**4. Getting started**

As pointed out above, the aim of this chapter is to help the reader understand and learn how to use the Smith chart describing step by step procedures based in *MATLAB* scripts that should teach the students to solve different kinds of transmission line problems by themselves in a paper chart using a pencil, a ruler and a compass. One of the first things they should know how to do, is how to mark a given reflection coefficient in a chart and read related data associated to this reflection coefficient, such the transmission coefficient, the normalized impedance, the normalized admittance and the voltage standing wave ratio. The authors developed a *MATLAB* script called *SmithChart\_InputRho\_Eng\_FV.m* that display, step by step, how to do this exercise.

The graphical solution given by this script, is shown in Figure 5, for a reflection coefficient 0.6 120º .

Figure 5a) shows the first 2 steps:

1. Marking *<sup>L</sup>* , from the amplitude and phase;

2. Drawing the *<sup>L</sup>* constant circle;

Figure 5b) shows the last 3 steps:


Another basic thing students should know how to do, is to mark a given normalized impedance in a chart and read related data associated to it, such as the reflection coefficient, the normalized admittance and the voltage standing wave ratio. The authors developed a *MATLAB* script called *SmithChart\_InputZ\_Eng\_FV.m* that displays step by step, how to do this exercise.

The graphical solution given by this script, is shown in Figure 6, for a normalized impedance 0.3 0.5 *Lz j* .

Figure 6a) shows the first 2 steps:


Figure 6b) shows the last 3 steps:


of the simulation results given by commercial software about antennas and microwave devices. Most modern computer based automatic network analyzers rely on the Smith chart

This chart is a unique diagram which has been used nearly for ninety years and we believe that it will be in use for many years to come not only as a pedagogically perfect analogue display, but also as an aid to professionals in obtaining quick answers to many line

As pointed out above, the aim of this chapter is to help the reader understand and learn how to use the Smith chart describing step by step procedures based in *MATLAB* scripts that should teach the students to solve different kinds of transmission line problems by themselves in a paper chart using a pencil, a ruler and a compass. One of the first things they should know how to do, is how to mark a given reflection coefficient in a chart and read related data associated to this reflection coefficient, such the transmission coefficient, the normalized impedance, the normalized admittance and the voltage standing wave ratio. The authors developed a *MATLAB* script called *SmithChart\_InputRho\_Eng\_FV.m* that

The graphical solution given by this script, is shown in Figure 5, for a reflection coefficient

*t Lt e*

 ;

Another basic thing students should know how to do, is to mark a given normalized impedance in a chart and read related data associated to it, such as the reflection coefficient, the normalized admittance and the voltage standing wave ratio. The authors developed a *MATLAB* script called *SmithChart\_InputZ\_Eng\_FV.m* that displays step by step, how to do

The graphical solution given by this script, is shown in Figure 6, for a normalized

*L L e*

 ;

*t Lt e*

 ;

for data display.

problems which they meet.

display, step by step, how to do this exercise.

*<sup>L</sup>* , from the amplitude and phase;

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

3. Getting the transmission coefficient 1 *<sup>j</sup>*

5. Getting the normalized admittance, by inverting *zL* to *yL*;

3. Getting the normalized admittance, by inverting *zL* to *yL*; 4. Getting the corresponding reflection coefficient *<sup>j</sup>*

5. Getting the corresponding transmission coefficient 1 *<sup>j</sup>*

Figure 5a) shows the first 2 steps:

Figure 5b) shows the last 3 steps:

6. Getting the associated SWR.

impedance 0.3 0.5 *Lz j* . Figure 6a) shows the first 2 steps: 1. Highlighting the curves *r*L and *x*L; 2. Marking the normalized impedance *zL*;

Figure 6b) shows the last 3 steps:

6. Getting the associated SWR.

4. Getting the normalized impedance *zL*;

**4. Getting started** 

0.6 120º .

1. Marking

this exercise.

2. Drawing the

a)

Fig. 5. Inputting a reflection coefficient. Display given by *SmithChart\_InputRho\_Eng\_FV.m*.

Using the Smith Chart in an E-Learning Approach 109

By definition, the reflection coefficient is the ratio of the phasors of the reverse and forward voltage waves. A voltage maximum occurs when these waves are in phase adding together constructively. If these waves are in opposite phase, a voltage minimum results. Therefore, travelling along the line, from the load toward the generator, if the right horizontal axis is reached first this corresponds to a reflection coefficient with a 0º phase, which means a voltage maximum. If, however the left horizontal axis is reached first, this corresponds to a

The graphical solution given by this script, is shown in Figure 7, for the normalized

reflection coefficient with a 180º phase, which means a voltage minimum.

Fig. 7. Display given by SmithChart\_InputZ\_FindVmin\_FindVmax\_Eng\_FV.m.

line, all the points separated by /2 have the same characteristics.

From Figure 7, one can perceive that travelling along the line, from the load toward the generator, the first particular point is a voltage maximum that occurs at a distance d=0.0587. This happens because the load is inductive. Continuing to travel along the line, after a voltage minimum is found. As it is well known, the maxima are separated by The same applies to the minima. This implies that a minimum is separated from the

For a lossless line the absolute value of the reflection coefficient remains constant along the line, however its phase changes and therefore the impedance along the line also changes. Since a complete turn on the chart corresponds to travel /2, this means that after traveling /2, from the load to the generator, the load point is reached. This means that the impedance at a distance of /2 from the load is equal to the load. Therefore, for a lossless

impedance 1.4 1.6 *Lz j* .

consecutive maxima by

b)

Fig. 6. Inputting a normalized impedance. Display given by the script *SmithChart\_InputZ\_Eng\_FV.m*.

