To calculate the characteristic impedance ($Z_0$) and the propagation velocity ($v$) of the coaxial transmission line, we use the following formulas:

1. Characteristic Impedance Formula for a Coaxial Cable:

$Z_0 = \frac{60}{\sqrt{\epsilon_r}} \ln \left( \frac{r_{\text{outer}}}{r_{\text{inner}}} \right)$ Where:

• $\epsilon_r$ is the relative permittivity of the dielectric.
• $r_{\text{outer}}$ is the radius of the outer conductor.
• $r_{\text{inner}}$ is the radius of the inner conductor.

2. Propagation Velocity Formula:

$v = \frac{c}{\sqrt{\epsilon_r}}$ Where:

• $c = 3 \times 10^8 \, \text{m/s}$ is the speed of light in vacuum.
• $\epsilon_r$ is the relative permittivity of the dielectric.

Given values:

• $r_{\text{inner}} = 5 \, \text{mm} = 0.005 \, \text{m}$
• $r_{\text{outer}} = 1 \, \text{cm} = 0.01 \, \text{m}$
• $\epsilon_r = 2.25$

Now let's calculate the characteristic impedance and the propagation velocity.The characteristic impedance of the coaxial transmission line is approximately $27.73 \, \Omega$, and the propagation velocity is $2 \times 10^8 \, \text{m/s}$.

Would you like further details or an explanation of the calculations? Here are five related questions to expand your understanding:

1. How would the characteristic impedance change if the inner conductor radius was larger?
2. What is the effect of using a dielectric with a higher permittivity on the propagation velocity?
3. How does the characteristic impedance change with the thickness of the dielectric?
4. What would be the propagation velocity in vacuum (without dielectric)?
5. How can we minimize signal loss in coaxial transmission lines?

Tip: The characteristic impedance depends on the ratio of the radii of the inner and outer conductors and the permittivity of the dielectric material between them.