**5.1.1 Graphical procedure**

As explained above, the admittance at the load side of the stub must be 1 *A A y jb* . This means that the locus of *Ay* in the Smith chart is the *g* 1 circle. Since lossless transmission lines are assumed, when travelling from the load toward the generator the absolute value of the reflection coefficient remains constant, that is, *<sup>L</sup>* .

Therefore, the possible values of *Ay* must be given by the intersection of the two circles: the *g* 1 circle, and the *<sup>L</sup>* constant circle. Except for loads with real part equal to zero, there are two intersection points and therefore two solutions.

The solutions are found starting at the value of *<sup>L</sup> y* in the Smith chart and moving toward the generator (clockwise), along the corresponding constant *<sup>L</sup>* circle, until the intersection with the *g* 1 circle is obtained.

After choosing a solution for *Ay* , since 1 *A A y jb* it is possible to get the corresponding value of *ssA y jb jb* .

To determine the length of a short-circuited terminated stub, with an input admittance *<sup>s</sup> y* , one should move, from the admittance SC point in the Smith chart, toward the generator (clockwise) to the point corresponding to the input admittance, along the *g* 0 circle and read the required distance in wavelengths. To determine the length of an open-circuited terminated stub, with an input admittance *<sup>s</sup> y* , one should move from the admittance OC point in the Smith chart, toward the generator (clockwise) to the point corresponding to the input admittance, along the *g* 0 circle and read the required distance in wavelengths.

The authors developed a *MATLAB* script called *SingleStubMatching\_Eng\_FV.m* that displays step by step, the graphical procedure described above. Four cases are studied:


The graphical solution given by this script for the first case, is shown in Figure 10, for a line with a characteristic impedance Z0=50Ω and the load ZL=100+j60 Ω.

Figure 10a) shows the first 7 steps:


Figure 10b) shows the last 2 steps:


As explained above, the admittance at the load side of the stub must be 1 *A A y jb* . This means that the locus of *Ay* in the Smith chart is the *g* 1 circle. Since lossless transmission lines are assumed, when travelling from the load toward the generator the absolute value of

Therefore, the possible values of *Ay* must be given by the intersection of the two circles: the

The solutions are found starting at the value of *<sup>L</sup> y* in the Smith chart and moving toward

After choosing a solution for *Ay* , since 1 *A A y jb* it is possible to get the corresponding

To determine the length of a short-circuited terminated stub, with an input admittance *<sup>s</sup> y* , one should move, from the admittance SC point in the Smith chart, toward the generator (clockwise) to the point corresponding to the input admittance, along the *g* 0 circle and read the required distance in wavelengths. To determine the length of an open-circuited terminated stub, with an input admittance *<sup>s</sup> y* , one should move from the admittance OC point in the Smith chart, toward the generator (clockwise) to the point corresponding to the input admittance, along the *g* 0 circle and read the required distance in wavelengths. The authors developed a *MATLAB* script called *SingleStubMatching\_Eng\_FV.m* that displays

 Intersection with the upper half of the *g* 1 circle and a short-circuited stub; Intersection with the lower half of the *g* 1 circle, and a short-circuited stub; Intersection with the upper half of the *g* 1 circle, and a open-circuited stub; Intersection with the lower half of the *g* 1 circle, and a open-circuited stub; The graphical solution given by this script for the first case, is shown in Figure 10, for a line

step by step, the graphical procedure described above. Four cases are studied:

3. Transforming the normalized impedance in admittance, by inverting *zL* to *yL*;

*<sup>L</sup>* constant circle, until *Ay* .

point corresponding to the input admittance of the stub (point B**).**

7. Finding the distance *d*, in wavelengths, moving from *Ly* , toward the generator

9. Determining the length *Ls* of the stub, in wavelengths, moving, from the admittance SC point in the Smith chart, toward the generator (clockwise) along the *g* 0 circle to the

*<sup>L</sup>* constant circle with the *g* 1 circle,

with a characteristic impedance Z0=50Ω and the load ZL=100+j60 Ω.

6. Finding the admittance *Ay* from the chosen intersection point;

 *<sup>L</sup>* .

*<sup>L</sup>* constant circle. Except for loads with real part equal to zero, there

*<sup>L</sup>* circle, until the intersection

**5.1.1 Graphical procedure** 

*g* 1 circle, and the

with the *g* 1 circle is obtained.

Figure 10a) shows the first 7 steps:

4. Drawing the *g* 1 constant circle;

(clockwise) along the

Figure 10b) shows the last 2 steps:

2. Drawing the

(point A);

1. Marking the normalized impedance *zL*;

*<sup>L</sup>* constant circle;

5. Choosing one of the intersection points of the

8. Getting the value of *<sup>s</sup> y* from the value of *Ay* ;

value of *ssA y jb jb* .

the reflection coefficient remains constant, that is,

are two intersection points and therefore two solutions.

the generator (clockwise), along the corresponding constant

Fig. 10. Graphical solution given by the *SingleStubMatching\_Eng\_FV.m* script.

Using the Smith Chart in an E-Learning Approach 115

Point A moves away from the load and point S away from SC. *ym* moves away from the center of chart (perfect matching). With this solution a VSWR 2:1 bandwidth of 38% is

a)

b)

Figure 12 shows the graphical evolution of the matching values given by the script *SingleStubMatching\_Eng\_BW\_FV2b.m* for the example illustrated in Figure 10. In this script

Fig. 11. Graphical solution given by the *SingleStubMatching\_Eng\_BW\_FV2a.m* script.

obtained.

Following the display produced by this script, students should be able to solve in a paper chart any single-stub impedance matching problem using a ruler and a compass.

The other cases given by script *SingleStubMatching\_Eng\_FV.m* can be explored by the reader.
