A classic microwave-circuit design tool that matches a mismatched load to a transmission line with a single parallel stub. Change the load impedance and characteristic impedance to see the stub position d, stub length ℓ and the VSWR before and after matching update in real time.
Parameters
Load resistance R_L
Ω
Real part of the load impedance
Load reactance X_L
Ω
Imaginary part (positive: inductive / negative: capacitive)
Line characteristic impedance Z₀
Ω
Characteristic impedance of the main line and the stub
Frequency f
GHz
Design frequency (sets the wavelength λ)
Stub type
Whether the stub far end is shorted or left open
Results
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Stub position d (λ)
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Stub length ℓ (λ)
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Load reflection |Γ|
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VSWR before matching
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VSWR after matching
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Stub position d (mm)
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Line & stub layout — standing-wave animation
The main line runs from the source on the left to the load on the right. The stub branches off at distance d; the load side stays mismatched and rippled, the source side is matched and flat.
Line admittance y(d) vs distance
Reflection coefficient |Γ| vs frequency (narrowband)
Normalised input admittance a distance d from the load. y_L: normalised load admittance, β=2π/λ: phase constant. The smallest d with Re[y(d)] = 1 is the stub position.
Stub input susceptance. When y=1+jb at position d, choose the length ℓ so the stub supplies −b and the match is complete.
What is the Single-Stub Matching Simulator?
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I keep hearing the term "stub matching" — but what exactly is a "stub"? Some kind of branch growing off the line?
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Yes, it really is a branch. A stub is "a short offcut of transmission line, shorted or left open at its far end". You tap it onto the main line in parallel. The key point is that this branch acts not as a resistance but as a pure reactance — a susceptance. Change its length and you can make any susceptance you want. Stub matching is the idea of building the "tuning knob" for impedance matching out of the line itself.
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Wait, couldn't you just match with a coil and a capacitor? That's what I was taught at school.
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At low frequencies, sure. But push the frequency slider up to 2 GHz or 10 GHz. At that point the wavelength is only a few centimetres, and real coils and capacitors are full of parasitic inductance and capacitance — they no longer behave as "ideal elements". So in the microwave world you use the transmission line itself as the reactive component. Drawing a single trace on a circuit board gives you a stub: it is cheap and accurate.
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I see. But how do you decide "where" to put the stub and "how long" it should be?
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Think of it in two stages. First, view the load as an admittance — the reciprocal of impedance. As you walk back up the line from the load, the input admittance keeps changing. That is what the "Line admittance" chart below shows. There is always a point where the real part is exactly 1 — the characteristic admittance. That point is the stub position d. There y = 1 + jb, with only a leftover susceptance b. Then you make the stub produce −b to cancel it, and upstream the line looks perfectly matched. The stub length ℓ is set to deliver exactly that −b.
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On the chart on the right, the VSWR was 2.6 before matching and snapped to exactly 1.00 after! Amazing. But what happens if I shift the frequency a little?
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Good question — that is the weak spot of stub matching. Both d and ℓ are fixed physical lengths, but when the frequency changes their electrical lengths (βd, βℓ) shift. The tangent values change and the matching condition breaks. Look at the "reflection vs frequency" chart: it drops to a sharp notch at the design frequency and the reflection rises quickly away from it. That is the "narrowband" behaviour. When you need a wide band you move on to two stubs (double-stub), or combine a quarter-wave transformer or a tapered line.
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There's also a choice between a short-circuit stub and an open-circuit stub. What's the difference?
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The "starting phase" of the susceptance they produce is different. A short-circuit stub has input susceptance −cot(βℓ), an open-circuit stub +tan(βℓ). So even to produce the same −b, the required length ℓ differs. As for implementation: in coaxial lines and metal waveguides it is reliable to short the far end with metal, so short-circuit stubs are preferred. In board circuits such as microstrip, shorting the end needs a via and is a nuisance, so open-circuit stubs that need nothing at the end are used more often. Switch the stub type and see how ℓ changes.
Frequently Asked Questions
Single-stub matching connects one short piece of transmission line, called a stub, in parallel between a mismatched load and the main line, so that the line becomes perfectly matched. At the right distance d from the load, the real part of the line input admittance equals the characteristic admittance, leaving only a residual susceptance. A stub shorted (or opened) at its far end is then tapped in parallel at that point, and its length is set so its input susceptance cancels the residual one. Upstream of the stub the line then sees Z0.
At microwave frequencies of several GHz the wavelength becomes comparable to component sizes, and real inductors and capacitors are dominated by parasitics, so they no longer behave as ideal reactive elements. Instead the transmission line itself is used as the reactive element. A fixed length of line that is shorted or opened at its far end presents any required susceptance depending on its length, so matching is achieved simply by tapping a stub on at the right position and length. It can also be made cheaply and accurately as a board trace or a coaxial section.
