XR-EE-ETK_2007_0111

XR-EE-ETK 2007:011 Royal Institute of Technology Department of Electromagnetic Engineering

Teknikringen 33 100 44 Stockholm, Sweden

Calculation models for estimating DC currents impact on power transformers

Master of Science Thesis By

Sergei Egorov

Supervisor Kurt Gramm ABB AB, Power transformers Ludvika, Product Development department Examiner Göran Engdahl Professor at Electro Technical Design School of Electrical Engineering Acknowledgments This master thesis has been carried out at ABB AB Power transformers department in Ludvika. I want to acknowledge some of the people who have provided me with information and supported me during the work process.

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First of all I would like to direct my foremost gratitude to my supervisor Kurt Gramm from the Product Development department at ABB, Sören Peterson the former transformer expert at ABB, Leif Andersson calculation expert at ABB and my examiner Göran Engdahl professor at Electro Technical Design at KTH for supporting me in my work. Moreover I would like to thank Pavel Stuchl and Jan Johansson for their contribution to my work on finalizing state.

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Abstract The geomagnetic disturbances (magnetic storms) which can give rise to electrical potentials on the surface of the earth causing Geomagnetically Induced Currents (GIC), some times even called solar induced currents, are of great concern of many companies since many years ago. The frequency of GIC is very low - typically between 0,001 and 0,01Hz and can cause peak currents in the neutral of power system in the range of 200-300 A. There are lots of devices that get affected by GIC. This determined the selection of the theme for this research. The report concentrates on GIC influence on power transformers. In earlier estimations only some rough approximations were made in order to handle the DC problem for specific designs, but those approximations were not accurate enough, not sufficient for all the different power transformer types. New investigations are really needed, to increase general precision and time efficiency of calculations. This research concentrates on the investigation of the influence of Geomagnetically Induced Currents (GIC: which is basically a DC current created by solar storms) on the power transformers. The addition of a DC current to the normal AC load current affects the magnetic flux in the transformer core and surrounding in such a way that it becomes shifted by some constant value directly related to the DC current. This shift influences the performance of the core and winding. This report will show the results of the investigations with respect to increased losses, permitted load time, noise calculations and the impacts on the winding e.g. possible hotspots in windings or constructive parts during the impact of dc current. ABB has many different power transformer types, each of them having two different design types: Trafostar (characterized by non split core frame) and TCA (often split core frames). Those two transformer types react differently to the DC current influence due to chosen core type and particular design features. The DC calculation models of the different transformers core types are created in Microsoft Excel and include some simplifying approximations. The DC calculations models of Microsoft Excel are created for Trafostar and TCA cases as well as for different type of magnetic materials. The treated models in this report are D, EY, DY, TY-3, TY-1 and T transformers.

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FIGURES AND TABLES Figures FIGURE 1: THE BASIC CIRCUIT (SCHOOL BOOK EXAMPLE)........................................................................................ 8 FIGURE 2: MAGNETIC CIRCUIT OF THE SCHOOL BOOK EXAMPLE .............................................................................. 9 FIGURE 3: D-CORE TRANSFORMER ......................................................................................................................... 11 FIGURE 4: AIR AREA APPROXIMATION IN THE WINDING. ........................................................................................ 12 FIGURE 5: MAGNETIC EQUIVALENT OF THE D-CORE POWER TRANSFORMER.......................................................... 12 FIGURE 6: THE DC CURRENTS INFLUENCE ON THE FLUX OF THE WINDING............................................................. 14 FIGURE 7: CURRENT PLOT FOR THE SYMMETRICAL RESPECTIVE UNSYMMETRICAL CASE....................................... 15 FIGURE 8: WINDING AND DC CURRENT PLOT. ....................................................................................................... 16 FIGURE 9: EY-CORE TRAFOSTAR........................................................................................................................... 17 FIGURE 10: EY-CORE TCA CASE ........................................................................................................................... 18 FIGURE 11: MAGNETIC EQUIVALENT OF THE HALF OF THE DY-CORE POWER TRANSFORMER WHICH IS VALID FOR

BOTH THE TRAFOSTAR AND TCA DESIGNS................................................................................................... 18 FIGURE 12: DY-CORE TRAFOSTAR......................................................................................................................... 20 FIGURE 13: THE MAGNETIC EQUIVALENT OF THE DY-CORE TRAFOSTAR............................................................... 21 FIGURE 14: DY-CORE TCA CASE........................................................................................................................... 23 FIGURE 15: MAGNETIC EQUIVALENT OF OUR CIRCUIT ........................................................................................... 24 FIGURE 16: TY-3 CORE TRAFOSTAR ...................................................................................................................... 25 FIGURE 17: MAGNETIC EQUIVALENT OF THE TY-3 TRANSFORMER TRAFOSTAR TYPE ........................................... 26 FIGURE 18: TY-1 CORE TRANSFORMER TRAFOSTAR CASE ..................................................................................... 29 FIGURE 19: MAGNETIC EQUIVALENT OF THE TY-1 TRANSFORMER CORE TRAFOSTAR........................................... 30 FIGURE 20: APPROXIMATION OF THE PERMITTED LOAD TIME COMPARED WITH LINE APPROXIMATION.................. 35 FIGURE 21: TEMPERATURE RELATION OF THE MAIN PARAMETERS IN THE HOT SPOT TEMPERATURE DETERMINATION

