Tapered-Line Matching
A gradually tapered line matches two impedances over a broad band without discrete steps, behaving like a high-pass matching section: below a cutoff (set by the taper length in wavelengths) it barely matches, above it the reflection stays small. The exponential and triangular tapers are simple; the Klopfenstein taper is optimum — the shortest taper for a specified passband ripple. Enter the impedances and length to see the reflection and cutoff for each.
Equations & Parameters ▸
\(\Gamma_0=\tfrac12\ln\dfrac{R_L}{Z_0},\quad \theta=\beta L\)
exp: \(|\Gamma|=\Gamma_0\Big|\dfrac{\sin\theta}{\theta}\Big|\); tri: \(|\Gamma|=\Gamma_0\Big(\dfrac{\sin(\theta/2)}{\theta/2}\Big)^2\)
Klopfenstein: \(A=\cosh^{-1}\!\dfrac{\Gamma_0}{\Gamma_m}\); passband \(\theta\ge A\), \(|\Gamma|=\Gamma_m|\cos\sqrt{\theta^2-A^2}|\)
exp: \(|\Gamma|=\Gamma_0\Big|\dfrac{\sin\theta}{\theta}\Big|\); tri: \(|\Gamma|=\Gamma_0\Big(\dfrac{\sin(\theta/2)}{\theta/2}\Big)^2\)
Klopfenstein: \(A=\cosh^{-1}\!\dfrac{\Gamma_0}{\Gamma_m}\); passband \(\theta\ge A\), \(|\Gamma|=\Gamma_m|\cos\sqrt{\theta^2-A^2}|\)
| Z0, RL | Source impedance and real load resistance (Ω). |
| L | Taper length in wavelengths at the operating frequency (θ = 2πL/λ). |
| Γm | Klopfenstein passband ripple (max reflection). Sets the taper parameter A. |
| Cutoff | Minimum length for the passband: exp λ/2, triangular λ, Klopfenstein A·λ/2π. |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §5.8; R. W. Klopfenstein, Proc. IRE 44, 31 (1956).
Inputs
Ω
FeedΩ
ResistiveProfile
λ
at f₀Klopfenstein
Results
At length L
Reflection |Γ|—
VSWR—
Return loss—
Design
Total Γ₀ (low-f)—
Passband cutoff length—
Taper parameter A—
Reflection vs length