3.4 Analysis of the single-phase transformer parameters
3.4.1 Simulation of the single-phase transformer no load condition The purpose of the simulation in no load condition is the calculation of the
and by means of the surface integral, the value of the A.J is calculated: It is 0.8334454 HA2(Henry Amperes2). Then the inductance can be calculated as:
L¼ψ I ¼
Ð
S∇�A�dS
I ¼∮A�dl
I ¼∮A�JdV
I2 ¼0:834454
2:252 ¼0:164 H It is seen that the results obtained by both ways of calculation are identical.
If the ferromagnetic circuit was saturated, the results would be different.
Other calculations can be made for any armature position and any current.
It depends on the reader needs.
and by means of the surface integral, the value of the A.J is calculated: It is 0.8334454 HA2(Henry Amperes2). Then the inductance can be calculated as:
L¼ψ I ¼
Ð
S∇�A�dS
I ¼∮A�dl
I ¼∮A�JdV
I2 ¼0:834454
2:252 ¼0:164 H It is seen that the results obtained by both ways of calculation are identical.
If the ferromagnetic circuit was saturated, the results would be different.
Other calculations can be made for any armature position and any current.
It depends on the reader needs.
3.4 Analysis of the single-phase transformer parameters
The FEM is used for the analysis of the single-phase transformer parameters.
As it is known, the single-phase transformer can be designed into two main configurations: core-type construction (Figure 54a) and shell-type construction (Figure 54b). In the transformers of small powers and low voltage, the primary winding can be wound on the core and on it the secondary winding, as it is seen in Figure 54. Transformers for higher voltage have usually the secondary winding wound closer to the core and on it the primary winding.
The no load test can be simulated by means of FEM. The parameters of the square branch of the equivalent circuit, mainly magnetizing inductance, can be calculated. Second, also the short circuit test can be simulated by FEM. The param- eters of the direct axis of the equivalent circuit, mainly leakage inductances, can be calculated. The usual equivalent circuit of the single-phase transformer is in Figure 55.
The calculation of the equivalent circuit parameters is made for a real trans- former, nameplate and rated values of which are inTable 7. An illustration figure of
¼ of transformer cross-section area is inFigure 56.
3.4.1 Simulation of the single-phase transformer no load condition The purpose of the simulation in no load condition is the calculation of the magnetizing inductance Lμ.
The procedure is the same as in the case of electromagnet analysis. The calcula- tion is started with a preprocessor, where the magnetostatic analysis and the planar
Figure 54.
Winding and core arrangements of the single-phase transformers: (a) core-type construction and (b) shell-type construction.
type of the problem is set up. Frequency is zero, because only one time instant is solved. The z-coordinate (depth) is an active thickness of ferromagnetic core. The total thickness of the ferromagnetic core is obtained from the measurement lFetotal= 49.6 mm. This must be reduced by the value of sheet insulation thickness and air layers between them. This reduction is respected by correction factors kFe= lFe/lFetotal= 0.866 ->lFe= 43 mm, and this value was used in the calculation.
Based on the transformer dimensions, a cross-section area is drawn and the materials are allocated to the blocks as follows: air around the coils; ferromagnetic material of the core; and primary and secondary windings. These are divided into two parts, right and left, and their indication is as follows: primary winding right side (pwr), primary winding left side (pwl), secondary winding right side (swr), and secondary winding left side (swl). The geometry of the cross-section area with the marked parts and setting up of the parameters are inFigure 59. Magnetic energy Wmis absorbed mainly into ferromagnetic core; therefore it is suitable to increase density of the mesh nodes in the core. In the other parts, the magnetic energy is neglected.
Figure 55.
Single-phase transformer equivalent circuit.
Rated voltage of the primary side U1N 230 V
Rated voltage of the secondary side U2N 24 V
Rated power SN 630 VA
Rated frequency f 50 Hz
Rated current of the primary side I1N 2.75 A
Rated current of the secondary side I2N 26.3 A
Number of turns of the primary side N1 354
Number of turns of the secondary side N2 39
Shell-type construction —
Total thickness of the core lFetotal 49.6 mm
Transformer sheets EI 150 N —
No load current I0obtained from no load measurement 0.274 A
Magnetizing inductance Lμobtained from no load measurement 2.82 H Total leakage inductance Lσobtained from short circuit measurement 5.2 mH
Table 7.
Nameplate and rated values of the investigated transformer.
In the no load condition, the secondary winding is open-circuited, and rated voltage at rated frequency is applied to the primary terminals. Under this condition, the primary current, the so-called no load current I0(Figure 55), flows in the primary winding. It is not necessary to draw the individual turns but the whole coil side is replaced by one block in the FEMM program. It is seen inFigure 59that the block (pwl) corresponds to all conductors of the left side of the primary winding.
A calculation based on the magnetostatic analysis is very popular, and many authors recommend this kind of simulation [11].