First find the load admittance yL, then sweep the input admittance a distance d from the load, y(d) = (yL + j·tan(βd)) / (1 + j·yL·tan(βd)), finely over 0 to 0.5λ. The smallest d at which Re[y(d)] = 1 is the stub position. There y = 1 + j·b, so the stub must supply a susceptance of −b. A short-circuit stub has input susceptance −cot(βℓ) and an open-circuit stub +tan(βℓ); setting either equal to −b and solving for ℓ in (0, 0.5λ] gives the stub length. This tool performs that sweep automatically.
Stub matching chooses d and ℓ so that Re[y(d)]=1 and the susceptance cancellation both hold at the design frequency. But d and ℓ are fixed physical lengths, so when the frequency changes the electrical lengths βd and βℓ shift, the tangent values change and the matching condition breaks. As a result the reflection coefficient |Γ| dips deeply at the design frequency and rises rapidly away from it. This narrowband behaviour is intrinsic to stub matching; for wide bandwidth, multi-section matching, quarter-wave transformers or tapered lines are used.
Real-World Applications
Antenna feeds and impedance matching: The input impedance of an antenna is almost always offset from the design value of 50 Ω or 75 Ω. Inserting a single stub between the transmitter and the antenna restores the match, reducing the reflected wave, raising the radiation efficiency and protecting the transmitter output stage. Stub matching, or its more advanced forms, is widely used in mobile-network base-station antennas, radar and satellite-communication feeds.
Microstrip board circuits: In microwave circuits on a printed board the matching stub can be drawn directly as a copper trace. Open-circuit stubs appear constantly in amplifier input/output matching, mixers and the matching stages of filters. The fact that adding one trace to the board costs nothing in components is why they are so valued in volume production.
High-power and waveguide systems: In high-power circuits such as broadcast transmitters, industrial RF heating and the RF systems of accelerators, lumped elements cannot be used because of heating and dielectric breakdown. Matching is done with a short-circuit stub — including sliding adjustable stubs — on a coaxial line or a metal waveguide. The "stub tuner", whose stub position and length can be moved mechanically, is also a standard instrument in measurement labs.
RF design education and Smith-chart exercises: Single-stub matching is the classic problem solved on the Smith chart by combining "constant-conductance circles" with "constant-radius rotation". Comparing the numerical solution of this tool with the d and ℓ obtained by hand on a paper Smith chart is a good way to confirm how to read the chart. It is an ideal first topic for learning matching theory.
Common Misconceptions and Pitfalls
A common assumption is that "stub matching has only one pair of d and ℓ as a solution". In reality, the d values where Re[y(d)]=1 recur with a half-wavelength period, each with a corresponding short and open solution. This tool shows the "smallest d", but choosing a different solution can make the stub shorter or change the bandwidth. Proper design selects from several solutions based on layout space, bandwidth and loss — the smallest d is not always the best.
Next, treating "the line wavelength as the free-space wavelength". This tool simplifies things with λ ≈ free-space wavelength (c/f), but a real transmission line is filled with dielectric, so the wavelength on the line — the effective wavelength — is shorter than the free-space value. On microstrip it shrinks by √εeff with the effective relative permittivity. When you cut the real d and ℓ in millimetres, always convert using the line's effective wavelength; building them at the free-space wavelength shifts the matching frequency.
Finally, the misconception that "as long as you get a match, bandwidth does not matter". Stub matching can drive the reflection to zero at the design frequency, but at the cost of a narrow band. As the "reflection vs frequency" chart shows, the deeper the notch — and the larger the load mismatch (VSWR) — the narrower the allowable frequency range. If you handle a wideband signal, always check the bandwidth over which |Γ| stays below the specified value (for example −10 dB), and if it is insufficient, decide to switch the design to a double stub or multi-section matching.
How to Use
Enter the load impedance as resistance (R) and reactance (X) in ohms—for example, R=50Ω, X=25Ω for a mismatched antenna load.
Set the characteristic impedance Z₀ of your transmission line (typically 50Ω or 75Ω for coaxial cable).
Input the operating frequency in GHz (e.g., 2.4 GHz for Wi-Fi or 10 GHz for X-band radar).
Click Simulate to compute the stub position distance d in wavelengths and the required stub length ℓ, both normalized and in millimeters.
Worked Example
Load impedance Z_L = 40 − j30 Ω connected to 50Ω line at 5 GHz. The simulator calculates wavelength λ = 60 mm. Initial VSWR = 2.8 from reflection coefficient |Γ| = 0.47. The algorithm places a parallel shunt stub at distance d = 0.132λ = 7.9 mm from the load, with stub length ℓ = 0.184λ = 11.0 mm (short-circuited). After matching, VSWR reduces to 1.0, achieving perfect impedance match for maximum power transfer.
Practical Notes
Smith chart visualization confirms that the load point rotates along the constant-conductance circle to the center as you move toward the stub location.
Stub length is highly sensitive to frequency; a 2% shift in frequency can require 0.01λ stub-length adjustment in narrow-band systems.
Use short-circuited stubs in practice rather than open stubs to avoid radiation losses in microwave PCB layouts above 3 GHz.
For broadband matching, cascade two or three stubs; single-stub matching typically maintains VSWR < 1.5 over ±5% bandwidth around design frequency.