Stub Matching — Transmission Line Stubs for Antenna Impedance Matching
A stub is a short section of transmission line connected in parallel (shunt) with a feed line or antenna system, with its far end either open-circuited or short-circuited. Stubs do not contain any lumped components — no capacitors, no inductors, no resistors — yet they can present any value of reactance from very high inductive to very high capacitive, depending on their length and termination. They are pure transmission-line elements, and their reactance is entirely determined by the stub length, the characteristic impedance of the line, and whether the far end is open or shorted.
Stub matching is widely used in ham radio for matching antenna feed points, for constructing sharp notch filters, for diplexers that combine two transmitters on one antenna, and for implementing the quarter-wave balun and traps that appeared in earlier modules. Understanding stubs is fundamental to understanding how transmission lines behave as reactive elements — knowledge that becomes essential when working with the Smith chart in the next lesson.
How Stubs Behave
To understand stubs, recall the behavior of transmission lines from the Transmission Lines module. A transmission line terminated in its characteristic impedance Z0 carries a pure traveling wave with no reflections. But when a line is terminated in anything other than Z0 — including an open circuit or a short circuit — standing waves appear on the line. The impedance seen looking into the line at any point is a function of the termination impedance, the line's Z0, and the electrical length of the line.
A stub, being terminated in either an open circuit (infinite impedance) or a short circuit (zero impedance), presents a purely reactive impedance at its input. No real (resistive) component appears because there is nothing to dissipate energy — the energy bounces back and forth between the stub termination and the feed line junction. The sign and magnitude of the reactance depend entirely on the stub's electrical length (its physical length relative to the signal wavelength, corrected for velocity factor).
Stub Reactance Formulas
For a stub of characteristic impedance Z0 and electrical length θ (in degrees, where 90° = quarter wavelength):
XSC = +Z0 × tan(θ)
(positive = inductive for 0 < θ < 90°; negative = capacitive for 90° < θ < 180°)
Open-circuited stub reactance:
XOC = −Z0 / tan(θ) = −Z0 × cot(θ)
(negative = capacitive for 0 < θ < 90°; positive = inductive for 90° < θ < 180°)
Electrical length from physical length:
θ (degrees) = 360° × l / λ = 360° × l × f / (vp × c)
where l = physical length, vp = velocity factor, c = speed of light
Or equivalently: θ = 360° × l × fMHz / (300 × vp) [l in meters]
Notice the symmetry: a short-circuited stub behaves exactly like an open-circuited stub that is 90° longer, and vice versa. This means you can always substitute one for the other if you have the space — or if physical constraints (like a floating end needing weatherproofing) favor one type.
Stub reactance vs. electrical length for 50 Ω stubs. Short-circuit stub (blue): inductive for 0–90°, capacitive for 90–180°. Open-circuit stub (red): capacitive for 0–90°, inductive for 90–180°. Both have infinite (resonant) reactance at 90° and at 180°.
View LargerSpecial Stub Lengths
Certain stub lengths produce particularly useful or notable behavior:
| Electrical length | Short-circuit stub | Open-circuit stub |
|---|---|---|
| Approaching 0° | X → 0 (short circuit) | X → −∞ (capacitive, very high) |
| 30° (λ/12) | X = +0.577 × Z₀ (inductive) | X = −1.732 × Z₀ (capacitive) |
| 45° (λ/8) | X = +Z₀ (inductive) | X = −Z₀ (capacitive) |
| 90° (λ/4) | X → +∞ (open circuit) | X = 0 (short circuit) |
| 135° (3λ/8) | X = −Z₀ (capacitive) | X = +Z₀ (inductive) |
| 180° (λ/2) | X = 0 (short circuit) | X → −∞ (open circuit) |
The quarter-wave (90°) stubs are especially important: a shorted quarter-wave stub presents an open circuit (very high impedance) at its input, and an open quarter-wave stub presents a short circuit. This is the principle behind the quarter-wave trap and the quarter-wave sleeve balun covered in previous modules.
Single-Stub Matching
Single-stub matching is the most common stub-based impedance matching technique. The goal is to transform a complex load impedance ZL = RL + jXL to a purely resistive 50 Ω at the feed line.
The procedure has two steps: first, find the point along the transmission line (the stub position) where the real part of the admittance equals 1/Z0 (the line's characteristic admittance); second, connect a stub at that point whose susceptance (imaginary part of admittance) cancels the imaginary part of the line admittance at that position.
1. Normalize the load: zL = ZL / Z0 (e.g., for ZL = 25+j15 Ω on 50 Ω line: zL = 0.5 + j0.3)
2. Find the stub position d (distance from load): the two solutions are where the real part of the line admittance equals 1 (normalized). Use the formula:
d/λ = (1/2π) × arctan(XL ± √(RL(Z0−RL)+XL²)) / RL ... [simplified for lossless line]
In practice: use a Smith chart (next lesson) or an online stub calculator for this step.
3. At position d, the line admittance has a normalized real part of 1 and an imaginary part ±jB.
4. Connect a stub with susceptance ∓jB to cancel the imaginary part.
For a short-circuit stub: lstub = (λ/2π) × arctan(Z0 × B)
For an open-circuit stub: lstub = (λ/2π) × arctan(1/(Z0 × B)) + λ/4 if needed
Worked example: Match a 25 Ω resistive load to a 50 Ω line at 14 MHz using a shorted stub.
The load is purely resistive: ZL = 25 + j0 Ω, Z0 = 50 Ω
Normalized load admittance: yL = Z0/ZL = 50/25 = 2 + j0
We need to move along the line until the real part of the admittance = 1. Starting at yL = 2, moving toward the generator on the admittance Smith chart, the real part equals 1 at two positions. For this purely resistive case, the first position is at d = λ/8 from the load (45° of electrical length).