A constant value of the no load current, which is the magnitude of the sinusoi- dal waveform, is entered in this analysis. The secondary winding is opened, in which no current flows ; therefore, a zero value of the current density is allocated to the blocks corresponding to the secondary winding. The value of the current density in the primary winding can be obtained from the analytical calculation during the transformer design or by measurements on a real transformer, which is this case.
From the no load test, the no load current at rated voltage UNis I0= 0.274 A, which is an effective (rms) value. Its magnitude at the sinusoidal waveform is I0 max
= ffiffiffi p2
I0¼0:387A, but it is true that the waveform of the no load current at rated voltage is not sinusoidal, which is caused by the iron saturation [1, 13]. For illustra- tion inFigure 57, there is no load current waveform of the analyzed transformer.
The current density Jpof the magnitude of the no load current in the primary winding is calculated based on the primary winding number of turns:
Jp¼N1I0 max
Sp ¼354�0:387
0:00065 ¼0:210766 MA m2
where Spis a surface of one part of primary winding, right or left side (pwr or pwl), and can be calculated by means of the geometrical dimensions (Figure 59).
Figure 56.
An illustration figure of¼of transformer cross-section area.
Then the current density is introduced to the block which corresponds to the right side of primary winding (pwr) Jpwr= +0.210766 MA/m2and to the block corresponding to the left side of the primary winding (pwl) Jpwl=0.210766 MA/m2. In the section where materials are defined, B-H curve is introduced, which was obtained by a producer, by the values of magnetic flux density, and by the magnetic field intensity of the employed sheets. The B-H curve is shown inFigure 58.
After a definition of all geometrical dimensions, materials, and current densities, it is necessary to define boundary conditions (Figure 59). Here Dirichlet boundary conditions can be applied, where constant value of the magnetic vector potential A = 0 is defined. It is defined on the circumference of the transformer, and the dialog window of the FEMM program is given as Boundary Property, as seen in Figure 60.
In the last step before starting the calculation, it is necessary to create a mesh of the finite elements. As was mentioned before, in the ferromagnetic core, it is necessary to create a denser mesh, because there is concentrated dominant part of electromagnetic energy. The mesh is created by means of the command
Figure 57.
No load current waveform of the analyzed transformer.
Figure 58.
B-H curve of the sheets employed in the investigated transformer, given by its producer.
In the no load condition, the secondary winding is open-circuited, and rated voltage at rated frequency is applied to the primary terminals. Under this condition, the primary current, the so-called no load current I0(Figure 55), flows in the primary winding. It is not necessary to draw the individual turns but the whole coil side is replaced by one block in the FEMM program. It is seen inFigure 59that the block (pwl) corresponds to all conductors of the left side of the primary winding.
A calculation based on the magnetostatic analysis is very popular, and many authors recommend this kind of simulation [11].
A constant value of the no load current, which is the magnitude of the sinusoi- dal waveform, is entered in this analysis. The secondary winding is opened, in which no current flows ; therefore, a zero value of the current density is allocated to the blocks corresponding to the secondary winding. The value of the current density in the primary winding can be obtained from the analytical calculation during the transformer design or by measurements on a real transformer, which is this case.
From the no load test, the no load current at rated voltage UNis I0= 0.274 A, which is an effective (rms) value. Its magnitude at the sinusoidal waveform is I0 max
= ffiffiffi p2
I0 ¼0:387A, but it is true that the waveform of the no load current at rated voltage is not sinusoidal, which is caused by the iron saturation [1, 13]. For illustra- tion inFigure 57, there is no load current waveform of the analyzed transformer.
The current density Jpof the magnitude of the no load current in the primary winding is calculated based on the primary winding number of turns:
Jp¼N1I0 max
Sp ¼354�0:387
0:00065 ¼0:210766 MA m2
where Spis a surface of one part of primary winding, right or left side (pwr or pwl), and can be calculated by means of the geometrical dimensions (Figure 59).
Figure 56.
An illustration figure of¼of transformer cross-section area.
Then the current density is introduced to the block which corresponds to the right side of primary winding (pwr) Jpwr= +0.210766 MA/m2and to the block corresponding to the left side of the primary winding (pwl) Jpwl=0.210766 MA/m2. In the section where materials are defined, B-H curve is introduced, which was obtained by a producer, by the values of magnetic flux density, and by the magnetic field intensity of the employed sheets. The B-H curve is shown inFigure 58.
After a definition of all geometrical dimensions, materials, and current densities, it is necessary to define boundary conditions (Figure 59). Here Dirichlet boundary conditions can be applied, where constant value of the magnetic vector potential A = 0 is defined. It is defined on the circumference of the transformer, and the dialog window of the FEMM program is given as Boundary Property, as seen in Figure 60.
