Quarter-Wave Impedance Transformer: Matching 50 Ω to 100 Ω
A quarter-wave section matches a real 100 Ω load to a 50 Ω system if that section is 70.71 Ω. It works because a lossless 90° transmission line inverts a load impedance. The result is exact at its design frequency in the ideal line model, but the line must have the right electrical length and its match changes with frequency. [1, 2]
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The impedance transformation
For a lossless line of characteristic impedance Zt terminated by ZL, its input impedance at a quarter wavelength is Zin = Zt²/ZL. To make Zin equal to a real source impedance Z0, choose: [1, 3]
For 50 Ω and 100 Ω, Zt = √5000 = 70.71 Ω. At 1 GHz with εeff = 2.25, the length is 49.97 mm. At the design frequency Zin = 70.71²/100 = 50 Ω. At twice that frequency the section is half a wavelength and repeats the 100 Ω load instead of matching it. [1, 3]
Check the 50 Ω to 100 Ω transformer and its frequency response →
Why the physical length is not always λ/4 in free space
The wavelength is shorter in a dielectric. For a stripline or uniform coax, εeff is close to the filling dielectric's εr; a microstrip's fields run partly in air, so its εeff is lower and depends on the geometry. Enter a solved trace width and effective permittivity for the actual stackup rather than dividing the free-space wavelength by four. Bends, steps and open ends add parasitic reactance. [3]
When one section is too narrow-band
A single section produces a match at its centre frequency and a limited band around it. Adding sections trades board area for bandwidth. A binomial design makes the response flat near the centre; a Chebyshev design accepts controlled ripple to gain a wider specified passband. The tool shows both the designed impedances and an exact cascade response, because the simple design equations use a small-reflection approximation. [1, 3]
A complex load is not matched by the real-to-real √(Z0ZL) formula alone. For that case, use a stub or an L-network and verify the full frequency sweep. [3]
Design a lumped-element match for a complex load →
Tools used in this guide
- Quarter-Wave Transformer & Stub Calculator – Single and multisection binomial or Chebyshev λ/4 transformers with exact response and bandwidth, open and short stubs, and single-stub matching, with physical lengths and trace widths.
- L-Network Impedance Matching & Smith Chart – Two-element L-section matching of a complex load: every low-pass and high-pass solution with L and C values, node Q, swept return loss, bandwidth, nearest E-series parts and a Smith chart.
- Impedance Calculator – Single-ended and differential microstrip, coated and embedded microstrip, stripline, coplanar lines and broadside differential pairs, solved with a 2D field solver.
Sources
- Keysight, RF Design Software Learning Kit, chapter 4: single- and multisection quarter-wave transformers, including the 50 Ω to 100 Ω, 70.71 Ω example.
- Rohde & Schwarz, Cable Impedance Measurement: the quarter-wave impedance-inversion principle.
- D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012, §2.3 and §§5.4–5.7: terminated-line input impedance and quarter-wave, binomial and Chebyshev transformers.
Related guides
- L-Network Impedance Matching with a Smith Chart
- Coaxial Cable Impedance: Diameter Ratio, Dielectric and Delay
- Controlled Impedance Explained: Microstrip, Stripline and Solder Mask
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