Microwave resonator Q factors – Relationship between loaded Q, external Q, and unloaded Q For a single-resonator coupled system, relate the loaded quality factor QL, the external (coupling) quality factor QE, and the unloaded (intrinsic) quality factor QU.

Introduction / Context:
Quality factors quantify energy storage versus loss in resonators. In practical microwave filters and oscillators, a resonator is coupled to external circuitry, so the observed (loaded) Q includes both intrinsic losses and coupling losses. Knowing how these combine is critical for bandwidth, insertion loss, and sensitivity calculations.

Given Data / Assumptions:

• Single resonator coupled to a port or network.
• Intrinsic loss modeled by QU; coupling loss modeled by QE.
• Loaded resonance includes both loss mechanisms.

Concept / Approach:

The rate of energy loss adds in terms of 1/Q. Therefore the standard relation is 1/QL = 1/QU + 1/QE. Solving for QL gives QL = (QE * QU) / (QE + QU). This expression directly relates coupling strength (QE) to the observed 3 dB bandwidth and informs design trade-offs between selectivity and insertion loss.

Step-by-Step Solution:

Verification / Alternative check:

Limiting cases: (i) Weak coupling QE → ∞ ⇒ QL → QU. (ii) Strong coupling QE ≪ QU ⇒ QL ≈ QE. Both limits match intuition, confirming correctness.

Why Other Options Are Wrong:

Option A is an intermediate relation but is not the requested expression for QL; option B and C are incorrect arithmetic combinations; option E is not physically derived for additive loss channels.

Common Pitfalls:

Confusing series vs. parallel addition of Qs; mixing bandwidth definitions without relating them back to Q via Q = f0 / Δf.

Final Answer:

QL = (QE * QU) / (QE + QU).