Quarter-Wave Lines, Transformers & Stubs

Fig 1 — The three line elements of microwave circuits: the quarter-wave transformer, the open-circuited stub and the short-circuited stub

Introduction to Transmission-Line Length Effects

Part 7 derived the input impedance of a terminated lossless line, Z_in = Z₀(Z_L + jZ₀ tan βℓ)/(Z₀ + jZ_L tan βℓ), and Part 6 showed that the reflection coefficient rotates by twice the electrical length along it. Those two results turn a length of line into a circuit component. By choosing its characteristic impedance and electrical length, a section of line can transform an impedance, act as an inductor or capacitor, resonate, or block a frequency — with no lumped parts at all.

Stage 4 of this series develops those components. This article covers the most important ones: the quarter-wave line, the half-wave line and the stub. Part 13 deals with the junctions and steps that connect them, and Part 14 with resonators.

Quarter-Wavelength Transmission Lines

A line that is a quarter of a guided wavelength long has electrical length βℓ = π/2. Its most important property follows directly from the input-impedance formula.

Quarter-Wave Impedance Transformation

Derivation — impedance inversion

Divide the numerator and denominator of the input-impedance formula by tan βℓ and let βℓ → π/2, so that tan βℓ → ∞:

Z_{i n} = Z_{0} \frac{Z_{L} / \tan β ℓ + j Z_{0}}{Z_{0} / \tan β ℓ + j Z_{L}} \overset{β ℓ \to π / 2}{\to} Z_{0} \frac{j Z_{0}}{j Z_{L}}

Quarter-wave impedance inversion

Z_{i n} = \frac{Z_{0}^{2}}{Z_{L}} (β ℓ = \frac{π}{2}, \frac{3 π}{2}, \dots)

Equivalently, in admittance: Y_in = Z_L/Z₀² — the line turns an impedance into a proportional admittance. That is why a quarter-wave line is called an impedance inverter.

The inversion turns a short into an open, an open into a short, an inductance into a capacitance, a series resonance into a parallel one, and a small resistance into a large one. It underlies quarter-wave transformers, stub filters, Doherty amplifiers, branch-line couplers and quarter-wave bias feeds.

Quarter-Wave Impedance Transformer Design

To match a resistive load R_L to a line of impedance Z₀, insert a quarter-wave section of impedance Z_T chosen so that its input impedance is Z₀:

Quarter-wave transformer

Z_{i n} = \frac{Z_{T}^{2}}{R_{L}} = Z_{0} \Rightarrow Z_{T} = \sqrt{Z_{0} R_{L}}

ℓ = \frac{λ_{g}}{4} = \frac{c}{4 f_{0} \sqrt{ε_{eff, T}}}

Z_T is the geometric mean of the two impedances. The section must be a quarter of the guided wavelength in the transformer line itself, whose εeff differs from that of the Z₀ line on microstrip.

A complex load cannot be matched by the transformer directly, because Z₀²/Z_L is then complex. Two standard remedies exist:

Move to a real point first. Along the line between the load and the transformer the impedance becomes real at every voltage maximum (R = Z₀S) and minimum (R = Z₀/S), λ/4 apart (Part 7). Insert the transformer there, with Z_T = Z₀√S or Z₀/√S (Example 3).
Cancel the reactance first with a series or shunt stub, then transform the remaining resistance.

Bandwidth of the Quarter-Wave Transformer

Derivation — reflection of a quarter-wave transformer away from f₀

Let θ = βℓ, equal to π/2 at f₀. Substituting Z_T = √(Z₀R_L) into Z_in and forming Γ = (Z_in − Z₀)/(Z_in + Z₀) gives, after some algebra (Pozar §5.5),

Reflection near the design frequency

| Γ | = \frac{1}{\sqrt{1 + [4 Z_{0} R_{L} / (R_{L} - Z_{0})^{2}] \sec^{2} θ}} \approx \frac{| R_{L} - Z_{0} |}{2 \sqrt{Z_{0} R_{L}}} | \cos θ |

\frac{Δ f}{f_{0}} = 2 - \frac{4}{π} \cos^{- 1} [\frac{Γ_{m}}{\sqrt{1 - Γ_{m}^{2}}} \frac{2 \sqrt{Z_{0} R_{L}}}{| R_{L} - Z_{0} |}]

Γ vanishes at θ = π/2 and grows as cos θ away from it. Setting |Γ| = Γ_m, the maximum tolerable reflection, and using θ = (π/2)(f/f₀) for a TEM line gives the fractional bandwidth.

