Task 3 - Harvey Meza PDF

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Task 3 - Electromagnetic waves in guided media Individual work format

Student name Harvey Meza Jacome Group 203058_20 Identification number 1091661053

UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA UNAD Escuela de Ciencias Básicas, Tecnología e Ingeniería Teoría Electromagnética y Ondas 2021 16-01

Exercises development

Activity Answers: (write with your own words) 1. What do you understand by a transmission line? A transmission line is a physical medium through which information is transmitted by means of frequencies and electrical impulses. Today there are several transmission lines that, depending on their materials, use impedance to carry different data. The most common transmission lines are the ion lines, the fiber optic line, this transmission line is the most used for transporting data over long distances used by the Internet operator. - Copper: this transmission line is the most used in LAN or WAN networks, such as UTP and STP cables. - Coaxial or concentric transmission line: widely used for high frequency applications to reduce losses and isolate transmission paths. - Coaxial: this type of cable is the most widely used by urban cable operators. - Balanced or differential signal: in balanced two-wire lines, both conductors carry a current, one conductor carries the signal and the other is the return. 2. Define the following electrical parameters of transmission lines: a. Input impedance 𝑍𝑖𝑛. Answer: it is understood as the relationship between total voltage and total current.

𝑍𝑖𝑛 =

𝑉𝑠1 𝑉1 𝑒 −𝑗𝛽𝑥 + 𝑉1− 𝑒 𝑗𝑏𝑥 = + −𝑗𝛽𝑥 𝐼𝑠1 𝐼1 𝑒 + 𝐼1− 𝑒 𝑗𝑏𝑥

In terms of the impedance of the line and the load, the equation of the burial impedance is: 𝑍𝑖𝑛 =

𝑍2 cos(𝛽𝑥) + 𝑗𝑍1 sin(𝛽𝑥) 𝑍1 cos(𝛽𝑥) + 𝑗𝑍2 sin(𝛽𝑥)

b. Stationary wave ratio 𝑉𝑆𝑊𝑅.

Answer: standing waves represent a power that is not accepted by the load and reflected along the transmission or power line, ion line or power.

𝑉𝑆𝑊𝑅 =

𝑉𝑚𝑎𝑥 1 + |Γ| = 𝑉𝑚𝑖𝑛 1 − |Γ|

Although standing waves and are very important, theory and analysis can often mask a view of what is happening. Fortunately, it is possible to get a good overview of the topic, without going too deep in VSWR theory. c. Physical length 𝐿 and electrical length 𝓁. Answer: Electrical length 𝓁: it is a unit of measurement that is used in the study of transmission lines of electrical energy, which expresses the distance to the load as a function of the wavelength. 𝓁=

𝑥 𝜆

Physical length L: the distance traveled by a periodic disturbance that propagates through a medium in a cycle. 𝜆 = 𝑓𝑣 𝜆0 3. What is the purpose of Smith's Letter in the study of the propagation of waves?

The purpose of the smith chart is that laborious calculations with complex numbers are avoided. Complex numbers are avoided for kwon the input impedance to the line or the reflection coefficient, so they are reflection coefficient, so they are very useful in the coupling of transmission lines and in the reverse transmission lines and in the inverse calculation of a complex number. Application exercises: For the development of the following exercises, note that 𝐺𝐺 corresponds to the

group number and 𝐶𝐶𝐶 to the last 3 digits of the identification number. 1. A coaxial line has the following characteristics: Geometric parameters: 𝑎 = 0.1𝑚𝑚 𝑏 = 10𝑚𝑚 𝑡 = (20 + 10)𝜇𝑚 Conductor properties: 𝜎𝑐 = 4.3 ∗ 106 𝑆𝑚/𝑚

Properties of the insulator: 𝜎𝑑 = 1𝑥 10−10 𝑆𝑚/𝑚 𝜖𝑟 = 2.3 𝜇𝑟 = 1

Applied signal frequency: 𝑓 = 053𝐾𝐻𝑧 = 53 ∗ 103 𝐻𝑧

Magnetic permeability: 𝜇0 = 1.257 ∗ 10−6 Empty Permittivity:𝜀0 = 8.854 ∗ 10−12

a. Calculate the electrical parameters R L C G. Data 𝐺𝐺 = 20𝐶𝐶𝐶 = 53 First we calculate the depth penetration 𝛿𝑝 =

1 1 = 𝑎 √𝜋𝑓𝜎𝑐 𝜇0

We replace 𝛿𝑝 =

1

√𝜋 ∗ 53𝑥 103 𝐾𝐻𝑧 ∗ 4.3𝑥 106 𝑆𝑚 ⁄𝑚 ∗ 1.257𝑥 10−6 𝛿𝑝 = 1.054108371𝑥10−3

According to the result of the penetration depth, we compare it in the table and we realize that we use the low frequency formulas

