Numerical7:Double Stub(3λ/8)ZL=109.5−j120Ω, Z0=75Ω
\[
Z_L = 109.5 - j120\,\Omega
\]
\[
Z_0 = 75\,\Omega
\]
\[
\text{A 75-}\Omega \text{ air-filled coaxial line is terminated with a complex load of }
(109.5 - j120)\,\Omega.
\]
\[
\text{Design a double stub matching system using coaxial lines of characteristic impedance }
75\,\Omega.
\]
\[
\text{The separation between the stubs is }
\frac{3\lambda}{8}
\text{ and the signal frequency is }
3\,\text{GHz}.
\]
\[
\text{Determine the lengths of the open-circuited and short-circuited stubs using the Smith Chart.}
\]
\[
Z_L = 109.5 - j120\,\Omega
\]
and
\[
Z_0 = 75\,\Omega
\]
Step 1: Normalized the load impedance by dividing it by characteristic impedance of line. Plot Zn in the Smith Chart. Construct SWR circle for load line recording the wave length at Yn
Zn= 1.46-j1.6
SWR = 3.6:1
\[
\text{WTG}_B = 0.057\lambda
\]
\[
Y_B = 0.31 + j0.34
\]
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: Since the distance of 1st stub is not given, we can assume a distance d1 that is d1 distance apart from the load overlapped within the spacing circle or we could assume a point E, and based on WTGE distance can be calculated. For point E, the point should be inside the overlapping region between 1st SWR and spacing Circle.
Overlapping Region:
• The region where the SWR circle and the spacing circle intersect.
• Any point within this overlapping region satisfies both the SWR and the spacing requirements.
Arbitrarily Choosing Point E:
• Within the overlapping region, you can choose any point for the stub placement.
• This point is chosen based on practical considerations or ease of implementation.
For Ease of implementation, Point A (Zn) lies between Spacing Circle and on SWR circle, we chose arbitrarily Point E somewhere between Point A and Point C.
Let us consider a distance to the WTGE as 0.35 λ so that the distance d1 can be calculated. The placement distance of 1st Stub from the load is equal to the clockwise distance from Point B To Point Assuming the distance doesn’t cross the 0.5 λ point
\[
\text{Point E is } 0.7 - j1.1
\]
\[
D_1 = \text{WTG}_E - \text{WTG}_B
= 0.35\lambda - 0.056\lambda
= 0.294\lambda
\]
Alternatively,
you can assume D1 as 0.4 λ apart from the load and calculate the value of WTGE. Assuming the distance doesn’t cross the 0.5 λ point.
\[
D_1 = \text{WTG}_E + \left(-\text{WTG}_B\right)
\]
\[
0.4\lambda = \text{WTG}_E - 0.056\lambda
\]
\[
\text{WTG}_E = (0.4 + 0.056)\lambda
\]
\[
\text{WTG}_E = 0.456\lambda
\]
Step 4: Follow the reactance circle through point E (0.7) in the direction of a smaller reactance. In the case, move left to the point C at the edge of spacing circle and note the coordinates i.e. move anticlockwise towards the edge of the spacing circle. Label the point as F
\[
Z_F = 0.7 - j0.057
\]
Step 5: Find the difference in reactance between point F and E.
\[
\text{The amount of susceptance that must be cancelled is}
\]
\[
X_E - X_F
\]
\[
= -j1.1 - (-j0.05)
\]
\[
= -j1.05
\]
\[
\therefore \text{Required susceptance to be cancelled} = -j1.05
\]
\[
\text{To cancel } -j1.05,\ \text{locate the point } +j1.05
\]
\[
\text{and mark it as Point H.}
\]
\[
\text{Measure and record } \text{WTG}_H.
\]
WTGH = 0.1293 λ
Step 6: The Stub length is found in the same way as single stub matching. The length of the 1st Stub is calculated as
\[
\text{For the open-circuited stub, measure clockwise from the zero-admittance point.}
\]
\[
L_{1O} = 0.1293\lambda
\]
\[
\text{For the short-circuited stub, measure clockwise from the infinite-admittance point.}
\]
\[
L_{1S} = 0.1293\lambda + 0.25\lambda
\]
\[
L_{1S} = 0.3793\lambda
\]
Step 7: From Point F, again draw 2nd SWR Circle. Using the point F as a circumference location and prime center of the short as a pivot point. Construct a second SWR circle
SWR2: 1.5:1
Step 9: Extend the line from 2nd SWR circle to clockwise direction till in intersects R=1 circle on inside of the spacing circle, mark that point as K.
Find the point +j0.375 to cancel -j0.375 and note it as Point I 
\[
\text{WTG}_I = 0.058\lambda
\]
\[
\text{For the open-circuited stub, measure clockwise from the zero-admittance point.}
\]
\[
L_{2O} = 0.058\lambda
\]
\[
\text{For the short-circuited stub, measure clockwise from the infinite-admittance point.}
\]
\[
L_{2S} = 0.058\lambda + 0.25\lambda
\]
\[
L_{2S} = 0.308\lambda
\]
