Single Stub Matching

Single stub matching is one of the most practical impedance matching techniques used in microwave engineering. It uses a short section of transmission line (the stub) connected either in shunt (parallel) or in series at a specific distance from the load to cancel the reactive part of the input admittance or impedance, achieving a perfect match at the design frequency. This technique is widely implemented in microstrip circuits on printed circuit boards.

Core Concept Explanation

When a transmission line is connected to a mismatched load, reflections arise and the input impedance at any point along the line varies with position. At most positions, this impedance has both real and imaginary parts. The key insight behind single stub matching is that at a specific distance d from the load, the real part of the normalized input admittance (for shunt stub) becomes exactly 1, meaning g = 1. At this point, only the reactive (susceptance) part remains as the source of mismatch.

A stub is a short section of transmission line that is terminated in either a short circuit or an open circuit. Because of its termination, a stub is purely reactive — it presents no resistive component. By choosing its length appropriately, the stub provides exactly the susceptance needed to cancel the residual susceptance at position d, making the total admittance at that point equal to Y₀ (normalized: y = 1 + j0), which is the matched condition.

For a shunt stub, the stub is connected in parallel (shunt) with the main line at position d. Since parallel elements add in admittance, the susceptance of the stub directly cancels the residual susceptance of the line. This makes the shunt stub configuration the most common and fabrication-friendly, particularly in microstrip where shunt stubs are easy to implement as open-circuited branches.

For a series stub, the stub is inserted in series with the main transmission line at position d. In this case, the design is performed in impedance form: the position d is chosen where the normalized resistance r = 1, and the series stub provides a reactance that cancels the residual series reactance. Series stubs are less common in microstrip but are used in coaxial and other media.

Mathematical Expression

For shunt stub design, the input admittance at distance d from the load is:

y_in(d) = y_L ·[1 + j·tan(βd)] / [y_L + j·tan(βd)] ... (in normalized admittance form)

where y_L = Y_L/Y₀. The distance d is chosen so that the real part of y_in equals 1. This gives two possible positions per period, leading to two solutions. For a short-circuited stub, the stub susceptance is:

b_stub = -cot(βl) (for SC stub)

For an open-circuited stub:

b_stub = tan(βl) (for OC stub)

The required stub susceptance b_stub = -b_residual, where b_residual is the imaginary part of y_in at position d. Solving for l:

For SC stub: l = (1/β) · arccot(-b_stub) For OC stub: l = (1/β) · arctan(b_stub)

Practical Understanding

In microstrip design, open-circuited stubs are preferred over short-circuited ones because they do not require drilling through the substrate to make a via-hole ground connection. However, open-circuited stubs can radiate at their open end, particularly at millimeter-wave frequencies. Short-circuited stubs are preferred in high-power applications and coaxial systems where a clean ground is available.

The single stub tuner is a narrowband matching network: both d and l are optimized at the design frequency, and the match degrades at other frequencies because the electrical lengths of both the line section and the stub change with frequency. This limits its application to circuits where wide bandwidth is not required.

Practical stub design also uses the Smith chart graphically. The load admittance is plotted, the VSWR circle is drawn, and the chart is rotated until the g = 1 circle is reached. The WTG scale reading at this intersection gives d/λ, and the stub length l/λ is found from the outer edge of the chart where the required susceptance is located.

Given:
Z₀ = 50Ω
Z_L = 100 + j100Ω
Design: shunt short-circuit stub, f₀ = 2 GHz (λ = 15 cm in free space)

Why this formula applies:
Shunt stub matching uses admittance. Find d where g=1, then cancel b with SC stub.

Formula:
y_L = Z₀/Z_L
g = 1 condition gives stub position d
b_stub = −cot(βl), set equal to −b_residual

Substitution:
y_L = 50/(100+j100) = 50(100−j100)/[(100²+100²)] = 50(100−j100)/20000
    = (5000−j5000)/20000 = 0.25 − j0.25

|Γ| = |(z_L−1)/(z_L+1)|, z_L = (100+j100)/50 = 2+j2
|Γ| = |(1+j2)/(3+j2)| = √5/√13 = 0.620

At position d, g=1 intersection of VSWR circle:
y_in = 1 + j1.0 (one solution) or 1 − j1.0 (second solution)
For y_in = 1 − j1.0: b_residual = −1.0

Required stub susceptance: b_stub = −(−1.0) = +1.0
SC stub: b_stub = −cot(βl) = +1.0 → cot(βl) = −1.0 → βl = 3π/4
l = (3π/4) / (2π/λ) = (3/8)λ = 3/8 × 15 cm = 5.625 cm

Stub position d from Smith chart WTG reading ≈ 0.088λ = 1.32 cm

Final Answer:
Shunt SC stub placed at d = 1.32 cm from load, stub length l = 5.625 cm provides perfect match at 2 GHz.

Exam Tip: In single stub matching, there are always two positions d per wavelength where g=1 (or r=1 for series stub). Each position gives a different stub length l. For GATE problems, check which solution gives a shorter stub length — that is usually the preferred answer. Also remember: OC stub susceptance = +tan(βl), SC stub susceptance = -cot(βl).

Mechanism Summary

• Shunt stub matching: convert load to admittance y_L = Y_L/Y₀, rotate on Smith chart toward generator until g = 1, then cancel residual susceptance b with a short or open-circuited stub in parallel.
• Series stub matching: work in impedance form z_L = Z_L/Z₀, rotate until r = 1, cancel residual reactance x with a series stub.
• Short-circuit stub susceptance: b_SC = -cot(βl). Open-circuit stub susceptance: b_OC = tan(βl). Both are purely imaginary and vary between -∞ and +∞.
• Two solutions exist per λ period for single stub matching, corresponding to two g = 1 (or r = 1) circle intersections. Choose the solution with shorter stub and line lengths.
• Single stub matching is narrowband; both d and l are optimized only at f₀. At other frequencies, the electrical lengths change and the match degrades.

Quick Revision

• Single stub matching uses a stub at position d to cancel the residual reactance/susceptance and achieve y_in = 1+j0 (or z_in = 1+j0).
• Shunt stub design: admittance form, find g=1 intersection, stub provides b_stub = -b_residual.
• Series stub design: impedance form, find r=1 intersection, stub provides x_stub = -x_residual.
• SC stub: b = -cot(βl). OC stub: b = tan(βl). SC stub: x = j·tan(βl). OC stub: x = -j·cot(βl).
• Two solutions per wavelength; shorter solution preferred for compact circuit design.
• Narrowband technique: degrades away from design frequency f₀.
• Common trap: using impedance (r, x) form for shunt stub instead of admittance (g, b) form — leads to wrong stub position.