An engineer needs a second-order low-pass filter with a maximally flat passband and no resonance peak. Which Q value achieves this, and what is this condition called?
AQ >> 1 — higher Q gives a sharper rolloff and a flatter passband
BQ = 1/√2 ≈ 0.707 — this is the Butterworth condition for maximally flat response
CQ = 1 — unity Q guarantees no peaking by definition
DQ = 0 — no resonance factor means no possibility of peaking
Q = 1/√2 (equivalently, damping ratio ζ = 1/√2) produces the Butterworth response: maximally flat in the passband with no amplitude peak before rolloff, and the −3 dB point falls exactly at ω₀. For Q > 1/√2, the magnitude rises above 0 dB near resonance before falling — the filter is peaking. For Q < 1/√2, the response is overdamped: no peak, but the rolloff begins before ω₀. The Butterworth condition is the boundary between these regimes and is widely used as a default design target.
Question 2 Multiple Choice
In a series RLC circuit driven by a voltage source, which output configuration produces a notch (band-reject) filter?
ATaking the output across the resistor R
BTaking the output across the capacitor C alone
CTaking the output across the series combination of L and C together
DTaking the output across the inductor L alone
At resonance, a series LC combination has impedances that cancel: the inductive reactance jωL equals and opposes the capacitive reactance 1/jωC in magnitude. The net impedance of the series LC is zero at ω₀, making it a short circuit. Therefore no voltage appears across the LC pair at resonance — the output is zero (deep notch) while passing frequencies above and below resonance normally. This is distinct from taking the output across R (bandpass) or across C alone (low-pass).
Question 3 True / False
A second-order passive RLC filter with Q > 1/√2 can produce output voltages near resonance that exceed the source voltage, even though no active amplifying elements are present.
TTrue
FFalse
Answer: True
Passive resonance allows voltage magnification across individual reactive components (L or C), even though the total energy in the circuit is conserved. At resonance in a series RLC, the voltage across the capacitor (or inductor) can be Q times the source voltage — this is sometimes called the 'Q-factor voltage magnification.' A high-Q circuit with Q = 10 can produce 10× the source voltage across the capacitor at resonance. This is not a violation of energy conservation; energy oscillates between L and C, with only small losses through R each cycle.
Question 4 True / False
Taking the output across the resistor in a series RLC circuit produces a low-pass filter response.
TTrue
FFalse
Answer: False
Output across R in a series RLC gives a bandpass response — at DC (ω = 0), the capacitor blocks and no current flows, so no voltage appears across R. At very high frequencies, the inductor blocks and again no current flows. Current (and therefore voltage across R) is maximum at resonance, where the reactive impedances cancel. For a low-pass response, take the output across C (its impedance is high at low frequencies, allowing voltage to develop there). For a high-pass response, take the output across L.
Question 5 Short Answer
Explain why the quality factor Q determines both the sharpness of a second-order filter's rolloff and whether a resonance peak appears in the passband.
Think about your answer, then reveal below.
Model answer: Q = ω₀/(R/L) for a series RLC, representing the ratio of energy stored to energy dissipated per radian of oscillation. High Q means little energy is lost each cycle, so the circuit can sustain oscillations near resonance — producing a sharp frequency selectivity and a large amplitude peak before rolloff. In filter terms: high Q creates a tall, narrow resonance peak and a steep transition band. Low Q means heavy damping (energy dissipates quickly), suppressing oscillation — the response rolls off smoothly without any peak. The Butterworth condition Q = 1/√2 is the precise boundary where peaking just disappears while maximizing rolloff steepness.
The same Q governs the time-domain transient: high Q → underdamped ringing; low Q → overdamped sluggish return. The frequency-domain peak and time-domain ringing are two manifestations of the same underlying physics — a system that stores energy relative to its losses will both oscillate in time and selectively amplify near its natural frequency in steady state.