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Chapter 3: Problem 22

A load has a reflection coefficient of 0.5 when referred to \(50 \Omega\). The load is at the end of a line with a \(50 \Omega\) characteristic impedance. (a) If the line has an electrical length of \(45^{\circ}\), what is the reflection coefficient calculated at the input of the line? (b) What is the VSWR on the \(50 \Omega\) line?

Short Answer

Expert verified
For (a), \ \Gamma_{in} = -0.5j \. For (b), \ VSWR = 3 \.

Step by step solution

01

Understanding the Problem

Identify and write down given data: reflection coefficient at the load \(\Gamma_L = 0.5\), characteristic impedance \(Z_0 = 50 \Omega\), and electrical length of the line \(\theta = 45^\circ\).
02

Reflection Coefficient at the Input

The reflection coefficient at the input of the line \(\Gamma_{in}\) is related to the reflection coefficient at the load \(\Gamma_L\) and the electrical length of the line \(\theta\): \[\Gamma_{in} = \Gamma_L e^{-j2\theta}\] Apply the given values: \[\Gamma_{in} = 0.5 e^{-j2 \times 45^\circ} = 0.5 e^{-j90^\circ}\] Simplify using Euler's formula: \[0.5 e^{-j90^\circ} = 0.5 \times (-j) = -0.5j\] Therefore, \ \Gamma_{in} = -0.5j \.
03

Calculating the VSWR

The Voltage Standing Wave Ratio (VSWR) is calculated using the magnitude of the reflection coefficient: \[VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}\] Substituting \ |\Gamma| = 0.5 \, we get: \[VSWR = \frac{1 + 0.5}{1 - 0.5} = \frac{1.5}{0.5} = 3\] Therefore, the VSWR on the \ 50 \Omega \ line is \ 3 \.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection Coefficient
The reflection coefficient, often denoted as \(\Gamma\), is a measure of how much of an electromagnetic wave is reflected back at an impedance discontinuity in a transmission line. It is calculated as the ratio of the reflected wave amplitude to the incident wave amplitude when a wave encounters a load different from the characteristic impedance of the line.

Mathematically, it is expressed as: \[ \Gamma = \frac{Z_L - Z_0}{Z_L + Z_0} \] where \(Z_L\) is the load impedance and \(Z_0\) is the characteristic impedance of the line.

In the given problem, the reflection coefficient at the load is \(\Gamma_L = 0.5\). This tells us that 50% of the wave's amplitude is reflected back towards the source.
Characteristic Impedance
Characteristic impedance \(Z_0\) is a fundamental property of a transmission line, representing the impedance that the line would have if it were infinite in length. It is imperative for understanding wave propagation in the line.

For a transmission line, the characteristic impedance can be found using the relation: \[ Z_0 = \sqrt{\frac{R + j\omega L}{G + j \omega C }} \] where:
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Most popular questions from this chapter

A lossless transmission line is driven by a 1 GHz generator having a Thevenin equivalent impedance of \(50 \Omega\). The transmission line is lossless, has a characteristic impedance of \(75 \Omega,\) and is infinitely long. The maximum power that can be delivered to a load attached to the generator is \(2 \mathrm{~W}\). (a) What is the total (phasor) voltage at the input to the transmission line? (b) What is the magnitude of the forwardtraveling voltage wave at the generator side of the line? (c) What is the magnitude of the forwardtraveling current wave at the generator side of the line?

A load consists of a shunt connection of a capacitor of \(10 \mathrm{pF}\) and a resistor of \(25 \Omega\). The load terminates a lossless \(50 \Omega\) transmission line. The operating frequency is \(1 \mathrm{GHz}\). [Parallels Example 3.6\(]\) (a) What is the impedance of the load? (b) What is the normalized impedance of the load (normalized to the characteristic impedance of the line)? (c) What is the reflection coefficient of the load? (d) What is the current reflection coefficient of the load? (e) What is the standing wave ratio (SWR)? (f) What is the current standing wave ratio (ISWR)?

A transmission line has the per-unit length parameters \(L=85 \mathrm{nH} / \mathrm{m}, G=1 \mathrm{~S} / \mathrm{m},\) and \(C=150 \mathrm{pF} / \mathrm{m}\). Use a frequency of \(1 \mathrm{GHz}\). [Parallels Example 3.3\(]\) (a) What is the phase velocity if \(R=0 \Omega / \mathrm{m}\) ? (b) What is the group velocity if \(R=0 \Omega / \mathrm{m}\) ? (c) If \(R=10 \mathrm{k} \Omega / \mathrm{m}\) what is the phase velocity? (d) If \(R=10 \mathrm{k} \Omega / \mathrm{m}\) what is the group velocity?

A line has a characteristic impedance \(Z_{0}\) and is terminated in a load with a reflection coefficient of 0.8 . A forward-traveling voltage wave on the line has a power of \(1 \mathrm{~W}\). (a) How much power is reflected by the load? (b) What is the power delivered to the load?

What is the electrical length of a line that is a quarter of a wavelength long, (a) in degrees? (b) in radians?
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