Group 2030580 24 John Jairo Valencia Rojas

Universidad Nacional Abierta y a Distancia – UNAD Escuela de Ciencias C iencias Básicas Tecnologías e Ingenierías-ECBTI Curso: Teoria electromagnetica y Ondas Unity III

Step Step 4- Wave Wavegui des - E lect lectrr i c pa par ameter ter s in i n tra tr ansmi nsmi ssio ssi on line li ness -

John Jairo Valencia Valencia Rojas Cod: 94326428 94326428 Jonny Zuñiga Suarez Cod: 16799106 16799106 Andrés Felipe Ortega Ortega Aristizábal Cód. 1.116.245.923 1.116.245.923 Jhonnatan Ordoñez Piamba Cod. 1114818 1114818912 912 Jonny Zuñiga Cód. Cód. 16799106 16799106

G r oup: up: 2030580_2 2030580_24 4

Tutor Tutor : Wilm Wi lme er H er nan nan Gutié Guti ér r ez

- UNAD OPEN AND DI STANCE STANCE NATI ONAL UNI UNI VE RSI TY -

Palmi Palmi r a ( V alle) lle) Mayo Mayo /10/ 10/2018 2018

Universidad Nacional Abierta y a Distancia – UNAD Escuela de Ciencias Básicas Tecnologías e Ingenierías-ECBTI Curso: Teoria electromagnetica y Ondas Unity III

INTRODUCCTION

The transmission lines confine the electromagnetic energy to a region of the space limited by the physical medium that constitutes the line itself, unlike the waves that propagate in the air, with no other barrier than the obstacles they encounter on their way. The line is formed by electrical conductors with a determined geometric layout that conditions the characteristics of the waves electromagnetic in it. In communication systems, transmission lines find numerous applications not only in the transport of signals between a source and a load, but also as resonant circuits, filters and impedance couplers. Some of the most common applications include the transport of telephone signals, data and television, as well as the connection between transmitters and antennas and between these and receivers.


1. What is the practical implications associated to a line with only reactive components or only resistive components?

According to the power triangle, it is obtained that the power factor depends on the relationship between the active power and the apparent power:

TRIANGLE OF POWERS Its value depends on the characteristics of the circuit itself, and is an important parameter in installations with an important demand for electrical power. According to its definition, the power factor is dimensionless and can only take values between 0 and 1. In a pure resistive circuit traveled by an alternating current, the intensity and voltage are in phase (φ = 0), that is, they change polarity at the same time in each cycle, the unit factor being therefore the power factor. On the other hand, in a pure reactive circuit, the intensity and voltage are in quadrature (φ = 90º), the value of fp being zero. In practice, the circuits cannot be purely resistive or reactive, observing phase shifts, more or less significant, between the wave forms of current and voltage. Thus, if the fp is close to the unit, it will be said that it is a strongly resistive circuit so its fp. it is high, while if it is close to zero it is strongly reactive and its fp is low. When the circuit is of inductive character, the most common case, we will speak of a fp. in delay, while it is said in advance when it is of a capacitive nature.

A low power factor means energy losses, which affects the efficiency of the operation o f the electrical system. It is penalized with an additional surcharge in the electric bill to companies that have a power factor lower than 0.9 or 0.95 according to their demanded power.


When you have a low power factor, you have additional costs that negatively affect the customer's billing, so the problem must be solved by installing electric capacitor banks. Correcting the low power factor in a facility is good business, not only because fines will be avoided in electricity bills, but because equipment will operate more efficiently, reducing costs for energy consumption.

2. In a practical transmission system. What is a good value for the reflection coefficient and the VSWR? Explain.

The ROE is always real and positive, in the range 1 ≤  ≤ ∞. When the line is coupled,  = 0 and there is no reflected wave. In these conditions,  = 1 . If the line is terminated in open circuit or in short circuit, there is total reflection and  = ∞ . The standing wave ratio is important, since, unlike the reflection coefficient, it is an easy parameter to measure and gives an indication of the operating conditions of the line and its coupling to the load and the generator. The maximum voltage of the standing wave, Vmax, occurs when the incident and reflected voltages are in phase, ie: (Vega, 2000, pág. 279)

| | = | | + | | Relationship between the reflection coefficient and ROE. From the definition of the coefficient of reflection, Γ, in we obtain that:

|Г| =

| | − | | | | + | |

=

 − 1  + 1

The equation gives the magnitude of the reflection coefficient, but not its phase.

3. What occurs with the voltage and current in a line with the following conditions: line terminated in its characteristic impedance, line terminated in a short and line terminated in an open?

