Formulas

Let Z1, Z2 be the impedances at the entrance and exit, obtained from the diameters via the chosen scaling Z ∝ dn, so Z2/Z1 = (d2/d1)n.

In the small-reflection approximation the lumped step reflection is Γ0 = ½ ln(Z2/Z1). Phase along the taper of length L at design frequency f with wave speed v: θ = βL, β = 2πf/v.

• Linear taper (diameter changes linearly): |Γlin(θ)| = |Γ0| · |sinθ/θ|. Envelope decays like 1/θ.
• Klopfenstein taper with design max-ripple Γm: define A = arccosh(|Γ0|/Γm). Then
  • For θ ≥ A (passband): |ΓK(θ)| = |Γ0| · |cos√(θ²−A²)| / cosh A, bounded by Γm.
  • For θ < A (stopband): |ΓK(θ)| = |Γ0| · cosh√(A²−θ²) / cosh A.
  Passband cutoff: Lpb = A·v/(2πf) = Aλ/(2π).
• Transmission (lossless): |T|² = 1 − |Γ|², so the insertion loss in dB is 10 log10(1 − |Γ|²).
• Underlying Z(z) profile (not plotted here): ln(Z(z)/√(Z1Z2)) = (Γ0/cosh A) · A² φ(2z/L − 1, A) with φ(x,A) = ∫0x I1(A√(1−y²))/(A√(1−y²)) dy.

Limits of validity & what to do for large transitions

Both the Klopfenstein and the linear-taper formulas come from a perturbation expansion in the local reflection coefficient. They are quantitative while |Γ0| = ½|ln(Z2/Z1)| stays roughly below 0.5. Beyond that, two assumptions break down:

• The expressions add up single (first-order) reflections coherently and ignore multiple reflections inside the taper; for a large step, the multiply-reflected paths become as large as the direct ones, so simply summing the first-order term mis-estimates the answer.
• The analysis treats the taper as a 1D transmission line with a single propagating mode. A large cross-section change excites higher-order modes that aren't part of the model, so |Γ| from the formulas no longer represents the physical reflection.

For large impedance steps, or for the typical case of a single-mode waveguide opening into a much larger (multi-mode) cavity, more appropriate approaches are:

• Stepped quarter-wave transformer (binomial or Chebyshev, Pozar §5.6–5.7) — a chain of discrete λ/4 sections handles arbitrary impedance ratios with each step analysed exactly.
• Mode-matching method — expand the field in modes at each cross-section and enforce boundary conditions at the interfaces. Standard for waveguide step junctions and multi-mode transitions.
• Full-wave numerical simulation (HFSS, CST, COMSOL, openEMS) — solves Maxwell's equations on the 3D geometry without any of the small-reflection or single-mode assumptions.
• Coupling apertures (irises), pins, or loops — for a waveguide entering a large cavity, the standard engineering practice is not a smooth geometric taper but a deliberate small-aperture or probe coupling whose dimensions set the external coupling strength directly. The Klopfenstein analysis assumes a single mode propagates throughout, which fails when the cavity supports many modes.
• Corrugated or stepped horn antennas — for free-space couplings, designed horns achieve low reflection over wide bandwidths via engineered mode control rather than impedance tapering.

Reference: R. W. Klopfenstein, "A transmission line taper of improved design," Proc. IRE 44, 31 (1956); see also Pozar, Microwave Engineering §5.8.