Selection - ResearchGate

Single Phase E-core Transformer

Introduction

• This is a 3D model of an E-core transformer.

• The core is made out of a pair of E-cores to form a closed magnetic flux path.

• The primary and secondary windings are around the central leg of the core.

References:

1. http://en.wikipedia.org/wiki/Transformer

2. http://en.wikipedia.org/wiki/Magnetic_core#.22E.22_core

© 2010 COMSOL Inc. All rights reserved

E-core transformer

Primary winding

Secondary winding

E-core


Model Features

• Full non-linear time domain analysis at a constant frequency of 50 Hz is modeled.

• Non-linear magnetic material (with saturation effect) is used for magnetic core.

• The primary and secondary windings are modeled as homogenized current carrying domains. Individual wires are not resolved.

• Skin effect in windings and core is not included.


Principle of Transformer (1/2)

Electro-magnetic Induction

Faraday’s Law – Induced voltage ( V ) across a coil is proportional to the rate of change of magnetic flux

( d

Φ

/dt ) and number of turns ( N ) in a coil.

V

= −

N d

φ

dt


Principle of Transformer (2/2)

Mutual Induction

V s

V p

=

N s

N p

V p

= voltage across primary coil

V s

= voltage across secondary coil

N p

= number of turns in primary coil

N s

= number of turns in secondary coil


Start from here – Chose the Space

Dimension

Click the Next icon


Add the physics interface – Magnetic Fields

Click the Next icon


Add the study type – Time Dependent

Click the Finish icon


Define Parameters

Variable name area

Rp

Rs

Np

Ns nu omega

Description

Longitudinal sectional area of winding = length x thickness

Resistance of primary coil

Resistance of secondary coil

Number of turns in primary coil

Number of turns in secondary coil

Frequency of power line (60 Hz in

USA, 50 Hz in Europe)

Angular frequency = 2 x

π x nu


Define Parameters

You can either type these or upload from the file ecore_parameters.txt


Geometry Steps

• The next few slides show step-by-step instructions on how to create the geometry.

• The user can also import the CAD object as a .mphbin file (COMSOL native geometry format).


Create concentric circles

1.

Right-click on Model 1 > Geometry 1 and select Workplane

2.

Right-click on Workplane 1 > Geometry and select Circle

3.

Specify a radius of 0.015

4.


5.


6.


7.


8.

In the Settings window, click Build Selected

9.

In the Graphics window, click on Zoom Extents

10. Right-click on Workplane 1 > Geometry and select Boolean

Operations > Compose


Composite object

11. Click anywhere in the Graphics window. Hit Ctrl+A to select all three circles and then right-click to add all three objects as input objects in the Compose settings.


Create an annulus

12. In the Settings windows, locate the Set formula edit field and type the formula c2+c3-c1

13. In the Settings window, click Build Selected .


15.

16.

17.

Step

14.

Draw the following shapes to create the

E-core

Shape

Square (SQ1)

Size

Width = 0.02

Base Position

Center (x = 0, y = 0)

Rectangle (R1)

Rectangle (R2)

Rectangle (R3)

Width = 0.01

Height = 0.02

Width = 0.01

Height = 0.02

Width = 0.08

Height = 0.02

Corner (x = -0.04, y = -0.01)

Corner (x = 0.03, y = -0.01)

Corner (x = -0.04, y = -0.01)

18. Click on Zoom Extent .


This is how the geometry should look like at this point


Extrusion

19. Right-click on Model 1 > Geometry 1 > Workplane 1 and select

Extrude

20. Deselect wp1.co1

and wp1.r3

from the list of objects to be extruded


Extrude the three arms

21. Type an extrusion distance of 0.05 to extrude the square and other two rectangles


Extrude the coils


Extrude

23. Select only wp1.co1

and extrude it by 0.04


Extrude one end of the E-core


Extrude

23. Select only wp1.r3

and extrude it by 0.01


Positioning the objects

24. Right-click on Model 1 > Geometry 1 and select Transforms > Move

25. Select all geometries and move by z = -0.02


27. Select the three arms and move by z = -0.005

Before After


Positioning the lower arm


29. Select the lower arm and move by z = -0.015

Before After


The final E-core geometry

30. Right-click on Model 1 > Geometry 1 and select Transforms > Copy

31. Select the lower arm and move by z = 0.06.

Make sure the Keep input objects box is checked.


