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Cavity and coupling
The resonance depends on the width and length for TE10p modes, and the dip depth depends on the coupling coefficient beta. Beta = 1 is critical coupling: all the power is accepted.
Mode
Model checks
The science
What is modelled
A rectangular metal box resonates when its walls fit standing waves. For the TEm0p modes the resonant frequency depends only on the width a and length d, and the field has m half-waves across a and p along d. Near one resonance the box acts as a single resonator coupled to a line with coupling coefficient beta; this fixes how much of the incident power is reflected.
Equations
- Resonance:
f0 = c/(2 √εr) √((m/a)² + (p/d)²). - Reflection:
Γ = (1 - β + j x)/(1 + β + j x), withx = 2 Q0 (f - f0)/f0. - Loaded Q and bandwidth:
QL = Q0/(1 + β),Δf = f0(1 + β)/Q0. - Accepted power fraction:
1 - |Γ|².
Checks run in the page
- Resonance of a square air-filled cavity against c/(√2 a).
- The reflection at resonance, (1 - beta)/(1 + beta).
- The accepted power at the half-power frequencies.
- The bandwidth measured on a 2001-point sweep against f0/Q_L.
Honest simplifications
A narrowband, one-port, steady-state model of a single isolated mode: no full-wave solution, no neighbouring modes, no loss mechanism beyond the quality factor and no frequency pulling by the probe. The field colours show the mode shape only.
What to try
- Press Critical coupling and then Overcoupled: the dip becomes shallower and the Gamma circle encloses the origin.
- Detune the source by the half bandwidth and read the accepted power (half of the maximum).
- Press TE102 and note the second half-wave along the length.