At d = λ/8, the line admittance is y = 1 + j1 (the imaginary part is +j1 in normalized form = +j/50 = +j0.02 S).
We need a stub with susceptance −j0.02 S (capacitive) to cancel the +j0.02 S. A shorted stub of characteristic impedance 50 Ω with capacitive reactance X = −1/0.02 = −50 Ω requires electrical length 135° (from the table: shorted stub at 135° has X = −Z₀).
Physical stub length at 14 MHz with velocity factor 0.66 (RG-8 coax):
l = θ × λ × vp / 360° = 135° × (300/14) m × 0.66 / 360° = 135° × 21.43 × 0.66 / 360° = 5.31 m (17.4 ft)
Physical main line section from load to stub position at 14 MHz:
ld = 45° × 21.43 × 0.66 / 360° = 1.77 m (5.8 ft)
Double-Stub Matching
Single-stub matching requires the stub to be placed at a specific distance from the load — which means cutting the feed line at a precise location. In some installations this is impractical; the stub connection point may be difficult to access or the exact load impedance may not be known in advance.
The double-stub tuner solves this by using two stubs at a fixed spacing (typically λ/8 or 3λ/8 apart), where each stub length is independently adjustable. By adjusting both stub lengths, a match can be achieved for a wide range of load impedances without knowing the exact position on the feed line. Double-stub tuners are the basis of most commercial antenna tuners that use coaxial-line sections rather than lumped LC networks.
The limitation of the double-stub tuner is that there exists a region of load impedances that cannot be matched regardless of the stub lengths. This forbidden region depends on the stub spacing. For λ/8 spacing, loads with normalized conductance G > 2 cannot be matched; for 3λ/8 spacing, the forbidden region is G > 2/tan²(3π/4) which is different. This limitation can be addressed by placing a fixed λ/8 line section between the load and the first stub, shifting the load into a matchable region.
Practical Stub Construction
Stubs for HF and VHF are most commonly constructed from coaxial cable. The procedure is straightforward:
- Calculate the required stub length using the formulas above, accounting for the velocity factor of the coaxial cable you are using.
- For a shorted stub: cut the cable to the calculated length, strip and tin both conductors at the far end, and solder them together. The near end connects (in shunt) to the feed line.
- For an open stub: cut the cable to length and leave the far end open, but seal it with self-amalgamating tape or heat-shrink tubing to prevent moisture ingress. The near end connects (in shunt) to the feed line.
- The shunt connection to the main feed line is made with a T-connector (UHF or N-type tee, as appropriate for the frequency) or by cutting the main feed line and soldering a pigtail connection.
For accurate stub construction, use the exact velocity factor of the specific cable you have on hand — not the nominal figure. RG-58 velocity factor varies from 0.65 to 0.69 between manufacturers; RG-213 from 0.66 to 0.68; LMR-400 at 0.85. A 2% error in velocity factor causes a 2% error in electrical length, which at 144 MHz (where even small errors matter) corresponds to a 2° phase error — acceptable for most purposes but significant for precision matching.
Ham Radio Applications of Stubs
Trap antennas: A coaxial trap for a multi-band dipole is a parallel resonant circuit formed by a section of coaxial cable coiled into an inductor (the coiled cable provides inductance) with the coaxial capacitance providing the parallel capacitance. At the trap's resonant frequency, it presents very high impedance, electrically shortening the antenna to the resonant length for that band.
Diplexer/triplexer stubs: Two stub filters of different lengths can separate two transmitted frequencies that share a common antenna. A shorted quarter-wave stub at frequency f1 appears as an open circuit at f1 (blocking that frequency from going into the second transmitter's port) but presents low impedance at frequencies other than f1. Carefully designed stub combinations allow two or even three transmitters to share one antenna with adequate isolation between them.
Phased vertical arrays: Many amateur phased vertical antenna systems (like the 4-square receiving array for 160m) use stubs to provide the precise phase shifts needed between array elements. A quarter-wave shorted stub connected in series provides a 90° phase shift at the design frequency with very low insertion loss — far better than a lumped LC phase-shift network at these low impedance levels.
Impedance measurement: A shorted coaxial stub of adjustable length can be used to measure the electrical length of a cable or to measure an unknown capacitance or inductance. By trimming the stub length until the reactance at a test frequency equals the negative of the unknown reactance, the stub length directly gives the equivalent component value.
Frequently Asked Questions
If a shorted quarter-wave stub presents an open circuit, why not just leave the feed line disconnected instead?
A quarter-wave shorted stub presents an open circuit only at the design frequency — at all other frequencies, it presents a reactance that can be quite low (at half and full multiples of the design frequency, it presents a short circuit again). This frequency-selective behavior is the entire point: the stub blocks specific frequencies while passing others, which a physical disconnection cannot do. In trap antennas, the stub blocks signal above the trap frequency from reaching the outer portion of the element, allowing the inner element to resonate on the higher band. A simple break in the wire would stop all frequencies equally.
Does it matter whether I use an open stub or a shorted stub?
Both types can provide any required reactance value, but there are practical differences. Open stubs are easier to build (just leave the end open) but the open end can collect moisture, leading to impedance errors. They also radiate slightly from the open end at VHF and above. Shorted stubs are mechanically clean, handle weather better, and do not radiate, but require soldering the conductors together at the far end. At HF for outdoor installations, shorted stubs are generally preferred. For indoor bench use or VHF work where precision matters, open stubs can work well if moisture is not an issue. The stub lengths for the two types differ by λ/4 for the same reactance value.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.