Show field patterns at 1420 MHz, derive wave impedance vs frequency, explain why feed probes behave oddly near cutoff, and compare WR-650 vs WR-770 specifically for H-line work.

Excellent — I’ll go through all four, keeping it practical and tied to 1420.405 MHz H-line work with WR-650.

1. Field patterns in WR-650 at 1420 MHz

For the dominant TE10 mode, the electric field is:

E_y(x,z) = E0 * sin(pi * x / a) * e^(-j*beta*z)

Key points:

• One half-cycle of E-field across the broad wall a
• No field variation along the narrow wall b
• Maximum E-field at the center (x = a/2)
• Zero at the side walls (x = 0, a)

At 1420 MHz, because you are not far above cutoff:

• The field bulges more strongly toward the center
• Energy is less tightly confined
• Phase velocity is higher than c (normal for waveguides)

This is why probes placed slightly off-center can behave strangely near cutoff.

2. Wave impedance vs frequency (this matters a lot)

Wave impedance for TE modes is:

Z_TE = eta / sqrt(1 - (fc/f)^2)

where:

eta ≈ 377 ohms
fc = 0.91 GHz (WR-650 TE10)
f  = 1.420 GHz

Compute the ratio:

(fc/f)^2 ≈ (0.91/1.42)^2 ≈ 0.41

So:

Z_TE ≈ 377 / sqrt(1 - 0.41)
     ≈ 377 / 0.77
     ≈ 490 ohms

Compare this to “comfortable” waveguide operation:

• Deep single-mode region → ~400 Ω
• Near cutoff → impedance shoots upward

Implication:
Your feed probe is trying to couple 50 Ω coax into ~490 Ω wave impedance — this is why tuning is delicate at H-line.

3. Why feed probes misbehave near cutoff

Three coupled effects bite you at 1420 MHz:

(a) Probe reactance increases

Probe length is set by guide wavelength, not free-space wavelength:

lambda_g = lambda_0 / sqrt(1 - (fc/f)^2)

At 1420 MHz:

lambda_0 ≈ 0.211 m
lambda_g ≈ 0.211 / 0.77 ≈ 0.274 m

A “quarter-wave” probe becomes:

~68 mm inside the guide

Small length errors → large reactance changes.

(b) Probe position sensitivity

Coupling is proportional to local E-field:

E(x) ∝ sin(pi * x / a)

Near cutoff, field curvature is steep →
1–2 mm probe movement noticeably changes match.

(c) Backshort depth becomes critical

Shorting plane distance ideally:

d ≈ lambda_g / 4

At 1420 MHz:

d ≈ 68 mm

But because impedance is high, the match bandwidth is narrow — millimetre-level tolerance matters.

This is why WR-650 probes feel “touchy” at H-line compared to higher bands.

4. WR-650 vs WR-770 for 1420 MHz (honest comparison)

Why professionals like WR-770:

• Further from cutoff → easier matching
• Lower dispersion
• Less probe reactance sensitivity
• Cleaner horn patterns

Why WR-650 still works well:

• Smaller horns
• Less wind loading
• Cheaper surplus
• Perfectly valid single-mode at 1420 MHz

Practical takeaway for hydrogen-line feeds

At 1420 MHz:

WR-650 is physically valid but electrically sensitive
WR-770 is electrically relaxed but mechanically bulky

If you already have WR-650:

• Use adjustable probe length
• Adjustable backshort
• Expect narrow SWR bandwidth
• Measure, don’t trust formulas alone