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Chapter 10: Problem 23

The normalized load on a lossless transmission line is \(2+j 1\). Let \(\lambda=20 \mathrm{~m}\) and make use of the Smith chart to find \((a)\) the shortest distance from the load to a point at which \(z_{\text {in }}=r_{\text {in }}+j 0\), where \(r_{\text {in }}>0 ;\) (b) \(z_{\text {in }}\) at this point. (c) The line is cut at this point and the portion containing \(z_{L}\) is thrown away. A resistor \(r=r_{\text {in }}\) of part \((a)\) is connected across the line. What is \(s\) on the remainder of the line? \((d)\) What is the shortest distance from this resistor to a point at which \(z_{\text {in }}=2+j 1 ?\)

Short Answer

Expert verified
The purpose of plotting the given load impedance on the Smith chart is to locate the point representing the load impedance, which is essential for finding the shortest distance to a purely resistive input impedance. 2. How do you find the shortest distance to a purely resistive input impedance on the Smith chart? To find the shortest distance to a purely resistive input impedance on the Smith chart, follow the constant resistance circle counterclockwise from the load point until the imaginary part becomes zero. The distance traveled along the circle gives the shortest distance in terms of electrical degrees. 3. Once you have the shortest distance in electrical degrees, how do you calculate the physical distance using the given wavelength? To calculate the physical distance using the given wavelength, multiply the rotation angle (in degrees) by the wavelength and divide the result by 360. 4. After cutting the line at the resistive impedance point, how do you calculate the reflection coefficient on the remaining part of the line? To calculate the reflection coefficient on the remaining part of the line, use the formula: \(s = \frac{r_{in} - 1}{r_{in} + 1}\), where \(r_{in}\) is the real input impedance. 5. How do you find the shortest distance from the resistor to the given impedance value on the Smith chart? To find the shortest distance from the resistor to the given impedance value on the Smith chart, start from the point corresponding to the connected resistor with input impedance \(r_{in}\) and follow the constant resistance circle counterclockwise until the imaginary part becomes 1. Calculate the physical distance using the wavelength as done previously.

Step by step solution

01

Plot the given load impedance on the Smith chart

To begin, plot the given load impedance \(Z_L = 2+j1\) on the Smith chart. Locate the point corresponding to a real part of 2 and an imaginary part of 1. This point represents the load impedance.
02

Find the shortest distance to a purely resistive input impedance

Now, we need to find the shortest distance from the load point (found in Step 1) to a point on the Smith chart where the input impedance is purely resistive and positive. Follow the constant resistance circle (the circle centered on the Smith chart) counterclockwise from the load point, until the imaginary part becomes zero. The distance traveled along the circle will give us the shortest distance in terms of electrical degrees.
03

Calculate the physical distance using the wavelength

We are given a wavelength of \(\lambda = 20\) meters. We can compute the physical distance using the relationship between electrical degrees and the wavelength. Let \(\theta\) be the rotation angle in degrees found in Step 2. Then, the physical distance \(d\) can be calculated as $$ d = \frac{\lambda \cdot \theta}{360} $$
04

Find the input impedance at this point

To find the input impedance at this point, simply read off the value of the real part of the impedance (denoted as \(r_{in}\)) at the point where the imaginary part is zero (found in Step 2).
05

Calculate the reflection coefficient after cutting the line

After cutting the line at the resistive impedance point, the remainder of the line will have a real input impedance \(r_{in}\). Since the line is lossless, the characteristic impedance remains the same. Use the formula for the reflection coefficient, \(s\), as $$ s = \frac{r_{in} - 1}{r_{in} + 1} $$
06

Find the shortest distance to the given impedance value

Now, we need to find the shortest distance from the resistor to a point at which the input impedance is equal to the given value of \(Z_L = 2+j1\). Starting from the point corresponding to the connected resistor with input impedance \(r_{in}\), follow the Smith chart's constant resistance circle counterclockwise until the imaginary part becomes 1. The distance traveled along the circle will give us the shortest distance in terms of electrical degrees. Finally, calculate the physical distance using the wavelength as we did in Step 3.

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Key Concepts

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

Transmission Line Impedance
In electrical engineering, transmission line impedance is a fundamental concept that characterizes how alternating current (AC) signals propagate through a transmission line. It is essential for designing systems that transmit signals over distances with minimal loss and distortion. The characteristic impedance of a line, often represented as 'Z0', is determined by the line's physical construction, such as the conductors' size and spacing and the properties of the insulating material between them. Impedance matching involves ensuring the load impedance (the impedance at the end of the transmission line where power is consumed) matches the characteristic impedance to maximize power transfer and reduce reflections. In the given exercise, impedance matching is utilized to adjust a load impedance of \(2+j 1\) to match the line's characteristic impedance to minimize reflection and power loss.

