Parallel resonant band-pass design: A 6.8 Ω series resistor feeds a parallel tank of L = 8 µH and C = 120 pF (RW ≈ 0 Ω). The output is taken across the LC. What is the center frequency f0?

Introduction / Context:This item tests the core resonance relation for a parallel LC tank, commonly used in RF band-pass filters. The series source resistor sets loading, but the ideal center frequency is set by L and C alone when winding resistance is negligible.

Given Data / Assumptions:

• Inductance L = 8 µH.
• Capacitance C = 120 pF.
• Series feed resistor = 6.8 Ω (does not change f0 ideally).
• Inductor winding resistance RW = 0 Ω (assumed).

Concept / Approach:The center (resonant) frequency for an LC tank is f0 = 1 / (2 * pi * sqrt(L * C)). The exact f0 is largely independent of a small series feed resistor; resistive elements mainly affect bandwidth and Q, not the ideal resonant frequency.

Step-by-Step Solution:

Verification / Alternative check:A quick RF check: Using the rule-of-thumb f0(MHz) ≈ 159 / sqrt(L(µH) * C(pF)) gives 159 / sqrt(8 * 120) ≈ 159 / 31.06 ≈ 5.12 MHz, very close to 5.14 MHz.

Why Other Options Are Wrong:

• 514 kHz and 503 kHz: Off by a factor of 10 (wrong power-of-ten handling).
• 5.03 MHz: Slightly low due to rounding; 5.14 MHz matches precise calculation.

Common Pitfalls:

• Mistaking series vs. parallel resonance formulas.
• Mixing µH and mH, or pF and nF, leading to decade errors.

Final Answer:5.14 MHz