A series RLC circuit has L = 10 mH, C = 100 nF, and R = 50 Ω. An engineer wants to shift the resonant frequency to a higher value. Which single change will accomplish this?
AIncrease R to 100 Ω
BDecrease L to 5 mH
CIncrease C to 200 nF
DDecrease R to 25 Ω
The resonant frequency ω₀ = 1/√(LC) depends only on L and C — resistance R has no effect on ω₀. To increase ω₀, you must decrease L, decrease C, or both. Decreasing L to 5 mH gives a higher ω₀ (since ω₀ ∝ 1/√L). Options A and D only change R, leaving ω₀ unchanged. Option C increases C, which decreases ω₀. This question directly tests the key insight that resonant frequency is set by the energy storage elements (L and C), not by the dissipative element (R), which only affects bandwidth and Q.
Question 2 Multiple Choice
A circuit with a very high Q-factor is most useful for which application?
ABroadband amplification over a wide range of input frequencies
BPrecisely selecting a narrow band of frequencies, as in a radio tuner or bandpass filter
CMaximizing the rate of energy dissipation in the resistive element
DRapidly suppressing oscillations after a transient input to reach steady state quickly
Q-factor equals ω₀/BW, so a high Q means a narrow bandwidth — the circuit responds strongly only to frequencies very near ω₀ and rejects others sharply. This selectivity is ideal for frequency selection applications like radio tuners, crystal oscillators, and bandpass filters. Option A (broadband) requires LOW Q. Option C is wrong — high Q means low energy dissipation per cycle relative to stored energy, not high dissipation. Option D (quick settling) requires LOW Q; high-Q circuits oscillate for many cycles before settling, which is the opposite of rapid transient decay.
Question 3 True / False
In a series RLC circuit at resonance, the current through the circuit is at its maximum possible value.
TTrue
FFalse
Answer: True
At resonance in a series circuit, inductive and capacitive reactances cancel exactly (jω₀L = 1/jω₀C in magnitude, opposite in sign), leaving total impedance equal to just R — its minimum possible value. Since current I = V/Z and Z is minimized, the current is maximized. This is why series resonance is sometimes called 'current resonance.' The voltage across the inductor and capacitor individually can be much larger than the source voltage (by factor Q) even though they cancel in the series loop — a counterintuitive consequence of resonance that has practical implications for component ratings.
Question 4 True / False
A high-Q resonant circuit has a wider bandwidth than a low-Q resonant circuit at the same resonant frequency.
TTrue
FFalse
Answer: False
Bandwidth BW = ω₀/Q, so Q and bandwidth are inversely related: high Q means NARROW bandwidth; low Q means WIDE bandwidth. A high-Q circuit stores a large amount of energy relative to what it dissipates per cycle, so it responds sharply and selectively to frequencies near ω₀ — tall, narrow peak. A low-Q circuit dissipates energy quickly relative to storage, giving a broad, flat response. Confusing high Q with wide bandwidth is among the most common errors in resonance problems. The correct intuition: high Q → high selectivity → narrow bandwidth.
Question 5 Short Answer
Explain why the quality factor Q unifies the time-domain and frequency-domain descriptions of a resonant circuit.
Think about your answer, then reveal below.
Model answer: Q is the ratio of energy stored to energy dissipated per cycle. In the time domain: a high-Q circuit oscillates many times before its transient response decays (underdamped, slow envelope decay), because little energy leaks through the resistor each cycle. In the frequency domain: that same high-Q circuit has a narrow, sharply peaked frequency response (small bandwidth BW = ω₀/Q), because it only responds strongly near ω₀ where reactances cancel. Both behaviors arise from the same physical ratio — energy stored vs. energy lost per cycle. The circuit that rings longest in the time domain also filters most selectively in the frequency domain.
This connection is a specific instance of the time-bandwidth uncertainty relationship in signal processing: a long-duration impulse response corresponds to a narrow frequency response, and vice versa. For resonant circuits, Q = ω₀L/R connects these: changing R simultaneously changes the transient decay rate (damping coefficient α = R/2L) and the bandwidth (BW = R/L = ω₀/Q). This is why Q appears identically in formulas for both the decay envelope of transient oscillations and the 3-dB bandwidth of the frequency response — they are two measurements of the same underlying physical quantity.