Transmission Line Maximum Input Impedance Calculation
This problem involves calculating the maximum possible magnitude of the input impedance ($|Z_{in}|$) seen looking towards the load on a lossless transmission line. We are given the characteristic impedance ($Z_0$) and the magnitude of the reflection coefficient ($|\Gamma|$).
Key Concepts
- Characteristic Impedance ($Z_0$): The ratio of voltage to current for a traveling wave on the line. Given as $Z_0 = 50 \,\Omega$.
- Reflection Coefficient ($|\Gamma|$): Represents the ratio of the reflected voltage wave magnitude to the incident voltage wave magnitude at the load. Given as $|\Gamma| = 0.6$.
- Input Impedance ($Z_{in}$): The impedance seen looking into the transmission line at a certain distance from the load.
- Lossless Line: A transmission line where resistance and conductance are assumed to be zero, simplifying calculations.
Input Impedance Formula
For a lossless transmission line, the input impedance ($Z_{in}$) at a distance '$l$' from the load is given by:
$ Z_{in}(l) = Z_0 \frac{1 + \Gamma e^{-j2\beta l}}{1 - \Gamma e^{-j2\beta l}} $
Where $\Gamma$ is the complex reflection coefficient and $\beta$ is the phase constant. The magnitude of the input impedance varies with distance '$l$'. The maximum value of the input impedance magnitude occurs when the denominator term is minimized. This happens when the reflection coefficient term $\Gamma e^{-j2\beta l}$ is positive real and aligned with the '+1' term.
The maximum possible magnitude of the input impedance is given by the formula:
$ |Z_{in,max}| = Z_0 \frac{1 + |\Gamma|}{1 - |\Gamma|} $
Calculation Steps
- Identify Given Values:
- Characteristic Impedance, $Z_0 = 50 \,\Omega$
- Magnitude of Reflection Coefficient, $|\Gamma| = 0.6$
- Apply the Maximum Impedance Formula: Substitute the given values into the formula for $|Z_{in,max}|$. $ |Z_{in,max}| = 50 \,\Omega \times \frac{1 + 0.6}{1 - 0.6} $
- Simplify the Expression: $ |Z_{in,max}| = 50 \,\Omega \times \frac{1.6}{0.4} $ $ |Z_{in,max}| = 50 \,\Omega \times 4 $
- Calculate the Final Result: $ |Z_{in,max}| = 200 \,\Omega $
Conclusion
The maximum value of the input impedance magnitude looking towards the load on this lossless transmission line is $200 \,\Omega$. This value occurs at specific points along the line where the incident and reflected waves combine constructively to create a voltage maximum.
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