If the load impedance is much larger than the characteristic impedance (ZL » Z0), what is the approximate relationship between ZL, Z0, and VSWR?

Introduction:This item checks your facility with the relationship between load mismatch and standing wave ratio (VSWR) on a line terminated in a real load that is much larger than the line impedance. In that limit, the VSWR connects directly to the impedance ratio.

Given Data / Assumptions:

• Purely resistive load ZL » Z0.
• Lossless or low-loss line for simplicity.
• Standard definitions: Γ = (ZL − Z0) / (ZL + Z0), VSWR = (1 + |Γ|) / (1 − |Γ|).

Concept / Approach:

When ZL » Z0, |Γ| ≈ (ZL − Z0) / (ZL + Z0) → (ZL / Z0 − 1) / (ZL / Z0 + 1) ≈ 1 − 2 Z0 / ZL. The corresponding VSWR ≈ (1 + |Γ|) / (1 − |Γ|) ≈ (2 − 2 Z0 / ZL) / (2 Z0 / ZL) ≈ ZL / Z0. Hence, ZL ≈ Z0 * VSWR.

Step-by-Step Solution:

Verification / Alternative check:

Numerical example: Z0 = 50 Ω, ZL = 1000 Ω ⇒ exact VSWR ≈ 1000/50 = 20 (very close), confirming the approximation.

Why Other Options Are Wrong:

• Proportionalities using VSWR − 1 or inverses do not match the high-ratio limit.
• Equations setting Z0 = ZL(VSWR) invert the relationship incorrectly.

Common Pitfalls:

Applying the approximation outside its range; for modest mismatches, use exact formulas rather than the ZL/Z0 ≈ VSWR shortcut.

Final Answer:

ZL ≈ Z0 * VSWR (for ZL » Z0)