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# Three-Phase Transformer

# Page Content

# List of Figures

**Figure 1:** Delta winding of the transformer**Figure 2:** Star winding of the transformer**Figure 3:** Representation of a three-phase transformer**Figure 4:** Equivalent scheme of the single-phase transformer**Figure 5:** Kapp diagram**Figure 6:** Transformer with RL load

# 1. Objective

The first part of the experiment aims to predetermine the loaded 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 verify the predeterminations through load tests, given the relatively low power of the tested transformers.

# 2. Theory : Presentation of the Transformer

## 2.1 Winding Configuration

The primary and secondary windings can be connected in a star or delta configuration.

**Figure 1:**Delta winding of the transformer

**Figure 2:**Star winding of the transformerBy convention, a direct three-phase circuit is connected to the input windings e1, e2, and e3.**In our case, a star configuration is used**

## 2.2 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 this nominal voltage.

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

**2.3 **Representation

A three-phase transformer can be represented as follows:

## 2.4 Equivalent Scheme

Regardless of the coupling mode of a three-phase transformer, it can be represented by the equivalent star single-phase diagram shown in Figure 4. This diagram involves line voltages and currents

with:

: magnetizing resistance

: magnetizing reactance

: winding resistance, referred to the secondary side

: total leakage inductance, referred to the secondary side

: phase lag of the secondary voltage compared to the primary voltage.

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

### a) Open-circuit Test

With the secondary of the transformer as an open-circuit, 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 the transformer secondary short-circuited, 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 the rated secondary current , it follows:

## 2.5 Load Operation, Kapp Diagram

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

Using the reduced values and defining:

which gives:

which is associated with a Fresnel diagram called the Kapp diagram (Figure 5).

Considering the low values of **𝑟** and **𝑛𝜔** on one hand, and the fact thatcannot theoretically exceed 1, the vector (𝑟+𝑗𝑛𝜔) has a module much smaller than that of . Therefore, we can write:

## 2.6 The Efficiency of a Transformer

When supplied with a voltage at the primary and delivering a current 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

The primary and secondary windings will be connected in a star configuration. These connections will not be changed during the experiment.

## 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 three-phase network with the phase A voltage of 460V (since it's a three-phase network, the phases are set as balanced) 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.

## 3.3 Short-circuit Test

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

### 3.3.1 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 three-phase network with phase A voltage of **0V** (since it's a three-phase network, the phases are set as balanced) 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.

### 3.3.2 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**.

## 3.4 Load Operation

The setup used is that shown in Figure 5:

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

### 3.4.1 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 , when:

With:

### 3.4.2 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 network with the phase A voltage of 460V (since it's a three-phase network, the phases are set as balanced) and no phase shift.

c) Activate the transformer.

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

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.

### 3.4.3 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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