The bandwidth shrinks as the impedance ratio grows: matching 100 Ω to 50 Ω with |Γ| ≤ 0.05 gives 18% bandwidth, matching 200 Ω gives 9% (Part 6, Fig 3).

Multi-Section Transformers

Bandwidth can be traded for length by splitting the impedance step into several smaller steps, each a quarter-wave long. In the theory of small reflections, each step reflects a small Γₙ, and the total reflection is the sum of the partial reflections delayed by the round trips between them:

Multi-section transformer (small-reflection approximation)

Γ (θ) \approx Γ_{0} + Γ_{1} e^{- j 2 θ} + Γ_{2} e^{- j 4 θ} + \dots + Γ_{N} e^{- j 2 N θ}

binomial : \ln \frac{Z_{n + 1}}{Z_{n}} = 2^{- N} (\binom{N}{n}) \ln \frac{R_{L}}{Z_{0}}

Choosing the Γₙ as binomial coefficients makes Γ(θ) ∝ cos^Nθ — maximally flat at f₀ (binomial transformer). Choosing them to give a Chebyshev polynomial gives equal ripple and the widest bandwidth for a given N (Chebyshev transformer).

Fig 2 — |Γ| against frequency for 50 Ω-to-100 Ω transformers: single section (70.7 Ω), two-section binomial (59.5, 84.1 Ω) and three-section binomial (54.5, 70.7, 91.7 Ω). The |Γ| ≤ 0.05 bandwidth grows from 18% to 49% and 70%

1 section2-section binomial3-section binomial

For larger ratios or wider bandwidths, Chebyshev designs give more bandwidth for the same number of sections at the cost of a ripple in the passband, and tapered lines (exponential, Klopfenstein) extend the idea to a continuous change of impedance. All of these are standard in Pozar's Microwave Engineering and implemented in circuit simulators.

Half-Wavelength Transmission Lines

A half-wave line (βℓ = π) gives tan βℓ = 0, so Z_in = Z_L whatever the line's own impedance.

Half-Wave Impedance Repetition

The half-wave line is "invisible" at its design frequency: it repeats the load impedance at its input, and in general the input impedance of any line repeats every half wavelength. Uses include:

Moving a load. A feeder cut to an integer number of half wavelengths presents the antenna's impedance at the transmitter, so that a measurement at the radio shows the true antenna impedance (at one frequency).
Half-wave resonators. An open-circuited half-wave line is resonant, with voltage maxima at both ends; it is the basic element of edge-coupled and hairpin filters (Part 14).
The half-wave coaxial balun. A half-wave loop of coax reverses the phase of the signal at one terminal of a balanced load, giving a balanced output at four times the impedance — the classic 4:1 balun for a 300 Ω folded dipole fed from 75 Ω (Example 5).
DC paths and spacing. Stubs and branches spaced by λ/2 do not interact through the main line at the design frequency.

Note that a half-wave line inverts the voltage (a phase shift of 180°) even though it repeats the impedance — a property the half-wave balun uses.

Open-Circuit and Short-Circuit Stubs

A stub is a section of line connected at one end to a circuit and terminated at the other end in an open or a short. Because its load is a pure reactance, its input impedance is a pure reactance too (Part 7, Fig 2):

Stub input impedance (stub impedance Z_s, electrical length θ)

Z_{o c} = - j Z_{s} \cot θ Z_{s c} = j Z_{s} \tan θ

Below λ/4 an open stub is capacitive and a short stub inductive; they swap at every quarter wavelength. A stub of impedance Z_s connected in shunt adds a susceptance B = tan θ/Z_s (open) or −cot θ/Z_s (short).

