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A transmission line of characteristic impedance z0 = 50 Ω. Phase Velocity = 2 x 108 m/sec, and length l = 1 m is terminated by a load z1 = (30 - j 40) ohms. The input impedance of the line for a frequency of 100 MHz will be
  • a)
    (30 + j x 40) Ω
  • b)
    (30 - j 40) Ω
  • c)
    (50 + j x 40) Ω
  • d)
    (50 - j 40) Ω
Correct answer is option 'B'. Can you explain this answer?
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A transmission line of characteristic impedance z0 = 50 Ω. Phase Veloc...
Given:

  • z0 = 50 Ω

  • Phase Velocity = 2 x 108 m/sec

  • length l = 1 m

  • Load z1 = (30 - j 40) ohms

  • Frequency f = 100 MHz



To find: Input impedance of the line for the given frequency

Solution:

  • First we need to calculate the wavelength (λ) of the signal on the transmission line

  • Phase Velocity (Vp) = λ x Frequency (f)

  • λ = Vp / f = (2 x 108) / (100 x 106) = 2.0 meters

  • Next, we need to calculate the electrical length (β) of the line

  • β = 2π / λ = 2π / 2.0 = π radians

  • Using the given load impedance and the characteristic impedance of the line, we can calculate the reflection coefficient (Γ) at the load

  • Γ = (z1 - z0) / (z1 + z0) = (30 - j40 - 50) / (30 - j40 + 50) = -0.2 - j0.4

  • The input impedance (Zin) of the line can be calculated as:



Zin = z0 x (ZL + jz0 x tanβ) / (z0 + jZL x tanβ)


  • Substituting the values, we get:



Zin = 50 x ((30 - j40) + j50 x tanπ) / (50 + j(30 - j40) x tanπ)


  • tanπ = 0, so we get:



Zin = 50 x (30 - j40) / (50 + j30)


  • Simplifying, we get:



Zin = (30 - j40) Ω

Answer: Option B

Explanation:

  • The input impedance of the line is the impedance seen by the source when looking into the line

  • In this case, the line is terminated by a load impedance of (30 - j40) Ω

  • The input impedance is calculated using the reflection coefficient and the transmission line equation

  • After simplifying the equation, we get the input impedance as (30 - j40) Ω

  • Therefore, the correct answer is option B

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A transmission line of characteristic impedance z0 = 50 Ω. Phase Velocity = 2 x 108 m/sec, and length l = 1 m is terminated by a load z1 = (30 - j 40) ohms. The input impedance of the line for a frequency of 100 MHz will bea)(30 + j x 40) Ωb)(30 - j 40) Ωc)(50 + j x 40) Ωd)(50 - j 40) ΩCorrect answer is option 'B'. Can you explain this answer?
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