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Teoria Electromagnetica y Ondas 203058A_471 Unit 3 - Step 4 DAGLY SÁNCHEZ CARVAJAL, CÓDIGO 79185472 Group number:

203058_33

Presented to: WILMER HERNAN GUTIERREZ

UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA ESCUELA DE CIENCIAS BÁSICAS, TECNOLOGÍA E INGENIERÍA M OF 2017

Activities to develop

Each student in the group has to answer the following questions using academic references to support the research: 1. What is the practical implications associated to a line with only reactive components or only resistive components? A transmission line is a pair of electrical conductors carrying an electrical signal from one place to another. Coaxial cable and twisted pair cable are examples. The two conductors have inductance per unit length, which we can calculate from their size and shape. They have capacitance per unit length, which we can calculate from the dielectric constant of the insulation. In the early days of cable-making, there would be current leaking through the insulation, but in modern cables, such leakage is negligible. Transmission lines: they are formed, at least, by two drivers

We consider a generator and a load connected across one transmission line (e. g. a coaxial cable)

Characteristic Impedance - Z0 We're now ready to introduce a fundamental parameter of every transmission line: its characteristic impedance. This is defined as the ratio of the magnitude of the forward traveling voltage wave to the magnitude of the forward traveling current wave:

In terms of the transmission line per-length parameters, the characteristic impedance is given by:

Z0 will be extensively used in determining other transmission line parameters. This page will end with special cases of the characteristic Impedance. If R'=G'=0, then the conductors of the transmission line are perfectly conducting (so R'=0) and the dielectric medium that separates the conductors has zero conductivity (so that G'=0). In this case, the line is referred to as a Lossless Line. The characteristic impedance becomes:

Another type of line of interest is the distortion less line This type of line may contain loss (so that the voltage dies off somewhat as it propagates down the line), but the magnitude of the attenuation is frequency-independent, and the phase constant varies linearly with frequency. This is desirable; similar to filter theory, this would be considered "linear phase" - that is, signals that come out of the transmission line might be attenuated, but have the same shape. The criteria for this is:

http://www.antenna-theory.com/tutorial/txline/transmission2.php

2. In a practical transmission system. What is a good value for the reflection coefficient and the VSWR? Explain. We are now aware of the characteristic impedance of a transmission line, and that the tx line gives rise to forward and backward travelling voltage and current waves. We will use this information to determine the voltage reflection coefficient, which relates the amplitude of the forward travelling wave to the amplitude of the backward travelling wave. To begin, consider the transmission line with characteristic impedance Z0 attached to a load with impedance ZL:

At the terminals where the transmission line is connected to the load, the overall voltage must be given by:

Recall the expressions for the voltage and current on the line (derived on the previous page):

If we plug this into equation [1] (note that z is fixed, because we are evaluating this at a specific point, the end of the transmission line), we obtain:

The ratio of the reflected voltage amplitude to that of the forward voltage amplitude is the voltage reflection coefficient. This can be solved for via the above equation:

The reflection coefficient is usually denoted by the symbol gamma. Note that the magnitude of the reflection coefficient does not depend on the length of the line, only the load impedance and the impedance of the transmission line. Also, note that if ZL=Z0, then the line is "matched". In this case, there is no mismatch loss and all power is transferred to the load. At this point, you should begin to understand the importance of impedance matching: grossly mismatched impedances will lead to most of the power reflected away from the load.

Note that the reflection coefficient can be a real or a complex number.

Standing Waves

We'll now look at standing waves on the transmission line. Assuming the propagation constant is purely imaginary (lossless line), We can re-write the voltage and current waves as:

If we plot the voltage along the transmission line, we observe a series of peaks and minimums, which repeat a full cycle every half-wavelength. If gamma equals 0.5 (purely real), then the magnitude of the voltage would appear as:

Similarly, if gamma equals zero (no mismatch loss) the magnitude of the voltage would appear as:

Finally, if gamma has a magnitude of 1 (this occurs, for instance, if the load is entirely reactive while the transmission line has a Z0 that is real), then the magnitude of the voltage would appear as:

One thing that becomes obvious is that the ratio of Vmax to Vmin becomes larger as the reflection coefficient increases. That is, if the ratio of Vmax to Vmin is one, then there are no standing waves, and the impedance of the line is perfectly matched to the load. If the ratio of Vmax to Vmin is infinite, then the magnitude of the reflection coefficient is 1, so that all power is reflected. Hence, this ratio, known as the Voltage Standing Wave Ratio (VSWR) or standing wave ratio is a measure of how well matched a transmission line is to a load. It is defined as:

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? 4. What is the voltage reflection coefficient and what is an ideal value for a transmission system? 5. What is the effect of Lossy line on voltage and current waves? 6. In the Smith Chart identify a 𝑍𝐿 =∝, a 𝑍𝐿 = 0, two resistive loads and two complex loads. You have to assume the characteristic impedance.

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 𝑍0 = 60Ω and the load at the end of the line has an impedance of 𝑍𝐿 = 45 + 𝑗95Ω. Using the Smith Chart, find:

a. Reflection coefficient Γ (magnitude and phase), and the VSWR. b. The input impedance if the line is 0.75𝜆 long. c. The length of the line, necessary to make the input impedance real and the value of the impedance in this point. 2. A transmission line has the following parameters:, 𝐿 = 2𝑚𝐻/𝑘𝑚, 𝐶 = 45 𝑛𝐹/𝑘𝑚, 𝐺 = 1.2𝜇𝑆/𝑘𝑚 and 𝑅 = 4Ω/𝑘𝑚. It has a generator supplying 200𝑉 𝑟𝑚𝑠 at 𝜔 = 5 𝑥 103 𝑟𝑎𝑑/𝑠 and in series with a resistance of 200Ω. The load has an impedance of 300Ω. Find the input voltage and current.

3. A 45Ω lossless transmission line has a 𝑍𝐿 = 45 − 𝑗60Ω. If it is 200𝑚 long and the wavelength is 23𝑚. Find and probe with the smith chart: a. Input impedance. b. Reflection coefficient. c. VSWR

4. A transmission line of length 𝑙 = 0.35𝜆 has an input impedance 𝑍𝑖 = 25 + 𝑗45Ω. Using the Smith Chart, find the load impedance if 𝑍0 = 75Ω. 5. A load 𝑍𝐿 = 45 − 𝑗60Ω is connected to a transmission line with 𝑍0 = 75Ω. The line is 𝑙 = 0.35𝜆. Find the input impedance and at least two line lengths where the input impedance is real. Use the Smith Chart to Solve the exercise.