Impedance matching with a transformer — compute required turns ratio: Find the primary-to-secondary turns ratio needed so that a 5 kΩ source (to be matched) properly drives an 8 Ω loudspeaker. Use the relationship (Np/Ns)^2 = Zp/Zs and report the ratio as Np:Ns.

Introduction / Context:
Audio and RF systems often need to match a relatively high source impedance to a low speaker or antenna impedance. Transformers provide impedance transformation where the impedance ratio equals the square of the turns ratio.

Given Data / Assumptions:

• Desired match: Zp = 5 kΩ (5000 Ω) to Zs = 8 Ω.
• Ideal transformer assumption for first-pass calculation (no leakage, no losses).
• Definition: (Np/Ns)^2 = Zp/Zs.

Concept / Approach:
Impedance seen from the primary is related to the load by Z_reflected = (Np/Ns)^2 * Z_load. To make the source “see” 5 kΩ, choose Np/Ns so that (Np/Ns)^2 = 5000/8.

Step-by-Step Solution:

Verification / Alternative check:
If Np/Ns = 25, then Z_reflected = 25^2 * 8 = 625 * 8 = 5000 Ω, exactly the target source impedance. This confirms the computation.

Why Other Options Are Wrong:
5:1 and 8:1 yield impedance ratios of 25 and 64, far from 625.

625:1 produces an absurd impedance ratio of 390,625, which is not appropriate here.

Common Pitfalls:
Forgetting to square the turns ratio when converting impedances. Mixing up the direction (primary vs secondary) and thereby inverting the ratio. Ignoring non-idealities (leakage, winding resistance), which slightly alter real-world matching but not the fundamental ratio.

Final Answer:
25:1