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Chapter 10: Problem 22

A lossless \(75-\Omega\) line is terminated by an unknown load impedance. VSWR of 10 is measured, and the first voltage minimum occurs at \(0.15\) wavelengths in front of the load. Using the Smith chart, find \((a)\) the load impedance; \((b)\) the magnitude and phase of the reflection coefficient; \((c)\) the shortest length of line necessary to achieve an entirely resistive input impedance.

Short Answer

Expert verified
Answer: The load impedance is \(138.75 + j127.5\Omega\), the magnitude of the reflection coefficient is 0.818, the phase of the reflection coefficient is \(-108^\circ\), and the shortest length of line necessary to achieve a resistive input impedance is \(0.45\lambda\).

Step by step solution

01

a. Find the normalized load impedance

Using the given VSWR value of 10, we can find the reflection coefficient (Γ) as: \(\Gamma = \frac{VSWR-1}{VSWR+1} = \frac{10-1}{10+1} = 0.818\) Now, we can use the first voltage minimum distance (0.15 wavelengths) to determine the angle of the reflection coefficient: \(\angle{\Gamma} = - 2\pi(0.15) = - 0.3\pi \, \text{radians}\) or \(-108^\circ\) Next, plot the point corresponding to the angle on the constant VSWR circle with a radius of 0.818 on the Smith chart. By interpolation and reading the chart, we find the normalized load impedance: \(Z_L' = 1.85 + j1.7\) Then, to find the actual load impedance, multiply the normalized load impedance by the characteristic impedance: \(Z_L = Z_0 * Z_L' = 75\Omega * (1.85 + j1.7) = 138.75 + j127.5\Omega\)
02

b. Calculate the magnitude and phase of the reflection coefficient

We already found the magnitude and phase of the reflection coefficient in step (a): Magnitude of reflection coefficient: \(\Gamma = 0.818\) Angle of reflection coefficient: \(\angle{\Gamma} = -108^\circ\)
03

c. Determine the shortest length of line necessary to achieve a resistive input impedance

To find the length of the transmission line needed to achieve a resistive input impedance, we need to extend the line until it reaches the real axis of the Smith chart. From step (a), the normalized load impedance is: \(Z_L' = 1.85 + j1.7\) Drag the point representing the load impedance inwards along the line connecting to the center of the Smith chart. Read the angle where the line crosses the real axis. This angle represents the amount of transmission line necessary to be added or subtracted to the existing \(\frac{0.15\lambda}{2}\) section to cancel the reactance and make the input impedance purely resistive: \(\angle{\text{TL}} = 0.1\lambda\) Thus, the added length necessary to achieve a resistive input impedance is: \(\alpha = 0.1\lambda - 0.15\lambda = -0.05\lambda\) If we want an entirely resistive input impedance, we can add or subtract multiples of \(\frac{\lambda}{2}\): \(\alpha' = -0.05\lambda \pm n\frac{\lambda}{2} = 0.45\lambda(n=1), 0.95\lambda(n=2), ...\) The shortest length of line necessary to achieve an entirely resistive input impedance is \(0.45\lambda\).

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Key Concepts

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

Voltage Standing Wave Ratio (VSWR)
The Voltage Standing Wave Ratio (VSWR) is a measure that indicates the efficiency of the power transfer from a transmission line to a load. It's the ratio of the maximum voltage on the line to the minimum voltage. A VSWR of 1:1 represents a perfect match, where all power is transmitted to the load with no reflections. However, as the VSWR number increases, it indicates a greater mismatch and, consequently, more power is reflected back towards the source. High VSWR values can lead to inefficient power transfer and potential damage to the system. Understanding VSWR is critical for RF engineers to optimize transmission lines for maximum power delivery.

When a VSWR of 10 is measured, it implies that the mismatch between the transmission line and the load is considerable, thus leading to a reflection coefficient with a large magnitude. Additional equipment, such as impedance matching networks, may be used to reduce the VSWR, thereby improving the efficiency of the system.
Reflection Coefficient
The reflection coefficient, typically denoted as \(\Gamma\), is a complex number that quantifies how much of an electromagnetic wave is reflected by an impedance discontinuity in the transmission line. The magnitude of the reflection coefficient varies between 0 and 1, where 0 indicates no reflection (perfect match) and 1 represents total reflection (open or short circuit). The angle of the reflection coefficient provides information about the phase shift that the reflected wave undergoes.

