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# Single-Phase Transformer

# Page Contents

# List of Figures

**Figure 1:** Equivalent scheme of the single-phase transformer**Figure 2:** Kapp diagram**Figure 3:** Transformer with RL load

# 1. Objective

The first part of the experiment aims to establish the preset characteristics of a transformer by conducting tests at reduced power. This part corresponds to a common industrial practice. In fact, for high-power transformers, there is rarely a power installation available to test the transformer under conditions close to its normal operation.

The second part of the experiment will validate the preset characteristics through load tests, given the relatively low power of the tested transformers.

# 2. Theory: Presentation of the Transformer

## 2.1 Nominal Values

The nominal power corresponds to the rated (secondary) use.

The primary nominal voltage corresponds to iron losses or no-load losses:

The transformer is designed for maximum flux density and therefore maximum iron losses corresponding to its nominal voltage.

The secondary nominal voltage corresponds to the no-load voltage when the primary is supplied at nominal voltage.

The primary and secondary nominal currents can be calculated from the nominal power and the primary and secondary nominal voltages.

The nominal apparent power is greater than the nominal power. It corresponds to the nominal values (voltage and current) allowable considering the corresponding iron and Joule losses:

The subscript n is characteristic of any nominal quantity.

The nominal primary impedance is:

The nominal secondary impedance is:

The transformation ratio is defined by:

**2.2 Equivalent Scheme**

An equivalent diagram of the transformer is provided in Figure 1.

**Figure 1:**Equivalent scheme of the single-phase transformerwith:

: magnetizing resistance

: magnetizing reactance

: winding resistance, referred to the secondary side

: total leakage inductance, referred to the secondary side

The elements of the equivalent diagram are determined from two tests:

### a) Open-circuit Test

With the transformer's secondary circuit left open, the primary is supplied with a voltage .

Measurements are taken for , , , and . This leads to:

Where and are the measured active and reactive powers during the no-load test.

The reduced quantities then have the following expressions:

If the no-load test is conducted at the primary rated voltage , it follows:

Measuring the ratio when allows determining the absolute value of the transformation ratio m:

### b) Short-circuit Test

With a short-circuit at the transformer secondary, the primary is energized with reduced voltage.

Measure , , , and . This leads to:

Where and are the measured active and reactive powers during the short-circuit test.

The reduced quantities then have the following expressions:

If the short-circuit test is conducted at rated secondary current , it follows:

## 2.3 Load Operation, Kapp Diagram

The load operation of the transformer is characterized by the complex equation:

which is associated with a Fresnel diagram called the Kapp diagram (Figure 2).

**Figure 2:**Kapp diagramSince is small, it follows:

Using the reduced values and defining:

Then we get:

## 2.4 The Efficiency of a Transformer

When supplied with a voltage at the primary and delivering a current at a voltage with a power factor equal to , can be expressed as follows:

Here, represents the no-load losses of the transformer when supplied with the voltage .

# 3. Exercises

## 3.1 Open Circuit Test

In this test, the secondary winding of the transformer remains open, and the primary winding is powered with a variable voltage.

**3.1.1 Setup Initialization**

When the virtual laboratory is launched, the initial parameters in the "Network and Transformer" tab under "Fundamental" and "Harmonic" are set as follows:

a) Set the frequency to 60Hz.

b) Choose the network as a single-phase network with the phase A voltage of 460V (since it's a single-phase network, the b) and no phase shift .

c) Activate the transformer.

d) The circuit breaker is not used; the “Without Protection” button is activated.

e) Under the "Harmonic" tab, no harmonics are added. The "A" button is deactivated.

f) Under the "Passive Load" tab, choose the "Open circuit" configuration.

**3.1.2 Questions**

Collect the following parameters for various primary voltages :

Secondary voltage

Primary current

Active power

Power factor

Then, plot the following curves:

Using this test, determine the values of and , as well as their reduced values and . Then calculate the transformation ratio

**m**.

**Note:** Since the current is much lower than the rated current , the active power represents the transformer losses.

## 3.2 Short-circuit Test

In this test, the secondary of the transformer is short-circuited, and the primary is supplied with variable voltage.

### 3.2.1 Setup Initialization

When the virtual laboratory is launched, the initial settings in the "Network and Transformer" tab under "Fundamental" and "Harmonic" are configured as follows:

a) Set the frequency to 60Hz.

b) Choose the network as a single-phase with phase A voltage of **0V** (since it's a single-phase network, the button related to balance phases is not activated) and no phase shift.

c) Activate the transformer.

d) The circuit breaker is not used; the "Without Protection" button is enabled.

e) Under the "Harmonic" tab, no harmonics are added. The "A" button is disabled.

f) Under the "Passive load" tab, select the "Short-circuit" configuration.

### 3.2.2 Questions

Begin this test with a voltage and gradually increase this voltage until .

Record the following values for this current value :

Primary voltage

Primary current

Active power absorbed

Determine the power factor from these measurements.

Using this test, record the values of and as well as their reduced values and .

Calculate the short-circuit transformation ratio and compare it to **m**.

## 3.3 Load Operation

The setup used is that shown in Figure 3:

In this test, the secondary of the transformer consists of connecting a variable resistor R and a variable inductance L in parallel, and the primary is supplied with a fixed voltage .

### 3.3.1 Theoretical Result

Using the Kapp method and the elements of the transformer's equivalent circuit:

Predict the relative voltage drop:

With:

With the transformer being supplied at the rated voltage , calculate the efficiency of the transformer for

With:

### 3.3.2 Setup Initialization

When the virtual laboratory is launched, the initial settings in the "Network and Transformer" tab under "Fundamental" and "Harmonic" are configured as follows:

a) Set the frequency to 60Hz.

b) Choose the network as a single-phase with phase A voltage of 460V (since it's a single-phase network, the button related to balance phases is not activate) and no phase shift.

c) Activate the transformer.

d) The circuit breaker is not used; and the "Without Protection" button is enabled.

e) Under the "Harmonic" tab, no harmonics are added. The "A" button is deactivated.

f) Under the "Passive load" tab, choose the "A1 + B1//C1" configuration.

### 3.3.3 Questions (Using The Virtual Laboratory)

Record the values of the secondary voltage when the current successively takes on the values for:

**Note:** It is required to choose the values of **R** and **L** in order to achieve the correct power factor and current.

Plot the relative voltage drop curves:

Compare the virtual laboratory results to the theoretical ones. Explain any potential discrepancies.

At what fraction of the rated current does the transformer achieve its maximum efficiency?

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