For a 50 Ω resistor intended for use at 3 GHz, what should be the approximate maximum stray capacitance so that its reactance does not significantly disturb operation?

Introduction / Context:
At microwave frequencies, even tiny stray capacitances can produce reactances comparable to a 50 Ω system impedance, detuning networks and degrading matching. It is therefore useful to estimate an upper bound on allowable stray capacitance.

Given Data / Assumptions:

• System impedance around 50 Ω at f = 3 GHz.
• Design goal: capacitive reactance magnitude |Xc| should be much larger than 50 Ω (e.g., ≥ 10 × 50 Ω) so it is negligible.
• Use ideal capacitor model for a first estimate.

Concept / Approach:

Capacitive reactance is Xc = 1 / (2 * π * f * C). Choose a target |Xc|, solve for C. Taking |Xc| ≈ 500 Ω (ten times 50 Ω) is a reasonable engineering rule-of-thumb for negligible effect.

Step-by-Step Solution:

Verification / Alternative check:

If C were 1 pF, |Xc| ≈ 1 / (2 * π * 3e9 * 1e-12) ≈ 53 Ω, which is already comparable to 50 Ω and thus unacceptable. Therefore 0.1 pF is a sensible upper limit.

Why Other Options Are Wrong:

• 1 μF, 1 nF, 10 pF, 1 pF: These yield reactances far too small in magnitude at 3 GHz, badly disturbing a 50 Ω environment.

Common Pitfalls:

Ignoring lead inductance (also important), using DC component values without considering parasitics, or assuming 1 pF is 'tiny'—at GHz it is not.

Final Answer:

0.1 pF