In context to the provided exercise, the reflection coefficient's magnitude (0.818) and phase (-108°) are calculated based on the given VSWR and the position of the first voltage minimum on the transmission line. The magnitude tells us the degree of mismatch, and the phase helps to locate the position of the load impedance on the Smith chart.
Load Impedance
Load impedance refers to the complex impedance presented by the load at the end of a transmission line, which determines how the load will interact with the incident signal. It comprises a resistive component and a reactive component (capacitive or inductive). The reactive component can cause reflections and standing waves along the transmission line. Load impedance is critical to consider in designing RF systems since mismatches can result in power reflections and inefficient signal transmissions.

Using the Smith chart, engineers can visually determine the load impedance and make necessary adjustments. In the exercise, the load impedance calculated, 138.75 + j127.5 Ω, shows that the load has a significant reactive part, which contributes to the high VSWR value mentioned. By matching this impedance to the characteristic impedance of the transmission line, a better power transfer can be achieved.
Resistive Input Impedance
A resistive input impedance is an impedance at the input of a transmission line that is purely resistive, with no reactive (capacitive or inductive) components. Achieving a purely resistive input impedance is critical in RF and microwave circuits to maximize power transfer and minimize reflections. Reactive components can store energy and cause fluctuations in power delivery, while a resistive load consumes power continuously.

In the exercise, the goal is to manipulate the length of the transmission line such that the reactance is canceled. The Smith chart facilitates this process by showing a graphical path to move from the complex impedance to the real axis, indicating a purely resistive impedance. By adding or subtracting appropriate lengths of line, one can achieve the resistive input impedance. The calculation shows that by adding a specific length to the current line, we can adjust the input impedance to be resistive, optimizing the system's performance.

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Most popular questions from this chapter

A lossless transmission line having characteristic impedance \(Z_{0}=50 \Omega\) is driven by a source at the input end that consists of the series combination of a 10 -V sinusoidal generator and a \(50-\Omega\) resistor. The line is one- quarter wavelength long. At the other end of the line, a load impedance, \(Z_{L}=50-j 50 \Omega\) is attached. (a) Evaluate the input impedance to the line seen by the voltage source-resistor combination; \((b)\) evaluate the power that is dissipated by the load; \((c)\) evaluate the voltage amplitude that appears across the load.

A standing wave ratio of \(2.5\) exists on a lossless \(60 \Omega\) line. Probe measurements locate a voltage minimum on the line whose location is marked by a small scratch on the line. When the load is replaced by a short circuit, the minima are \(25 \mathrm{~cm}\) apart, and one minimum is located at a point \(7 \mathrm{~cm}\) toward the source from the scratch. Find \(Z_{L}\).

The characteristic admittance \(\left(Y_{0}=1 / Z_{0}\right)\) of a lossless transmission line is \(20 \mathrm{mS}\). The line is terminated in a load \(Y_{L}=40-j 20 \mathrm{mS}\). Use the Smith chart to find \((a) s ;(b) Y_{\text {in }}\) if \(l=0.15 \lambda ;(c)\) the distance in wavelengths from \(Y_{I}\) to the nearest voltage maximum.

A sinusoidal voltage wave of amplitude \(V_{0}\), frequency \(\omega\), and phase constant \(\beta\) propagates in the forward \(z\) direction toward the open load end in a lossless transmission line of characteristic impedance \(Z_{0}\). At the end, the wave totally reflects with zero phase shift, and the reflected wave now interferes with the incident wave to yield a standing wave pattern over the line length (as per Example 10.1). Determine the standing wave pattern for the current in the line. Express the result in real instantaneous form and simplify.

In order to compare the relative sharpness of the maxima and minima of a standing wave, assume a \(\operatorname{load} z_{L}=4+j 0\) is located at \(z=0 .\) Let \(|V|_{\min }=1\) and \(\lambda=1 \mathrm{~m}\). Determine the width of the \((a)\) minimum where \(|V|<1.1 ;(b)\) maximum where \(|V|>4 / 1.1\).
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