Double Stub Matching Using (λ/8 Spacing)

Why \(\lambda/8\) and \(3\lambda/8\) Stub Spacing are Used in Double Stub Matching

In double stub matching, the spacing between the two stubs is fixed. The spacing determines how the admittance is transformed from the first stub to the second stub and directly affects the size of the forbidden region on the Smith Chart.

The most commonly used spacings are:

\[
S=\frac{\lambda}{8}
\]

and

\[
S=\frac{3\lambda}{8}
\]

because they provide practical matching solutions while keeping the stubs physically close together.

Why \(\lambda/8\) Spacing is Used

When the spacing between the stubs is

\[
S=\frac{\lambda}{8}
\]

the admittance rotates by

\[
2\beta S
\]

Since

\[
\beta=\frac{2\pi}{\lambda}
\]

we have

\[
2\beta S
=
2\left(\frac{2\pi}{\lambda}\right)
\left(\frac{\lambda}{8}\right)
\]

\[
=
\frac{\pi}{2}
\]

\[
=
90^\circ
\]

Thus the admittance undergoes a \(90^\circ\) rotation on the Smith Chart between the two stubs.

Advantages of \(\lambda/8\) Spacing

\[
\text{1. Stubs are physically closer together.}
\]

\[
\text{2. Compact circuit implementation.}
\]

\[
\text{3. Convenient for microwave integrated circuits.}
\]

\[
\text{4. Easier fabrication on microstrip and stripline boards.}
\]

When \(\lambda/8\) is Used

\[
\text{Compact RF circuits}
\]

\[
\text{Microwave integrated circuits}
\]

\[
\text{Limited available transmission-line length}
\]

\[
\text{PCB microstrip designs}
\]

Why \(3\lambda/8\) Spacing is Used

For

\[
S=\frac{3\lambda}{8}
\]

the phase shift becomes

\[
2\beta S
=
2\left(\frac{2\pi}{\lambda}\right)
\left(\frac{3\lambda}{8}\right)
\]

\[
=
\frac{3\pi}{2}
\]

\[
=
270^\circ
\]

Thus the admittance rotates by \(270^\circ\) on the Smith Chart.

Since a Smith Chart repeats every

\[
180^\circ
\]

a \(270^\circ\) rotation is equivalent to

\[
270^\circ - 180^\circ
=
90^\circ
\]

from a matching perspective.

Therefore, the mathematical behavior of the \(\lambda/8\) and \(3\lambda/8\) configurations is very similar.

Advantages of \(3\lambda/8\) Spacing

\[
\text{1. Same matching capability as } \lambda/8
\]

\[
\text{2. More physical separation between stubs}
\]

\[
\text{3. Easier mechanical adjustment}
\]

\[
\text{4. Reduced interaction between tuning elements}
\]

When \(3\lambda/8\) is Used

\[
\text{Waveguide tuners}
\]

\[
\text{Coaxial transmission lines}
\]

\[
\text{Laboratory tuning benches}
\]

\[
\text{Systems requiring easier access to stubs}
\]

Relationship Between \(\lambda/8\) and \(3\lambda/8\)

Because the Smith Chart repeats every

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

the two spacings differ by

\[
\frac{3\lambda}{8}
-
\frac{\lambda}{8}
=
\frac{\lambda}{4}
\]

which corresponds to

\[
180^\circ
\]

on the Smith Chart.

Therefore,

\[
\lambda/8
\]

and

\[
3\lambda/8
\]

produce nearly identical matching equations and identical forbidden-region boundaries.

Difference Between \(\lambda/8\) and \(3\lambda/8\)

\[
\begin{array}{|c|c|c|}
\hline
\text{Parameter}
&
\lambda/8
&
3\lambda/8
\\
\hline
\text{Stub Separation}
&
\text{Small}
&
\text{Large}
\\
\hline
\text{Physical Size}
&
\text{Compact}
&
\text{Larger}
\\
\hline
\text{Ease of Adjustment}
&
\text{Moderate}
&
\text{Better}
\\
\hline
\text{Stub Interaction}
&
\text{Higher}
&
\text{Lower}
\\
\hline
\text{Matching Capability}
&
\text{Same}
&
\text{Same}
\\
\hline
\text{Forbidden Region}
&
\text{Same}
&
\text{Same}
\\
\hline
\text{Typical Use}
&
\text{PCB / MIC}
&
\text{Waveguide / Coax}
\\
\hline
\end{array}
\]

Therefore, neither \(\lambda/8\) nor \(3\lambda/8\) is theoretically superior. Both provide essentially the same matching capability. The choice depends mainly on physical layout requirements, available space, and ease of tuning.

Double stub matching is an impedance matching technique used in RF and microwave engineering to match an arbitrary load impedance to a transmission line using two adjustable stubs separated by a fixed distance.

In the λ/8 spacing method, the distance between the two stubs is:

\[
D=\frac{\lambda}{8}
\]

The Smith Chart provides a graphical solution for determining the required stub susceptances and stub lengths needed to achieve a perfect match.

The procedure below assumes shunt stubs and therefore operates primarily in the admittance domain.

Steps of Double Stub Impedance matching (λ/8)

Step 1: Normalize the Load Impedance

Normalize the load impedance by dividing the load impedance by the characteristic impedance of the transmission line. Normalization converts actual impedance values into dimensionless quantities that can be plotted directly on a universal Smith Chart regardless of the value of line impedance.

Calculate the normalized impedance:

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

Plot the normalized impedance on the Smith Chart.

Construct the SWR circle passing through the normalized load point.