Adding an air domain

32. Right-click on Model 1 > Geometry 1 and select Block

33. Create a block of dimension 0.15 x 0.15 x 0.15 centered at (0,0,0)

34. In the Graphics window, click on Zoom Extents

35. In the Graphics window, click on Wireframe Rendering


Grouping the modeling domains

• Right-click on Model 1 > Definitions and select Selections >

Explicit

• Assign all the domains that belong to the E-core

• Right-click on Explicit 1 and rename it as Core


Direction of current in the windings

• Current flows through the coil along the circumferential direction.

• Define the direction of current in the windings (Domains 5 and 6) using vectors ix , iy and iz .

• Note that iz is set to zero as there is no axially flowing current.

ix

= − where y

, iy r r

=

= x

2 x

, iz r

+ y

2

=

0


Domain variables

• Right-click on Model 1 > Definitions and select Variables


Voltage across coil windings

V

=

L

∫

E

⋅

dl

V = Voltage drop across a coil

E = Electric field in the windings (vector)

E = ( E x

, E y

, E z

) dl = Differential length of coil (vector)

L = Total length of wire in a coil


Assumption – Coil with thin wire

• The model assumes that the primary and secondary windings are made of thin wire and multiple number of turns.

• If the wire diameter is less than the skin depth and there are “many” turns, then the coil can be assumed to be a homogeneous current carrying domain.

≈

Coil with thin wires and many turns

Equivalent homogenous current-carrying domain


Voltage across coil windings

Since we do not resolve each turn in a coil in this model and use a homogenized volume approach instead, the voltage drop across a coil can be approximated using the following formula for this model.

V i

=

=

N area

N area

V

∫

E

⋅ dV

V

∫

(

E x ix

+

E y iy

+

E z iz

) dV

N = Number of turns area = cross section of coil dV = Differential volume element

V = Total volume of a subdomain which depicts a coil.


Setting up volume integral

• Right-click on Model 1 > Definitions and select Model Couplings

> Integration . Assign domain 6 which is the primary coil.

• Setup another integration coupling operator and assign domain 5 which is the secondary coil.


Define variables to calculate induced voltage in primary and secondary coils

Variable name

Vip = intop1((mf.Ex*ix+mf.Ey*iy+mf.Ez*iz)*Np/area)

Vis = intop2((mf.Ex*ix+mf.Ey*iy+mf.Ez*iz)*Ns/area)

Description

Induced voltage in primary

Induced voltage in secondary

The operators intop1 and intop2 perform integration of the expression in parenthesis on the geometric entity on which they have been defined, i.e. domains 6 and 5 respectively.


Define variables to calculate current density in coils

Variable name

Vp

Description

Externally applied time-varying voltage across primary coil.

Ip = (Vp + Vip)/Rp Current in primary coil.

Jp = (Np x Ip)/area Current density in primary coil.

Is= Vis/Rs Current in secondary coil.

Js = (Ns x Is)/area Current density in secondary coil.


Define Variables

• Right-click on Model 1 > Definitions and select Variables

• You can either type these or upload from the file ecore_variables.txt


Check the discretization

• Before we proceed further, we need to check the option to view the discretization order


Use linear elements

• In the settings of Magnetic Fields (mf) , expand the Discretization section and set it to Linear .

• We do this for computational efficiency. COMSOL will now use

Linear Vector elements to discretize the Magnetic vector potential.


Set up the physics

• Right-click on Model 1 > Magnetic Fields (mf) and select

Ampère’s Law.

• Assign all the domains corresponding to the E-core to Ampère’s

Law 2 .

• This can be easily done by specifying the Domain Selection as

Core . This is the selection that we created earlier.


Nonlinear BH curve

• In the settings of Ampère’s Law 2 , locate the Magnetic field section and select the Constitutive relation as HB curve

• This will allow us to use a nonlinear BH curve (rather an HB curve) to represent the E-core.

• Note that when using the Magnetic Fields (mf) interface in

COMSOL, we need to define the H-field as a function of the B-field.


Assign Material Properties – Air

• Right-click Model 1 > Materials and select Open Material Browser .

• In the Material Browser , expand Built-In and select Air .

• Right-click on Air and select Add Material to Model .

• In the Settings window, browse to the Material Contents section.

• Type a value of 10 for Electrical conductivity . A small non-zero value is necessary to get numerical convergence


Material Properties – Soft iron

• Right-click Model 1 > Materials and select Open Material

Browser .

