RF & Radar Microwave Engineering, WR-90 Rectangular Waveguides & TE10 Dispersion
Rectangular Waveguide TE10 Cutoff & Wavelength Formula
Calculate Dominant TE10 Cutoff Frequency fc (GHz), Free-Space Wavelength lambda0 (mm), Stretched Guide Wavelength lambda_g (mm), and TE10 Wave Impedance Z_TE (ohms).
Inside a hollow metal waveguide, guide wavelength is LONGER than free-space wavelength—and phase velocity exceeds the speed of light!
Air-filled hollow rectangular radar/satcom waveguides of internal Broad Wall `a` (mm) and Narrow Wall `b` (`b <= a / 2`, e.g. X-band `WR-90`: `22.86` mm x `10.16` mm) act as high-pass electromagnetic filters! Below the **Dominant `TE_10` Cutoff Frequency `f_c,10 = c / (2 * a)`** (`c = 299,792,458` m/s), waves decay evanescently without propagating; above `2 * f_c,10` (`TE_20` onset), higher-order multi-mode distortion corrupts the signal! Across the clean single-mode band `f_c,10 < f < MIN(2*f_c,10, c/(2*b))`, bouncing plane waves off the side walls stretches **Guide Wavelength `lambda_g = lambda_0 / SQRT(1 - (f_c,10 / f)^2)`** LONGER than Free-Space Wavelength `lambda_0 = c / f` (crucial when spacing `lambda_g / 4` iris matching stubs or slotted-array radar slots!) and raises Transverse Wave Impedance `Z_TE = eta_0 / SQRT(1 - (f_c,10 / f)^2)` above free-space `eta_0 = 376.73` ohms!
Verify on a Standard X-Band WR-90 Radar Waveguide at 10.00 GHz—Broad_Wall `a` = 22.86 mm (`0.900"`, B2), Narrow_Wall `b` = 10.16 mm (`0.400"`, B3), and Operating_Frequency `f` = 10.00 GHz (B4): Dominant `TE_10` Cutoff `f_c` equals `6.557` GHz (E2), Free-Space Wavelength `lambda_0` is `29.98` mm (E3), Guide Wavelength `lambda_g` stretches to `39.71` mm (E4), and `TE_10` Wave Impedance equals `499.0` ohms (E5)!
| WR-90 Rectangular Waveguide KPI (a=22.86 mm, b=10.16 mm @ f=10.0 GHz) | Formula | Verified Output |
|---|---|---|
| Dominant TE10 Cutoff fc (GHz) | (299792458 / (2 * (a_mm / 1000))) / 1e9 | 6.557 GHz |
| Free-Space Wavelength lambda0 (mm) | (299792458 / (Freq_GHz * 1e9)) * 1000 | 29.98 mm (41.64 mm @ 7.20 GHz) |
| Stretched Guide Wavelength (mm) | lambda0_mm / SQRT(1 - ((fc / f)^2)) | 39.71 mm (100.81 mm near cutoff @ 7.2 GHz!) |
| TE10 Wave Impedance Z_TE (ohms) | 376.730313 / SQRT(1 - ((fc / f)^2)) | 499.0 ohms (912.1 ohms @ 7.20 GHz!) |
Single-Mode TE10 Waveguide Cutoff, Guide Wavelength & Wave Impedance Formula
Validate `Broad_Wall a > Narrow_Wall b > 0` mm and strictly `f > f_c,10` (evanescent cutoff below `f_c,10`!)!
=LET(aw_m, B2 / 1000, bw_m, B3 / 1000, fq_g, B4, fc_g, (299792458 / (2 * aw_m)) / 1000000000, IF(OR(B3 <= 0, B2 <= B3, fq_g <= fc_g), "Invalid or Evanescent Below Cutoff", ROUND(376.730313 / SQRT(1 - ((fc_g / fq_g) ^ 2)), 1)))Near-Cutoff 7.20 GHz Severe Dispersion & Impedance Surge Check (Set Frequency f B4 = 7.20 GHz)
Tune down to `7.20` GHz (`B4 = 7.2`, only `9.8%` above `6.557` GHz cutoff—why EIA recommends operating `WR-90` only across `8.20–12.40` GHz!): Free-Space Wavelength E3 is `41.64` mm, yet `SQRT(1 - (fc/f)^2) = 0.413` balloons Guide Wavelength E4 to `100.81` mm and spikes Wave Impedance E5 to `912.1` ohms!
=IF(ISTEXT(E5), E5, LET(fc_g, (299792458 / (2 * (B2 / 1000))) / 1000000000, lz_m, (299792458 / (B4 * 1000000000)) * 1000, ROUND(lz_m / SQRT(1 - ((fc_g / B4) ^ 2)), 2)))Try the example
Use the original values or change them to test your understanding.