Microwave/waveguide fundamentals: match each statement to its correct concept. List I (Statement) List II (Concept) A. Ratio of maximum stored energy to energy lost per cycle 1. Propagation constant (β = ω / v_p) B. TEM mode in a lossless medium 2. Cutoff frequency is zero C. Ratio of angular frequency to phase velocity 3. Quality factor (Q) of a cavity or resonator D. Lowest-cutoff mode statement relevant to 4. Cylindrical (circular) waveguide context

Introduction / Context:
This item links core microwave concepts: quality factor, propagation constant, cutoff, and waveguide modes. These fundamentals are repeatedly used in cavity design, transmission-line analysis, and mode identification.

Given Data / Assumptions:

• Lossless media are assumed unless stated otherwise.
• Propagation along z uses phase velocity v_p and angular frequency ω.
• Hollow circular waveguides support TE/TM modes; TEM does not propagate in single-conductor waveguides.

Concept / Approach:

Quality factor Q is defined as stored energy / energy lost per radian or per cycle (proportional definition used in exams). For TEM lines (e.g., coax), cutoff frequency is zero. The spatial phase constant β equals ω / v_p. Discussions about the lowest cutoff mode belong to specific waveguide geometries such as circular guides (TE11 is lowest).

Step-by-Step Solution:

Verification / Alternative check:

Standard formulas: β = ω / v_p; for TEM on ideal lines, β = ω√(L′C′) and no cutoff. For cavities/waveguides, Q = 2π * (energy stored / energy lost per cycle), consistent with statement A.

Why Other Options Are Wrong:

• Swapping Q with β confuses energy storage with phase progression.
• Assigning “cutoff zero” to non-TEM modes is incorrect for hollow waveguides.

Common Pitfalls:

Believing TEM can exist in a single-conductor hollow guide; TEM requires two conductors or a return path.

Final Answer:

A-3, B-2, C-1, D-4