Quarter-Wave Transformers
The quarter-wave transformer is one of the most elegant solutions in all of RF engineering. By inserting a short section of transmission line — exactly one quarter-wavelength long at the operating frequency — between a source and a load, you can match any two real impedances to each other with zero lumped components, zero loss (apart from the line's own conductor and dielectric loss), and no adjustments. The only requirement is that you can fabricate or obtain coaxial cable of the right characteristic impedance.
A quarter-wave transformer between a 50-ohm source and a 200-ohm load. The transformer section must have Z₀ = √(50 × 200) = 100 ohms. At exactly the design frequency, the source sees 50 ohms looking into the transformer.
View LargerThe Quarter-Wave Impedance Transformation
Recall from lesson M13D that a quarter-wave transmission line section has the impedance inversion property:
Zin = Z₀² / ZL
This means the input impedance of a quarter-wave line is inversely proportional to the load impedance, with the proportionality constant being Z₀². If you know what load impedance ZL you want to match to a source of impedance ZS, you need the transformer's input impedance to equal ZS:
ZS = Zin = Z₀² / ZL
Solving for Z₀:
Z₀ = √(ZS × ZL)
This is the fundamental quarter-wave transformer formula. It applies whenever both ZS and ZL are purely resistive (real) at the design frequency. If either impedance has a reactive component, the quarter-wave transformer does not produce a perfect match at any frequency — you would need to resonant the reactive component first, then apply the transformer.
The Transformer Formula
ZT = √(Z1 × Z2)
- ZT = characteristic impedance of the quarter-wave section (ohms)
- Z1 = source impedance (ohms, must be real)
- Z2 = load impedance (ohms, must be real)
Physical length of the transformer section:
Length = (c × VF) / (4 × f)- c = 3 × 10⁸ m/s
- VF = velocity factor of the transformer cable
- f = design frequency (Hz)
In practical units:
Length (feet) = 246 × VF / f(MHz)Length (meters) = 75 × VF / f(MHz)
Quarter-Wave Transformer Calculator
Quarter-Wave Transformer Calculator
Enter the source impedance, load impedance, frequency, and velocity factor of the transformer cable. The calculator returns the required transformer characteristic impedance and physical length.
Worked Examples
Example 1: 2-element Yagi to 50-ohm coax
A 2-element Yagi beam designed for 144 MHz has a feedpoint impedance of approximately 200 ohms (a common value for this configuration). You want to feed it with 50-ohm coaxial cable.
ZT = √(50 × 200) = √10,000 = 100 ohms
Physical length of 100-ohm quarter-wave section at 144.2 MHz, RG-62/U (93-ohm, VF = 0.84 — not available; need to find 100-ohm cable).
Alternative: make 100-ohm cable by connecting two 50-ohm cables in series (for high impedance) — no, that gives 100 ohms total impedance on a series connection. Actually, connecting two 50-ohm cables in parallel gives 25 ohms, which is not right either.
In practice, 100-ohm coaxial cable exists (RG-62, nominally 93 ohms, or custom) and is sometimes used. A more common approach for this application is to use two 50-ohm cables in parallel (which makes 25 ohms) to match 200 ohms to 25 ohms? No — Z_T = sqrt(200 × 50) = 100. But two 50-ohm cables in parallel = 25 ohms. That matches a 200-ohm load to 25-ohm lines.
The correct solution: Use two runs of 50-ohm coaxial cable in parallel (effective Z₀ = 25 ohms) as the quarter-wave transformer to match 200 ohms to 12.5 ohms? No — let's recalculate: two 50-ohm cables in parallel → 25 ohm effective Z₀. Z_in = 25²/200 = 625/200 = 3.125 ohms. That's wrong.
Actually, the correct answer: Single 100-ohm cable works. The common amateur approach for matching 200 ohms to 50 ohms is to use a matching network at the feedpoint. If you want to use a coax transformer: you need 100-ohm cable. Some 100-ohm cable exists (e.g. CXP13D or similar) or you can home-brew an air-spaced cable. Alternatively, use a 4:1 balun at the feedpoint to transform 200 ohms to 50 ohms.
Summary: Z_T = √(50 × 200) = 100 Ω. Use a quarter-wave section of 100-ohm coaxial cable cut for the operating frequency.
Length at 144.2 MHz, VF = 0.66 (solid PE): 75 × 0.66 / 144.2 = 49.5/144.2 = 0.343 m = 1.13 ft = 13.5 in
Example 2: 50-ohm dipole to 75-ohm cable
You want to convert a 50-ohm antenna system to work with the 75-ohm CATV coaxial cable already installed in your attic.
ZT = √(50 × 75) = √3750 = 61.2 ohms
61-ohm coaxial cable does not exist as a commercial standard. However, a single quarter-wave section of 75-ohm coax can be used as a mediocre transformer (it gives Z_in = 75²/50 = 112.5 ohms — not a match). For this application, a small broadband transformer (wound on a ferrite core) is more practical than a quarter-wave section.
Conclusion: Quarter-wave transformers work best when the required intermediate impedance is achievable with a real cable type. For 50-to-75 ohm conversion, a wound transformer is more practical.
Example 3: Vertical antenna to 50-ohm coax
A quarter-wave ground-plane vertical at 7.1 MHz has a feedpoint impedance of approximately 36 ohms (theoretical for a perfect vertical over a perfect ground plane). You want to match it directly to 50-ohm coax.
