Parallel resonant band-pass: A parallel RLC band-pass filter has a resonant frequency of 14 kHz and a bandwidth of 4 kHz. What is the lower critical frequency?

Introduction / Context:
Band-pass filters are characterized by a center (resonant) frequency f0 and a bandwidth BW. The lower (f1) and upper (f2) critical frequencies define the -3 dB points. For symmetric first-order approximations, f0 is the midpoint: f0 = (f1 + f2) / 2 and BW = f2 - f1.

Given Data / Assumptions:

• Resonant (center) frequency f0 = 14 kHz.
• Bandwidth BW = 4 kHz.
• Assume symmetric response about f0 (standard definition).

Concept / Approach:
For a narrowband filter, f1 = f0 - BW/2 and f2 = f0 + BW/2. This simple relation holds for many practical tuned circuits and is commonly used in quick design checks.

Step-by-Step Solution:
BW/2 = 4 kHz / 2 = 2 kHzLower frequency f1 = f0 - BW/2 = 14 kHz - 2 kHz = 12 kHzUpper frequency f2 = f0 + BW/2 = 16 kHz (not asked)

Verification / Alternative check:
Check midpoint: (12 kHz + 16 kHz)/2 = 14 kHz, and difference 16 kHz - 12 kHz = 4 kHz; both match the given values.

Why Other Options Are Wrong:

• 7 kHz or 10 kHz: Too far below f0 given the stated bandwidth.
• Cannot be determined: It can be determined directly from f0 and BW.

Common Pitfalls:
Confusing lower with upper frequency, or subtracting the full bandwidth instead of half the bandwidth from the center frequency.

Final Answer:
is 12 kHz