Numerical4:DoubleStub3λ/8 :Z L=100+j100 Ω,Z0=50 Ω
Doubly Terminated Double Stub Matching (3λ/8 Spacing) Using Smith Chart
\[
\text{The terminating impedance } Z_L = (100 + j100)\,\Omega \text{ and the characteristic impedance of the coaxial cable is } 50\,\Omega.
\]
\[
\text{The first stub is placed } 0.4\lambda \text{ away from the load and the spacing between the stubs is } \frac{3}{8}\lambda.
\]
\[
\text{Determine the lengths of the open-circuited and short-circuited double shunt stubs when matching is achieved.}
\]
GIven:
\[
Z_L = 100 + j100 \ \Omega
\]
\[
Z_0 = 50 \ \Omega
\]
Step 1: Normalized the load impedance by dividing it by the characteristic impedance of the line.
Plot Zn in the Smith Chart. Construct SWR circle for load line recording the wavelength at Yn
Zn= 2+j2
\[
\text{SWR} = 4.25 : 1
\]
\[
y_N = 0.25 - j0.25
\]
\[
\text{WTG}_B = 0.458\lambda
\]
Step 2: Construct the 3/8 Spacing Circle.
The Spacing circle cuts the stub circle at point c and d. Join c and d with the prime circle. Starting at the wavelength reading at the Yn. Move clockwise around the wavelength scale so that the line ends up anywhere between the dashed line C and D. Line C and D describes an arc between two radii that defines the position of SWR Circle reside of spacing circle.
Step 3: Let us consider a distance of \(0.4\lambda\) to the first matching stub. Starting from \[ \text{WTG}_B = 0.459\lambda \] move clockwise by \[ 0.4\lambda - 0.25\lambda = 0.15\lambda \] giving \[ 0.459\lambda - 0.15\lambda \approx 0.358\lambda \] This places the distance reading at Line E. Line E intersects the SWR circle at Point F.
Alternative,
Starting from the load admittance location,
\[
\text{WTG}_B = 0.459\lambda
\]
the first stub is located at a distance
\[
D_1 = 0.4\lambda
\]
towards the generator. Therefore, the new wavelength reading becomes
\[
\text{WTG}_F = \text{WTG}_B + D_1
\]
\[
\text{WTG}_F = 0.459\lambda + 0.4\lambda
\]
\[
\text{WTG}_F = 0.859\lambda
\]
Since the outer wavelength scale of the Smith Chart repeats every
\[
0.5\lambda
\]
a reading greater than \(0.5\lambda\) indicates that one complete revolution has been made around the chart and the movement continues further by the remaining distance.
Thus,
\[
0.859\lambda = 0.5\lambda + 0.359\lambda
\]
Therefore, after completing one full rotation of the Smith Chart, the final wavelength reading is
\[
\text{WTG}_F = 0.359\lambda
\]
Hence, Point \(F\) is located at
\[
0.359\lambda
\]
on the Wavelengths Toward Generator (WTG) scale.
\[
y_F = 0.55 - j1.08
\]
\[
D_1 = 0.4\lambda
\]
Step 4: Follow the reactance circle through point F in the direction of a smaller reactance. In the case, move left to the point C at the edge of spacing circe and note the coordinates.
\[
y_G = 0.55 - j0.11
\]
Step 5: Fint the difference in reactance between point F and G
\[
(-j1.08)-(-j0.11)
\]
\[
-j1.08 + j0.11
\]
\[
-j0.97
\]
To cancel -j0.94 fint the point j0.94 on reactance circle and note it as H. Also find WTGh
\[
\text{WTG}_H = 0.123\lambda
\]
Step 6: The Stub length is found in the same way as single stub matching
For Short Stub: Measure from infinite point clockwise direction.
\[
L_s = 0.25\lambda + 0.123\lambda
\]
\[
L_s = 0.373\lambda
\]
For open Stub: Measure clockwise from zero point
\[
L_o = 0.123\lambda
\]
Step 7: Using the point G as a circumference location and prime center of the short as a pivot point. Construct a second SWR circle
SWR: 1.8:1
Step 8: From point G, move around the second
SWR circle in clockwise direction until reaching to R=1 circle on the inside of the spacing circle, note the edge as I
\[
y_i = 1 - j0.6
\]
\[
\text{WTG}_i = 0.358\lambda
\]
\[
+j0.62
\]
\[
\text{WTG}_s = 0.091\lambda
\]
\[
L_s = 0.091\lambda + 0.25\lambda
\]
\[
L_s = 0.341\lambda
\]
\[
L_o = 0.091\lambda
\]