Open-Circuit Stubs

Open stubs are the easiest stubs to make in microstrip — no via is needed — and are the usual choice on PCBs. Their main uses:

Shunt capacitors (θ < 90°), used in stepped-impedance and stub low-pass filters and in matching networks.
Notches (bandstop traps) (θ = 90°): a quarter-wave open stub is a short circuit at its input, so connected in shunt across a line it shorts out that one frequency — used to suppress a harmonic or an interfering band (Fig 3).
Radial stubs, fan-shaped open stubs whose wide end gives a lower-impedance, broader-band RF ground in bias networks.

Their open end is not perfectly open: fringing at the end makes the stub electrically longer by Δℓ ≈ 0.4h on microstrip (Part 13), so stubs are drawn slightly short.

Fig 3 — |S₂₁| of a 50 Ω line with a shunt quarter-wave open stub of impedance 25, 50 or 100 Ω. The stub shorts the line at f₀; a lower-impedance stub gives a wider notch

Z_s = 25 ΩZ_s = 50 ΩZ_s = 100 Ω

Short-Circuit Stubs

Short stubs need a via (or a wire) to ground, but they have one property open stubs lack: they provide a DC path. Their main uses:

Shunt inductors (θ < 90°) in matching networks and filters.
Quarter-wave bias feeds and DC returns (θ = 90°): the stub is an open circuit at f₀, so it can feed DC to an amplifier or ground the DC on an antenna feed without disturbing the RF. A high-impedance stub gives the widest band of high impedance around f₀ (Example 6).
Harmonic shorts: a λ/4 short stub at f₀ is λ/2 at 2f₀, where it is a short circuit — so a bias feed also shorts the second harmonic, a property used in class-F and class-E amplifiers.
Metal-insulator support: a quarter-wave shorted stub can support the centre conductor of an air line mechanically with no effect at f₀.

Impedance Transformation Using Transmission-Line Sections

Combining a length of line with a stub gives a network that can match any load. In single-stub matching, a section of line of length d moves the load until the real part of its admittance equals Y₀, and a shunt stub then cancels the remaining susceptance:

Single shunt-stub match for a load Z_L = R_L + jX_L (Pozar §5.2)

t = \frac{X_{L} \pm \sqrt{R_{L} [(Z_{0} - R_{L})^{2} + X_{L}^{2}] / Z_{0}}}{R_{L} - Z_{0}}

\frac{B}{Y_{0}} = \frac{R_{L}^{2} t - (Z_{0} - X_{L} t) (X_{L} + Z_{0} t)}{R_{L}^{2} + (X_{L} + Z_{0} t)^{2}} Z_{0}

Two solutions for t give two distances d = (λ/2π) tan⁻¹t (add λ/2 if negative). At that point Y = Y₀ + jB; the stub must provide −jB: an open stub of length (λ/2π) tan⁻¹(−B/Y₀) or a short stub of length (λ/2π) tan⁻¹(Y₀/B), each plus λ/2 if negative.

The same problem is solved graphically on the Smith chart, which is how most engineers do it and how double-stub and balanced-stub variants are designed — see Single-Stub and Double-Stub Impedance Matching and Impedance Matching Using the Smith Chart. Other line-based matching networks use a series section of a different impedance in place of the stub, or two quarter-wave sections in cascade; all of them are combinations of the elements in this article.

