RLC damping change: A series RLC circuit is underdamped. To convert its response to overdamped, how must the resistance R be adjusted while keeping L and C fixed?

Introduction / Context:The transient response of a second-order series RLC circuit (to steps or impulses) depends on the damping ratio ζ, which is controlled by resistance R for fixed inductance L and capacitance C. This question asks how to move from underdamped (oscillatory) to overdamped (non-oscillatory) behavior by adjusting R.

Given Data / Assumptions:

• Series RLC with fixed L and C; only R is varied.
• Underdamped initially, so ζ < 1.
• Transition desired to overdamped where ζ > 1.

Concept / Approach:

For a series RLC, the characteristic equation is s^2 + (R/L)s + 1/(LC) = 0. The damping ratio is ζ = (R/2) * sqrt(C/L) in normalized units. The regimes are: underdamped if R < 2√(L/C), critically damped if R = 2√(L/C), and overdamped if R > 2√(L/C). Thus, increasing R beyond the critical value transitions the system through critical to overdamped behavior.

Step-by-Step Solution:

Verification / Alternative check:

Compare time responses: increasing R reduces oscillation amplitude and eventually eliminates oscillation when ζ ≥ 1, confirming the qualitative effect of higher resistance.

Why Other Options Are Wrong:

• Decrease R: Reduces damping → more oscillatory (further underdamped).
• Increase to infinity: Open circuit removes dynamics rather than producing a practical overdamped response.
• Reduce to zero: Yields an undamped LC oscillator (ζ = 0).

Common Pitfalls:

• Mixing up parallel vs. series RLC damping conditions; formulas differ but the trend—larger R in series increases damping—holds.

Final Answer:

has to be increased