**5.1 Single-stub impedance matching**

In any Transmission Line course, the concept of impedance matching is a topic that must be addressed. In transmission line context, impedance matching occurs when the characteristic impedance *Z*0 of the line is equal to the load impedance *ZL* .When this happens, the

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

Fig. 8. Impedance variation along a lossless line 0.35 long. Graphical solution given by the

In this section, some examples that integrate transmission line concepts are presented. The following examples are explained: (1) Single stub matching, (2) Bandwidth of a single-stub impedance matching system, (3) Quarter wavelength impedance matching, (4) Analysis of lossy lines. The authors developed *MATLAB* scripts that display, step by step, the graphical

An example illustrating the double-stub impedance matching problem was also developed

In any Transmission Line course, the concept of impedance matching is a topic that must be addressed. In transmission line context, impedance matching occurs when the characteristic impedance *Z*0 of the line is equal to the load impedance *ZL* .When this happens, the

generator plane.

*LosslessLine\_Eng\_FV.m* script.

**5. Applications examples** 

procedures used to solve these problems.

by the authors, (Pereira & Pinho, 2010).

**5.1 Single-stub impedance matching** 

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 is desirable to achieve the matching condition.

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

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

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 an admittance, using the graphical procedure explained before.

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

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

Using the Smith Chart in an E-Learning Approach 113

a)

b)

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