Transmission lines with finite attenuation: Under what condition is power transmission to the load most efficient?

Introduction:
Standing waves on a transmission line are a signature of mismatch. In practical lines with loss (finite attenuation), reflections increase total loss and reduce the net power delivered to the load. Therefore, understanding the connection between standing waves (VSWR) and efficiency is central to RF design.

Given Data / Assumptions:

• Finite-loss transmission line (α > 0).
• Load may or may not be matched to Z0.
• Single-frequency steady-state operation.

Concept / Approach:

Reflections caused by mismatch create forward and reflected waves that form standing waves. In a lossy line, reflected power makes a second pass back and forth, suffering attenuation each time and never fully reaching the load. Matching the load (VSWR = 1) eliminates reflections and maximizes power transfer to the load for the given line loss.

Step-by-Step Solution:

Verification / Alternative check:

Power at the load with mismatch can be written as a function of |Γ| and line loss; differentiating shows maximum at |Γ| = 0 (match). Empirically, system efficiency curves peak at VSWR = 1.

Why Other Options Are Wrong:

• Least efficient at VSWR = 1: opposite of reality.
• Depends only on phase constant: ignores mismatch and attenuation.
• Standing waves always exist: false—matching removes them.
• Very large VSWR: large reflections waste power.

Common Pitfalls:

Confusing the effect of line loss with matching; even with some loss, eliminating reflections still maximizes delivered power.

Final Answer:

power transmission is most efficient when there are no standing waves on the line