Practical RF Applications

Application	Element	How it works
Antenna and amplifier matching	λ/4 transformer, single stub	Transforms a resistive load to 50 Ω, or cancels reactance then transforms
Bias tee / DC feed	λ/4 short or high-Z λ/4 line with RF bypass capacitor	Open circuit to RF at f₀, DC path to the supply
Harmonic trap	λ/4 open stub at the harmonic	Short circuit at the harmonic, small capacitance at f₀
DC return / ESD protection	λ/4 short stub on an antenna feed	Grounds static charge, invisible at f₀
Filters	Open and short stubs, λ/2 resonators	Stubs as L and C; resonators coupled to each other (Part 14)
Power dividers and couplers	λ/4 sections of 70.7 Ω and 35.4 Ω	Wilkinson divider and branch-line coupler use impedance inversion
Doherty amplifier	λ/4 impedance inverter	Modulates the load of the main amplifier as the peaking amplifier turns on
Baluns	λ/2 coax loop, λ/4 sleeve (bazooka) balun	Phase inversion or common-mode choking

Numerical Design Examples

The examples use c = 299.79 mm/ns and the microstrip calculator tables of Part 9 for line widths.

A Quarter-Wave Transformer on FR-4

Problem: Match a 100 Ω resistive load to 50 Ω at 2.45 GHz on 1.6 mm FR-4. Find Z_T, its width and its length.

Transformer impedance

Z_{T} = \sqrt{50 \times 100} = 70.71 Ω

From the calculator's FR-4 table at 2.45 GHz

w = 1.618 mm

ε_{eff} = 3.079

ℓ = \frac{299.79}{4 \times 2.45 \times \sqrt{3.079}} = 17.43 mm

Answer: Z_T = 70.71 Ω, w = 1.62 mm, ℓ = 17.43 mm

The transformer is 0.48 mm longer than a 50 Ω quarter-wave line on the same board (16.95 mm), because the narrower line has a lower εeff. The width steps at both ends are discontinuities that add a small capacitance; at 2.45 GHz on this board their effect is minor (Part 13).

Bandwidth of the Transformer for VSWR ≤ 1.5

Problem: Find the fractional bandwidth of the transformer of Example 1 for VSWR ≤ 1.5 (Γ_m = 0.2), treating the line as TEM.

Bandwidth formula

\frac{Γ_{m}}{\sqrt{1 - Γ_{m}^{2}}} \cdot \frac{2 \sqrt{Z_{0} R_{L}}}{| R_{L} - Z_{0} |} = \frac{0.2}{0.980} \times \frac{141.4}{50} = 0.577

\frac{Δ f}{f_{0}} = 2 - \frac{4}{π} \cos^{- 1} (0.577) = 2 - \frac{4}{π} \times 0.9553 = 0.784

Answer: about 78% — 1.49 GHz to 3.41 GHz

A 2:1 impedance ratio with a modest VSWR target gives a very wide band. For |Γ| ≤ 0.05 (26 dB return loss) the same transformer covers only 18%; for a 10:1 ratio, a few per cent.

Matching a Complex Load with a Transformer

Problem: Match Z_L = 30 + j40 Ω to 50 Ω with a 50 Ω line section followed by a quarter-wave transformer. Find where to put the transformer and its impedance.

Reflection coefficient and VSWR

Γ_{L} = \frac{- 20 + j 40}{80 + j 40} = 0.5 ∠ 90^{\circ} S = \frac{1.5}{0.5} = 3

Real points: V_max at d = φλ/4π, V_min λ/4 further

d_{max} = \frac{90^{\circ}}{720^{\circ}} λ = 0.125 λ (R = 150 Ω)

d_{min} = 0.375 λ (R = 16.7 Ω)

Transformer at the first real point (V_max)

Z_{T} = \sqrt{50 \times 150} = 86.6 Ω

Answer: insert an 86.6 Ω λ/4 transformer 0.125λ from the load (or a 28.9 Ω one at 0.375λ)

The V_max solution is shorter and uses a convenient 86.6 Ω line. Both solutions are narrower-band than the purely resistive case, because the 50 Ω section's electrical length also changes with frequency.

A Two-Section Binomial Transformer

Problem: Design a two-section binomial transformer from 50 Ω to 100 Ω and compare its |Γ| ≤ 0.05 bandwidth with the single-section design.

