Single-Stub Series Matching is a widely used impedance matching technique in RF and microwave engineering. It employs a short-circuited or open-circuited transmission line stub connected in series with the main transmission line at a specific distance from the load. The objective is to transform an arbitrary load impedance into a matched condition, minimizing reflections and maximizing power transfer.

Since the matching element is connected in series, impedance is more convenient than admittance for analysis. Engineers can design a series stub matching network using either analytical calculations or the graphical Smith Chart method.

Understanding Single-Stub Series Matching

A series stub is inserted directly into the transmission line at a carefully selected location. The stub introduces a reactive impedance that cancels the unwanted reactance present at that point on the line.

The normalized impedance is:

\[
z = \frac{Z}{Z_0} = r + jx
\]

The objective is to achieve a matched condition:

\[
z_{in} = z_d + z_{stub} = 1 + j0
\]

When this condition is satisfied, the transmission line is perfectly matched.

Analytical Design Procedure

Step 1: Calculate the Normalized Load Impedance

Begin by dividing the load impedance by the characteristic impedance.

\[
z_L = \frac{Z_L}{Z_0} = r_L + jx_L
\]

This normalized impedance serves as the starting point for all matching calculations.

Step 2: Determine the Distance to the Stub

Move along the transmission line until the real part of the normalized impedance becomes unity.

The impedance transformation equation for a lossless transmission line is:

\[
z(d) = \frac{z_L + j\tan(\beta d)}
{1 + jz_L\tan(\beta d)}
\]

The matching condition requires:

\[
\text{Re}\{z(d)\} = 1
\]

which gives:

\[
z(d) = 1 + jx
\]

Solving this equation produces two valid solutions:

\[
d_1 \quad \text{and} \quad d_2
\]

within:

\[
0 \le d < \frac{\lambda}{2}
\]

Step 3: Determine the Required Stub Reactance

At the selected location, the transformed impedance becomes:

\[
z_d = 1 + jx
\]

To achieve matching, the stub must cancel the reactive component.

\[
x_{stub} = -x
\]

The total impedance then becomes:

\[
z_{in} = 1 + j0
\]

Step 4: Calculate the Stub Length

The required length depends on whether a short-circuited or open-circuited stub is used.

Short-Circuited Stub

The normalized impedance of a short-circuited stub is:

\[
z_{stub} = j\tan(\beta l)
\]

Since:

\[
z_{stub} = -jx
\]

the required length is obtained from:

\[
\tan(\beta l) = -x
\]

Open-Circuited Stub

The normalized impedance of an open-circuited stub is:

\[
z_{stub} = -j\cot(\beta l)
\]

Since:

\[
z_{stub} = -jx
\]

the required length is determined from:

\[
\tan(\beta l) = \frac{1}{x}
\]

The final solution must satisfy:

\[
0 \le l < \frac{\lambda}{2}
\]

If the calculated value is negative, add:

\[
\frac{\lambda}{2}
\]

Series Stub Matching: Smith Chart Graphical Procedure

The following procedure outlines the exact Smith Chart steps used for series stub matching.

Step 1: Normalize the Load Impedance

Calculate the normalized load impedance.

\[
z_N = \frac{Z_L}{Z_0}
\]

Step 2: Plot the Normalized Impedance

Locate the normalized impedance on the Smith Chart and mark it as Point A.

\[
z_A = z_N
\]

Step 3: Draw the SWR Circle

Draw the SWR circle using the chart center as the pivot point and the normalized impedance as the radius.

The Standing Wave Ratio is read from the positive real axis.

\[
\text{SWR}
\]

Step 4: Read Load Wavelength Scale (WTG_A)

Project Point A to the Wavelengths Toward Generator scale and record:

\[
\text{WTG}_A
\]

Step 5: Move Clockwise to the r = 1 Circle (Point B)

Move clockwise along the SWR circle until it intersects the unity resistance circle.

At Point B:

\[
z_B = 1 + jx_n
\]

Step 6: Read Wavelength for Point B

Project Point B to the wavelength scale and record:

\[
\text{WTG}_B
\]

Step 7: Calculate Distance to the Stub

The distance from the load to the stub location is:

\[
D = \text{WTG}_B - \text{WTG}_A
\]

If:

\[
\text{WTG}_B < \text{WTG}_A
\]

add:

\[
0.5\lambda
\]

Step 8: Find Stub Reactance Point (Point C)

The reactance obtained at Point B must be cancelled by an equal reactance of opposite polarity.

\[
x_{stub} = -jx_n
\]

Locate this value on the outer perimeter and mark it as Point C.

Record:

\[
\text{WTG}_C
\]

Step 9: Calculate Stub Length

Shorted Series Stub

Measure clockwise along the outer perimeter from the short-circuit point.

\[
L_s = \text{WTG}_C
\]

Open Series Stub

Measure clockwise along the outer perimeter from the open-circuit point.

\[
L_o = \text{WTG}_C - 0.25\lambda
\]

If the calculated value is negative, add:

\[
0.5\lambda
\]

Series Smith Chart Worksheet                                                                                                                                                                     

Parameter	Symbol / Variable	Value
Normalized Load Impedance	\(z_N\)	\(Z_L / Z_0\)
Standing Wave Ratio	\(\text{SWR}\)	Read on Real Axis
Load WTG Reading (Point A)	\(\text{WTG}_A\)	\(\text{?}\lambda\)
Matched Impedance (Point B)	\(z_B\)	\(1 + jx_n\)
Point B WTG Reading	\(\text{WTG}_B\)	\(\text{?}\lambda\)
Distance to Series Stub	\(D\)	\(\text{WTG}_B - \text{WTG}_A\)
Target Stub Reactance (Point C)	\(x_{stub}\)	\(-jx_n\)
Point C WTG Reading	\(\text{WTG}_C\)	\(\text{?}\lambda\)
Shorted Stub Length	\(L_s\)	\(\text{WTG}_C\)
Open Stub Length	\(L_o\)	\(\text{WTG}_C - 0.25\lambda\)

 

Comparison: Shunt Stub vs. Series Stub Matching   

 

Aspect	Shunt Stub Matching	Series Stub Matching
Connection Style	Parallel with main line	Series with main line
Primary Domain	Admittance \((y = g + jb)\)	Impedance \((z = r + jx)\)
Starting Point on Chart	Load Admittance \(y_N\)	Load Impedance \(z_N\)
Target Circle	\(g = 1\) Circle (Unity Conductance)	\(r = 1\) Circle (Unity Resistance)
Matching Condition	\(b_{stub} = -b_{line}\)	\(x_{stub} = -x_{line}\)
Shorted Stub Start Point	\(y = \infty\) (\(\text{WTG} = 0.25\lambda\))	\(z = 0\) (\(\text{WTG} = 0.00\lambda\))
Open Stub Start Point	\(y = 0\) (\(\text{WTG} = 0.00\lambda\))	\(z = \infty\) (\(\text{WTG} = 0.25\lambda\))
Physical Implementation	Easy to implement (coaxial/microstrip)	Difficult to implement in coaxial lines