Find the corresponding normalized admittance by rotating 180° through the center of the chart.

Record:

\[
z_N=?
\]

\[
\text{SWR}=?
\]

\[
y_N=?
\]

\[
\text{WTG}_B=?
\]

Step 2: Construct the λ/8 Spacing Circle

The spacing circle represents the fixed separation between the first and second matching stubs.

Construct the λ/8 spacing circle corresponding to a spacing of:

\[
D=\frac{\lambda}{8}
\]

The spacing circle intersects the stub circle at two points.

Join the intersection points through the prime center.

Starting from the wavelength reading recorded at the normalized admittance point, move clockwise around the wavelength scale.

The transformed admittance produced by the first stub must lie within the allowable region defined by the spacing-circle boundaries.

This region determines all possible admittances that can be transformed into a match by the second stub.

Step 3: Move to the First Stub Location

The first matching stub is connected at a specified distance from the load.

Moving clockwise on the Smith Chart corresponds to moving from the load toward the generator.

Assume the first stub location is:

\[
D_1=0.4\lambda
\]

Move clockwise along the SWR circle by this amount.

The new intersection point is labeled Point F.

Record the admittance at Point F:

\[
y_F=?
\]

\[
D_1=0.4\lambda
\]

Step 4: Move Along the Susceptance Circle

The purpose of the first stub is to modify the admittance so that the resulting point falls on the spacing circle.

From Point F, move along the constant conductance circle toward smaller susceptance values until the spacing circle is reached.

Mark this new location as Point G.

Record the admittance at Point G:

\[
y_G=?
\]

Point G represents the admittance after the first stub has been connected.

Step 5: Determine the Required First Stub Susceptance

Determine the difference in susceptance between Point F and Point G.

This difference represents the susceptance that must be supplied by the first stub.

Calculate:

\[
\Delta b=b_G-b_F
\]

Record:

\[
\text{Distance}=?
\]

To realize the required susceptance, locate the corresponding point on the outer susceptance scale.

For example, if the required correction is:

\[
+j0.94
\]

locate that point and label it as Point H.

Record:

\[
\text{WTG}_H=?
\]

Step 6: Calculate the First Stub Length

The first stub length is determined in the same manner as single-stub matching.

For a short-circuited stub, measure clockwise along the outer rim beginning from the infinite-admittance point.

Short-Circuited Stub:

\[
L_s=?
\]

For an open-circuited stub, measure clockwise along the outer rim beginning from the zero-admittance point.

Open-Circuited Stub:

\[
L_o=?
\]

These lengths provide the susceptance required to move from Point F to Point G.

Step 7: Construct the Second SWR Circle

The admittance at Point G now becomes the starting point for determining the second stub.

Using Point G as a point on the circumference and the prime center as the center of rotation, construct a second SWR circle.

This circle represents all possible admittances available after the first stub has been connected.

Record:

\[
\text{SWR}=?
\]

Step 8: Move to the Unity Conductance Circle

From Point G, move clockwise along the second SWR circle.

Continue until the circle intersects the unity conductance circle located inside the spacing-circle boundary.

Mark this location as Point I.

At Point I:

\[
y_I=?
\]

Record the wavelength reading:

\[
\text{WTG}_I=?
\]

The susceptance present at Point I must now be cancelled by the second matching stub.

Determine the equal and opposite susceptance required for cancellation.

Locate the corresponding susceptance point on the outer scale.

Record:

\[
\text{WTG}_S=?
\]

Step 9: Calculate the Second Stub Length

The second stub provides the final susceptance correction required to achieve a matched condition.

For a short-circuited stub, measure clockwise from the infinite-admittance point to the required susceptance location.

Short-Circuited Stub:

\[
L_s=?
\]

For an open-circuited stub, measure clockwise from the zero-admittance point to the required susceptance location.

Open-Circuited Stub:

\[
L_o=?
\]

After the second stub is connected, the overall normalized admittance becomes:

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

which corresponds to a perfect match.

Final Matching Condition

The objective of double stub matching is to transform the arbitrary load admittance into the characteristic admittance of the transmission line.

The final matched condition is:

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

Under this condition:

\[
\Gamma=0
\]

and

\[
\text{SWR}=1
\]

indicating maximum power transfer and zero reflected power.

Parameter	Symbol / Variable	Value
Normalized Load Impedance	\(z_N\)	\(Z_L/Z_0\)
Standing Wave Ratio	\(\text{SWR}\)	Read from Smith Chart
Normalized Admittance	\(y_N\)	\(1/z_N\)
Load WTG Reading	\(\text{WTG}_B\)	\(?\lambda\)
First Stub Location	\(D_1\)	\(0.4\lambda\)
Admittance at Point F	\(y_F\)	Read from Chart
Admittance at Point G	\(y_G\)	Read from Chart
WTG Reading at Point H	\(\text{WTG}_H\)	\(?\lambda\)
Second SWR Circle	SWR	Read from Chart
Admittance at Point I	\(y_I\)	Read from Chart
WTG Reading at Point I	\(\text{WTG}_I\)	\(?\lambda\)
WTG Reading of Stub Point	\(\text{WTG}_S\)	\(?\lambda\)
First Stub Length (Short)	\(L_s\)	Measured from \(y=\infty\)
First Stub Length (Open)	\(L_o\)	Measured from \(y=0\)
Second Stub Length (Short)	\(L_s\)	Measured from \(y=\infty\)
Second Stub Length (Open)	\(L_o\)	Measured from \(y=0\)
Final Matched Admittance	\(y_{in}\)	\(1+j0\)