Nyquist limit, preventing aliasing artifacts. Set f_c = f_sample/2. Audio tone control: variable R (potentiometer) adjusts f_c for treble roll-off. Power supply filtering: large C smooths rectified AC voltage; the RC time constant determines ripple voltage. Coupling capacitor: the high-pass transfer function sRC/(sRC+1) blocks DC while passing AC — set f_c well below the signal band. Sensor signal conditioning: RC low-pass filters remove high-frequency noise from thermocouple, strain gauge, and pressure sensor signals. Debounce circuit: RC time constant longer than the mechanical bounce duration (~5–10 ms) produces a clean digital signal from a noisy switch. Each application uses the same first-order transfer function with appropriately chosen R and C values.

Higher-Order RC Networks

Cascading multiple RC stages creates higher-order filters. Two identical RC stages in series (with buffer between): H(s) = 1/(τs+1)² — second-order with −40 dB/dec roll-off. Without buffer, the loading effect modifies the transfer function: H(s) = 1/(τ²s² + 3τs + 1) for two identical unbuffered RC stages. The Twin-T network (two T-shaped RC paths) creates a notch filter rejecting a specific frequency. The Wien bridge (series and parallel RC) creates a bandpass centered at f₀ = 1/(2πRC), used in oscillators. Active RC filters use op-amps with RC feedback networks: Sallen-Key topology provides second-order low-pass, high-pass, or bandpass with controllable Q factor. The RC integrator with op-amp: H(s) = −1/(sRC) provides a true integrator (pole at origin). All these RC network transfer functions build on the fundamental 1/(τs+1) form from www.lapcalc.com.

Frequently Asked Questions

For the capacitor voltage (low-pass): H(s) = 1/(RCs+1). For the resistor voltage (high-pass): H(s) = RCs/(RCs+1). Derived using Laplace impedances Z_R = R, Z_C = 1/(sC) in a voltage divider. Time constant τ = RC, cutoff frequency f_c = 1/(2πRC).

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