In the last step before starting the calculation, it is necessary to create a mesh of the finite elements. As was mentioned before, in the ferromagnetic core, it is necessary to create a denser mesh, because there is concentrated dominant part of electromagnetic energy. The mesh is created by means of the command
Figure 57.
No load current waveform of the analyzed transformer.
Figure 58.
B-H curve of the sheets employed in the investigated transformer, given by its producer.
Mesh – Create mesh. After a modification, there were 22,158 finite elements created (seeFigure 61).
Now the Processor can be started, and calculation of the transformer in the no load condition is launched. After the calculation, a distribution of magnetic flux lines in the cross-section area of transformer is seen in postprocessor. Also distribu- tion of magnetic flux density by means of command View density plot can be displayed (seeFigure 62).
A magnetizing inductance calculation is done by means of linkage magnetic flux, according to Eq. (480). It must be done this way, because in the no load condition at rated voltage, the B-H curve is in nonlinear region, which means that energy and coenergy is not the same [4], because usually coenergy is higher, e.g., in this case the electromagnetic energy is 0.163 J and coenergy 0.263 J. It corresponds also to the non-sinusoidal waveform of the no load current, as it is seen inFigure 57.
Therefore, it is better to employ Eq. (480), in which linkage magnetic flux appears. In the program FEMM value A.J, is obtained in such a way that only the
Figure 60.
Definition of the transformer boundary conditions.
Figure 59.
Geometry of the investigated transformer with marked blocks and settings of the parameters.
blocks corresponding to the coils of primary winding are marked (Figure 63) and by means of surface integral calculate the A�J = 0.425975 H/A2:
LμFEMM¼I0 maxψ ¼ Ð
S∇�A�dS
I0 max ¼∮I0 maxA�dl¼∮IA�JdV2
0 max ¼0:4259750:3872 ¼2:84 H: If this value is compared with the value obtained by no load measurement (Lμ= 2.82 H, seeTable 7), it is seen that the difference is less than 1%.
As it is known, during the measurements, it is possible to make measurement also at other voltages, not only rated, which results in various no load currents vs.
voltage and also magnetizing inductances vs. no load currents. Such characteristics (Lμvs. I0) can be employed in transient investigation during the no load trans- former switch onto the grid. For comparison inFigure 64, there are waveforms of such characteristics obtained by measurements and simulations. In the region of the rated voltage and corresponding no load currents I0N, the coincidence is almost perfect, but at the lower currents, the discrepancy is higher because of lower saturation and perhaps lower precision of the B-H curve.
Figure 62.
Magnetic flux lines and magnetic flux density distribution in the cross-section area of the transformer.
Figure 61.
Created mesh of the finite elements.
Mesh – Create mesh. After a modification, there were 22,158 finite elements created (seeFigure 61).
Now the Processor can be started, and calculation of the transformer in the no load condition is launched. After the calculation, a distribution of magnetic flux lines in the cross-section area of transformer is seen in postprocessor. Also distribu- tion of magnetic flux density by means of command View density plot can be displayed (seeFigure 62).
A magnetizing inductance calculation is done by means of linkage magnetic flux, according to Eq. (480). It must be done this way, because in the no load condition at rated voltage, the B-H curve is in nonlinear region, which means that energy and coenergy is not the same [4], because usually coenergy is higher, e.g., in this case the electromagnetic energy is 0.163 J and coenergy 0.263 J. It corresponds also to the non-sinusoidal waveform of the no load current, as it is seen inFigure 57.
Therefore, it is better to employ Eq. (480), in which linkage magnetic flux appears. In the program FEMM value A.J, is obtained in such a way that only the
Figure 60.
Definition of the transformer boundary conditions.
Figure 59.
Geometry of the investigated transformer with marked blocks and settings of the parameters.
blocks corresponding to the coils of primary winding are marked (Figure 63) and by means of surface integral calculate the A�J = 0.425975 H/A2:
LμFEMM¼I0 maxψ ¼ Ð
S∇�A�dS
I0 max ¼∮I0 maxA�dl¼∮IA�JdV2
0 max ¼0:4259750:3872 ¼2:84 H:
If this value is compared with the value obtained by no load measurement (Lμ= 2.82 H, seeTable 7), it is seen that the difference is less than 1%.
As it is known, during the measurements, it is possible to make measurement also at other voltages, not only rated, which results in various no load currents vs.
voltage and also magnetizing inductances vs. no load currents. Such characteristics (Lμvs. I0) can be employed in transient investigation during the no load trans- former switch onto the grid. For comparison inFigure 64, there are waveforms of such characteristics obtained by measurements and simulations. In the region of the rated voltage and corresponding no load currents I0N, the coincidence is almost perfect, but at the lower currents, the discrepancy is higher because of lower saturation and perhaps lower precision of the B-H curve.
Figure 62.
Magnetic flux lines and magnetic flux density distribution in the cross-section area of the transformer.
Figure 61.
Created mesh of the finite elements.