When a line is not coupled, that is, terminated in its characteristic impedance, part of the energy incident on the load is reflected towards the generator. The ratio between the reflected wave voltage and that of the incident is the coefficient of reflection. This fact gives rise to that along the line a standing wave is formed, with maximum and minimum voltage and current, to fixed distances along the line and having the form shown in the figure.


In lines of low losses,  ≈ 0, with which ℎ ≈  and ℎ ≈  and the input impedance is reduced to: Input impedance of a short-terminated line. In this case  = 0 and  = 1∠180º and from: Input impedance of a line terminated in open circuit. Under these conditions,  = ∞ and  = 1∠0º. The input impedance is: Where  and  denote the impedances in short circuit and open circuit, respectively. (Vega, 2000, pág. 289)

4. What is the voltage reflection coefficient and what is an ideal value for a transmission system?

The reflection coefficient is, in general, complex a nd, although it is expressed in terms of the situation in the load, it can be expressed at any distance point z from it as: Where ΓL is the value of the reflection coefficient in the load, given b y the magnitude; When the attenuation in the line is zero (α = 0), the reflection coefficient has the same magnitude in the whole line, but if α ≠ 0, the magnitude of the reflected wave decreases as the distance to the load increases. The relationship between voltage and current in the load and the voltag es of the incident and reflected waves can be deduced from the previous equations and is given by: (Vega, 2000, pág. 288)


5.

What is the effect of Lossy line on voltage and current waves?

Every transmission line internally has a finite resistance, which causes unavoidable power losses of the circulating signal through the line. This loss is directly proportional to the length of the line, meaning that the greater the length, the greater the internal resistance and the greater the loss of power. The losses in the conductor can vary from a small number of decibels per hundred meters in rigid coaxial cables with air dielectric, up to two hundred decibels per hundred meters in a flexible line of rigid dielectric. Because the resistance is distributed along the transmission line, the heating loss of the conductor is directly proportional to the square of length of the line. In addition, because the power dissipation is

directly proportional to the square of the current, the loss of the conductor is inversely proportional to the characteristic impedance.

6. In the Smith Chart identify a  =∝, a  = , two resistive loads and two complex loads. You have to assume the characteristic impedance.

Moving the cursor to the end of the left side of the horizontal axis is _ =  , and moving the cursor to the end of the right part of the horizontal axis is _ = 0 .


Examples of employment of the Smith Charter.

Suppose that it is necessary to determine the input impedance of a lossless line, whose length is λ / 10 (36º electric) and which is terminated by a load impedance of 21 + j24 Ω, if the characteristic impedance of the line is 60 Ω. (Parra, 2013, pág. 349)

Choose one of the following problems, solve it and share the solution in the forum. Perform a critical analysis on the group members’ contributions and reply this in the forum.

1. A lossless transmission line has a characteristic impedance of  = 60Ω and the load at the end of the line has an impedance of  = 45 + 95Ω. Using the Smith Chart, find: a. Reflection coefficient Γ (magnitude and phase), and the VSWR. b. The input impedance if the line is 0.75 long. c. The length of the line, necessary to make the input impedance real and the value of the impedance in this point.


SOLUCTION

Normalizing the load impedance  .

 =

 

=

45 + 95Ω 60Ω

= 0,8 + 1,6

We place the previous value in Smith's Letter, we draw a straight line that starts in 1.0 and go through  = 0,8 + 1.6 The straight line is extended until you cut the peripheral circle, we can read the "Origin of  Which in this case is

 = .



At this moment we draw a circle, centered on 1.0 and passing through  . This circle intersects the horizontal axis, on the right side at point 6 .0. This value is the system VSWR.

 = 5,27

To determine the value of de  = 0.75  , a distance of 0.75  Long can be moved over the peripheral circle from Z_ (N) and we arrive at the point 0.424  Long  A straight line is drawn that starts at en 0.424   and reaches the center of the chart at 1.0, the intersection with the VSWR circle determines the value of  = 0.75 .

.

 + .  = .  

Because a distance of. 0.5 is a complete return to the Charter, the integer multiples of. 0.5 of the value obtained, in this case:

.  − .   = .  

 0.75  = .  + . 


The previous value is normalized; therefore, the real value will be:

  0.75    0.5   0.25  60  

 

The reflection coefficient is read directly from Smith's Letter. The following is done: a vertical straight line is drawn, which starts at the intersection of the VSWR circle and the horizontal axis on the right side and ends in parameters of the radial scale. In this scale we obtain the magnitude and the phase is on the peripheral circle, graduated in degrees.

we have that the coefficient of reflection has the value:

Г = . 

