**5.2 Bandwidth of a single-stub impedance matching system**

Another important topic that should be explained to the undergraduate students is the concept of bandwidth of a system. The majority of the systems based in transmission lines are perfectly matched at just one frequency. However, since the system should be able to operate over a frequency band, it is important to find the frequency band for which the system is consider to be acceptable matched. This frequency band is called the bandwidth of the system. Usually the criterion used to consider a system matched is when the VSWR is less or equal to 2.

The lower and the upper frequencies of the bandwidth can be found analytically. However, the authors believe that using the Smith chart to find graphically the bandwidth, will give to the students a much better understanding of the effects on the admittance of the line, and therefore on the matching conditions, when the frequency changes. The authors developed two *MATLAB* scripts called *SingleStubMatching\_Eng\_BW\_FV2a.m* and *SingleStubMatching\_Eng\_BW\_FV2b.m* that displays graphically, step by step, the effects on the matching conditions, when the frequency changes. Both are for systems using shortcircuited stubs. One of the scripts is for one of the two possible solutions and the other for the other solution. It is important for the students to verify that the bandwidth is not the same for the two solutions. These scripts is available for download and should be explored when reading this section.

The single-stub impedance matching system explained in the previous section, gives a perfect matching at one frequency. Once constructed, if the frequency changes so does the electric length of the distance *d* and length *Ls*, and therefore a perfect matching is no longer achieved.

It is important to study the evolution of the matching values for a single-stub matching system, when the frequency varies around the central value for which a perfect matching was achieved. To observe this evolution, the authors developed *MATLAB* scripts that graphically display this evolution and give the VSWR 2:1 bandwidth. After choosing shortcircuited or open-circuited stubs, there are still two solutions for the single-stub matching system, the perfect matching can be achieved with two pairs of values of *d* and *Ls* and therefore two different evolution of the matching values are also obtained. The script called *SingleStubMatching\_Eng\_BW\_FV2a.m* is intended for one of the solutions and script called *SingleStubMatching\_Eng\_BW\_FV2b.m* is intended for the other. Both use short-circuited stubs. It is important to compare the VSWR 2:1 bandwidth obtained for both solutions.

Figure 11 shows the graphical evolution of the matching values given by the script *SingleStubMatching\_Eng\_BW\_FV2a.m* for the example illustrated in Figure 10. In this script the chosen solution is the one that corresponds to the intersection with the upper half of the *g* = 1 circle.

Figure 11a) shows the graphical evolution of the matching values when the frequency decreases. When the frequency decreases, then the electrical sizes of *d* and *Ls* also decreases. Point A moves toward the load and point S toward SC. *ym* moves away from the center of chart (perfect matching).

Figure 11b) shows the graphical evolution of the matching values when the frequency increases. When the frequency increases, then the electrical sizes of *d* and *Ls* also increases.

Following the display produced by this script, students should be able to solve in a paper

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

Another important topic that should be explained to the undergraduate students is the concept of bandwidth of a system. The majority of the systems based in transmission lines are perfectly matched at just one frequency. However, since the system should be able to operate over a frequency band, it is important to find the frequency band for which the system is consider to be acceptable matched. This frequency band is called the bandwidth of the system. Usually the criterion used to consider a system matched is when the VSWR is

The lower and the upper frequencies of the bandwidth can be found analytically. However, the authors believe that using the Smith chart to find graphically the bandwidth, will give to the students a much better understanding of the effects on the admittance of the line, and therefore on the matching conditions, when the frequency changes. The authors developed two *MATLAB* scripts called *SingleStubMatching\_Eng\_BW\_FV2a.m* and *SingleStubMatching\_Eng\_BW\_FV2b.m* that displays graphically, step by step, the effects on the matching conditions, when the frequency changes. Both are for systems using shortcircuited stubs. One of the scripts is for one of the two possible solutions and the other for the other solution. It is important for the students to verify that the bandwidth is not the same for the two solutions. These scripts is available for download and should be explored

The single-stub impedance matching system explained in the previous section, gives a perfect matching at one frequency. Once constructed, if the frequency changes so does the electric length of the distance *d* and length *Ls*, and therefore a perfect matching is no longer

