In a single-phase transformer, the ratio of transformation is 2 and the secondary resistance is 0.24 Ω. Find the resistance of secondary in terms of primary.
0.06 Ω
Transformer Resistance Calculation Overview
In the study of electrical machines, a single-phase transformer is a crucial device used to step up or step down AC voltages. When analyzing the performance of a transformer, it is often necessary to refer the impedance (or resistance) from one side of the transformer to the other. This process simplifies the equivalent circuit of the transformer, making calculations easier. The core concept behind this referral is the transformation ratio of the transformer.
Transformation Ratio and Resistance Referral Formula
The transformation ratio (often denoted by \(k\) or \(a\)) of a transformer is defined as the ratio of the secondary winding voltage to the primary winding voltage, or the ratio of secondary turns to primary turns.
- If the transformation ratio \(k\) is defined as \(\frac{V_2}{V_1}\) or \(\frac{N_2}{N_1}\) (secondary to primary), then the resistance \(R_2\) on the secondary side, when referred to the primary side (\(R_2'\)), is given by the formula:
\[R_2' = \frac{R_2}{k^2}\]
- Conversely, if the transformation ratio \(a\) is defined as \(\frac{V_1}{V_2}\) or \(\frac{N_1}{N_2}\) (primary to secondary), then the resistance \(R_2\) on the secondary side, when referred to the primary side (\(R_2'\)), is given by the formula:
\[R_2' = R_2 \times a^2\]
In this problem, the "ratio of transformation" is given as 2. In most contexts, especially when not specified otherwise, this typically refers to \(k = \frac{N_2}{N_1}\). We will proceed with this common interpretation to calculate the resistance of secondary in terms of primary.
Secondary Resistance Referral Step-by-Step Calculation
To determine the resistance of secondary in terms of primary, we apply the appropriate formula for referring secondary resistance to the primary side.
Given information:
- Ratio of transformation (\(k\)) = 2
- Secondary resistance (\(R_2\)) = 0.24 \(\Omega\)
Formula used:
\[R_2' = \frac{R_2}{k^2}\]
Step-by-step calculation:
- Substitute the given values into the formula:
\[R_2' = \frac{0.24 \, \Omega}{(2)^2}\]
- Calculate the square of the transformation ratio:
\[k^2 = 2^2 = 4\]
- Perform the division:
\[R_2' = \frac{0.24 \, \Omega}{4}\]
\[R_2' = 0.06 \, \Omega\]
Thus, the resistance of secondary in terms of primary is 0.06 \(\Omega\). This process of impedance transformation is fundamental for analyzing the equivalent circuit of a single-phase transformer.
Comparing Calculated Value with Options
Let's compare the calculated value of the secondary resistance referred to the primary side with the given options.
| Option | Value (\(\Omega\)) |
|---|---|
| 1 | 0.6 |
| 2 | 0.48 |
| 3 | 0.12 |
| 4 | 0.06 |
Our calculated value of 0.06 \(\Omega\) perfectly matches Option 4, confirming the correct application of the transformation ratio formula for referring secondary resistance to the primary side in a single-phase transformer.
Important Questions from Transformer Copper Losses
What is the condition at which a transformer gives maximum efficiency?
- Which of the following losses remains variable during normal operation of transformer?
For a single phase transformer, the maximum efficiency occurs at 75% of the load, then \(\rm \frac{iron \ loss \ at\ full \ load }{copper \ loss \ at\ full \ load}=?\)
A single-phase \(400\;V,\;50\;Hz\) transformer has an iron loss of \(5000\;W\) at the rated condition. When operated at \(200\;V,\;25\;Hz\), the iron loss is \(2000\;W\). When operated at \(416\;V,\;52\;Hz\), the value of the hysteresis loss divided by the eddy current loss is ______.
The full load copper loss and iron loss of a transformer are 6400 W and 5000 W respectively the copper loss and iron loss at half load will be respectively.
