E-Core Transformer

Transformers are electrical components that are used for power transmission. Most transformers work on the principle of electromagnetic induction. A typical transformer consists of a primary coil, a secondary coil and a ferromagnetic core. The primary coil receives the AC electrical input signal. As a result of mutual induction, an induced voltage is obtained across the secondary coil. The ferromagnetic core serves the purpose of a magnetic flux concentrator thereby minimizing losses due to flux leakage.

Commercial transformers use several types of cores, which are named based on their geometric shapes, for example I-core, U-core, E-core, pot core, toroidal, and planar. This application uses a pair of E-cores for magnetic flux concentration.

This example demonstrates how to perform transient simulations of a single-phase E-core transformer. Including the effect of a nonlinear B-H curve in the soft-iron core, the model computes the spatial distribution of the magnetic and electric fields, the magnetic saturation effect, the transient response, and the flux leakage to the surroundings. Two different versions of the transformer are simulated: the first one with a turn ratio of unity and the second one with a turn ratio of 1000.

Model Definition

The core of the single-phase E-core transformer considered here consists of a pair of E-cores, which form a closed magnetic flux path. The primary and secondary coils in the transformer are placed around the central leg of the core as shown in Figure 1.

A nonlinear B-H curve that includes saturation effects is used to simulate the magnetic behavior of the soft-iron core. Hysteresis effects in the core are neglected. The model assumes that the primary and secondary windings are made of thin wire and have multiple turns. Using the assumptions that the wire diameter is less than the skin depth and that there are many turns, these windings are modeled with Coil features. Furthermore, the model does not account for eddy currents in the individual turns of the coil. The primary winding is connected to a primary resistor, Rp and the AC voltage source, Vac while the secondary winding is connected to the secondary load resistor, Rs as shown in Figure 2.

Figure 1: Model illustration of an E-core transformer.

Figure 2: A transformer connected to an external circuit with voltage source and resistors.

The model is solved in time domain for a line frequency of 50 Hz. Several important design parameters such as the magnitude of the input voltage, the line frequency, the number of turns in the coils, and the coil resistance are parameterized and can therefore easily be changed.

A transformer works by the principle of Faraday’s law of induction which states that the induced voltage (Vin) in a coil is proportional to the rate of change of magnetic flux (

) and the number of turns (N) in a coil as shown in Equation 1.

(1)

If two coils are coupled, Equation 1 can be used to deduce that the induced voltage in the secondary coil (Vs) is proportional to the induced voltage in the primary coil (Vp):

(2)

Here Ns and Np are the number of turns in the secondary and primary coils, respectively. Np/Ns is known as the turn ratio.

Results and Discussion

Figure 3 shows the surface plot of the magnetic flux density norm distribution and the arrow plots of the currents in the windings at t = 50 ms.

Figure 4 shows the slice and the arrow plot of the magnetic flux density norm in the core at t = 50 ms.

Figure 5 and Figure 6 display the induced voltage in the primary and secondary windings respectively. Since the number of turns on each winding is equal, the induced voltage in both windings is same as given by Equation 2.

The currents flowing through the primary and secondary windings are shown in Figure 7 and Figure 8, respectively.

Figure 9 displays the voltage induced in the primary winding for a step-down transformer with a turn ratio of Np/Ns = 1000 and supply voltage of 25 kV.

Finally, the induced voltage in the secondary winding for a step-down transformer is displayed in Figure 10. This induced voltage is 1000 times smaller compared to the voltage in the primary winding of Figure 9.

Figure 3: Magnetic flux density norm and the currents in the windings at t = 50 ms.

Figure 4: Magnetic flux density inside the transformer core at t = 50 ms.

Figure 5: The induced voltage in the primary winding versus time.

Figure 6: Induced voltage in the secondary windings versus time.

Figure 7: Current in the primary winding versus time.

Figure 8: Current in the secondary winding versus time.

Figure 9: Induced voltage in the primary winding versus time for a step-down transformer.

Figure 10: Induced voltage in the secondary winding versus time for a step-down transformer.

Notes About the COMSOL Implementation

Use the Magnetic Fields interface to model the magnetic fields of the transformer. Model the primary and secondary windings with Coil features. Connect the primary and secondary windings to an external circuit with the AC voltage source and resistors using an Electrical Circuit interface. Add a Coil Geometry Analysis study step to calculate the current in the coils. Perform a Time Dependent study to determine the voltage and currents in both the primary and secondary windings.

ACDC_Module/Other_Industrial_Applications/ecore_transformer

Modeling Instructions

From the menu, choose .

New

In the window, click .

Model Wizard

In the window, click .

In the tree, select .

Click .

In the tree, select .

Click .

In the tree, select .

Click .

Global Definitions

Define all the required parameters.

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Rp	100[ohm]	100 Ω	Primary side resistance
Rs	10[kohm]	10000 Ω	Secondary side resistance
Np	300	300	Number of turns in primary winding
Ns	300	300	Number of turns in secondary winding
nu	50[Hz]	50 Hz	Frequency of supply voltage
Vac	25[V]	25 V	Supply voltage

Geometry 1

Insert the geometry sequence from the ecore_transformer_geom_sequence.mph file.

In the toolbar, click .

Browse to the model’s Application Libraries folder and double-click the file ecore_transformer_geom_sequence.mph.

Extrude 1 (ext1)

Define the selections for the core and the windings and create a selection for the core domain.

In the window, under click .

In the window for , locate the section.

Find the subsection. Click .

In the dialog box, type Core in the text field.

Click .

Specify then a selection for the primary winding.

Primary Winding

In the toolbar, click and choose .

In the window for , type Primary Winding in the text field.

On the object , select Domains 5, 6, 8, and 9 only.

Locate the section. Find the subsection. Click .

In the dialog box, type Windings in the text field.

Click .

Add a selection for the secondary winding.

Secondary Winding

In the toolbar, click and choose .

On the object , select Domains 3, 4, 7, and 10 only.

In the window for , locate the section.

Find the subsection. From the list, choose .

In the text field, type Secondary Winding.

Form Union (fin)

In the toolbar, click .

Click the button in the toolbar.

Choose wireframe rendering to get a better view of the interior parts.

Click the button in the toolbar.

The geometry should look like the one in the figure above.

Definitions

Now set up the physics for the magnetic fields. Start by hiding the outer boundaries to visualize the results only in the transformer core and windings.

Hide for Physics 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 1–5 and 56 only.

Magnetic Fields (mf)

Choose linear vector elements for discretizing the magnetic vector potential. This will make the model computationally efficient. Typically, the default quadratic elements are recommended.

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Apply in the core and the air domain.

Ampère’s Law 2

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Add features to model the primary and the secondary windings using the conductor model.

Coil 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the N text field, type Np.

Locate the section. From the list, choose .

Input 1

In the window, expand the node, then click .

Select Boundary 35 only.

Coil 2

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the list, choose .

Locate the section. In the N text field, type Ns.

Locate the section. From the list, choose .

Input 1

In the window, expand the node, then click .

Select Boundary 31 only.

Gauge Fixing for A-field 1

In the toolbar, click and choose .

In the window for , locate the section.

From the list, choose .

Gauge fixing in its static form is applied on all domains in which the interface is active.

Electrical Circuit (cir)

Add the external circuit elements to the primary and the secondary side of the transformer as shown in Figure 2.

In the window, under click .

Voltage Source V1

In the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	1
n	0

Locate the section. From the list, choose .

In the Vsrc text field, type Vac.

In the f text field, type nu.

External I vs. U 1

In the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	2
n	1

Locate the section. From the V list, choose .

Resistor R1

In the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	0
n	2

Locate the section. In the R text field, type Rp.

External I vs. U 2

In the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	3
n	0

Locate the section. From the V list, choose .

Resistor R2

In the toolbar, click .

In the window for , locate the section.

In the table, enter the following settings:


Label	Node names
p	0
n	3

Locate the section. In the R text field, type Rs.

Assign materials for the model. Begin by specifying air for all domains.

Add Material

In the toolbar, click to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Air (mat1)

Next, assign the soft iron (without losses) material for the core.

Add Material

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click to close the window.

Materials

Soft Iron (Without Losses) (mat2)

In the window for , locate the section.

From the list, choose .

Optionally, use the following steps to visualize the nonlinear B-H curve of the soft iron. Note that the extrapolation of the B-H curve is set to ’constant’ only for visualization purpose. When solving the model, the extrapolation has to be changed to ’linear’.

In the window, expand the node.

Interpolation 1 (BH, BH_inv, BH_prim)

In the window, expand the node, then click .

In the window for , locate the section.

From the list, choose .

Click .

From the list, choose .

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Free Tetrahedral 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click .

In the dialog box, type 2-10 in the text field.

Click .

In the window for , locate the section.

Click the button.

Locate the section. Select the check box.

In the associated text field, type 8.

Click .

After adjusting the hiding settings by clicking on the button, the mesh should look like that shown in the figure above. A coarse mesh is used here to solve the model quickly. For real simulations, you are recommended to use a finer mesh.

Study 1

In the window, click .

In the window for , locate the section.

Clear the check box.

Solve the study step for the analysis of the geometry of the primary and the secondary windings which will be necessary for the next transient studies.

In the toolbar, click .

Add Study

In the toolbar, click to open the window.

Go to the window.

Find the subsection. In the tree, select .

Click in the window toolbar.

In the toolbar, click to close the window.

Study 2

In the window, click .

In the window for , locate the section.

Clear the check box.

Step 1: Time Dependent

Solve the problem in time domain from 0 to 50 milliseconds.

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,5e-4,0.05).

Click to expand the section. Follow the steps given below to tune the solver. Such tuning is typically necessary in order to successfully use a realistic nonlinear B-H curve in a large time-dependent model. As the equations are nonlinear the solver needs to be robust enough to handle such nonlinearities. It is worth noticing that as the problem does contains only nodes where conductivity is null (air and laminated iron) and describing conductors, it is computationally more efficient to enforce to . This allows to have zero conductivity in regions and it is making the transient solution easier to handle from the solver.