It is important to study the evolution of the matching values for a single-stub matching system, when the frequency varies around the central value for which a perfect matching was achieved. To observe this evolution, the authors developed *MATLAB* scripts that graphically display this evolution and give the VSWR 2:1 bandwidth. After choosing shortcircuited or open-circuited stubs, there are still two solutions for the single-stub matching system, the perfect matching can be achieved with two pairs of values of *d* and *Ls* and therefore two different evolution of the matching values are also obtained. The script called *SingleStubMatching\_Eng\_BW\_FV2a.m* is intended for one of the solutions and script called *SingleStubMatching\_Eng\_BW\_FV2b.m* is intended for the other. Both use short-circuited stubs. It is important to compare the VSWR 2:1 bandwidth obtained for both solutions. Figure 11 shows the graphical evolution of the matching values given by the script *SingleStubMatching\_Eng\_BW\_FV2a.m* for the example illustrated in Figure 10. In this script the chosen solution is the one that corresponds to the intersection with the upper half of the

Figure 11a) shows the graphical evolution of the matching values when the frequency decreases. When the frequency decreases, then the electrical sizes of *d* and *Ls* also decreases. Point A moves toward the load and point S toward SC. *ym* moves away from the center of

Figure 11b) shows the graphical evolution of the matching values when the frequency increases. When the frequency increases, then the electrical sizes of *d* and *Ls* also increases.

chart any single-stub impedance matching problem using a ruler and a compass.

**5.2 Bandwidth of a single-stub impedance matching system** 

less or equal to 2.

when reading this section.

achieved.

*g* = 1 circle.

chart (perfect matching).

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 obtained.

b)

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

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

Using the Smith Chart in an E-Learning Approach 117

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 16% is obtained. So, as mentioned before, different bandwidth are obtained depending on the chosen for pair of

As pointed out before, there are several methods to achieve impedance matching. One well known method consists in the insertion of a transmission line with a length of a quarter wavelength and an appropriate characteristic impedance, in a position where the impedance is real. This matching technique is known as a quarter wavelength impedance matching

The characteristic impedance of this quarter wavelength transformer is given by *Z ZZ* 1 12 *R R* , where *ZR*1 is the impedance at the right side of the transformer and *ZR*<sup>2</sup> is the impedance at the left side of the transformer. Since a common transmission line has a real characteristic impedance, *ZR*1 and *ZR*2 must both be real. If a line is terminated in a complex load, the quarter wavelength transformer cannot be inserted at the load plane. It is then necessary to move along the line, a distance *d*, toward the generator till a real impedance *ZR*1 is obtained. This is illustrated in Figure 13. Two values for *ZR*1 are possible. One greater than *Z*0 and another less than *Z*<sup>0</sup> , separated by /4. Once *ZR*1 is chosen, *Z*1 can be calculated bearing in mind that *ZR*2 must be equal *Z*0 to achieve a

values of *d* and *Ls*.

perfect matching.

impedance of the main line.

*Z*L=100+j60 Ω.

**5.3 Quarter wavelength impedance matching system** 

Fig. 13. Basic layout of the quarter wavelength impedance matching.

that graphically explains, the quarter wavelength matching mechanism.

In general the students learn this method in a analytical way, by the direct computation of the required characteristic impedance of the quarter wavelength line and the location for its insertion. However, the authors believe that the use of the Smith chart to solve graphically this problem will give to the students a much better insight of the several impedance transformations involved in this problem to achieve an impedance equal to the characteristic

The authors developed a *MATLAB* script called *QuarterWavelengthTransformer\_Eng\_FV.m*

This is shown in Figure 14, for a line with a characteristic impedance *Z*0=50Ω and a load

system or as the quarter wavelength transformer.

Fig. 12. Graphical solution given by the *SingleStubMatching\_Eng\_BW\_FV2b.m* script.

the chosen solution is the one that corresponds to the intersection with the lower half of the *g* = 1 circle.

Figure 12a) shows the graphical evolution of the matching values when the frequency decreases. When the frequency decreases, the electrical sizes of *d* and *Ls* also decreases. Point A moves toward the load and point S toward SC. *ym* moves away from the center of chart (perfect matching).

Figure 12b) shows the graphical evolution of the matching values when the frequency increases. When the frequency increases, the electrical sizes of *d* and *Ls* also increases. Point

a)

Fig. 12. Graphical solution given by the *SingleStubMatching\_Eng\_BW\_FV2b.m* script.

*g* = 1 circle.

chart (perfect matching).

the chosen solution is the one that corresponds to the intersection with the lower half of the

Figure 12a) shows the graphical evolution of the matching values when the frequency decreases. When the frequency decreases, the electrical sizes of *d* and *Ls* also decreases. Point A moves toward the load and point S toward SC. *ym* moves away from the center of

Figure 12b) shows the graphical evolution of the matching values when the frequency increases. When the frequency increases, the electrical sizes of *d* and *Ls* also increases. 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 16% is obtained. So, as mentioned before, different bandwidth are obtained depending on the chosen for pair of values of *d* and *Ls*.

