How does a quarter-wave transformer work?
A line a quarter-wavelength long with characteristic impedance Z1 = √(Z0·ZL) transforms a real load ZL to Z0 at its design frequency. To match 100 Ω to 50 Ω, use a 70.7 Ω λ/4 line. Bandwidth narrows as the impedance ratio grows; multisection binomial or Chebyshev transformers widen it.[1]
Why does it work?
At λ/4 the input impedance is Z1²/ZL. Physically, the reflections from the two impedance steps are equal in magnitude and arrive at the input 180° apart (round trip of λ/2), so they cancel.[1]
What bandwidth does it give?
Pozar gives the fractional bandwidth for a maximum tolerable reflection Γm (TEM line) as Δf/f0 = 2 − (4/π)·arccos[ (Γm/√(1 − Γm²))·(2√(Z0ZL)/|ZL − Z0|) ]. Small ratios such as 2:1 give wide bandwidth; large ratios such as 10:1 are narrow.[1]
What are multisection transformers?
Cascading N quarter-wave sections with gradually changing impedances gives a maximally flat (binomial) or equal-ripple (Chebyshev) passband. Chebyshev designs trade in-band ripple for more bandwidth. Tapered lines (exponential, Klopfenstein) are the continuous limit; the Klopfenstein taper is optimal for minimum length at a given ripple.[1][2]
What if the load is complex?
First add a line section or reactive element to make the impedance real (a point on the real axis of the Smith chart), then apply the quarter-wave section. Quarter-wave lines are also used as impedance inverters in Doherty amplifiers and Wilkinson dividers.
Frequently asked questions
What impedance matches 25 Ω to 50 Ω?
√(25×50) ≈ 35.4 Ω, a quarter wavelength long.
Does it work at harmonics?
It works again at odd multiples (3λ/4 etc.), not at even multiples.
Can it match complex loads?
Not directly; the load must first be made real.
See also
References
Online sources accessed October 21, 2026; printed works cited by edition. See all sources.
- “Microwave Engineering, 4th ed.”. D. M. Pozar, Wiley. 2012.
- “Microwave Filters, Impedance-Matching Networks, and Coupling Structures”. G. Matthaei, L. Young, E. M. T. Jones, McGraw-Hill / Artech House. 1964; Artech reprint 1980.