Transformer Impedance Transformation

Understanding how resistance transforms from the secondary to the primary side of a transformer is crucial in electrical engineering. An ideal transformer does not dissipate power; instead, it transforms voltage, current, and impedance according to its turn ratio.

Turn Ratio Defined

The turn ratio of a transformer, often denoted by \(n\), is the ratio of the number of turns in the primary coil (\(N_1\)) to the number of turns in the secondary coil (\(N_2\)).

In this problem, the turn ratio is given as 20 ∶ 1. This means:

\[ n = \frac{N_1}{N_2} = \frac{20}{1} = 20 \]

Effective Resistance Calculation Formula

When a load resistance is connected across the secondary coil, it appears as an "effective" or "reflected" resistance on the primary side. This effective primary resistance (\(R_P\)) is related to the secondary load resistance (\(R_L\)) and the turn ratio (\(n\)) by the following formula:

\[ R_P = \left(\frac{N_1}{N_2}\right)^2 \times R_L \]

Or, more simply:

\[ R_P = n^2 \times R_L \]

Step-by-Step Resistance Calculation

Let's apply the given values to the formula to find the effective resistance seen looking into the primary.

• Given Turn Ratio (\(n\)): 20
• Given Secondary Load Resistance (\(R_L\)): 10 Ω

Substitute these values into the formula:

\[ R_P = (20)^2 \times 10 \, \Omega \]

First, calculate the square of the turn ratio:

\[ (20)^2 = 400 \]

Now, multiply this by the secondary load resistance:

\[ R_P = 400 \times 10 \, \Omega \]

\[ R_P = 4000 \, \Omega \]

Converting Resistance to Kilohms

The resistance is typically expressed in kilohms (kΩ) for larger values. Since 1 kΩ = 1000 Ω, we can convert the calculated resistance:

\[ R_P = \frac{4000 \, \Omega}{1000 \, \Omega/k\Omega} \]

\[ R_P = 4 \, k\Omega \]

Therefore, the effective resistance seen looking into the primary is 4 kΩ.