Question

Two characteristics of a certain lossless transmission line are \(Z_{0}=50 \Omega\) and \(\gamma=0+j 0.2 \pi \mathrm{m}^{-1}\) at \(f=60 \mathrm{MHz}(a)\) find \(L\) and \(C\) for the line. \((b) \mathrm{A}\) load \(Z_{L}=60+j 80 \Omega\) is located at \(z=0 .\) What is the shortest distance from the load to a point at which \(Z_{\text {in }}=R_{\text {in }}+j 0 ?\)

Short answer

Question: Calculate the inductance (L) and capacitance (C) of a lossless transmission line given its characteristic impedance (Z0) and propagation constant (γ) at a specific frequency (f). Then, determine the shortest distance from the load to a point where the input impedance (Zin) becomes purely resistive (Rin). Answer: The inductance (L) of the transmission line is approximately \(77.35 \times 10^{-9} H\) and the capacitance (C) is approximately \(46.41 \times 10^{-12} F\). The shortest distance from the load to a point where the input impedance is purely resistive is approximately 2.31 meters.

Step-by-step solution

Relation between L, C, Z0, and γ

For a lossless transmission line, we can establish the following relations: - Characteristic impedance (Z0) = \(Z_{0}=\sqrt{\frac{L}{C}}\) - Propagation constant (γ) = \(0 + j\beta\), where \(β = 2πf\sqrt{LC}\)

Calculate L and C

We're given \(Z_{0} = 50\Omega\) and \(\gamma = 0 + j0.2\pi m^{-1}\). Now let's find L and C using the provided information and equations: From the propagation constant, we can write: \(β = 0.2\pi m^{-1}\) As mentioned before, \(β = 2\pi f\sqrt{L C}\) Plugging in the given values (f = 60 MHz) into the equation, we can find LC: \((0.2\pi) = 2\pi(60\times10^6)\sqrt{L C}\) \(LC = \frac{1}{(60\times10^6)^2 4\pi^2}\) Now, using the relation between Z0 and L, C: \(50 = \sqrt{\frac{L}{C}}\) Squaring both sides: \(50^2 = \frac{L}{C}\) Now we have a system of two equations with two unknowns (L and C). Solving this system, we get: \(L = 77.35 \times 10^{-9} H\) \(C = 46.41 \times 10^{-12} F\) Part (b):

Input impedance equation

The input impedance for a transmission line can be written as: \(Z_{in} = Z_{0} \frac{Z_L + jZ_{0}\tan(\beta l)}{Z_{0}+jZ_L\tan(\beta l)}\) In our case, \(Z_{0} = 50\Omega\) and \(Z_{L} = 60 + j80\Omega\)

Find the shortest distance for purely resistive Zin

We want to find the shortest distance (l) at which \(Z_{in} = R_{in} + j0\). In other words, the imaginary part of Zin should be zero. For this condition, the imaginary part of the denominator in Zin should be equal to the imaginary part of the numerator. So, \(Z_0\tan(\beta l) = 80\) Previously we found \(β = 0.2\pi m^{-1}\). Plugging this into the equation and solving for l, we get: \(\tan(0.2\pi l) = \frac{80}{50}\) \(l = \frac{1}{0.2\pi} \tan^{-1}\left(\frac{80}{50}\right)\) \(l \approx 2.31m\) The shortest distance from the load to a point where the input impedance is purely resistive is approximately 2.31 meters.

Key concepts

Characteristic Impedance

The characteristic impedance, commonly denoted as \( Z_0 \), is a fundamental concept in the study of transmission lines. It represents the impedance that a transmission line would have if it were infinitely long. This value is significant because it determines how signals will behave as they travel along the line. For a lossless transmission line, the characteristic impedance has a real value, which means it's purely resistive. The formula used to compute \( Z_0 \) is:\[ Z_0 = \sqrt{\frac{L}{C}} \]
where \( L \) is the inductance per unit length and \( C \) is the capacitance per unit length. The units of \( Z_0 \) are ohms (\( \Omega \)), and it tells us how the line will match with other components, like generators and loads. A mismatch can lead to reflections, impacting the efficiency of signal transmission.

Propagation Constant

The propagation constant, denoted as \( \gamma \), is used to describe how an electromagnetic wave propagates through a transmission line. For a lossless line, this constant is purely imaginary, expressed as \( \gamma = 0 + j\beta \), where \( \beta \) is the phase constant. The phase constant \( \beta \) relates to how the phase of the wave changes per unit length along the line, and it is calculated using the formula:

• \( \beta = 2\pi f \sqrt{LC} \)

where \( f \) is the frequency of the wave, \( L \) is the inductance per unit length, and \( C \) is the capacitance per unit length.
In this exercise, \( \beta \) was given as \( 0.2\pi \) m^-1, illustrating the phase shift a wave undergoes over one meter of the line. This concept helps in understanding the speed and manner in which information travels down the transmission line.

Input Impedance

Input impedance \( Z_{in} \) refers to the impedance that is "seen" from the input end of the transmission line. It depends on the line's characteristic impedance, the load impedance, and the electrical length of the line. The general equation for input impedance of a transmission line is:

• \( Z_{in} = Z_{0} \frac{Z_L + jZ_{0}\tan(\beta l)}{Z_{0}+jZ_L\tan(\beta l)} \)

The term \( \beta l \) represents the electrical length of the transmission line segment from the input point to the load. At a specific frequency, the value of \( Z_{in} \) may vary considerably based on the load \( Z_L \) attached at the other end and the distance \( l \).
Understanding \( Z_{in} \) is crucial for designing systems that minimize power loss and prevent signal reflections, ensuring that the power transferred between the source and the load is maximized.

Purely Resistive Impedance

A purely resistive impedance indicates that the impedance has no reactive (inductive or capacitive) part and consists entirely of resistance. In an ideal lossless transmission line, one can achieve a purely resistive impedance at a specific distance, known as the electrical quarter-wavelength, from the load. This is calculated when the imaginary component of the input impedance is nullified, leaving only a real, resistive component.
To achieve this condition, adjustments in the length of the transmission line are made, ensuring:

• The imaginary part of \( Z_{in} \) equals zero, simplifying the impedance to \( R_{in} + j0 \).

In practical terms, this means you reach a point where the line impedance is perfectly DC (resistive) without reactive elements, which is ideal for maximizing power transfer without phase shifts that could degrade signal quality.