• In the Material Browser , expand AC/DC and select Soft Iron

(without losses) .

• Right-click on Soft Iron (without losses) and select Add Material to Model .

• In the Settings window, locate the Geometric Entity Selection section. Change the Selection to Core .

• In the Settings window, browse to the Material Contents section.

• Type a value of 10 for Electrical conductivity . A small non-zero value is necessary to get numerical convergence


Soft iron settings


How to visualize the nonlinear HB curve

Click on the

Plot icon


Nonlinear HB curve


Magnetic flux density, B (T)

Gauge fixing

• Right-click on Model 1 > Magnetic Fields (mf) > Ampère’s Law 1 and select Gauge Fixing for A-field .

• Repeat the same for Ampère’s Law 2 .

• Gauge fixing is used to numerically stabilize the model. It consumes more computational memory but helps in the convergence of the model.


Assign current density to the primary coil


External Current Density .

• Assign domain 6.

• Type ix*Jp , iy*Jp and iz*Jp in the J e edit fields.


Assign current density to the secondary coil


External Current Density .

• Assign domain 5.

• Type ix*Js , iy*Js and iz*Js in the J e edit fields.


Note on Exterior Boundary Conditions

• Boundary 1-5 and 54 – Boundary condition is Magnetic insulation .

• This boundary condition sets the tangential component of the magnetic vector potential A equal to zero. This is an approximation for a boundary that really should be infinitely far away. The user may also use infinite elements for a more high fidelity model. See the

AC/DC Module User's Guide for more information on infinite elements.


Note on Interior Boundary Conditions

• All other boundaries are automatically set to Continuity .

• Note that this is a default boundary condition for all internal boundaries in COMSOL so the user will not have to make any changes.


Mesh – Global settings

• Right-click on Model > Mesh 1 and select Free Tetrahedral .

• Click on the Mesh 1 > Size branch.

• In the Settings window, choose Extremely coarse as the Predefined

Element size.

• Click on the Custom button. Set the Maximum element size to 0.03

.


Mesh – Local settings

• Right-click on Model > Mesh 1 > Free Tetrahedral and select Size .

• Click on the Mesh 1 > Free Tetrahedral > Size 1 branch.

• In the Settings window, select and assign Domains 2-8.

• In the Element Size section, click on the Custom button. Set the

Maximum element size to 0.007

.


Meshed geometry

• On building the mesh you should see about 13770 elements.

• You can click on the Transparency icon in the Graphics window to see this picture


Time-dependent Study

• Select Study 1 > Step 1: Time Dependent .

• In the Settings window, locate the Times edit field and type in range(0,0.002,0.06) .

• Right-click on Study 1 and select Show Default Solver .


Tuning the solver

• The following slides show how to tune the solvers.

• Such tuning is necessary in order to successfully use a realistic nonlinear BH curve in a large time-dependent model.

• The model is nonlinear is space and time.

• The solver needs to be robust enough to handle such nonlinearities.


Absolute Tolerance

Need to specify the absolute tolerances of different variables being solved because of the significant difference in their order of magnitude


Time stepping method

• BDF time stepping is more robust

• Lower the maximum BDF order to

2 to speed up the computation


Use a Direct Solver

• Right-click on Study 1 > Solver Configurations > Solver 1 >

Time-Dependent Solver 1 > Direct and select Enable .

• A direct solver is more robust and hence a better choice for solving highly nonlinear problems.

• We will use the default Direct solver, i.e. MUMPS.


Tuning the Fully coupled Solver

A higher Maximum number of iterations will minimize the risk that the nonlinear solver will terminate prematurely due to the high nonlinearity


Solving the model

• Right-click on Study 1 and select Compute .

• It takes about 30 minutes to solve the model on a 64-bit computer with dual core processor and 4 GB RAM.


Results (Case 1)

• Number of turns in the coils Np = Ns

• Only induction but no step up/down of voltages


Results (Case 1)

• Ip and Is are different by roughly 4 orders of magnitude which is also the difference between Rp and Rs

• Is/Ip

≠ Rs/Rp because Is depends only on Vis but Ip depends on both Vip and Vp


Magnetic flux density distribution (slice plots)


Magnetic flux density distribution

Arrow plot showing magnetic flux concentration through transformer core


Results (Case 2)


Vip/Vis = Np/Ns = 1000

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