Understanding the concept of transmission line impedance helps you comprehend why mismatches cause reflections, which lead to standing wave patterns on the line, and how these standing waves can affect signal integrity and power transfer. For instance, if the load impedance differs from the line's characteristic impedance, part of the signal will reflect back towards the source, resulting in an undesirable effect known as standing waves.
Reflection Coefficient Calculation
The reflection coefficient, commonly denoted by 's' or '\(\Gamma\)', quantifies the magnitude and phase of a reflected wave on a transmission line compared to the incident wave. It's a dimensionless value that ranges from -1 to +1, where '0' indicates no reflection (perfect matching), and '-1' or '+1' indicates total reflection. To calculate the reflection coefficient for a point at which the transmission line is terminated in an impedance \(Z_L\), you can use the formula:\[s = \frac{Z_L - Z_0}{Z_L + Z_0}\]where \(Z_0\) is the characteristic impedance of the transmission line.

In the given problem, the transmission line is cut at the point of purely resistive input impedance, and a resistor equal to this resistive input impedance is connected, effectively becoming the new load impedance. Hence, the reflection coefficient can be calculated as the ratio of the difference between this new load impedance (\(r_{in}\)) and the characteristic impedance (assumed to be normalized here to 1) to the sum of the load impedance and the characteristic impedance.
Impedance Transformation
Impedance transformation occurs when the electrical impedance of a load, viewed through a length of transmission line, appears to change due to the reactive elements introduced by the transmission line's inductance and capacitance. This can often be visualized and analyzed using the Smith chart, a graph which allows complex impedance transformations to be represented graphically through movements on the chart.

The Smith chart provides a powerful visual tool to perform various calculations and to understand how the impedance at the load (the far end of the transmission line) transforms as you move toward the generator or source. When the line is cut and terminated in a resistor as in the exercise, this resistor's value affects the entire system's input impedance, and this is what we call impedance transformation — the line's remaining section now presents a different impedance to the source.
Standing Wave Ratio (SWR)
The Standing Wave Ratio (SWR) is a measure that indicates how well a transmission line is impedance-matched. SWR is defined as the ratio of the amplitude of the voltage maximum (antinode) to the amplitude of the voltage minimum (node) along the line. It's expressed as:\[SWR = \frac{1 + |s|}{1 - |s|}\]where \(|s|\) is the magnitude of the reflection coefficient.

An SWR of 1:1 means that there is no reflected wave and thus perfect impedance matching, resulting in the most efficient power transmission. On the other hand, a higher SWR indicates greater mismatch and, therefore, less efficient power transmission, with more energy reflected back towards the source. In the exercise, determining the SWR after the transmission line is cut and a resistor is connected gives insight into how well the remaining system is matched.

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Most popular questions from this chapter

A \(100-\Omega\) lossless transmission line is connected to a second line of \(40-\Omega\) impedance, whose length is \(\lambda / 4\). The other end of the short line is terminated by a \(25-\Omega\) resistor. A sinusoidal wave (of frequency \(f\) ) having \(50 \mathrm{~W}\) average power is incident from the \(100-\Omega\) line. \((a)\) Evaluate the input impedance to the quarter-wave line. (b) Determine the steady-state power that is dissipated by the resistor. \((c)\) Now suppose that the operating frequency is lowered to one-half its original value. Determine the new input impedance, \(Z_{i n}^{\prime}\), for this case. \((d)\) For the new frequency, calculate the power in watts that returns to the input end of the line after reflection.

In Figure \(10.39, R_{L}=Z_{0}\) and \(R_{g}=Z_{0} / 3\). The switch is closed at \(t=0\). Determine and plot as functions of time \((a)\) the voltage across \(R_{L} ;(b)\) the voltage across \(R_{g} ;(c)\) the current through the battery.

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.

A transmission line having primary constants \(L, C, R\), and \(G\) has length \(\ell\) and is terminated by a load having complex impedance \(R_{L}+j X_{L}\). At the input end of the line, a dc voltage source, \(V_{0}\), is connected. Assuming all parameters are known at zero frequency, find the steady-state power dissipated by the load if \((a) R=G=0 ;(b) R \neq 0, G=0 ;(c) R=0\), \(G \neq 0 ;(d) R \neq 0, G \neq 0 .\)

A sinusoidal voltage source drives the series combination of an impedance, \(Z_{g}=50-j 50 \Omega\), and a lossless transmission line of length \(L\), shorted at the load end. The line characteristic impedance is \(50 \Omega\), and wavelength \(\lambda\) is measured on the line. \((a)\) Determine, in terms of wavelength, the shortest line length that will result in the voltage source driving a total impedance of \(50 \Omega .(b)\) Will other line lengths meet the requirements of part \((a)\) ? If so, what are they?
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