Binomial weights 1/4, 2/4, 1/4 of ln 2 = 0.693

Z_{1} = 50 e^{0.693 / 4} = 59.46 Ω Z_{2} = 50 e^{3 \times 0.693 / 4} = 84.09 Ω

Bandwidth (exact cascade, Fig 2)

single section: 0.91 – 1.09 f_{0} (18 %)

two sections: 0.755 – 1.245 f_{0} (49 %)

Answer: Z₁ = 59.5 Ω, Z₂ = 84.1 Ω; bandwidth increases from 18% to 49%

The price is twice the length. A three-section binomial (54.5, 70.7, 91.7 Ω) reaches 70%. The binomial response is maximally flat: its first N − 1 derivatives are zero at f₀.

A Half-Wave Coaxial 4:1 Balun

Problem: A 300 Ω folded dipole for 100 MHz is to be fed from 75 Ω coax (VF = 0.66) with a half-wave balun loop. How long is the loop?

Electrical half-wavelength in the coax

ℓ = VF \times \frac{λ_{0}}{2} = 0.66 \times \frac{2.998 m}{2} = 0.99 m

Answer: a 0.99 m loop of the same coax

The loop delays the signal on one side of the dipole by 180°, so the two dipole terminals are driven in antiphase (balanced). Each terminal sees 150 Ω, which is twice the cable's 75 Ω, and the two together present 300 Ω: a 4:1 transformation. Being a half-wave device, it works over a narrow band.

A Quarter-Wave Bias Feed for a 2.45 GHz Amplifier

Problem: A shorted (at RF, by a bypass capacitor) 100 Ω quarter-wave line on 0.813 mm RO4003C feeds DC to an amplifier output at 2.45 GHz. Find its length and the impedance it presents to the RF line at 2.2 and 2.7 GHz.

Length (calculator: 100 Ω line has εeff = 2.496)

ℓ = \frac{299.79}{4 \times 2.45 \times \sqrt{2.496}} = 19.36 mm

Off-frequency input reactance, Z_sc = j100 tan(90° f/f₀)

f = 2.2 GHz : + j 619 Ω f = 2.7 GHz : - j 619 Ω

Answer: 19.4 mm; it still presents more than 600 Ω across ±10% of f₀

A shunt reactance of 619 Ω across a 50 Ω line causes less than 0.01 dB of loss, so this feed is effectively invisible over the band. A 50 Ω stub would present only 310 Ω at the band edges; a narrow, high-impedance line is preferred for bias feeds.

An Open-Stub Trap for the Second Harmonic

Problem: A 2.45 GHz transmitter must suppress its 4.9 GHz harmonic with a 50 Ω open stub on RO4003C (εeff ≈ 2.79 at 4.9 GHz). Find the stub length and its effect at 2.45 GHz.

Quarter wave at 4.9 GHz

ℓ = \frac{299.79}{4 \times 4.9 \times \sqrt{2.79}} = 9.16 mm

At 2.45 GHz the stub is λ/8: Z_oc = −j50 cot 45°

Z_{o c} = - j 50 Ω C = \frac{1}{2 π \times 2.45 \times 10^{9} \times 50} = 1.30 pF

Answer: a 9.16 mm stub (shortened slightly for end fringing) that adds 1.3 pF of shunt capacitance at f₀

The stub shorts the harmonic but also loads the fundamental. The designer absorbs the 1.3 pF into the output matching network — or places the stub a quarter-wave (at f₀) away from the point where it must not load.

Single-Stub Match for a 100 Ω Load

Problem: Match R_L = 100 Ω to 50 Ω with a 50 Ω line and a shunt 50 Ω stub. Find d and the stub lengths.

t (X_L = 0)

t = \pm \frac{\sqrt{100 \times (50 - 100)^{2} / 50}}{100 - 50} = \pm \frac{70.71}{50} = \pm 1.414

Solution 1: d = tan⁻¹(1.414)/2π λ, B/Y₀ = +0.707

d = 0.152 λ open stub : 0.098 λ short stub : 0.348 λ

Solution 2: d = 0.348λ, B/Y₀ = −0.707

open stub : 0.402 λ short stub : 0.152 λ

Answer: d = 0.152λ with a 0.098λ open stub (shortest), or d = 0.348λ with a 0.152λ short stub

The shortest solution (0.152λ of line and 0.098λ open stub) has the widest bandwidth, because the shortest lines change least with frequency. The same answer is read from the Smith chart by moving along the S = 2 circle to the 1 + jb circle.