We calculate R 𝑅=

1 1 1 ( 2+ ) 𝜋𝜎𝑐 𝑎 2𝑏𝑡

We replace 𝑅=

1 1 1 ) ∗( + −3 6 −3 2 𝜋(4.3 ∗ 10 ) (0.1 ∗ 10 ) 2(10 ∗ 10 )(3 ∗ 10−5 ) 𝑅 = 7.5 𝑜ℎ𝑚⁄𝑚

We calculate G 𝐺=

2𝜋𝜎𝑑 𝑏 𝑙𝑛 𝑎

We replace

𝐺=

2𝜋1 ∗ 10−10 (10 ∗ 10−3 ) 𝑙𝑛 (0.1 ∗ 10−3 )

𝐺 = 1.36 ∗ 10−10 𝑆𝑚⁄𝑚 We calculate L 𝐿=

𝜇0 𝑏 [1 + 𝑙𝑛 ( )] 𝑎 2𝜋

We replace 𝐿=

1.257𝑥 10−6 10 ∗ 10−3 )] [1 + 𝑙𝑛 ( 0.1 ∗ 10−3 2𝜋 𝐿 = 1.2 ∗ 10−6 𝐻⁄𝑚 We calculate C 𝐶=

2𝜋𝜀 𝑏 𝑙𝑛 ( ) 𝑎

We replace 𝐶=

2𝜋(2.3 ∗ 8.854 ∗ 10−12 ) 10 ∗ 10−3 𝑙𝑛 ( ) 0.1 ∗ 10−3

𝐶 = 2.78 ∗ 10−11 𝐹⁄𝑚

We check in Geogebra

b. Using the distributed model, calculate the propagation parameters 𝛼, 𝛽, 𝛾𝑎𝑛𝑑𝑍0 .

First we calculate 𝜔𝐶

𝜔𝐶 = 2𝜋𝑓𝐶 We replace 𝜔𝐶 = 2𝜋(53 ∗ 103 )(2.78 ∗ 10−11) 𝜔𝐶 = 0.0000092524613 First we calculate 𝜔𝐿 𝜔𝐿 = 2𝜋𝑓𝐿

We replace 𝜔𝐿 = 2𝜋(53 ∗ 103 )(1.2 ∗ 10−6 ) 𝜔𝐿 = 0.373

Now with those values we find𝑍0 𝑅 + 𝑗𝜔𝐿 𝑍0 = √ 𝐺 + 𝑗𝜔𝐶 We replace 𝑍0 = √

7.5 + 𝑗(0.373) 1.36 ∗ 10−10 + 𝑗(0.0000092524613)

𝑍0 = 653.74675504832 − 622.1041822677962𝑖𝑜ℎ𝑚 Then we calculate 𝛼𝑦𝛽

𝛾 = ±√(𝑅 + 𝑗𝜔𝐿 )(𝐺 + 𝑗𝜔𝐶 ) = 𝛼 + 𝑗𝛽 We replace 𝛾 = ±√(7.5 + 𝑗0.373)(1.36 ∗ 10−10 + 0.0000092524613) 𝛾 = 0.0057560840448 + 0.0060486816497𝑗 Then we have 𝛼 = 0.0057560840448

𝑁𝑝⁄ 𝑚

𝛽 = 0.0060486816497 𝑅𝑎𝑑⁄ 𝑚 We check in Geogebra

a. Calculate the propagation velocity𝑉𝑝 , the wavelength 𝜆 and the attenuation 𝛼𝑑𝐵/𝐾𝑚 .

Figure 1: Geometrical parameters in coaxial line. We first calculate the velocity of propagation 𝑉𝑝 =

2𝜋𝑓 𝛽

We replace

2𝜋53 ∗ 103 𝑉𝑝 = 0.0060486816497 𝑉𝑝 = 5.50 ∗ 107 𝑚 ⁄𝑠

Then we calculate the wavelength 𝜆=

2𝜋 𝛽

We replace 𝜆=

2𝜋 0.0060486816497 𝜆 = 1038.77𝑚

And finally we will calculate the attenuation 𝛼𝑑𝐵/𝑚 = −8.68𝛼 We replace 𝛼𝑑𝐵/𝑚 = −0.0499628095089 𝑑𝐵⁄𝑚 We check in Geogebra

Interpretation: According to the concepts explored, explain the meaning of the value obtained for 𝑉𝑝 , 𝜆 and 𝛼𝑑𝐵/𝐾𝑚 .

Velocity of propagation: it is the speed with which a wave travels from one point to another, and the result we had in our exercise is 𝑉𝑝 = 5.50 ∗ 107 𝑚⁄𝑠

Wavelength: it is the distance that a wave travels in a period of time, and

it is represented by the Greek letter 𝜆 and as a result of our exercise it gave us a wavelength of 𝜆 = 1038.77𝑚

Attenuation: It is known as the loss of power suffered by a signal when it is transmitted by some transmission medium. The value he gave us in our exercise is 𝛼𝑑𝐵/𝑚 = −0.0499628095089 𝑑𝐵⁄𝑚

2. A 𝑍𝑜 = 75Ω lossless transmission line has a 𝑍𝐿 = 35 − 𝑗75Ω. If it is 𝐺𝐺 𝑚 long and the wavelength is 𝐶𝐶𝐶 𝑚𝑚, Calculate:

3. 4. Figure 2: Graphic representation of the transmission line.

