ECE4122 Chap.7 Rectangular Waveguide Modes
Rectangular Waveguide Modes
A rectangular metallic waveguide supports transverse electric and transverse magnetic modes indexed by their field variation across the broad and narrow dimensions. TE modes may have one zero index but not both. TM modes require both indices to be nonzero because the longitudinal electric field must satisfy conducting-wall boundaries.
These validity rules should be applied before numerical cut-off calculations.
The cut-off frequency depends on the transverse wavenumber. For an air-filled guide, the dominant TE10 cut-off is set by the broad dimension. Higher modes are ordered by evaluating their index pairs, not by assuming a universal sequence.
Single-mode operation requires the frequency to be above the desired cut-off and below the lowest competing cut-off.
Above cut-off, the guide propagation constant grows from zero toward the unbounded-medium value. Guide wavelength exceeds the ordinary medium wavelength.
Phase velocity exceeds the medium wave speed, while group velocity is below it; their product gives the square of the medium wave speed in the ideal nondispersive filling. This superluminal phase velocity does not carry information faster than light.
TE and TM wave impedances behave differently near cut-off and must not be swapped.
Field reconstruction begins with the longitudinal component appropriate to the mode, applies transverse derivatives and then checks wall boundary conditions. Power flow follows from the transverse E cross H conjugate integral. Every answer should state the operating band and mode label.
What this chapter covers
- 01
TE and TM index rules
- 02
Rectangular-guide cut-off
- 03
Dominant and competing modes
- 04
Guide wavelength and velocity
- 05
Modal impedance and power flow
Worked example · free
Establish a dominant-mode window
- +2For TE10, fc = c/(2a) = 3.00e8/(0.060).
- +1This gives fc10 = 5.00 GHz.
- +1For TE20, fc20 = c/a = 10.0 GHz.
- +1For TE01, fc01 = c/(2b) = 12.5 GHz.
- +1The lower competitor is TE20 at 10.0 GHz.
- +1The basic dominant-mode window is 5.00 GHz < f < 10.0 GHz.
Key terms
- TE mode
- A mode with zero longitudinal electric-field component.
- TM mode
- A mode with zero longitudinal magnetic-field component.
- Dominant mode
- The allowed mode with the lowest cut-off frequency.
- Guide wavelength
- The longitudinal phase-repeat distance inside a guide.
- Group velocity
- The velocity of a narrowband energy or envelope disturbance.
- Modal impedance
- The transverse electric-to-magnetic field ratio for a mode.
Rectangular Waveguide Modes FAQ
Can a TM mode have a zero index?
Not in a rectangular conducting guide. Both transverse indices must be nonzero for the longitudinal electric field to satisfy the walls nontrivially. A zero index would make the required longitudinal electric-field eigenfunction vanish everywhere in the guide.
Why is guide wavelength longer?
Part of the total wavenumber is committed to transverse standing-wave variation, leaving a smaller longitudinal phase constant. Therefore beta is smaller than the filling-medium wavenumber and two pi divided by beta is larger.
Does phase velocity violate relativity?
No. Phase fronts can move faster than the medium speed, but information and energy are associated with group behaviour. A phase crest carries no independently controllable message; modulation and power follow the slower group velocity.
Exam move
Create a cut-off ladder for every guide dimension set. Cross out invalid mode indices, calculate the lowest candidates, mark the operating frequency and only then compute beta, wavelength, impedance or power. Begin with a dimension and mode-validity table. Mark a as the broad side and b as the narrow side, then list candidate TE and TM index pairs.
Reject TE00 and every TM mode containing a zero index before calculating anything. Evaluate and sort the remaining cut-offs rather than relying on a memorised order. Place the operating frequency on this ladder and write both inequalities required for a dominant-mode window. Only after eligibility is established should propagation quantities be calculated.
Form the ratio f_c/f and predict the square-root factor between zero and one. Use it consistently for beta, guide wavelength, phase velocity and group velocity. Check that guide wavelength exceeds the wavelength in the filling medium, phase velocity exceeds the medium speed, group velocity is smaller, and the ideal product relation is satisfied.
If a dielectric fills the guide, replace free-space speed and impedance with material values throughout rather than changing only one formula. Keep TE and TM field reconstruction separate. Start from the appropriate nonzero longitudinal component, apply conducting-wall conditions to select sine and cosine dependence, and differentiate across the correct dimension to recover transverse fields.
Label relative phase and propagation factor. Verify that tangential electric field vanishes at every metal wall and that the longitudinal power component of E cross H points in the declared travel direction. These checks expose swapped axes and signs without requiring memorisation of every component. Use limiting cases to understand dispersion.
Near cut-off beta and group velocity approach zero while guide wavelength and phase velocity grow. Far above cut-off the guide approaches the unbounded-medium result. Do not interpret superluminal phase motion as energy or information velocity. In design questions, state margin from cut-off because the near-threshold region is highly sensitive to dimensional tolerance and frequency.
Finish with the scope of the model: an ideal guide calculation establishes modal phase and eligibility, while conductor loss, surface roughness, bends and discontinuities require additional attenuation or scattering analysis.
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