Transformer Effective Resistance Calculation Explained

Understanding how resistance transforms across a transformer is a fundamental concept in electrical engineering. When a load is connected to the secondary side of a transformer, an equivalent resistance is "seen" or "reflected" back to the primary side. This reflected resistance, also known as the effective resistance, is crucial for analyzing the overall circuit behavior of the transformer.

Transformer Turns Ratio Definition

A transformer's turns ratio is the ratio of the number of turns in the primary winding to the number of turns in the secondary winding. It is often denoted as \(a\), or \(N_1:N_2\). In this problem, the transformer has a 15 ∶ 1 ratio, meaning the primary winding has 15 times more turns than the secondary winding. Therefore, the turns ratio \(N_1/N_2 = 15\).

Resistance Transformation Principle in Transformers

For an ideal transformer, the relationship between the voltages, currents, and turns in the primary and secondary windings allows us to determine how resistance (or impedance) is transformed. The effective resistance seen at the primary side (\(R_{in}\)) is directly related to the load resistance (\(R_L\)) connected to the secondary side by the square of the turns ratio. This principle is key to understanding the effective resistance that the source "sees" from the transformer's primary.

Formula for Reflected Resistance

The formula to calculate the effective resistance seen looking into the primary of a transformer is:

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

Where:

• \(R_{in}\) is the effective resistance seen at the primary side of the transformer.
• \(N_1\) is the number of turns in the primary winding of the transformer.
• \(N_2\) is the number of turns in the secondary winding of the transformer.
• \(R_L\) is the load resistance connected to the secondary side of the transformer.

Applying the Transformer Parameters

Let's identify the given values from the question for the transformer and its load:

• The turns ratio (\(N_1/N_2\)) is given as 15 ∶ 1, which means \(N_1/N_2 = 15\).
• The load resistance (\(R_L\)) connected to the secondary of the transformer is 8 Ω.

Calculating the Effective Resistance

Now, we can substitute these values into the formula to find the effective resistance \(R_{in}\) seen looking into the primary of the transformer:

\[ R_{in} = (15)^2 \times 8 \, \Omega \]

\[ R_{in} = 225 \times 8 \, \Omega \]

\[ R_{in} = 1800 \, \Omega \]

To convert this resistance from ohms (\(\Omega\)) to kilo-ohms (\(k\Omega\)), we divide by 1000:

\[ R_{in} = \frac{1800}{1000} \, k\Omega \]

\[ R_{in} = 1.8 \, k\Omega \]

Final Effective Resistance Value

The effective resistance seen looking into the primary of the 15 ∶ 1 transformer, when connected to an 8 Ω load, is 1.8 kΩ.