Quarter-Wave Transformer & Stub Calculator

Method, formulas and references

Lines

All lines are lossless and TEM, so the electrical length θ = βl grows in proportion to frequency. A line of impedance Z0 and length l terminated in ZL has the input impedance

For microstrip, εeff and an approximate width come from the quasi-static Hammerstad–Jensen formulas for each impedance, so sections of different impedance have different λ/4 lengths. For stripline εeff = εr and the width comes from Wheeler's formula. Both are closed-form estimates for bare lines; the Solve width link opens the field-solver impedance calculator with the target impedance and the stackup filled in. Microstrip dispersion, solder mask and conductor loss are not included.

Quarter-wave transformer

A λ/4 section of impedance Z1 transforms a resistive load to Z1²/ZL, so it matches ZL to Z0 when

Pozar (§5.4) gives the fractional bandwidth over which |Γ| ≤ Γm, exact for TEM lines:

Binomial transformer

The small-reflection response of N equal-length sections is Γ(θ) ≈ Σ Γn e^−2jnθ with Γn ≈ ½ ln(Zn+1/Zn) (Pozar §5.5). The binomial design makes it maximally flat at f0, Γ(θ) = A(1 + e^−2jθ)^N, and Pozar recommends computing the sections from

which ends exactly on ZL. With A ≈ 2^−(N+1) ln(ZL/Z0), the bandwidth for |Γ| ≤ Γm is

Chebyshev transformer

The equal-ripple design (Pozar §5.7) sets the response to a Chebyshev polynomial with ripple Γm in the passband θm ≤ θ ≤ π − θm:

T_N(sec θm cos θ) is expanded into cos(N − 2n)θ terms using cos^kθ = 2^−k Σ C(k, j) cos(k − 2j)θ and matched term by term to 2[Γ0 cos Nθ + Γ1 cos(N − 2)θ + …] (for even N the last term is ΓN/2), giving the symmetric Γn; then ln(Zn+1/Zn) = 2Γn. The bandwidth is Δf/f0 = 2 − 4θm/π. This works for any N, not only the tabulated N ≤ 4.

Exact response

Equations (4)–(6) rest on the small-reflection approximation. The plot and the exact bandwidth instead cascade the actual sections with equation (1), from the load to the input. For a Chebyshev design the exact ripple can exceed Γm slightly; the page reports it when it does. The response of real-impedance TEM sections is symmetrical about f0 and repeats every 2f0.

Stubs

With equation (1), a short-circuited stub has Zin = jZ0 tan βl and an open-circuited stub Zin = −jZ0 cot βl (Pozar §2.3). The shortest lengths for a reactance X are

A target L or C is converted with X = ωL or X = −1/(ωC) at f0; the stub only mimics it near f0. At the quarter-wave frequency f = c/(4l√εeff) an open stub becomes a short (series resonance) and a short stub becomes an open (parallel resonance).

Single-stub shunt matching

A shunt stub is placed at the distance d from the load where the line admittance is Y0 + jB, and cancels jB (Pozar §5.2). With t = tan βd and ZL = RL + jXL on a line of impedance Z0:

d/λ = tan⁻¹(t)/2π, plus ½ when t < 0. The susceptance B at d is computed from equation (1), and the stub length follows from jY0s tan βl = −jB (open) or −jY0s cot βl = −jB (short), where Y0s is the stub admittance, which may differ from the line's. Each solution is checked by computing the input impedance. For the sweep, the load is its resistance in series with the L or C that gives XL at f0.

Limits

Lines are lossless and dispersion-free and junctions are ideal. Real step discontinuities, T-junctions, open-end fringing (which makes an open stub electrically longer) and the via inductance of a short stub shift the response; check the final layout with an EM simulator.

Data sources

1. D. M. Pozar, Microwave Engineering, 4th ed., Wiley, 2012
   §2.3 (input impedance of a terminated line, open and short stubs), §5.2 (single-stub shunt tuning), §5.4 (quarter-wave transformer and its bandwidth), §5.5 (theory of small reflections), §5.6 (binomial) and §5.7 (Chebyshev multisection transformers). Worked examples of §5.2, §5.4, §5.6 and §5.7 are reproduced in the tests.
2. E. Hammerstad, Ø. Jensen, “Accurate Models for Microstrip Computer-Aided Design”, IEEE MTT-S International Microwave Symposium Digest, pp. 407–409, 1980
   Quasi-static microstrip impedance and εeff with thickness correction, inverted here by bisection to give an approximate width.
3. H. A. Wheeler, “Transmission-Line Properties of a Strip Line Between Parallel Planes”, IEEE Trans. Microwave Theory Tech., vol. 26, no. 11, pp. 866–876, 1978
   Symmetric stripline impedance with thickness, inverted by bisection for the approximate stripline width.