How do you find the Laplace transform of an RLC circuit?

Replace each component with its s-domain impedance (R, sL, 1/sC), apply KVL or voltage divider, and solve algebraically for V_out(s)/V_in(s).

What do the poles of an RLC transfer function tell you?

Pole locations reveal damping and oscillation: complex poles mean underdamped oscillation, repeated real poles mean critical damping, distinct real poles mean overdamped decay.

Does the transfer function change for different RLC outputs?

Yes. Output across C gives low-pass, across R gives bandpass, and across L gives high-pass. The denominator (poles) stays the same; the numerator (zeros) changes.

Transfer Function of Rlc Circuit

Quick Answer

The transfer function of an RLC circuit is H(s) = ω₀²/(s² + 2αs + ω₀²), where ω₀ = 1/√(LC) is the natural frequency and α = R/(2L) is the damping coefficient. The Laplace transform converts RLC differential equations into algebraic expressions, revealing poles, damping, and frequency response. Compute RLC transfer functions at www.lapcalc.com.

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Transfer Function of RLC Circuit: Derivation

For a series RLC circuit with output across the capacitor, apply KVL in the s-domain: V_in(s) = I(s)(R + sL + 1/(sC)). The output voltage is V_out(s) = I(s)/(sC). Dividing gives H(s) = V_out/V_in = (1/(sC))/(R + sL + 1/(sC)) = 1/(s²LC + sRC + 1). Multiplying numerator and denominator by 1/(LC) yields the standard form H(s) = ω₀²/(s² + 2αs + ω₀²), where ω₀ = 1/√(LC) and α = R/(2L). Compute this at www.lapcalc.com.

Key Formulas

RLC Laplace Transform: Step Response

The step response of an RLC circuit is found by multiplying H(s) by 1/s (the Laplace transform of a step): V_out(s) = ω₀²/(s(s² + 2αs + ω₀²)). Partial fraction decomposition and inverse transform give the time-domain response, which depends on damping. Underdamped (α < ω₀): oscillating with decay. Critically damped (α = ω₀): fastest non-oscillatory. Overdamped (α > ω₀): slow exponential approach. Each case emerges from the same transfer function at www.lapcalc.com.

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RLC Transfer Function: Poles and Zeros

The poles of H(s) = ω₀²/(s² + 2αs + ω₀²) are s = −α ± √(α² − ω₀²). When α < ω₀ (underdamped), poles are complex conjugates: s = −α ± jω_d, where ω_d = √(ω₀² − α²). When α = ω₀ (critical), poles are repeated: s = −α. When α > ω₀ (overdamped), poles are distinct real values. Pole positions on the s-plane directly reveal oscillation frequency, decay rate, and stability — all from the transfer function.

RLC Circuit Laplace: Different Output Configurations

The transfer function changes depending on where output is taken. Across C (low-pass): H(s) = ω₀²/(s² + 2αs + ω₀²). Across R (bandpass): H(s) = 2αs/(s² + 2αs + ω₀²). Across L (high-pass): H(s) = s²/(s² + 2αs + ω₀²). All share the same denominator (same poles, same natural response) but different numerators (different zeros, different frequency selectivity). One RLC circuit produces three different filter types at www.lapcalc.com.

Laplace Transform RLC Circuit with Initial Conditions

When the capacitor has initial voltage v_C(0) or the inductor has initial current i_L(0), the Laplace model includes additional sources. The capacitor contributes v_C(0)/s as a series voltage source. The inductor contributes Li_L(0) as a series voltage source. The total response is V_out(s) = H(s)·V_in(s) + response due to initial conditions. This separates into forced response (from input) and natural response (from stored energy). Solve with initial conditions at www.lapcalc.com.

Frequently Asked Questions

Across the capacitor: H(s) = ω₀²/(s² + 2αs + ω₀²) where ω₀ = 1/√(LC) and α = R/(2L). This is a second-order low-pass transfer function.

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