Series RLC Natural Response

When a resistor, inductor, and capacitor share one loop, the resulting differential equation is second-order — and its characteristic roots tell the whole story.

Circuit AnalysisElectrical Engineering Year 2Free preview

⏱️ About 16 min

A charged capacitor, an inductor, and a resistor walk into a single loop — and the current that flows is governed by an equation no first-order circuit could ever produce.

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The big idea: In a series RLC circuit with no source, KVL around the loop yields a second-order linear differential equation whose characteristic roots s1 and s2 are set by just two parameters: the Neper frequency α = R/(2L) and the undamped natural frequency ω0 = 1/√(LC).

🎯 By the end, you'll be able to

Write the KVL equation for a source-free series RLC loop and reduce it to a second-order homogeneous ODE
Derive the characteristic equation s² + (R/L)s + 1/(LC) = 0 and identify the Neper frequency α and resonant frequency ω0
Compute the characteristic roots s1,2 = −α ± √(α² − ω0²) for given component values
Recognize that the nature of the roots (real vs. complex) depends on the relationship between α and ω0

📎 Helpful to know first

From one energy-storage element to two

In first-order circuits — RC and RL — a single energy-storage element meant a single state variable and a first-order ODE. The solution was always an exponential decay with one time constant. Now we add a second storage element: both an inductor and a capacitor in the same loop. The system can exchange energy between its magnetic field (L) and electric field (C), while the resistor bleeds that energy away as heat. The resulting equation is second-order, and its behavior is far richer than a simple exponential.

Setting up the series RLC loop

Consider a series loop containing an inductor L, resistor R, and capacitor C, with no independent source. The capacitor is initially charged to some voltage V0 and the inductor may carry an initial current I0. Applying KVL around the loop (clockwise, with current i flowing through all three elements), the voltage drops are v_L = L(di/dt), v_R = Ri, and v_C = (1/C)∫i dt. Setting their sum to zero and differentiating once to eliminate the integral gives:

\[ L\frac{d^2i}{dt^2} + R\frac{di}{dt} + \frac{1}{C}\,i = 0 \]

KVL for a source-free series RLC loop, written in terms of loop current i(t). This is a second-order homogeneous linear ODE with constant coefficients.

The same equation in terms of capacitor voltage

Because the three elements are in series, the same current flows through all of them, and the capacitor voltage v is related to the current by i = C(dv/dt). Substituting into the current equation yields an equivalent ODE in terms of v(t):

\[ LC\frac{d^2v}{dt^2} + RC\frac{dv}{dt} + v = 0 \]

Series RLC natural response expressed in terms of capacitor voltage v(t). Whether you solve for i or v, the characteristic equation is identical.

The characteristic equation

Assuming a solution of the form e^(st) and substituting into either ODE yields the characteristic equation. Dividing the voltage-form ODE by LC and collecting terms:

\[ s^2 + \frac{R}{L}\,s + \frac{1}{LC} = 0 \]

Characteristic equation of the series RLC circuit. The two roots of this quadratic, s1 and s2, completely determine the form of the natural response.

Defining α and ω0

It is customary (and extremely useful) to rewrite the coefficients using two named parameters. Dividing the characteristic equation by LC and matching to the canonical form s² + 2αs + ω0² = 0 reveals:

\[ \alpha = \frac{R}{2L}, \qquad \omega_0 = \frac{1}{\sqrt{LC}} \]

The Neper frequency α (rad/s) measures how fast the response decays — it is set entirely by R and L. The undamped natural frequency ω0 (rad/s) is the frequency at which the circuit would oscillate if R were zero; it depends only on L and C.

With these definitions, the characteristic equation becomes s² + 2αs + ω0² = 0, and the quadratic formula gives the two roots:

\[ s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} \]

The characteristic roots of the series RLC circuit. Whether the quantity under the radical is positive, zero, or negative determines the character of the response — a topic we explore in the next lesson.

A note on the nature of the roots

The expression under the square root, α² − ω0², can be positive, zero, or negative depending on the component values. When α > ω0 the roots are real and distinct; when α = ω0 they are real and repeated; when α < ω0 they are complex conjugates. Each case produces a qualitatively different waveform. For now, we focus on computing α, ω0, and the roots themselves — classifying the response in detail is the subject of the next lesson.

📝 Worked example: A source-free series RLC circuit has R = 40 Ω, L = 0.5 H, and C = 0.01 F. Find the Neper frequency α, the undamped natural frequency ω0, and the characteristic roots s1 and s2.

Compute the Neper frequency: α = R/(2L) = 40/(2 × 0.5) = 40/1 = 40 rad/s.
Compute the undamped natural frequency: ω0 = 1/√(LC) = 1/√(0.5 × 0.01) = 1/√0.005 ≈ 14.14 rad/s.
Since α (40) > ω0 (14.14), the quantity under the radical is positive, so the roots are real and distinct.
Compute the radical: √(α² − ω0²) = √(1600 − 200) = √1400 ≈ 37.42 rad/s.
s1 = −α + √(α² − ω0²) = −40 + 37.42 ≈ −2.58 rad/s.
s2 = −α − √(α² − ω0²) = −40 − 37.42 ≈ −77.42 rad/s.

✓ α = 40 rad/s, ω0 ≈ 14.14 rad/s, s1 ≈ −2.58 rad/s, s2 ≈ −77.42 rad/s.

Why two roots matter

Each root corresponds to one exponential mode e^(s1t) or e^(s2t) in the solution. The general natural response is a linear combination: x(t) = A1e^(s1t) + A2e^(s2t), where x is any of the circuit variables (current, capacitor voltage, etc.) and A1, A2 are constants determined by the two initial conditions — the initial capacitor voltage and initial inductor current. Having two roots and two initial conditions is the hallmark of a second-order system.

✏️ Practice: For a series RLC circuit with R = 10 Ω, L = 1 H, and C = 0.04 F, compute the Neper frequency α in rad/s.

rad/s

Solution

α = R/(2L) = 10/(2 × 1) = 10/2 = 5 rad/s.

Check your understanding

1. In a series RLC circuit, the Neper frequency is given by which expression?

For the series topology, α = R/(2L). The expression 1/(2RC) is the Neper frequency for the parallel RLC topology — a key duality we explore next.

2. The undamped natural frequency ω0 = 1/√(LC) depends on which components?

ω0 depends only on L and C — the two energy-storage elements. This is true for both series and parallel RLC circuits.

3. If α > ω0 in a series RLC circuit, the characteristic roots are:

When α > ω0, the quantity under the radical (α² − ω0²) is positive, so both roots are real and distinct.

✅ Key takeaways

A source-free series RLC loop satisfies L(d²i/dt²) + R(di/dt) + i/C = 0, or equivalently LC(d²v/dt²) + RC(dv/dt) + v = 0 in terms of capacitor voltage.
The characteristic equation is s² + (R/L)s + 1/(LC) = 0, with roots s1,2 = −α ± √(α² − ω0²).
The Neper frequency α = R/(2L) governs the rate of decay; the undamped natural frequency ω0 = 1/√(LC) sets the oscillation scale.
Whether the roots are real or complex depends on the relationship between α and ω0 — to be classified in detail in the next lesson.

➡️ Now that we have the series RLC characteristic equation in hand, we turn to its dual — the parallel RLC topology — where the role of resistance flips and α takes on a different form.

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