Data 𝑍𝐿 = 35 − 𝑗75Ω 𝑍0 = 75Ω 𝐿 = 20𝑚

a. Input impedance 𝑍𝑖𝑛 .

𝜆 = 053𝑚𝑚 → 0.053𝑚 Then we calculate 𝑧𝑖𝑛

2𝜋 𝜆 𝐿) 𝑍𝐿 + 𝑗𝑍0 𝑡𝑎𝑛 ( 2𝜋 𝑍𝑖𝑛 = 𝑍0 𝑍0 + 𝑗𝑍𝐿 𝑡𝑎𝑛 ( 𝐿) 𝜆 We replace

2𝜋 (35 − 𝑗75Ω) + 𝑗75Ω𝑡𝑎𝑛 ( 20𝑚) 0.053𝑚 𝑍𝑖𝑛 = 75Ω 2𝜋 75Ω + 𝑗(35 − 𝑗75Ω)𝑡𝑎𝑛 ( 0.053𝑚 20𝑚) 𝑍𝑖𝑛 = 229.12 + 153.44𝑗

b. Reflection coefficient Γ (magnitude and phase). Γ=

𝑍𝐿 − 𝑍𝑂 𝑍𝐿 + 𝑍𝑂

We replace

Γ=

(35 − 𝑗75Ω) − 75Ω (35 − 𝑗75Ω) + 75Ω

Γ = 0.069 − 0.63i

We went from rectangular to a polar Γ = 0.638∢ − 83.785°𝑂ℎ𝑚

c. VSWR. 𝑉𝑆𝑊𝑅 =

1 + |Γ| 1 − |Γ|

We replace 𝑉𝑆𝑊𝑅 =

1 + |0.638| 1 − |0.638|

𝑉𝑆𝑊𝑅 = 4.53

We check in Geogebra

.

Interpretation: According to the concepts explored, explain the meaning of the value obtained for 𝑍𝑖𝑛 , Γ and VSWR.

Input impedance: this value is the relationship between the voltage and the current of the transmission line, and the result that the exercise gave us is 𝑍𝑖𝑛 = 229.12 + 153.44𝑗

Reflection coefficient: it is the one that tells us the amplitude of the reflected wave and the angle formed at the moment of reflection for our

exercise the result was Γ = 0.638∢ − 83.785°𝑂ℎ𝑚

VSWR: It means Stationary Wave Ratio, it is the one that shows us the behavior of the voltage (minimum and maximum) in a standing wave phenomenon between a transmission line and its load at the end. And as a result of this exercise it gives us 𝑉𝑆𝑊𝑅 = 4.53

5. Bearing in mind that Smith's letter is used to determine parameters of the transmission lines, use the "Smith 4.1" software to check the results obtained in point 2.

a. Input impedance 𝑍𝑖𝑛 .

b. Reflection coefficient Γ.

c. VSWR.

Application example

Example: The depth of penetration of a signal in a transmission medium was learned. Through mathematical calculations. The area where the signal works in the transmission medium is analyzed in a practical way by the Smith software. We have the case of the application of conductive guided waves, which are used to designate tubes of a conductive material of rectangular, circular, or elliptical section, in which the direction of the electromagnetic energy must be mainly conducted along the guide and limited at its borders, which allow the transmission of electromagnetic waves from one point to another.

Video link

URL: https://www.youtube.com/watch?v=ZOwHQRAYTeE

References

Chen, W. (2005). The Electrical Engineering Handbook. Boston: Academic Press, (pp. 525-551). Recovered from http://bibliotecavirtual.unad.edu.co:2048/login?url=http://searc h.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=117152&lang =es&site=ehost-live&ebv=EB&ppid=pp_525 Joines, W., Bernhard, J., & Palmer, W. (2012). Microwave Transmission Line Circuits. Boston: Artech House, (pp. 23-68). Recovered from http://bibliotecavirtual.unad.edu.co:2051/login.aspx?direct=true &db=nlebk&AN=753581&lang=es&site=edslive&ebv=EB&ppid=pp_23 Hierauf, S. (2011). Understanding Signal Integrity. Boston: Artech House, Inc. Chapter 6, 7, 11. Recovered from http://bibliotecavirtual.unad.edu.co:2051/login.aspx?direct=true &db=nlebk&AN=345692&lang=es&site=edslive&ebv=EB&ppid=pp_49 Impedance Matching Networks. (2001). Radio-Frequency & Microwave Communication Circuits, (pp. 146-188). Recovered from http://bibliotecavirtual.unad.edu.co:2051/login.aspx?direct=true &db=aci&AN=14528229&lang=es&site=eds-live...