One Stub, Two Jobs: Bias Feed and Harmonic Short

Problem: Show that the quarter-wave bias stub of Example 6 is a short circuit at the second harmonic, and find its impedance at the third harmonic.

Electrical length scales with frequency

2 f_{0} : θ = 180^{\circ}, Z_{s c} = j 100 \tan 180^{\circ} = 0

3 f_{0} : θ = 270^{\circ}, Z_{s c} \to \infty

Answer: a short at 2f₀ (and every even harmonic), an open at 3f₀ (and every odd harmonic)

This is exactly the harmonic termination a class-F power amplifier needs at the drain: short even harmonics, open odd ones. A single quarter-wave line performs both the bias and the waveform-shaping functions.

How the Stub Impedance Sets the Notch Width

Problem: For the shunt quarter-wave open-stub trap on a 50 Ω line (Fig 3), find the band over which the rejection exceeds 20 dB for stub impedances of 25, 50 and 100 Ω.

|S₂₁| = 2/|2 + jZ₀ tan θ/Z_s|, solved for |S₂₁| = −20 dB

Z_{s} = 25 Ω : \pm 6.4 % Z_{s} = 50 Ω : \pm 3.2 % Z_{s} = 100 Ω : \pm 1.6 %

Answer: the 20 dB notch width is inversely proportional to the stub impedance

A low-impedance (wide) stub has a larger susceptance, tan θ/Z_s, for a given detuning from f₀, so it keeps the line shorted over a wider band. Choose a low-impedance stub for a broad trap and a high-impedance one for a sharp notch that leaves nearby frequencies untouched.

Summary

Quarter-wave line: Z_in = Z₀²/Z_L (impedance inverter). Transformer: Z_T = √(Z₀R_L), ℓ = λ_g/4 in the transformer line.
Complex loads are matched by placing the transformer at a voltage maximum or minimum, or by cancelling the reactance first.
Bandwidth: |Γ| ≈ (|R_L − Z₀|/2√(Z₀R_L)) |cos θ|; multi-section binomial and Chebyshev transformers trade length for bandwidth.
Half-wave line: Z_in = Z_L with 180° phase reversal — repetition, resonators and baluns.
Stubs: Z_oc = −jZ_s cot θ, Z_sc = jZ_s tan θ — capacitors, inductors, traps, bias feeds and harmonic terminations.
A line plus a shunt stub matches any load (single-stub matching).

Every one of these elements has to be joined to the rest of the circuit, and the joints are not ideal. Part 13: Transmission-Line Discontinuities and T-Junctions examines steps in width, open ends, bends, gaps and T-junctions — and how to model and compensate them.

📘 Go deeper: Stub matching on the Smith chart

Single- and double-stub matching are much quicker to design graphically. Open Smith Chart & Impedance Matching series →

Put It Into Practice

Use the NEXT ANTENNAS calculators to apply the ideas from this article to real lines and circuits.

Impedance Matching Smith Chart Visualiser Plot impedances, move along lines and stubs, load Touchstone files and design matching networks. Open calculator → Transmission Lines Microstrip & Stripline Impedance Calculator Field-solver backed design of microstrip, embedded microstrip, stripline, GCPW and differential pairs: width, εeff, Z₀, loss, dispersion and tolerance. Open calculator → Link & Power VSWR & Return Loss Calculator Convert between VSWR, return loss, reflection coefficient and mismatch loss. Open calculator → All Tools All RF Calculators Browse every NEXT ANTENNAS calculator for RF, microwave and antenna design. Open calculator →

Quarter-Wave and Half-Wave Lines and Stubs