Chapter 13: Problem 1
The conductors of a coaxial transmission line are copper \(\left(\sigma_{c}=5.8 \times\right.\) \(\left.10^{7} \mathrm{~S} / \mathrm{m}\right)\), and the dielectric is polyethylene \(\left(\epsilon_{r}^{\prime}=2.26, \sigma / \omega \epsilon^{\prime}=0.0002\right) .\) If the inner radius of the outer conductor is \(4 \mathrm{~mm}\), find the radius of the inner conductor so that \((a) Z_{0}=50 \Omega ;(b) C=100 \mathrm{pF} / \mathrm{m} ;(c) L=0.2 \mu \mathrm{H} / \mathrm{m}\). A lossless line can be assumed.
Short Answer
Step by step solution
Characteristic impedance equation
Calculating inner conductor radius for given Z0
Capacitance and inductance equations
Calculating inner conductor radius for given C and L values
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Characteristic Impedance
In the realm of coaxial cables, the characteristic impedance can be calculated using the formula: \[Z_0 = \frac{1}{2\pi}\sqrt{\frac{\mu}{\epsilon^{\prime}}{\sigma} }\ln{\frac{b}{a}}\] where,
- \(\mu\) is the magnetic permeability of the medium, which is typically that of free space, \(\mu_0\), in a lossless line.
- \(\epsilon^{\prime}\) is the relative permittivity of the dielectric material.
- \(\sigma\) is the conductivity parameter, and \(b\) and \(a\) represent the radii of the outer and inner conductors respectively.
Capacitance per Unit Length
The capacitance per unit length for a coaxial cable is calculated using the formula: \[C = 2\pi\epsilon_r^{\prime}\epsilon_0\frac{1}{\ln{\frac{b}{a}}}\] where
- \(\epsilon_r^{\prime}\) is the relative permittivity of the dielectric material between the conductors,
- \(\epsilon_0\) is the vacuum permittivity, and
- \(a\) and \(b\) are the radii of the inner and outer conductors, respectively.
Inductance per Unit Length
The formula to determine the inductance per unit length in a coaxial cable is: \[L = \frac{\mu_0}{2\pi}\ln{\frac{b}{a}}\] Where,
- \(\mu_0\) is the permeability of free space,
- \(a\) is the radius of the inner conductor, and
- \(b\) is the inner radius of the outer conductor.
Dielectric Material Properties
Key properties of dielectric materials include:
- Relative Permittivity (\(\epsilon_r^{\prime}\)): This measure indicates how much the dielectric material can store electrical energy compared to a vacuum. A higher \(\epsilon_r^{\prime}\) means more energy storage but can also lead to slower signal propagation.
- Conductivity (\(\sigma\)): For ideal dielectrics, this should be extremely low. However, all dielectrics have slight conductivity that affects signal quality over long distances.
- Loss Tangent (\(\tan(\delta)\)): Reflects how much energy from the signal is lost as heat within the dielectric. A lower loss tangent indicates better performance for high-frequency signals.