ZT = √(36 × 50) = √1800 = 42.4 ohms
No commercial 42-ohm coax exists. However, RG-59/U (73-ohm cable, VF = 0.66) can be used as approximately 73 ohms. Not quite 42.4 ohms either.
In practice, a more commonly-used approach is to bend the antenna radials downward at 45° to raise the feedpoint resistance from 36 ohms to approximately 50 ohms — matching directly without a transformer. Or use a shunt-feed matching network at the base.
Physical length at 7.1 MHz for a 42-ohm transformer, VF 0.66: 75 × 0.66 / 7.1 = 6.97 m = 22.9 ft
Available Impedances and Approximations
The quarter-wave transformer requires a cable with a specific characteristic impedance, which may or may not be commercially available. Here are some useful tricks:
| Target ZT | Approximation Method | Notes |
|---|---|---|
| 25 Ω | Two 50-ohm cables in parallel | Splice two equal lengths of RG-213 in parallel |
| 35 Ω | Not easily available — use L-network instead | Some 35-ohm cable exists for special applications |
| 50 Ω | Standard RG-213, LMR-400, etc. | Matches Z1 = Z2 = 50 (no transformation) |
| 75 Ω | RG-6, RG-11 CATV cable | Matches 50 ohms to 112.5 ohms: Z_T = √(50×112.5) = 75 |
| 100 Ω | Two 50-ohm cables in series (for high-frequency isolation sections); or dedicated 100-ohm coax | Matches 50 ohms to 200 ohms |
| 150 Ω | Two 75-ohm CATV cables in series; or dedicated 150-ohm cable | Matches 50 ohms to 450 ohms (useful for some ladder-line sections) |
For the specific case of ZT = 75 ohms (widely available as CATV cable): it matches Z1 to Z2 where Z1 × Z2 = 75² = 5625. So 50 × 112.5 = 5625 — a 75-ohm quarter-wave section matches a 50-ohm source to a 112.5-ohm load.
Bandwidth Limitations
A quarter-wave transformer produces a perfect match at exactly the design frequency. At frequencies above and below the design frequency, the transformer section is no longer exactly 90° long, and the match degrades. The SWR at the input of the transformer rises as you move away from the design frequency.
The bandwidth of a quarter-wave transformer — defined as the frequency range over which SWR stays below 2:1 at the input — depends on the impedance ratio being matched. The larger the impedance ratio, the narrower the bandwidth:
| Impedance Ratio (Z2/Z1) | Approximate 2:1 SWR Bandwidth |
|---|---|
| 1.5:1 | Very wide (~50% of center frequency) |
| 2:1 | Wide (~35–40%) |
| 4:1 | Moderate (~20–25%) |
| 10:1 | Narrow (~10–15%) |
| 100:1 | Very narrow (<5%) |
For the typical amateur radio case of matching 50 ohms to 200 ohms (4:1 ratio), the 2:1 SWR bandwidth of a quarter-wave transformer is approximately 20–25% of the center frequency. On the 20-meter band (14 MHz), this means the transformer works well from about 12.6 to 15.4 MHz — comfortably covering the entire 14.0 to 14.35 MHz amateur allocation. For narrower bands or applications where the impedance ratio is larger, a single quarter-wave transformer is sufficient.
Cascaded Sections for Wider Bandwidth
If you need wider bandwidth than a single quarter-wave transformer provides, you can cascade multiple quarter-wave sections with different characteristic impedances. Each section performs a partial impedance transformation, and the sections together produce the full transformation over a wider bandwidth.
The impedances of the intermediate sections are chosen so that the overall transfer function approximates a Chebyshev or Butterworth response. The mathematics are covered in microwave engineering texts; for most amateur radio applications, a single quarter-wave transformer with moderate impedance ratio is sufficient, and wider matching bandwidth is achieved more easily by using a wideband ferrite transformer or an L-network antenna tuner.
Frequently Asked Questions
Can I use a quarter-wave transformer with a reactive (non-resistive) load?
Not directly — the quarter-wave transformer formula Z_T = √(Z1 × Z2) only works when both Z1 and Z2 are real (purely resistive). If your load impedance has a reactive component (such as Z = 50 − j30 ohms), you must first cancel the reactive component using a stub or a series inductor/capacitor, making the load resistive, and then apply the quarter-wave transformer to match the remaining resistive component to the source. Alternatively, use a completely different matching approach such as an L-network, T-network, or Pi-network, which can handle complex loads directly.
What happens if I can't find cable with exactly the right impedance for my transformer?
In practice, you work with what is available. The two most accessible coaxial impedances are 50 ohms and 75 ohms. By combining them in parallel or using other configurations, you can approximate a range of intermediate impedances. If your required impedance is 61.2 ohms (to match 50 to 75), neither 50-ohm nor 75-ohm cable alone will give a perfect match — but a single section of 75-ohm cable matches 50 ohms to 112.5 ohms, which may be close enough for some applications. For precise matching when no suitable cable exists, use a wound transmission-line transformer (a 1:1, 4:1, or 9:1 ferrite balun/unun) instead of a quarter-wave coax section. Ferrite transformers are broadband and do not require a specific cable impedance.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.