Justification for λ/8 and 3λ/8 Spacing

Justification for Double Stub Matching (λ/8 and 3λ/8 Spacing)

Double stub matching is used when the location of a single matching stub cannot be freely selected. Instead of adjusting both the stub position and stub length, two stubs separated by a fixed distance are employed to achieve impedance matching.

For the λ/8 method, the spacing between stubs is:

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

For the 3λ/8 method, the spacing between stubs is:

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

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

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

Physical and Network Justification

Since both matching stubs are connected in parallel with the transmission line, admittances add directly at the connection points.

\[ y_{in}=y_{line}+y_{stub1}+y_{stub2} \]

For this reason, all calculations are performed using admittance rather than impedance.

\[ y=g+jb \]

Step 1: Load Normalization

The load impedance is normalized by dividing it by the characteristic impedance of the transmission line. Normalization converts actual values into dimensionless quantities that can be plotted directly on a universal Smith Chart.

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

The corresponding normalized admittance is:

\[ y_N=\frac{1}{z_N} \]

Step 2: Spacing Circle Construction

The spacing circle represents the fixed separation between the two matching stubs.

For the λ/8 method:

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

For the 3λ/8 method:

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

The spacing circle defines all admittance locations that can be transformed into a matched condition by the second stub.

Step 3: First Stub Location

Moving clockwise on the Smith Chart corresponds to moving from the load toward the generator. The transformed admittance at the first stub location becomes:

\[ y_F=g_F+jb_F \]

Step 4: Action of the First Stub

The first stub modifies only the susceptance component of the admittance. The conductance remains unchanged.

\[ y_G=g_F+jb_G \]

The purpose of the first stub is to move the operating point onto the spacing circle.

Step 5: First Stub Susceptance Requirement

The susceptance supplied by the first stub equals the difference between the susceptance coordinates of Points F and G.

\[ \Delta b=b_G-b_F \]

Therefore, the first stub must provide:

\[ b_{stub1}=\Delta b \]

Its normalized admittance becomes:

\[ y_{stub1}=j\Delta b \]

Step 6: First Stub Length Calculation

For a short-circuited stub:

\[ y_{stub1}=-j\cot(\beta L_{s1}) \]

For an open-circuited stub:

\[ y_{stub1}=j\tan(\beta L_{o1}) \]

The corresponding stub length is determined from the Smith Chart susceptance scale.

Step 7: Construction of the Second SWR Circle

After the first stub has been connected, Point G becomes the new operating point. A second SWR circle is drawn through this point.

The magnitude of the reflection coefficient remains unchanged along a lossless transmission line.

\[ |\Gamma|=\text{constant} \]

Step 8: Move to the Unity Conductance Circle

Move clockwise along the second SWR circle until it intersects the unity conductance circle.

At Point I:

\[ y_I=1+jb_I \]

The conductance is already matched because:

\[ g=1 \]

Only the reactive component remains.

Step 9: Second Stub Cancellation

The second stub must provide a susceptance equal in magnitude and opposite in sign to the remaining susceptance at Point I.

\[ b_{stub2}=-b_I \]

Therefore:

\[ y_{stub2}=-jb_I \]

After cancellation, the total normalized admittance becomes: