3. Single- and three-phase transformer

#### 3.1. Introduction

Transformer allows developing different voltage levels across the system for the most costeffective price. Transformer functioning principle is based on the idea that energy can be transferred by means of magnetic induction from one winding at the primary side to another winding at the secondary side. This is done by varying the magnetic field produced by alternating current [2, 3].

In this section, graphical user interface (GUI) on MATLAB software will be used to calculate the circuit parameters, efficiency, and voltage regulation of single-phase and three-phase ac transformer. The MATLAB results have been verified and compared with manual calculation in order to ensure they are correct and reliable.

Using GUI in electrical simulation, the instructor/teacher could show the effect of variation for different parameters and then permit to analyze and conclude without the need of manual solving.

#### 3.2. Single-phase transformer model

A single-phase transformer consists of one primary winding and one secondary winding. The exact equivalent circuit with its parameter is shown in the figure below [4].

The parameters of this transformer are as follows (Figure 5):

Primary side:


Secondary side:

Also, the user should add the parameters of the ferromagnetic core such as length, area, air gap, and fringing percentage of each leg of the core; the ferromagnetic core is displayed after

Transformer allows developing different voltage levels across the system for the most costeffective price. Transformer functioning principle is based on the idea that energy can be transferred by means of magnetic induction from one winding at the primary side to another winding at the secondary side. This is done by varying the magnetic field produced by

In this section, graphical user interface (GUI) on MATLAB software will be used to calculate the circuit parameters, efficiency, and voltage regulation of single-phase and three-phase ac transformer. The MATLAB results have been verified and compared with manual calculation

Using GUI in electrical simulation, the instructor/teacher could show the effect of variation for different parameters and then permit to analyze and conclude without the need of manual

A single-phase transformer consists of one primary winding and one secondary winding. The

exact equivalent circuit with its parameter is shown in the figure below [4].

entering the inputs. Push buttons are added to load, save data, clear, and quit.

3. Single- and three-phase transformer

Figure 4. Graphical user interface for ferromagnetic core.

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in order to ensure they are correct and reliable.

3.2. Single-phase transformer model

3.1. Introduction

solving.

alternating current [2, 3].


These parameters can be calculated by open-circuit test and short-circuit test procedure.

#### 3.3. Transformer test

Two tests are applied on the transformer in order to determine its parameters: short-circuit and open-circuit tests [2].

The results permit to determine the equivalent circuit of the transformer, its voltage regulation, as well as its efficiency.

Figure 5. Exact model of transformer.

#### 3.3.1. Short-circuit test

A voltmeter, ammeter, and wattmeter are connected in the HV side of the transformer. Then, the voltage at rated frequency is applied to that HV side using a variable ratio autotransformer. We will then short circuit the LV side of the transformer. Keep increasing the applied voltage, slowly, till reaching the rated current of the HV side (ammeter reading).

Once the rated current is reached on the HV side, the readings extracted on all three instruments, voltmeter, ammeter, and wattmeter, are recorded. The full-load current equivalent corresponds to the ammeter reading.

The transformer core losses could be neglected in this test. In fact, the voltage applied during the short-circuit test on the transformer is very small when compared to the rated voltage of the transformer.

The copper losses in the transformer could be read on the wattmeter. In fact, the wattmeter indicates the input power during the short-circuit test, when the voltmeter is showing the short-circuit voltage VSC. At this time, no output power will appear (short circuited), the core losses are neglected due to the low applied voltage, and, thus, the copper losses in the transformer correspond to the input power.

The extracted values, when the test is accomplished on the transformer's HV side, are referred to the HV side. We can also refer these values to the LV side dividing by the squared turn ratio of the transformer.

Let us consider that the wattmeter reading is PSC:

$$P\_{\rm SC} = R\_{\rm t} I^2 \tag{16}$$

If Ze is the equivalent impedance of the transformer, then

$$R\_e = \frac{V\_{\text{SC}}}{I\_L} \tag{17}$$

Therefore, if the equivalent reactance of transformer is Xe, then

$$X\_{\epsilon}^{2} = Z\_{\epsilon}^{2} - \mathcal{R}\_{\epsilon}^{2} \tag{18}$$

Power factor of the current and angle of power factor are shown below:

$$PF = \cos\theta = \frac{P\_{\text{SC}}}{V\_{\text{SC}}I\_{\text{SC}}} \Rightarrow \theta = \cos^{-1}\frac{P\_{\text{SC}}}{V\_{\text{SC}}I\_{\text{SC}}} \tag{19}$$

#### 3.3.2. Open-circuit test

The open-circuit test consists of connecting an ammeter, a voltmeter, and a wattmeter to the LV side of the transformer. At rated frequency, a voltage is applied to the LV side using a variable ratio autotransformer.

Increasing this applied voltage until the LV side rated voltage is reached (using the voltmeter readings). The HV side of the transformer is kept open. Now, the three readings, voltage, current, and power, are recorded.

The recorded current is the no-load current Ie. It has a small value when compared to the transformer's rated current, and, thus, we can neglect the voltage drop due to this electric current. The recorded voltage V is now equal to the transformer's secondary induced voltage.

The wattmeter indicates the input power, which corresponds to the core and copper losses in the transformer, since no output power will appear (open circuit). Copper losses could be neglected since the no-load current is very small compared to the full-load current, and, thus, the core losses in the transformer are considered equal to the wattmeter reading, Po:

$$P\_o = \frac{V\_1^2}{R\_m} \tag{20}$$

where Rm is the transformer's shunt branch resistance.

3.3.1. Short-circuit test

of the transformer.

of the transformer.

3.3.2. Open-circuit test

ratio autotransformer.

corresponds to the ammeter reading.

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transformer correspond to the input power.

Let us consider that the wattmeter reading is PSC:

If Ze is the equivalent impedance of the transformer, then

Therefore, if the equivalent reactance of transformer is Xe, then

Power factor of the current and angle of power factor are shown below:

PF <sup>¼</sup> cos <sup>θ</sup> <sup>¼</sup> PSC

A voltmeter, ammeter, and wattmeter are connected in the HV side of the transformer. Then, the voltage at rated frequency is applied to that HV side using a variable ratio autotransformer. We will then short circuit the LV side of the transformer. Keep increasing the applied voltage,

Once the rated current is reached on the HV side, the readings extracted on all three instruments, voltmeter, ammeter, and wattmeter, are recorded. The full-load current equivalent

The transformer core losses could be neglected in this test. In fact, the voltage applied during the short-circuit test on the transformer is very small when compared to the rated voltage

The copper losses in the transformer could be read on the wattmeter. In fact, the wattmeter indicates the input power during the short-circuit test, when the voltmeter is showing the short-circuit voltage VSC. At this time, no output power will appear (short circuited), the core losses are neglected due to the low applied voltage, and, thus, the copper losses in the

The extracted values, when the test is accomplished on the transformer's HV side, are referred to the HV side. We can also refer these values to the LV side dividing by the squared turn ratio

PSC ¼ ReI

Re <sup>¼</sup> VSC IL

<sup>e</sup> � <sup>R</sup><sup>2</sup>

) <sup>θ</sup> <sup>¼</sup> cos �<sup>1</sup> PSC

VSCISC

X2 <sup>e</sup> <sup>¼</sup> <sup>Z</sup><sup>2</sup>

VSCISC

The open-circuit test consists of connecting an ammeter, a voltmeter, and a wattmeter to the LV side of the transformer. At rated frequency, a voltage is applied to the LV side using a variable

<sup>2</sup> <sup>ð</sup>16<sup>Þ</sup>

<sup>e</sup> ð18Þ

ð17Þ

ð19Þ

slowly, till reaching the rated current of the HV side (ammeter reading).

If Zm is the shunt branch impedance of the transformer, then

$$Z\_{\rm m} = \frac{V\_1}{I\_\varepsilon} \tag{21}$$

Therefore, if shunt branch reactance of transformer is Xm, then

$$\left(^{1/\_{\chi\_m}}\right)^2 = \left(^{1/\_{\chi\_m}}\right)^2 - \left(^{1/\_{\chi\_m}}\right)^2\tag{22}$$

The test is applied on the LV side of the transformer, so the calculated values are referred to the LV side. We could calculate the referred HV side values by multiplying these values with the squared turn's ratio of the transformer. The open-circuit test on transformer is used to determine the parameters of the shunt branch of the equivalent circuit of transformer:

$$PF = \cos\theta = \frac{P\_{\text{OC}}}{V\_{\text{OC}}I\_{\text{OC}}} \Rightarrow \theta = \cos^{-1}\frac{P\_{\text{OC}}}{V\_{\text{OC}}I\_{\text{OC}}} \tag{23}$$

The excitation admittance is therefore

$$Y\_E = \frac{I\_{\text{OC}}}{V\_{\text{OC}}} \Delta - \theta\_{\text{OC}} \tag{24}$$

The equivalent series impedance is therefore

$$Z\_{\rm SE} = \frac{V\_{\rm SC}}{I\_{\rm SC}} \mathcal{L} \theta\_{\rm SC} \tag{25}$$

The voltage regulation is

$$VR = \frac{V\_P/a - V\_{s,\theta}}{V\_{s,\theta}} \times 100\% \tag{26}$$

And the efficiency is

$$
\eta = \frac{P\_{out}}{P\_{in}} \times 100\% \tag{27}
$$

#### 3.4. Three-phase transformer

A three-phase transformer is made of three transformers that are either separated or combined in one core. The primary side and secondary side of any given three-phase transformer can be connected independently in either delta (Δ) or wye (Y) [2].

#### 3.5. Implementation on GUI MATLAB

The user will enter certain values into the GUI interface, and then the result will be displayed with respect to this flow chart (Figure 6).

Figure 6. Flow chart for GUI.

The graphical user interface for single-phase transformer is shown in Figure 7.

VR <sup>¼</sup> VP=<sup>a</sup> � Vs,fl Vs,fl

> <sup>η</sup> <sup>¼</sup> Pout Pin

A three-phase transformer is made of three transformers that are either separated or combined in one core. The primary side and secondary side of any given three-phase transformer can be

The user will enter certain values into the GUI interface, and then the result will be displayed

And the efficiency is

3.4. Three-phase transformer

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Figure 6. Flow chart for GUI.

3.5. Implementation on GUI MATLAB

with respect to this flow chart (Figure 6).

connected independently in either delta (Δ) or wye (Y) [2].

� 100% ð26Þ

� 100% ð27Þ

The user will add the inputs which are values of short-circuit test and open-circuit test. And then, choose between leading and lagging load. The results of the equivalent circuits referred to primary and secondary side are displayed after adding the parameter and clicking on to calculate the equivalent circuit, and the equivalent circuit of the transformer referred to the primary side and secondary side are displayed with their parameter.

The user may also choose the type of core of the transformer whether circular or rectangular in shape (Figure 8).

Push buttons were used to load and save data as well as to display the performance of the transformer (Figure 9).

Figure 7. Graphical user interface for single-phase transformer.

Figure 8. Transformer core shape calculated.

The graphical user interface for three-phase transformer is shown in Figure 10.

Here, the user has to choose the type of connection. An example of calculation is shown in Figure 11.

Figure 9. Single-phase transformer performance.

Figure 10. Graphical user interface for three-phase transformer.


Figure 11. Per unit equivalent circuit of three-phase transformer.
