In case the characteristic impedance of a transmission line is equal to the load impedance

Question

Examoogle · Accepted Answer

Option B: All the energy sent will be absorbed by the load

Explanation & Concepts:
When the load impedance ($Z_L$) is exactly equal to the characteristic impedance ($Z_0$) of a transmission line, the line is said to be 'matched'. In this state, the reflection coefficient at the load becomes zero, meaning no power is reflected back toward the source, and the entire incident power is absorbed by the load.

Working Principle: The reflection coefficient is defined as $\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$. When $Z_L = Z_0$, then $\Gamma = 0$. Consequently, the Standing Wave Ratio (SWR) becomes 1:1, representing a traveling wave condition with no reflected waves and pure active power transfer.

Key Formulas: $\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}$ — Reflection Coefficient; $SWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}$ — Standing Wave Ratio

Key Points: Maximum Power Transfer Theorem: Maximum power is delivered when load impedance equals the complex conjugate of source impedance.; Matched lines exhibit no standing waves, preventing potential overvoltage at the source due to reflections.; Return loss becomes infinite (or extremely high) when the line is perfectly matched.; Characteristic impedance ($Z_0$) depends only on the physical geometry and material properties of the line.

Advantages: Elimination of reflected waves preventing standing wave interference; Maximum efficiency of power transfer from source to load

Disadvantages: Requires precise impedance matching components at high frequencies; Changes in load impedance result in immediate signal reflection

Applications: Radio frequency (RF) transmission line matching; Antenna feed systems; High-speed digital signal integrity (PCIe, USB)

Additional Info: When $Z_L = Z_0$, the transmission line appears as if it were infinitely long to the source because there is no wave reflection.. Option A is incorrect because resonance occurs when reflections interfere constructively; matching prevents reflections.. Option C is incorrect as energy is absorbed by the load, not diverted to earth.. Option D is incorrect because transmission losses are independent of the matching condition; they are inherent to the line's resistance.

Answer: B is correct — When the line is terminated with its characteristic impedance, the reflection coefficient is zero, ensuring all incident energy is consumed by the load.

Exam Tip: Always remember that $Z_0$ is purely resistive for lossless lines, but $Z_0$ is complex for lossy lines; matching in lossy lines must account for both real and reactive components.

Examoogle · Answer

Option A: The system will resonate badly

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Option C: All the energy sent will pass to the earth

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Option D: All the energy will be lost in transmission line as transmission losses

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