Quarter-Wave Transformer

A λ/4 transmission line section transforms impedance: Z_in = Z_t²/Z_L. Choose the transformer impedance Z_t = √(Z₁·Z₂) to match Z₁ to Z₂. Works at one frequency (and odd harmonics) only.

Equations & Parameters ▸

\(Z_t = \sqrt{Z_1 Z_2} \qquad \ell = \dfrac{c}{4f\sqrt{\varepsilon_r}}\)

Z₁	Source impedance (Ω) — what you're feeding from.
Z₂	Load impedance (Ω) — what you're matching to.
f	Matching frequency. Physical length = λ/4 at this frequency.
εᵣ	Substrate dielectric constant. Sets physical length: ℓ = c/(4f√εᵣ).
Z_t	Transformer characteristic impedance = √(Z₁·Z₂).
ℓ	Physical length of the λ/4 section.

Physical constants used

c	Speed of light = 2.998×10⁸ m/s
µ₀	Permeability of free space = 4π×10⁻⁷ H/m ≈ 1.2566×10⁻⁶ H/m
ε₀	Permittivity of free space = 8.854×10⁻¹² F/m

Inputs

Source impedance, Z₁

e.g. 50 Ω coax

Load impedance, Z₂

e.g. 200 Ω antenna

Frequency, f

Dielectric constant, εᵣ

1 = air/coax, 4.4 = FR4

Results

Transformer

Transformer impedance, Z_t—

Physical length, ℓ—

Diagram

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