Find \(R, L, C\), and \(G\) for a two-wire transmission line in polyethylene at \(f=800 \mathrm{MHz}\). Assume copper conductors of radius \(0.50 \mathrm{~mm}\) and separation \(0.80 \mathrm{~cm}\). Use \(\epsilon_{r}=2.26\) and \(\sigma /\left(\omega \epsilon^{\prime}\right)=4.0 \times 10^{-4}\).

Short Answer

Expert verified
Question: Calculate the transmission line parameters (Resistance, Inductance, Capacitance, and Conductance per unit length) for a two-wire transmission line, given the following information: - Frequency: \(800\) MHz - Radius of copper conductors: \(0.50\) mm - Separation between conductors: \(0.80\) cm - Relative permittivity of polyethylene: \(2.26\) - Loss tangent value: \(\frac{\sigma}{\omega \epsilon'} = 4.0 \times 10^{-4}\) Answer: The transmission line parameters are as follows: 1. Resistance per unit length (R): \(0.00683 \, \Omega/m\) 2. Inductance per unit length (L): \(3.65 \times 10^{-7} \, H/m\) 3. Capacitance per unit length (C): \(2.37\times 10^{-11} \, F/m\) 4. Conductance per unit length (G): \(4.63\times 10^{-12} \, S/m\)

Step by step solution

01

Calculate the Resistance per unit length (R)

First, we need to find the conductivity of copper, which is \(\sigma_c = 5.8 \times 10^7 S/m\). Now, we'll use the given radius (\(r = 0.50 \times 10^{-3}\) m) to calculate the resistance per unit length using the above formula: $$ R = \frac{1}{\pi (0.50\times10^{-3})^2 (5.8 \times 10^7)} = 0.00683 \, \Omega/m $$
02

Calculate the Inductance per unit length (L)

To find the inductance per unit length, we'll use the permeability of free space, \(\mu_0 = 4\pi \times 10^{-7} \, H/m\). We plug in the conductor separation \(d = 0.80 \times 10^{-2}\) m and conductor radius \(r = 0.50 \times 10^{-3}\) m in the formula: $$ L = \frac{4\pi \times 10^{-7}}{\pi} \ln\frac{0.80 \times 10^{-2}}{0.50 \times 10^{-3}} = 3.65 \times 10^{-7}\,H/m $$
03

Calculate the Capacitance per unit length (C)

For the capacitance per unit length, we'll use the permittivity of free space, \(\epsilon_0 = 8.854 \times 10^{-12} \, F/m\), as well as the given relative permittivity for polyethylene, \(\epsilon_r = 2.26\). Plug these values, along with \(d\) and \(r\), into the formula: $$ C = \frac{\pi (8.854\times10^{-12})(2.26)}{\ln\frac{0.80 \times 10^{-2}}{0.50 \times 10^{-3}}} = 2.37\times 10^{-11} \, F/m $$
04

Calculate the Conductance per unit length (G)

Finally, let's find the conductance per unit length. We need the angular frequency \(\omega\), which is \(2\pi f\), with the given frequency \(f = 800\times10^6\) Hz: $$ \omega = 2\pi (800\times10^6) = 5.027\times10^9 \, rad/s $$ Now, using the given formula and values: $$ G = \frac{\pi(5.027\times10^9)(8.854\times10^{-12})(2.26)(4.0 \times 10^{-4})}{\ln\frac{0.80 \times 10^{-2}}{0.50 \times 10^{-3}}} = 4.63\times 10^{-12}\,S/m $$ To summarize our findings: 1. Resistance per unit length (R): \(0.00683 \, \Omega/m\) 2. Inductance per unit length (L): \(3.65 \times 10^{-7} \, H/m\) 3. Capacitance per unit length (C): \(2.37\times 10^{-11} \, F/m\) 4. Conductance per unit length (G): \(4.63\times 10^{-12} \, S/m\)

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Most popular questions from this chapter

In a symmetric slab waveguide, \(n_{1}=1.50, n_{2}=1.45\), and \(d=10 \mu \mathrm{m}\). (a) What is the phase velocity of the \(m=1 \mathrm{TE}\) or \(\mathrm{TM}\) mode at cutoff? (b) How will your part ( \(a\) ) result change for higher-order modes (if at all)?

The cutoff frequency of the \(m=1 \mathrm{TE}\) and TM modes in an air-filled parallel-plate guide is known to be \(f_{c 1}=7.5 \mathrm{GHz}\). The guide is used at wavelength \(\lambda=1.5 \mathrm{~cm}\). Find the group velocity of the \(m=2 \mathrm{TE}\) and \(\mathrm{TM}\) modes.

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