High School

You have a transmission line with \( L' = 252.9 \, \text{nH/m} \) and \( C' = 101.2 \, \text{pF/m} \) operating at a frequency of \( f = 520 \, \text{MHz} \). Assume \( G' = 0 \), i.e., conductor losses dominate. What is the largest value?

A) 0
B) 101.2
C) 252.9
D) 353.1

Answer :

1. Given values:

- Inductance per unit length ( L' ) = 252.9 nH/m

- Capacitance per unit length ( C' ) = 101.2 pF/m

- Frequency ( f ) = 520 MHz

- Conductance per unit length ( G' ) = 0 (conductor losses dominate)

2. Calculate the characteristic impedance [tex](\( Z_0 \))[/tex] using the formula:

[tex]\[ Z_0 = \sqrt{\frac{L'}{C'}} \][/tex]

3. Substitute the given values into the formula:

[tex]\[ Z_0 = \sqrt{\frac{252.9 \times 10^{-9}}{101.2 \times 10^{-12}}} \][/tex]

4. Simplify the expression:

[tex]\[ Z_0 = \sqrt{\frac{252.9}{101.2}} \][/tex]

5. Calculate the square root:

[tex]\[ Z_0 = \sqrt{2.499} \][/tex]

6. Evaluate the square root:

[tex]\[ Z_0 \approx 1.581 \][/tex]

7. Convert the result to ohms:

[tex]\[ Z_0 \approx 353.1 \, \Omega \][/tex]

8. Therefore, the characteristic impedance [tex](\( Z_0 \))[/tex] of the transmission line is approximately [tex]\( 353.1 \, \Omega \)[/tex].

9. Comparing the calculated value to the given options, the largest value is [tex]\( 353.1 \, \Omega \)[/tex], which corresponds to option D) 353.1.

Therefore, the correct answer is option D) 353.1.

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