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, very little loss (just the line's own conductor and dielectric loss), and no adjustments. The only requirement is a line of the right characteristic impedance — which on a circuit board simply means a trace of the right width. You will find quarter-wave transformers on almost every Wi-Fi, Bluetooth and GPS board, where they match chips, filters and amplifiers to 50-ohm traces.
A quarter-wave transformer as a PCB trace: a 70.7-ohm section a quarter-wavelength long makes a 100-ohm load look like 50 ohms at the design frequency.
View LargerThe Quarter-Wave Impedance Transformation
Recall from lesson the Electrical Length lesson 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: Matching a 100-ohm load on a 2.4 GHz board
A 2.4 GHz RF module's output must drive a component whose input is 100 ohms resistive at 2.44 GHz. The board uses 50-ohm microstrip everywhere else.
ZT = √(50 × 100) = √5,000 = 70.7 ohms
On the board, a 70.7-ohm microstrip is simply a narrower trace than the 50-ohm traces around it (use the microstrip formula from the Characteristic Impedance lesson or the board maker's calculator).
Length at 2,440 MHz with an effective VF of about 0.55: 75 × 0.55 / 2,440 = 0.0169 m = 16.9 mm
A narrower trace has a slightly higher effective velocity factor than a wide one, so in practice the length is fine-tuned with a field-solver tool or by measuring a prototype board on a network analyzer.
Example 2: Connecting 50-ohm test equipment to a 75-ohm video system
You want to drive a 75-ohm video input from a 50-ohm signal generator without reflections.
ZT = √(50 × 75) = √3,750 = 61.2 ohms
61-ohm coaxial cable does not exist as a commercial standard, and a quarter-wave section only works at one frequency, while a video signal covers everything from near DC to several megahertz. A quarter-wave section of 75-ohm coax would give Zin = 75²/50 = 112.5 ohms — not a match either.
Conclusion: For broadband signals such as video, the practical solution is a resistive minimum-loss matching pad (two resistors, 43.3 Ω in series and 86.6 Ω in shunt, which costs 5.7 dB of signal) or a wound broadband transformer. Quarter-wave transformers are for narrow-band, high-frequency signals.
Example 3: Matching a low-impedance amplifier output
A small RF power amplifier transistor at 900 MHz needs to see a load of about 12.5 ohms (resistive) for best output, but it must drive a 50-ohm filter.
ZT = √(12.5 × 50) = √625 = 25 ohms
On a circuit board, a 25-ohm microstrip is a wide trace. In a coaxial test fixture, two equal lengths of 50-ohm coax connected in parallel make a 25-ohm line.
Length at 900 MHz in the board (effective VF about 0.5 for a wide trace): 75 × 0.5 / 900 = 0.0417 m = 41.7 mm; in solid-PE coax (VF 0.66): 75 × 0.66 / 900 = 0.055 m = 55 mm.
Any reactance in the transistor's output (usually its output capacitance) must be tuned out first, as described in the FAQ below.
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 50-ohm coax in parallel, or use a wide PCB trace |
| 35 Ω | Easy as a PCB trace; rare as cable — use an 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 | Dedicated 93/100-ohm coax, or a narrow PCB trace | Matches 50 ohms to 200 ohms |
| 150 Ω | Very narrow PCB trace, or parallel-wire line | Matches 50 ohms to 450 ohms |
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 VSWR at the input of the transformer rises as you move away from the design frequency.
The bandwidth of a quarter-wave transformer — defined here as the frequency range over which the VSWR 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 VSWR 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 a typical case such as matching 50 ohms to 200 ohms (a 4:1 ratio), the 2:1 VSWR bandwidth of a quarter-wave transformer is approximately 20–25% of the center frequency. At 2.44 GHz this is roughly 500 MHz — comfortably covering the whole 2.400–2.4835 GHz Wi-Fi and Bluetooth band. For signals that cover a wide range of frequencies, or where the impedance ratio is large, a single quarter-wave transformer is not enough.
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 practical applications, a single quarter-wave transformer with a moderate impedance ratio is sufficient, and wider matching bandwidth is achieved more easily with a tapered line, a broadband ferrite transformer or a lumped L-network.
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. As cable, the two most accessible coaxial impedances are 50 ohms and 75 ohms; by combining them in parallel you can approximate some intermediate impedances. On a circuit board you have much more freedom, because any impedance from about 25 to 120 ohms is just a matter of trace width. 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 broadband matching, or when no suitable line exists, use a wound transmission-line transformer (1:1, 4:1 or 9:1 on a ferrite core) or a resistive matching pad instead of a quarter-wave section.
Test Your Knowledge
Answer the questions below to check your understanding of this lesson. Every answer can be found in the lesson above.