It is also important to know how to locate the voltage maxima and minima along the line, given a normalized impedance using the chart. The *MATLAB* script called *SmithChart\_InputZ\_FindVmin\_FindVmax\_Eng\_FV.m* displays, step by step, how to locate the first voltage maximum and minimum.

a)

b)

It is also important to know how to locate the voltage maxima and minima along the line, given a normalized impedance using the chart. The *MATLAB* script called *SmithChart\_InputZ\_FindVmin\_FindVmax\_Eng\_FV.m* displays, step by step, how to locate the

Fig. 6. Inputting a normalized impedance. Display given by the script

*SmithChart\_InputZ\_Eng\_FV.m*.

first voltage maximum and minimum.

By definition, the reflection coefficient is the ratio of the phasors of the reverse and forward voltage waves. A voltage maximum occurs when these waves are in phase adding together constructively. If these waves are in opposite phase, a voltage minimum results. Therefore, travelling along the line, from the load toward the generator, if the right horizontal axis is reached first this corresponds to a reflection coefficient with a 0º phase, which means a voltage maximum. If, however the left horizontal axis is reached first, this corresponds to a reflection coefficient with a 180º phase, which means a voltage minimum.

The graphical solution given by this script, is shown in Figure 7, for the normalized impedance 1.4 1.6 *Lz j* .

Fig. 7. Display given by SmithChart\_InputZ\_FindVmin\_FindVmax\_Eng\_FV.m.

From Figure 7, one can perceive that travelling along the line, from the load toward the generator, the first particular point is a voltage maximum that occurs at a distance d=0.0587. This happens because the load is inductive. Continuing to travel along the line, after a voltage minimum is found. As it is well known, the maxima are separated by The same applies to the minima. This implies that a minimum is separated from the consecutive maxima by

For a lossless line the absolute value of the reflection coefficient remains constant along the line, however its phase changes and therefore the impedance along the line also changes. Since a complete turn on the chart corresponds to travel /2, this means that after traveling /2, from the load to the generator, the load point is reached. This means that the impedance at a distance of /2 from the load is equal to the load. Therefore, for a lossless line, all the points separated by /2 have the same characteristics.

Using the Smith Chart in an E-Learning Approach 111

characteristic impedance and load impedance are said to be matched. In this situation, the reflection coefficient is zero and no standing waves exist. In transmission line applications, it

There are several methods to achieve impedance matching. One of the simplest methods to match a transmission line to a given load is to connected a reactive element in parallel with the line at a point where the real part of the line admittance is equal to the characteristic admittance. This reactive element can be realized by a short piece of line, called stub. That is

Although the location of the stub and its length can be found analytically using a computer or even a calculator, the authors believe that the use of the Smith chart to solve graphically this problem will give to the undergraduate students a much better insight of the aspects involved in this problem. The authors developed a *MATLAB* script called *SingleStubMatching\_Eng\_FV.m* that displays step by step, this graphical procedure. This

The basic layout of the single-stub impedance matching is illustrated in Figure 9. The parameters to be evaluated are the distance *d*, measured from the load, at which the stub must be placed and, the stub length *Ls*. The stub is connected in parallel with the line. The stub can be short-circuited terminated or open-circuited terminated. Since the stub is connected in parallel with the line, the solution of this problem must be approached in terms of admittance. If the load is inputted as an impedance, then it is necessary to transform it in

The goal is to match the load *ZL* to the line with characteristic impedance *Z*<sup>0</sup> . Therefore the normalized admittance *ym* , at the generator side of the stub, must be equal 1. On the other hand, this normalized admittance is equal to the sum of *Ay* and *<sup>s</sup> y* . Since by definition, assuming lossless transmission lines, the input admittance of a stub has no real part, that is, *s s y jb* , that implies that the admittance *Ay* must be 1 *<sup>A</sup> <sup>A</sup> y jb* . Furthermore, the value

is desirable to achieve the matching condition.

why this method is known by the single-stub impedance matching.

an admittance, using the graphical procedure explained before.

Fig. 9. Basic layout of the single-stub impedance matching.

of *<sup>s</sup> b* must be the symmetrical of *Ab* in order to cancel each other out.

script is available for download and should be used when reading this section.

The authors developed a *MATLAB* script called *LosslessLine\_Eng\_FV.m* that graphically illustrates how the impedance along the line changes, as shown in figure 8. In this example a lossless line 0.35 long with a characteristic impedance Z0=50Ω, terminated with the load ZL=100+j60 Ω, has an input impedance Zin=21.88+j17.43 Ω. It is important to note the changes in nature of the impedance along the line when one moves from the load to the generator. At the load the impedance is inductive, then became real greater than Z0. After that, and for the next /4, became capacitive and then again inductive until it reached the generator plane.

Fig. 8. Impedance variation along a lossless line 0.35 long. Graphical solution given by the *LosslessLine\_Eng\_FV.m* script.
