Match the following in the context of RLC series circuit : Codes :List - I List - II (a) Under damped (i) \(\xi=1\) (b) Critically damped (ii) \(\xi \gt 1\) (c) Quality factor (iii) \(\dfrac{1}{2\xi}\) (d) Over damped (iv) \(\xi \lt 1\)
(a)-(iv), (b)-(i), (c)-(iii), (d)-(ii)
Everything follows from the standard second-order form. A series RLC circuit obeys
\(\dfrac{d^{2}i}{dt^{2}}+\dfrac{R}{L}\dfrac{di}{dt}+\dfrac{i}{LC}=0\)
which is written canonically as \(s^{2}+2\xi\omega_{n}s+\omega_{n}^{2}=0\) with
\(\omega_{n}=\dfrac{1}{\sqrt{LC}}\qquad \xi=\dfrac{R}{2}\sqrt{\dfrac{C}{L}}\)
The damping ratio alone decides the character of the roots \(s=-\xi\omega_{n}\pm\omega_{n}\sqrt{\xi^{2}-1}\):
| Condition | Roots | Response |
|---|---|---|
| \(\xi \lt 1\) | Complex conjugate | Under damped — decaying oscillation with overshoot |
| \(\xi=1\) | Real and equal | Critically damped — fastest approach without overshoot |
| \(\xi \gt 1\) | Real and distinct | Over damped — sluggish, no oscillation |
So (a)-(iv), (b)-(i) and (d)-(ii).
The quality factor is the reciprocal link. For a series RLC circuit
\(Q=\dfrac{\omega_{n}L}{R}=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}\)
and substituting \(\xi=\dfrac{R}{2}\sqrt{\dfrac{C}{L}}\) gives directly
\(Q=\dfrac{1}{2\xi}\)
so (c)-(iii). High Q therefore means low damping: a sharply resonant, lightly damped circuit that rings for many cycles. The two quantities are simply two languages — the filter designer's Q and the control engineer's \(\xi\) — for the same physical fact.
A useful landmark. \(\xi=0.707\) gives \(Q=0.707\), the maximally flat Butterworth response — the usual compromise between speed and overshoot in practical designs.
Reading off the codes, the required order is (iv), (i), (iii), (ii), which is option 3.
Hence, the correct match is (a)-(iv), (b)-(i), (c)-(iii), (d)-(ii).
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