Numerical6:λ/8 Zl:60 + j40 Ω Zo:50 Ω
\[
Z_L = 60 + j40\,\Omega
\]
and
\[
Z_0 = 50\,\Omega
\]
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= 1.2+j0.8 Ω
SWR = 2.18:1
\[
\text{WTG}_B = 0.172\lambda
\]
\[
Y_B = 0.57 - j0.38
\]
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.
Method 1: Assuming WTGE
Step3: 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 0.18 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 E. Assuming d1 crosses one rotation.
Point E is
\[ Z_E = 0.7 + j1.86 \] \[ D_1 = \text{WTG}_E + \left(0.5\lambda - \text{WTG}_B\right) \] \[ D_1 = 0.18\lambda + \left(0.5\lambda - 0.349\lambda\right) \] \[ D_1 = 0.18\lambda + 0.151\lambda \] \[ D_1 = 0.331\lambda \]
Step 4: Follow the reactance circle through point E (0.7) in the direction of a smaller reactance. In the case, move left on the edge of SWR circle in anticlockwise direction and note the coordinates i.e. move anticlockwise towards the edge of the spacing circle. Label the point as F.
\[
Z_F = 0.73 + j0.05
\]
Step 5: Find the difference in reactance between point F and E.
\[
\text{The amount of susceptance that must be cancelled is}
\]
\[
X_E - X_F
\]
\[
= j1.86 - j0.05
\]
\[
= j1.81
\]
\[
\therefore \text{Required susceptance to be cancelled} = j1.81
\]
\[
\text{To cancel } +j1.81,\ \text{locate the point } -j1.81
\]
\[
\text{and mark it as Point G.}
\]
\[
\text{WTG}_G = 0.442\lambda
\]
\[
\text{Step 6: The stub length is found in the same manner as single-stub matching.}
\]
\[
\text{For the open-circuited stub, measure clockwise from the zero-admittance point.}
\]
\[
L_{1O} = 0.442\lambda
\]
\[
\text{For the short-circuited stub, measure clockwise from the infinite-admittance point.}
\]
\[
L_{1S} = 0.442\lambda - 0.25\lambda
\]
\[
L_{1S} = 0.192\lambda
\]
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: 1.6:1
Step 8: From point F, move around the 2nd SWR circle in clockwise direction until reaching to R=1 circle on the inside of the spacing circle, note the edge as H.
ZH = 1+j0.38
Step 9: Find the point +j0.38 to cancel -j0.38 and note it as Point I
\[
\text{WTG}_I = 0.442\lambda
\]
\[
\text{Step 9: The stub length is found in the same manner as single-stub matching.}
\]
\[
\text{The length of the second stub is calculated as:}
\]
\[
\text{For the open-circuited stub, measure clockwise from the zero-admittance point.}
\]
\[
L_{2O} = 0.442\lambda
\]
\[
\text{For the short-circuited stub, measure clockwise from the infinite-admittance point.}
\]
\[
L_{2S} = 0.442\lambda - 0.25\lambda
\]
\[
L_{2S} = 0.192\lambda
\]
Method 2: Assuming D1 as 0
Step3: The placement distance of 1st Stub from the load is equal to Point B. Assuming distance of 1st stub is 0 distance apart from the load
D1 = 0 λ
Step 4: Follow the reactance circle through point B (0.57) in the direction of a smaller reactance. In the case, move along SWR circle to the point until it meets the spacing circle Move clockwise to the direction of SWR circle until it intersect the spacing circle. Label the point as E.
ZE = 0.57+j0.06
Step 5: Find the difference in reactance between point B and E.
\[
\text{The amount of susceptance that must be cancelled is}
\]
\[
X_B - X_E
\]
\[
= -j0.38 - (-j0.06)
\]
\[
= -j0.32
\]
\[
\therefore \text{Required susceptance to be cancelled} = -j0.32
\]
\[
\text{To cancel } -j0.32,\ \text{locate the point } +j0.32
\]
\[
\text{and mark it as Point H.}
\]
\[
\text{Measure and record } \text{WTG}_H.
\]
WTGF = 0.066 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
\[
\text{For the open-circuited stub, measure clockwise from the zero-admittance point.}
\]
\[
L_{1O} = 0.066\lambda
\]
\[
\text{For the short-circuited stub, measure clockwise from the infinite-admittance point.}
\]
\[
L_{1S} = 0.066\lambda + 0.25\lambda
\]
\[
L_{1S} = 0.316\lambda
\]
Step 7: From Point F, again draw 2nd SWR Circle. Using the point E as a circumference location and prime center of the short as a pivot point. Construct a second SWR circle
SWR2: 1.8:1
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 G.
\[
Z_G = 1 + j0.6
\]
\[
\text{Locate the point } +j0.6 \text{ to cancel } -j0.6
\]
\[
\text{and mark it as Point H.}
\]
\[
\text{WTG}_H = 0.412\lambda
\]
\[
\text{Step 9: The stub length is found in the same manner as single-stub matching.}
\]
\[
\text{The length of the second stub is calculated as:}
\]
\[
\text{For the open-circuited stub, measure clockwise from the zero-admittance point.}
\]
\[
L_{2O} = 0.412\lambda
\]
\[
\text{For the short-circuited stub, measure clockwise from the infinite-admittance point.}
\]
\[
L_{2S} = 0.412\lambda + 0.25\lambda
\]
\[
L_{2S} = 0.662\lambda
\]
