Q1Microwave Theory And Techniques

Governing equation: for a circular waveguide of radius a, using cylindrical coordinates (ρ,φ,z), the longitudinal field component (Hz for TE modes, Ez for TM modes) satisfies the scalar Helmholtz equation:

Separation of variables: assuming ψ(ρ,φ)=R(ρ)Φ(φ), the equation separates into a Φ-equation with solution Φ=cos(nφ) or sin(nφ) (n=0,1,2,... for single-valuedness around the guide), and a ρ-equation that is exactly Bessel's differential equation of order n, whose solution regular at ρ=0 (the guide axis, where the field must remain finite) is the Bessel function of the first kind:

Boundary condition and cutoff wavenumber — TM modes: for TM modes, Ez=0 at the conducting wall ρ=a (tangential E must vanish), so Jn(kca)=0, meaning kca must equal a zero of the Bessel function Jn. Denoting the m-th zero of Jn as pnm, the cutoff wavenumber for TMnm is:

Boundary condition and cutoff wavenumber — TE modes: for TE modes, the boundary condition is instead ∂Hz/∂ρ=0 at ρ=a (since the tangential E-field, here Eφ ∝ ∂Hz/∂ρ, must vanish on the wall), so Jn′(kca)=0, meaning kca must equal a zero of the derivative of the Bessel function. Denoting the m-th zero of Jn′ as p′nm, the cutoff wavenumber for TEnm is:

The lowest root overall is p′11≈1.841, giving the dominant mode of circular waveguide as TE11 (analogous to TE10 in rectangular guide), with fc(TE11)=1.841c/(2πa); the lowest TM-mode root is p01≈2.405, giving TM01 as the lowest TM mode, with fc(TM01)=2.405c/(2πa).

Cutoff frequency (general form, either mode family): converting the cutoff wavenumber kc to a cutoff frequency using kc=ωc√(με)=2πfc√(με):

Phase constant: for operation above cutoff (f > fc, k > kc), the phase constant follows exactly the same relation as in rectangular waveguide:

Phase velocity: likewise following the same general waveguide relation, exceeding the unbounded-medium value c:

Summary: the circular waveguide derivation follows an entirely parallel structure to the rectangular-waveguide derivation (same general β, vp, λg formulas), with the only substantive difference being that the discrete cutoff wavenumbers are now set by Bessel-function zeros (pnm for TM, p′nm for TE) rather than by the simple (mπ/a, nπ/b) sinusoidal eigenvalues of the rectangular case — a direct consequence of the circular geometry's boundary being naturally described in cylindrical rather than Cartesian coordinates.

Guided wavelength: exactly as in the rectangular case, once β is known the guided wavelength follows directly from λg=2π/β, which can equivalently be written in terms of the free-space wavelength λ0=c/f as λg=λ0/√(1−(fc/f)²) — the identical relation used throughout the rectangular-waveguide analysis elsewhere in this examination series, again confirming that circular and rectangular waveguides share the same overall propagation-parameter structure and differ only in how their specific mode cutoffs are computed.

Field pattern significance of the mode indices: the index n (the azimuthal order, appearing in cos(nφ) or sin(nφ)) determines how many full field-pattern variations occur once around the guide's circumference — n=0 modes are azimuthally symmetric, while n≥1 modes have 2n field maxima/minima distributed around the circumference. The index m (the radial order) determines how many field maxima occur along a radius from the centre to the wall — it is the m-th zero of the relevant Bessel function (or its derivative) that sets kc, so higher m always corresponds to a higher cutoff frequency for fixed n, exactly analogous to how higher m or n in the rectangular case (counting half-wavelength field variations across a or b) raises the cutoff there.

Practical application — circular waveguide components: the dominant TE11 mode is used in most circular-waveguide feed and antenna applications (since its field pattern most closely resembles the free-space linearly-polarized wave pattern radiated or received by a horn or dish antenna), while the TE01 mode (despite not being dominant) is specifically favoured for long-distance circular-waveguide transmission lines because its attenuation actually decreases with increasing frequency — an unusual and useful property not shared by any rectangular-waveguide mode — making TE01 circular waveguide historically important for very-low-loss, high-power millimetre-wave transmission runs before low-loss optical fibre and modern coaxial technology became available. The rotational symmetry of circular waveguide overall (as noted in the discussion of waveguide types in the companion electromagnetics-waves paper of this examination series) also makes it the natural choice for rotary joints and circularly-polarized antenna feeds, applications not well served by the inherently asymmetric rectangular waveguide cross-section.