Question

Pertinent dimensions for the transmission line shown in Figure \(13.2\) are \(b=\) \(3 \mathrm{~mm}\) and \(d=0.2 \mathrm{~mm}\). The conductors and the dielectric are nonmagnetic. (a) If the characteristic impedance of the line is \(15 \Omega\), find \(\epsilon_{r}^{\prime}\). Assume a low-loss dielectric. ( \(b\) ) Assume copper conductors and operation at \(2 \times 10^{8}\) \(\mathrm{rad} / \mathrm{s}\). If \(R C=G L\), determine the loss tangent of the dielectric.

Short answer

Question: Determine the relative permittivity and loss tangent of the dielectric for a given transmission line. Answer: The relative permittivity of the dielectric is approximately 2.07, and the loss tangent is approximately 0.00231.

Step-by-step solution

Determine the relative permittivity

Since we are assuming a low-loss dielectric, we can use the following expression for the characteristic impedance of the transmission line: \[Z_0 = \frac{120\pi}{\sqrt{\epsilon_r'}}\frac{d}{b}\] We know that the characteristic impedance of the line is \(15 \Omega\). We can write down the equation and solve for \(\epsilon_r'\) as follows: \[15 = \frac{120\pi}{\sqrt{\epsilon_r'}}\frac{0.2}{3}\]

Calculate the relative permittivity \(\epsilon_r'\)

Square both sides of the equation and solve for \(\epsilon_r'\): \[(15)^2 = \frac{(120\pi)^2}{\epsilon_r'}\frac{0.2^2}{3^2}\] Now, solve for \(\epsilon_r'\): \[\epsilon_r' = \frac{(120\pi)^2}{(15)^2 \cdot \frac{0.2^2}{3^2}} \approx 2.07\] So, the relative permittivity of the dielectric is approximately \(2.07\).

Use the loss tangent equation

We will now find the loss tangent of the dielectric. We are given the condition that \(RC = GL\). The resistance, R, and conductance, G, of the conductors of a line are related to its length, and the dielectric properties are related to its capacitance, C, and inductance, L: \[R = \frac{R_s}{b}\] \[G = \frac{2 \pi f C}{\epsilon_0 \tan\delta}\] \[C = \frac{\epsilon_0 \epsilon_r' b}{d}\] \[L = \frac{\mu_0 d}{b}\] We are given that \(f = 2 \times 10^8 \mathrm{rad/s}\), which means the copper's skin depth, \(\delta\), is: \[\delta= \sqrt{\frac{2}{\mu_0 \sigma f}}\] Multiplicity with \(\epsilon_r'\) in \(RC=GL\)

Calculate the loss tangent

Now, we will substitute the values to find \(\tan\delta\). Equate both sides and solve for \(\tan\delta\): \[\frac{R_s}{b}\frac{\epsilon_0 \epsilon_r' b}{d} = \frac{2 \pi f C}{\epsilon_0 \tan\delta}\frac{\mu_0 d}{b}\] \[\tan\delta = \frac{R_s}{2 \pi f \mu_0 \epsilon_r'}\] Now, using the copper conductivity \(\sigma = 5.8 \times 10^7 \mathrm{S/m}\) and the permeability of free space \(\mu_0 = 4\pi \times 10^{-7} \mathrm{Tm/A}\), we will calculate the loss tangent: \[\tan\delta = \frac{\sqrt{\frac{\pi}{2 \mu_0 \sigma f}}}{2 \pi f \mu_0 \epsilon_r'} \approx 0.00231\] The loss tangent of the dielectric is approximately \(0.00231\).

Key concepts

Characteristic Impedance

In transmission line analysis, the characteristic impedance (often denoted by \( Z_0 \)) is a crucial parameter. It represents the impedance that a transmission line would have if it were infinitely long. This is a significant concept as it determines how voltage and current waves propagate along the line.

For calculating the characteristic impedance of a line, we use the formula:
\[ Z_0 = \frac{120\pi}{\sqrt{\epsilon_r'}}\frac{d}{b} \]
where \( \epsilon_r' \) is the relative permittivity, \( d \) is the conductor separation, and \( b \) is the width. Understanding this formula is key to solving problems where you need to find the impedance given certain physical and material properties.

• The factor \(120\pi\) relates to the intrinsic impedance of free space.
• The term \(\sqrt{\epsilon_r'}\) involves the electrical permittivity of the dielectric material.
• The ratio \(\frac{d}{b}\) adjusts for the physical dimensions of the transmission line.

Relative Permittivity

Relative permittivity, often symbolized as \( \epsilon_r' \), describes how much a material can "permeate" electric field lines compared to vacuum. It is a dimensionless quantity indicating how well a dielectric material can store electrical energy within an electric field.

This value directly impacts the characteristic impedance of a transmission line. The greater the permittivity, the more the dielectric can hold charge, thus lowering the impedance. In practical applications:

• Higher \( \epsilon_r' \) means higher capacitance within the line.
• Impacts how efficiently signals travel through the dielectric.
• Values greater than 1 indicate the material is more permissive than a vacuum.

These insights help in designing and solving exercises related to the behavior of transmission lines within certain dielectrics.

Loss Tangent

Loss tangent (\( \tan\delta \)) is a measure of dielectric losses in a material. It tells us how much energy from the electric field is lost as heat within the dielectric. For engineers, minimizing the loss tangent is crucial to ensure efficient signal transmission.

It's essential to understand:

• Low loss tangents imply that the material is efficient in transmitting signals with minimal energy loss.
• Calculated using the formula:
  \[ \tan\delta = \frac{R_s}{2 \pi f \mu_0 \epsilon_r'} \]
• Dependent on frequency (\( f \)), conductivity (\( \sigma \)), and dimensions of the material.

By solving for the loss tangent, engineers can determine suitability of dielectric materials under specific frequency conditions for optimal performance.

Low-Loss Dielectric

A low-loss dielectric is a material that introduces minimal loss when signals are conveyed through it. The term "low-loss" refers to its ability to support electric fields with minimal heat generation, thereby preserving signal strength and integrity.

The efficiency of a low-loss dielectric is characterized by:

• A very small loss tangent, which indicates low energy dissipation as heat.
• High relative permittivity, which aids effective storage of electrical energy.

These properties are crucial for applications like high-frequency transmission lines
where maximum signal preservation with minimum distortion is desired. Materials that are classified as low-loss help in ensuring that the transmitted signal remains strong over long distances.

Copper Conductors

Copper is widely used in transmission lines due to its excellent electrical conductivity. This property reduces resistive losses, making it a favored choice for conductors in electrical circuits.

Key aspects of copper as a conductor include:

• High conductivity results in low resistance, minimizing energy dissipation.
• Low skin effect, which ensures current flows mostly on the surface at high frequencies.
• Durability and ability to withstand environmental conditions.

This makes copper conductors essential in minimizing transmission losses and ensuring reliability over various operation conditions within a variety of electronic devices and systems.