Numerical 3:Seriesstub:z=l00+j80Ω and zo=50Ω

Given:

\[
Z_L = 100 + j80 \ \Omega
\]

\[
Z_0 = 50 \ \Omega
\]

Step 1: Normalize the load by dividing it by the characteristic’s impedance by the line

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

\[
z_N = \frac{100 + j80}{50}
\]

\[
z_N = 2 + j1.6
\]

Step 2: Plot Zn in the smith Chart

 

Step 3: Draw the SWR Circle using prime center as the pivot point and Zn as a point of circumference of circle

                SWR = 3.5:1

 

Step 4: Draw a line from the normalized load impedance through the prime center of the chart out to the wavelength scale towards generator scale

WTGb = 0.209 λ

 

Step 5: From Zn, making along the SWR circle in clockwise direction until it cuts the k=1 circle point Zb = 1 - j 1.35

 

Step 6: Draw a line from the perimeter center of chart through the susceptance point B to the wave length scale. Note the reading on wavelength towards generator scale.

WTGb = 0.328 λ

 

Step 7: The distance from load towards the generator at which the stub should be placed is given by the distance between A and B in clockwise direction

\[
D = WTG_B - WTG_A = (0.328 - 0.209)\lambda = 0.119\lambda
\]

 

Step 8: The reactive portion recorded in step 5 normailized susceptance of (-j1.35) must be cancelled. Find the opposite value (j1.35) on the R- Circle on the chart noted as point. Record the wavelength towards the generator reading at point C

                WTGs = 0.344 λ

 

Step 9: For shorted stub (admittance case), the length of the shorted stub is measured from the point of zero (0) to point C

Ls = 0.147 λ

 

 

For open shunt stub the length of stub is measured from the point of 0 wavelength to point D in clockwise directions

\[
L_0 = (0.25 + 0.147)\lambda = 0.397\lambda
\]