Overdamped, Underdamped & Critically Damped Cases

The characteristic equation's roots tell you whether a second-order circuit returns to rest slowly, quickly, or with a decaying wobble.

Circuit AnalysisElectrical Engineering Year 2Free preview
⏱️ About 18 min

A pendulum released from rest can crawl back slowly, settle as fast as possible, or swing back and forth with shrinking amplitude. Circuits do the same thing — and the math predicts which.

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The big idea: The ratio of damping α to natural frequency ω0 determines whether the natural response decays monotonically (overdamped), decays as fast as possible without overshooting (critically damped), or oscillates while decaying (underdamped).
🎯 By the end, you'll be able to
  • Identify the three damping regimes from α and ω0
  • Write the correct form of the natural response for each regime
  • Compute the damped natural frequency ωd for the underdamped case
  • Solve for the two arbitrary constants A1 and A2 using initial conditions x(0) and dx/dt(0)
📎 Helpful to know first
  • Parallel RLC Natural Response

Three personalities of the same equation

Every second-order circuit we study — series RLC, parallel RLC, or any two-energy-storage-element network — leads to the same characteristic equation:

s² + 2αs + ω0² = 0

The roots are s1,2 = −α ± √(α² − ω0²). What matters is the sign of the expression under the square root. If α² > ω0² the discriminant is positive (two real roots). If α² = ω0² it is zero (repeated root). If α² < ω0² it is negative (complex roots). These three possibilities give rise to three qualitatively different responses.

\[ s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2} \]
Roots of the characteristic equation. The sign of α² − ω0² determines the damping regime.

Overdamped (α > ω0): slow and steady

When damping exceeds the natural frequency, the discriminant is positive and we get two distinct, real, negative roots s1 and s2. The natural response is a sum of two decaying exponentials:

x(t) = A1·e^(s1t) + A2·e^(s2t)

There is no oscillation. The response decays monotonically toward zero. However, because one of the exponents is typically small in magnitude (the slower root dominates the tail), the response can take a long time to settle. Overdamping is stable but sluggish.

\[ x(t) = A_1\, e^{s_1 t} + A_2\, e^{s_2 t} \]
Overdamped natural response. Two distinct real roots, both negative, no oscillation.

Critically damped (α = ω0): fastest without overshoot

At the exact boundary α = ω0 the discriminant vanishes and the two roots merge into a single repeated root s = −α. The two independent solutions are no longer e^(s1t) and e^(s2t) (they would be identical), but rather e^(−αt) and t·e^(−αt). The natural response becomes:

x(t) = (A1 + A2·t)·e^(−αt)

This is the fastest possible return to zero without oscillating. It is the design target in systems where overshoot is unacceptable — for example, a vehicle suspension tuned so the car settles quickly without bouncing.

\[ x(t) = (A_1 + A_2\, t)\, e^{-\alpha t} \]
Critically damped natural response. Repeated real root s = −α; fastest non-oscillatory decay.

Underdamped (α < ω0): decaying oscillation

When damping is light, the discriminant is negative and the roots become complex conjugates: s = −α ± jωd, where ωd = √(ω0² − α²) is the damped natural frequency. The natural response oscillates at ωd (slightly lower than ω0) while the amplitude decays as e^(−αt):

x(t) = e^(−αt)·(A1·cos(ωd t) + A2·sin(ωd t))

This can also be written as x(t) = A·e^(−αt)·cos(ωd t − φ), where A and φ are an equivalent pair of constants related to A1 and A2. The oscillation frequency ωd is always less than ω0 and approaches ω0 as α → 0 (the undamped limit).

\[ x(t) = e^{-\alpha t}\left(A_1 \cos\omega_d t + A_2 \sin\omega_d t\right) \]
Underdamped natural response. Complex roots s = −α ± jωd; decaying oscillation at the damped frequency ωd.
\[ \omega_d = \sqrt{\omega_0^2 - \alpha^2} \]
The damped natural frequency. Always ωd < ω0 when α > 0; ωd → ω0 as damping vanishes.

Finding A1 and A2: you always need two initial conditions

Each regime has two arbitrary constants, so you need two pieces of information: the value x(0) and the derivative dx/dt(0). The first is usually given directly (e.g., an initial capacitor voltage or inductor current). The second typically comes from a physical relationship at t = 0:

  • If x is an inductor current: di/dt(0) = v_L(0) / L.
  • If x is a capacitor voltage: dv/dt(0) = i_C(0) / C.

You evaluate x(0) and dx/dt(0), substitute t = 0 into the general response and its derivative, and solve the resulting two equations for A1 and A2.

📝 Worked example: A series RLC circuit has α = 3 rad/s and ω0 = 5 rad/s. The initial conditions are x(0) = 0 and dx/dt(0) = 10. Find the natural response x(t).
  1. Compare α and ω0: since 3 < 5, the circuit is underdamped.
  2. Compute the damped frequency: ωd = √(ω0² − α²) = √(25 − 9) = √16 = 4 rad/s.
  3. Write the general underdamped form: x(t) = e^(−3t)·(A1·cos(4t) + A2·sin(4t)).
  4. Apply x(0) = 0: e⁰·(A1·cos 0 + A2·sin 0) = A1 = 0, so A1 = 0.
  5. Simplify: x(t) = A2·e^(−3t)·sin(4t). Differentiate: dx/dt = A2·e^(−3t)·(4·cos(4t) − 3·sin(4t)).
  6. Apply dx/dt(0) = 10: A2·(4·1 − 3·0) = 4·A2 = 10, so A2 = 2.5.
  7. Final answer: x(t) = 2.5·e^(−3t)·sin(4t).
✓ x(t) = 2.5·e^(−3t)·sin(4t)
✏️ Practice: An underdamped circuit has α = 6 rad/s and a damped natural frequency ωd = 8 rad/s. What is the undamped natural frequency ω0 in rad/s? (Hint: ωd = √(ω0² − α²).)
rad/s
Solution
  1. Rearrange ωd = √(ω0² − α²) to get ω0² = α² + ωd².
  2. Substitute: ω0² = 36 + 64 = 100, so ω0 = 10 rad/s.

Check your understanding

1. A circuit has α = 4 and ω0 = 3. Which damping regime applies?
Since α (4) > ω0 (3), the discriminant α² − ω0² = 16 − 9 = 7 > 0, giving two distinct real roots — overdamped.
2. For a critically damped circuit, what is the form of the natural response?
The repeated root s = −α gives solutions e^(−αt) and t·e^(−αt), so the response is (A1 + A2·t)·e^(−αt).
3. In the underdamped case, what happens to ωd as α approaches zero?
Since ωd = √(ω0² − α²), as α → 0 we get ωd → ω0. With no damping, the oscillation frequency equals the natural frequency.
🔎 Related lessons
Second-Order Step Response · soon
✅ Key takeaways
  • The three damping regimes are determined by comparing α and ω0: overdamped (α > ω0), critically damped (α = ω0), underdamped (α < ω0).
  • Overdamped response is a sum of two decaying exponentials with distinct real roots — no oscillation, potentially slow.
  • Critically damped response has a repeated root and takes the form (A1 + A2·t)·e^(−αt) — fastest non-oscillatory decay.
  • Underdamped response oscillates at ωd = √(ω0² − α²) while decaying as e^(−αt) — two constants always found from x(0) and dx/dt(0).
➡️ So far the circuit has been relaxing on its own with no driving source. But what happens when a DC source is switched in? The natural-response forms stay the same — they just get added to a steady-state value.
Want to test yourself on this? Try the Electrical Aptitude test →