In filter and resonant-circuit design, if the bandwidth (BW) of a band-pass response becomes larger while the center (resonant) frequency f0 is held fixed, how does the quality factor Q = f0 / BW change?

Introduction:
Understanding the relationship between bandwidth (BW), center frequency (f0), and quality factor (Q) is essential in analog filter design and resonant circuits. Q is a measure of selectivity: high Q means a narrow passband around f0, while low Q means a wider passband. This question tests the direct inverse relationship between Q and BW when f0 is fixed.

Given Data / Assumptions:

• Band-pass response centered at f0.
• Bandwidth BW = f2 - f1 between the two half-power (−3 dB) cutoff frequencies.
• Center frequency f0 is held constant while BW increases.
• Standard definition: Q = f0 / BW.

Concept / Approach:
By definition, Q is inversely proportional to BW for a given f0. If BW grows larger, the ratio f0 / BW becomes smaller. Therefore, the circuit becomes less selective and more broadbanded, which is interpreted as a decrease in Q.

Step-by-Step Solution:
Start with the identity: Q = f0 / BWHold f0 fixed (constant).Increase BW → denominator increases.Result: Q decreases because the ratio f0 / BW becomes smaller.

Verification / Alternative check:
Consider two designs at the same f0. If Design A has BW_A = 100 Hz and Design B has BW_B = 200 Hz, then Q_A = f0 / 100 and Q_B = f0 / 200 = Q_A / 2. The broader design (larger BW) indeed has half the Q, confirming the inverse relationship.

Why Other Options Are Wrong:

• The roll-off rate increases: Roll-off mainly depends on filter order, not directly on BW.
• The half-power frequency decreases: f1 and f2 move apart as BW increases; this statement is incomplete/misleading.
• The center frequency decreases: f0 is specified constant in the scenario.
• Q increases: Opposite of the Q = f0 / BW definition when f0 is fixed.

Common Pitfalls:

• Confusing Q with gain; Q characterizes selectivity, not passband gain directly.
• Assuming roll-off steepness changes with BW independent of order.
• Mixing arithmetic mean with geometric mean definitions of center frequency in wideband cases.

Final Answer:
Q decreases