Transmission Line Parameters Explained

This explanation details the calculation process for determining the wave propagation velocity and characteristic impedance of a lossless transmission line. We are given the inductance per unit length ($L$) and capacitance per unit length ($C$) for the line.

Calculating Wave Propagation Velocity

The speed at which a wave travels along a transmission line, especially a lossless one, depends on its physical properties, specifically inductance ($L$) and capacitance ($C$) per unit length. The formula used to find this velocity ($v$) is:

$$ v = \frac{1}{\sqrt{LC}} $$

The problem provides the following values:

• Inductance per unit length, $L = 250 \text{ nH/m} = 250 \times 10^{-9} \text{ H/m}$
• Capacitance per unit length, $C = 0.1 \text{ nF/m} = 0.1 \times 10^{-9} \text{ F/m}

Let's substitute these values into the velocity formula:

$$ v = \frac{1}{\sqrt{(250 \times 10^{-9} \text{ H/m}) \times (0.1 \times 10^{-9} \text{ F/m})}} $$

First, calculate the product inside the square root:

$$ LC = (250 \times 0.1) \times (10^{-9} \times 10^{-9}) \text{ (H} \cdot \text{F)/m}^2 $$

$$ LC = 25 \times 10^{-18} \text{ (H} \cdot \text{F)/m}^2 $$

Now, take the square root:

$$ \sqrt{LC} = \sqrt{25 \times 10^{-18}} = 5 \times 10^{-9} \text{ s/m} $$

Finally, calculate the velocity:

$$ v = \frac{1}{5 \times 10^{-9} \text{ s/m}} = \frac{1}{5} \times 10^{9} \text{ m/s} $$

$$ v = 0.2 \times 10^{9} \text{ m/s} = 2 \times 10^{8} \text{ m/s} $$

The calculated velocity of wave propagation is $2 \times 10^{8}$ m/s.

Calculating Characteristic Impedance

The characteristic impedance ($Z_0$) represents the ratio of voltage to current for a traveling wave on the transmission line. For a lossless line, it is determined by the inductance ($L$) and capacitance ($C$) per unit length using the formula:

$$ Z_0 = \sqrt{\frac{L}{C}} $$

Using the same given values:

• $L = 250 \times 10^{-9} \text{ H/m}$
• $C = 0.1 \times 10^{-9} \text{ F/m}

Substitute these into the characteristic impedance formula:

$$ Z_0 = \sqrt{\frac{250 \times 10^{-9} \text{ H/m}}{0.1 \times 10^{-9} \text{ F/m}}} $$

Simplify the fraction inside the square root:

$$ \frac{L}{C} = \frac{250 \times 10^{-9}}{0.1 \times 10^{-9}} = \frac{250}{0.1} = 2500 $$

Now, calculate the square root:

$$ Z_0 = \sqrt{2500 \text{ } \Omega^2} $$

$$ Z_0 = 50 \text{ } \Omega $$

The calculated characteristic impedance is $50 \text{ } \Omega$.

Conclusion on Transmission Line Properties

By applying the standard formulas for a lossless transmission line, we found the velocity of wave propagation to be $2 \times 10^{8}$ m/s and the characteristic impedance to be $50 \text{ } \Omega$. This combination matches one of the provided options.