°

Input: John Jairo Valencia Rojas


3.To calculate what they request, we must first find the electric length with the following formula SOLUCTION

ℓ=

 

ℓ=

200 23

= 8.695 

With this value we can calculate the input impedance

 = 

 = 45

 +  tan(2ℓ)  +  tan(2ℓ)

(45 − 60) + 45 ∗ tan(2 ∗ 8.695) 4 5 + (45 − 60) ∗ tan(2 ∗ 8.695)

 = 45

(45 − 60) +  4 5 ∗ (1.408) 4 5 + (45 − 60) ∗ (1.408)

 = .  + .  Now we find the reflection coefficient

=

=

 −   + 

(45 − 60) − 45 (45 − 60) + 45

 = .  −  = .  ∡ − ° Now we find VSWR

 =

 =

1 + | | 1 − ||

1 + |108.135| 1 − |108.135|

= −1.0186

Input: Andres Felipe Ortega

5.

Universidad Nacional Abierta y a Distancia – UNAD Escuela de Ciencias Básicas Tecnologías e Ingenierías-ECBTI Curso: Teoria electromagnetica y Ondas Unity III A transmission line of length  = 0.35 has an input impedance  = 25 + 45Ω. Using the Smith Chart, find the load impedance if 0 = 75Ω.

6.

SOLUCTION

For this exercise we must clear ZL from the following equation:

 = 0

 + 0tan(2ℓ) 0 + (2ℓ)

( +  tan(2ℓ)) = 0 ( + 0 tan(2ℓ))

 .  + .  tan(2ℓ) = 0.  + 0 tan(2ℓ)

( tan(2ℓ) − 0) = −0.  + 0 tan(2ℓ)

0(− + 0 tan(2ℓ)  = ( tan(2ℓ) − 0)

Replacing the values, we have:

 =

75(−25 + 45 + 75 tan(2ℓ) (25 + 45  tan(2ℓ) − 75)  = 303 + 15

Simulation letter smith


In the simulation it is observed that by loading the values of the load impedance and the values of Z0, the input impedance can be found. Input: Jonny Zuñiga

5. A load  = 45 − 60Ω is connected to a transmission line with  = 75Ω. The line is  = 0.35. Find the input impedance and at least two-line lengths where the input impedance is real. Use the Smith Chart to Solve the exercise.

SOLUCTION

We add a line with a length of,  = 0.35

For a frequency of 500 ,  = 3.5 

So, making the simulation we found that,  = 164.27 + 52.162 Ω


Input: Jhonnatan Ordoñez Piamba

In group solve the following practical exercise

Using the software Smith V4.0, found in the practice learning environment, solve the next exercises and explain each step of the simulation.

1. For the following input impedance and load impedance, find the wavelength necessary to get a real input impedance. a. b. c. d.

   

= 45 − 60Ω = 25 − 36Ω = 98 − 46Ω = 56 − 29Ω

   

= = = =

75Ω. 45Ω. 35Ω. 58Ω.


SOLUTION

a.  =  −   = .

 ℎ    = Г = 0.500⁄90 ° ℎ   =  = 45 + 60Ω    =  = (8.00 − 10.67)  ℎ  =     = 3.00


b.  =  −   = 

 ℎ    = Г = 0.524⁄91.762 ° ℎ    =  = 24.95 + 36.09Ω    =  = (12.96 − 18.75)  ℎ  =     = 3.21


c.  =  −   = .

 ℎ    = Г = 0.557⁄17.255 ° ℎ    =  = 98.04 + 47.02Ω    =  = (8.29 − 18.75)  ℎ  =     = 3.52


d.  =  −   = 

 ℎ    = Г = 0.245⁄80.538 ° ℎ    =  = 55.65 + 28.65Ω    =  = (14.20 − 7.31)  ℎ  =     = 1.65


CONCLUSIONES

There are applications and online tools that allow us to learn to use the letter of smith, being this tool of valuable help for the development of the own exercises of the technology. With the solution of the proposed exercises, some behaviors of the transmission lines were learned and understood. The smith chart allows the adaptation of the imped ance in a very easy way to have to resort to complex equations. In addition to containing other functions such as finding the attenuation coefficient, reflection angle between others.


Parra, A. P. (2013). Electromagnetismo para Ingeniería Electrónica. CALI: Sello Editorial Javeriano. Vega, ©. P. (2000). LINEAS DE TRANSMISION. CANTABRIA: Universidad de Cantabria . 



Electromagnetic theory and waves. (April 2018). webconference unit 3 part 2. Retrieved: https://goo.gl/t1SQBD Electromagnetic Wave Propagation. (2003). Fixed Broadband Wireless . 25-70. Retrieved from http://bibliotecavirtual.unad.edu.co:2048/login?url=http://search.ebscohost.co m/login.aspx?direct=true&db=aci&AN=14505422&lang=es&site=ehost-live

Group 2030580 24 John Jairo Valencia Rojas

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