Impedance matching with an ideal transformer: What turns ratio (primary:secondary) is required to match a 1 kΩ source resistance to a 160 Ω load?

Introduction / Context:
Transformers can match impedances between a source and a load to maximize power transfer or set a desired loading. The impedance seen at the primary is the secondary load scaled by the square of the turns ratio. Selecting the correct turns ratio prevents reflections, improves efficiency, and protects the source from excessive loading.

Given Data / Assumptions:

• Source resistance R_s = 1 kΩ.
• Load resistance R_L = 160 Ω.
• Ideal transformer (lossless) for calculation.

Concept / Approach:

The impedance referred to the primary is Z_p = (N_p/N_s)^2 * Z_s, where Z_s is the secondary impedance (the load). For a perfect match, Z_p should equal the source resistance R_s. Solve for the ratio N_p/N_s using the equality (N_p/N_s)^2 * R_L = R_s.

Step-by-Step Solution:

Verification / Alternative check:

Check reflection: With 2.5:1, reflected load at primary is (2.5)^2 * 160 = 6.25 * 160 = 1000 Ω, exactly matching the source. Any other ratio would mis-match and change the power transfer conditions.

Why Other Options Are Wrong:

• 0.4:1 or 1:2.5: These imply stepping down primary turns, reflecting the load to 25.6 Ω or 64 Ω equivalent, not 1 kΩ.
• 6.25:1 or 16:1: These ratios overshoot, reflecting loads to 6.25^2 * 160 or 16^2 * 160, far from 1 kΩ.

Common Pitfalls:

• Using voltage ratio instead of the squared relationship for impedance.
• Inverting the ratio (secondary:primary) and obtaining the reciprocal result by mistake.

Final Answer:

2.5:1