### **5.3 Quarter wavelength impedance matching system**

As pointed out before, there are several methods to achieve impedance matching. One well known method consists in the insertion of a transmission line with a length of a quarter wavelength and an appropriate characteristic impedance, in a position where the impedance is real. This matching technique is known as a quarter wavelength impedance matching system or as the quarter wavelength transformer.

The characteristic impedance of this quarter wavelength transformer is given by *Z ZZ* 1 12 *R R* , where *ZR*1 is the impedance at the right side of the transformer and *ZR*<sup>2</sup> is the impedance at the left side of the transformer. Since a common transmission line has a real characteristic impedance, *ZR*1 and *ZR*2 must both be real. If a line is terminated in a complex load, the quarter wavelength transformer cannot be inserted at the load plane. It is then necessary to move along the line, a distance *d*, toward the generator till a real impedance *ZR*1 is obtained. This is illustrated in Figure 13. Two values for *ZR*1 are possible. One greater than *Z*0 and another less than *Z*<sup>0</sup> , separated by /4. Once *ZR*1 is chosen, *Z*1 can be calculated bearing in mind that *ZR*2 must be equal *Z*0 to achieve a perfect matching.

Fig. 13. Basic layout of the quarter wavelength impedance matching.

In general the students learn this method in a analytical way, by the direct computation of the required characteristic impedance of the quarter wavelength line and the location for its insertion. However, the authors believe that the use of the Smith chart to solve graphically this problem will give to the students a much better insight of the several impedance transformations involved in this problem to achieve an impedance equal to the characteristic impedance of the main line.

The authors developed a *MATLAB* script called *QuarterWavelengthTransformer\_Eng\_FV.m* that graphically explains, the quarter wavelength matching mechanism.

This is shown in Figure 14, for a line with a characteristic impedance *Z*0=50Ω and a load *Z*L=100+j60 Ω.

Using the Smith Chart in an E-Learning Approach 119

2. Finding the distance *d*, in wavelengths, moving from *Lz* , toward the generator (clockwise)

7. Renormalizing *<sup>R</sup>*<sup>2</sup> *z* impedance with reference to *Z*<sup>1</sup> an impedance *Z Z <sup>R</sup>*2 0 is obtained,

8. Normalizing *Z Z <sup>R</sup>*2 0 with reference to *Z*0 a normalized impedance of 1 is obtained, which means that the center of the chart is reached, confirming a perfect

In all the above examples, lossless transmission lines have been used. However all lines have some losses and this changes the results. One of the main influences of the losses is in the amplitude of the reflection coefficient and therefore in the impedance along the line. For

> <sup>2</sup> *<sup>d</sup> <sup>j</sup>*<sup>2</sup> *<sup>d</sup> <sup>L</sup> d ee*

From equation 10, we notice that the phase changing of the reflection coefficient is equal to a lossless line, however the amplitude decreases from the load to the generator according with

> <sup>2</sup> *<sup>d</sup> <sup>L</sup> d e*

 

Due to this amplitude decreasing, when travelling from the load to the generator the locus

 is a spiral approaching the center of the chart instead of a circle like in a lossless line. This means that at the input of a lossy line there is a better matching than at the load. Due to the loss of energy in the line, at the generator there is less returned energy to the generator and therefore a better matching. The authors developed a *MATLAB* script called *LossyLine\_Eng\_FV.m* that graphically explains these effects as illustrated in Figures 15 and 16. As shown in Figure 16 a long line terminated in a load with the high VSWR of 10.4, has

 

the propagation constant in rad/m.

3. Renormalize *Rz* impedance with reference to *Z*0 in order to get *ZR* ;

*<sup>L</sup>* constant circle, until the real normalized impedance *Rz* is obtained. In this

(10)

(11)

the attenuation

constant circle, until the

Figure 14a) shows the first 5 steps:

4. Calculate *Z ZZ* 1 0 *<sup>R</sup>* ;

Figure 14b) shows the last 3 steps:

along the

matching;

**5.4 Analysis of lossy lines** 

constant in Np/m and

the equation 11.

of 

1. Marking the normalized impedance *zL*;

example *Rz* greater than 1 was chosen;

5. Normalize *ZR* with reference to *Z*1 obtaining *<sup>R</sup>*<sup>1</sup> *z* ;

real normalized impedance *<sup>R</sup>*<sup>2</sup> *z* is obtained;

meaning that a perfect matching is achieved;

a lossy line the reflection coefficient is given by equation 10.

at the input a very acceptable VSWR of 1.7.

being *d* the distance measured from the load toward the generator,

6. Moving from *<sup>R</sup>*<sup>1</sup> *z* , toward the generator (clockwise) along the

a)

Fig. 14. Graphical solution given by the *QuarterWavelengthTransformer\_Eng\_FV.m* script.

