Impedance matching by transformer: What turns ratio is required to match an 80 Ω source to a 320 Ω load?

Introduction / Context:
Impedance matching via a transformer uses the square law between turns ratio and impedance. Correct turns ratio minimizes reflections and maximizes power transfer in audio and RF systems alike.

Given Data / Assumptions:

• Source resistance Rs = 80 Ω.
• Load resistance RL = 320 Ω.
• Ideal transformer; express turns ratio as Ns/Np.

Concept / Approach:
For ideal transformers, impedance scales as (Ns/Np)^2. To match, choose Ns/Np so that RL' = Rs, where RL' = RL * (Np/Ns)^2 when seen from the primary. Equivalently, (Ns/Np)^2 = RL / Rs.

Step-by-Step Solution:
(Ns/Np)^2 = RL / Rs = 320 / 80 = 4Ns/Np = sqrt(4) = 2

Verification / Alternative check:
With Ns/Np = 2, a 320 Ω load reflects to Rs' = 320 / 4 = 80 Ω. This equals the source resistance, confirming matching.

Why Other Options Are Wrong:

• 4, 20, 80: These would grossly mismatch the impedances.

Common Pitfalls:
Using the simple ratio RL/Rs rather than its square root for the turns ratio. Always take the square root when moving from impedance ratio to turns ratio.

Final Answer:
2