What turns ratio does a transformer need in order to match a source impedance of 500 ohms to a load of 10 ohms?

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Multiple Choice

What turns ratio does a transformer need in order to match a source impedance of 500 ohms to a load of 10 ohms?

Impedance seen on the primary side of a transformer scales with the square of the turns ratio. If the load is on the secondary, the input impedance is Z_in = Z_load × (N_p/N_s)^2, where N_p/N_s is the primary-to-secondary turns ratio. To match a 10-ohm load to a 500-ohm source, set 500 = 10 × (N_p/N_s)^2. This gives (N_p/N_s)^2 = 50, so N_p/N_s = sqrt(50) ≈ 7.07. That means a turns ratio of about 7.1 to 1. Using a ratio of 14.2 to 1 would reflect the load as 10 × (14.2)^2 ≈ 2014 ohms, and 50 to 1 would reflect as 10 × 50^2 = 25,000 ohms, neither of which matches 500 ohms. So the appropriate choice is about 7.1 to 1.

Impedance seen on the primary side of a transformer scales with the square of the turns ratio. If the load is on the secondary, the input impedance is Z_in = Z_load × (N_p/N_s)^2, where N_p/N_s is the primary-to-secondary turns ratio.

To match a 10-ohm load to a 500-ohm source, set 500 = 10 × (N_p/N_s)^2. This gives (N_p/N_s)^2 = 50, so N_p/N_s = sqrt(50) ≈ 7.07. That means a turns ratio of about 7.1 to 1.

Using a ratio of 14.2 to 1 would reflect the load as 10 × (14.2)^2 ≈ 2014 ohms, and 50 to 1 would reflect as 10 × 50^2 = 25,000 ohms, neither of which matches 500 ohms. So the appropriate choice is about 7.1 to 1.

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