Figure 14a) shows the first 5 steps:

118 E-Learning – Organizational Infrastructure and Tools for Specific Areas

a)

b)

Fig. 14. Graphical solution given by the *QuarterWavelengthTransformer\_Eng\_FV.m* script.


Figure 14b) shows the last 3 steps:


#### **5.4 Analysis of lossy lines**

In all the above examples, lossless transmission lines have been used. However all lines have some losses and this changes the results. One of the main influences of the losses is in the amplitude of the reflection coefficient and therefore in the impedance along the line. For a lossy line the reflection coefficient is given by equation 10.

$$
\rho \left( d \right) = \rho\_L \ e^{-2\alpha d} \ e^{-j2\beta d} \tag{10}
$$

being *d* the distance measured from the load toward the generator, the attenuation constant in Np/m and the propagation constant in rad/m.

From equation 10, we notice that the phase changing of the reflection coefficient is equal to a lossless line, however the amplitude decreases from the load to the generator according with the equation 11.

$$\left|\rho(d)\right| = \left|\rho\_L\right|e^{-2ad} \tag{11}$$

Due to this amplitude decreasing, when travelling from the load to the generator the locus of is a spiral approaching the center of the chart instead of a circle like in a lossless line. This means that at the input of a lossy line there is a better matching than at the load. Due to the loss of energy in the line, at the generator there is less returned energy to the generator and therefore a better matching. The authors developed a *MATLAB* script called *LossyLine\_Eng\_FV.m* that graphically explains these effects as illustrated in Figures 15 and 16. As shown in Figure 16 a long line terminated in a load with the high VSWR of 10.4, has at the input a very acceptable VSWR of 1.7.

Using the Smith Chart in an E-Learning Approach 121

If a long line is to be considered, it is important to know its attenuation constant and to

It is well known that many transmission line problems can easily be solved using graphical procedures based on the Smith chart. The authors still believe that the use of the Smith chart by the students is an important pedagogical tool even knowing that personal computers and

Since the main topic of this book is concerned with *e-learning*, the aim of this chapter is to help the reader understand and learn how to use the Smith chart following a step by step procedure based on *MATLAB* scripts, that should be used when reading this chapter. This approach should teach the students to solve several kinds of transmission line problems by

To exemplify this concept, the authors developed *MATLAB* scripts that display, step by step, the graphical procedure used in several applications. Using these scripts, many aspects of the transmission line theory such as: the voltage, current, impedance, Voltage Standing Wave Ratio (VSWR), reflection coefficient and matching design problems can be easily

History and use of the Smith chart and its importance in the resolution of classical

 Presentation of some examples that integrates the transmission line concepts. The authors developed a *MATLAB* scripts that display, step by step, the graphical procedure

Inan, A. S. (2005). Remembering Phillip H. Smith on his 100th Birthday. *IEEE Antennas and* 

Marinčić, A. (1997). The Smith Chart, *Microwave Review*, Vol. 4, No.2, (December 1997), pp.

Mak, F. & Sundaram, R. (2008). A *MATLAB*-Based Teaching Of The Two-Stub Smith Chart

*Propagation Society International Symposium,* Vol. 3B, (July 2005), pp. 129-132, ISBN:

Application For Electromagnetics Class, *38th ASEE/IEEE Frontiers in Education Conference*, pp. T2A-7-11, ISBN 978-1-4244-1969-2, Saratoga Springs, NY, October

Construction of this chart from the basic equations and concepts.

evaluate its implications in the problem being considered.

themselves in a paper chart using a pencil, a ruler and a compass.

interpreted and well visualized using the Smith chart.

All the *MATLAB* scripts can be download from the link:

*MATLAB*TM, The MathWorks, Inc., http://www.mathworks.com

The chapter was organized as follows:

transmission line problems.

How to use the Smith chart.

that must be used to solve these examples.

**7. References** 

1-7

0-7803-8883-6

22–25, 2008

http://www.av.it.pt/rochap/MatlabScripts.zip

calculators are commonly available nowadays.

**6. Conclusion** 

Fig. 16. Graphical solution given by the *LossyLine\_Eng\_FV.m* script for a long line.

If a long line is to be considered, it is important to know its attenuation constant and to evaluate its implications in the problem being considered.
