To find the proportion of incident power that is reflected back, we need to determine the reflection coefficient, $\Gamma$, using the formula:

$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$

where:

• $Z_L = 50 - j25 \, \Omega$ (load impedance)
• $Z_0 = 75 \, \Omega$ (characteristic impedance of the transmission line)

1. Calculate the reflection coefficient $\Gamma$:

   $\Gamma = \frac{(50 - j25) - 75}{(50 - j25) + 75}$

   $\Gamma = \frac{-25 - j25}{125 - j25}$

   To simplify, we need to find the magnitude of this complex number. Let's multiply the numerator and denominator by the complex conjugate of the denominator:

   $\Gamma = \frac{(-25 - j25)(125 + j25)}{(125 - j25)(125 + j25)}$

   This gives us:

   $\Gamma = \frac{-3125 - j625 + j3125 - 625j^2}{125^2 + 25^2}$

   Since $j^2 = -1$, the equation simplifies to:

   $\Gamma = \frac{-3125 + 625 + j(3125 - 625)}{15625 + 625}$

   $\Gamma = \frac{-2500 + j2500}{16250}$

   $\Gamma = -\frac{2500}{16250} + j\frac{2500}{16250}$

   $\Gamma = -0.1538 + j0.1538$

2. Find the magnitude of $\Gamma$:

   $|\Gamma| = \sqrt{(-0.1538)^2 + (0.1538)^2}$

   $|\Gamma| = \sqrt{0.0237 + 0.0237}$

   $|\Gamma| = \sqrt{0.0474} \approx 0.2177$

3. Calculate the reflected power percentage:

   The proportion of reflected power is given by:

   $\text{Reflected power} = |\Gamma|^2 \times 100\%$

   $\text{Reflected power} = (0.2177)^2 \times 100\% \approx 7.84\%$

Thus, the correct answer is C. 7.84%.

Would you like further details or have any questions about the calculation?

Related Questions:

1. How would the reflected power change if the line impedance was different?
2. What is the significance of a complex load impedance in transmission lines?
3. How is the reflection coefficient used to determine standing wave ratio (SWR)?
4. Can this method be used for any transmission line and load configuration?
5. What are the practical implications of reflected power in transmission line design?

Tip:

To minimize reflected power in a transmission line, it is essential to match the load impedance with the line impedance, ensuring efficient power transfer.