..................................................................................................................................................................... 38 FIGURE 22: FLUX, INDUCTION FIELD (B), CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR PURE

AC CASE (NO DC CURRENT CASE)................................................................................................................ 39 FIGURE 23: FLUX, INDUCTION FIELD (B), CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR DC

CURRENT CASE AT DC CURRENT OF 20A...................................................................................................... 40 FIGURE 24: LOAD FACTOR K VS TIME TO REACH A WINDING HOT SPOT TEMPERATURE OF 140, 130, 120 & 100 0C 42 FIGURE 25: FLUX, INDUCTION FIELD, CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR THE PURE

AC CASE (NO DC CURRENT CASE) FOR EY-CORE TRAFOSTAR. .................................................................... 43 FIGURE 26: FLUX, INDUCTION FIELD, CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR DC THE

CURRENT CASE FOR EY-CORE TRAFOSTAR AT DC CURRENT OF 20 A. ......................................................... 43 FIGURE 27: LOAD FACTOR K VS TIME TO REACH A WINDING HOT SPOT TEMPERATURE OF 140, 130, 120 & 100 0C

FOR EY-CORE TRAFOSTAR. .......................................................................................................................... 45 FIGURE 28: FLUX, INDUCTION FIELD (B), CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR PURE

AC CASE (NO DC CURRENT CASE) FOR DY-CORE TRAFOSTAR..................................................................... 46 FIGURE 29: FLUX, INDUCTION FIELD (B), CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR DC

CURRENT CASE FOR DY-CORE TRAFOSTAR FOR DC CURRENT OF 20A......................................................... 46 FIGURE 30: FLUX, INDUCTION FIELD (B), CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR PURE

AC CASE (NO DC CURRENT CASE) FOR DY-CORE TCA................................................................................ 48 FIGURE 31: FLUX, INDUCTION FIELD (B), CURRENT AND THE WINDING HOT SPOT TEMPERATURE PLOTS FOR THE DC

CURRENT CASE FOR DY-CORE TCA FOR A DC CURRENT OF 150 A. ............................................................. 49 FIGURE 32: PARTIAL LINEAR APPROXIMATION OF THE MAGNETIZING FIELD OF THE SELECTED ELECTRIC STEEL ... 52 FIGURE 33: DETERMINATION OF THE STARTING POINT OF THE CIRCULAR APPROXIMATION................................... 54 FIGURE 34: DETERMINATION OF THE ENDING POINT OF THE CIRCULAR APPROXIMATION OF THE H(B).................. 54 FIGURE 35: FINAL AND THE LINEAR APPROXIMATIONS OF H(B) COMPARED TO EACH OTHER ................................ 55 FIGURE 36: PARTIAL LINEAR APPROXIMATION OF THE PERMEABILITY DEPENDENT OF THE INDUCTION FIELD ....... 57 FIGURE 37: THE FINAL APPROXIMATION COMPARED WITH THE LINE APPROXIMATION .......................................... 58

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Tables TABLE 1: TRANSFORMER RESIGNATIONS FOR D-CORE........................................................................................... 11 TABLE 2: MAGNETIC RESIGNATIONS D-CORE ........................................................................................................ 13 TABLE 3: TRANSFORMERS RESIGNATIONS EY-CORE.............................................................................................. 18 TABLE 4: MAGNETIC RESIGNATIONS EY-CORE...................................................................................................... 19 TABLE 5: TRANSFORMER RESIGNATIONS DY-CORE TRAFOSTAR ........................................................................... 20 TABLE 6: MAGNETIC RESIGNATIONS DY-CORE TCA............................................................................................. 21 TABLE 7: TRANSFORMER RESIGNATIONS DY-CORE TCA ...................................................................................... 23 TABLE 8: TRANSFORMER RESIGNATIONS TY-3 CORE TRAFOSTAR......................................................................... 26 TABLE 9: MAGNETIC RESIGNATIONS DY-CORE TRAFOSTAR.................................................................................. 27 TABLE 10: THE RECOMMENDED THERMAL CHARACTERISTICS FOR EXPONENTIAL EQUATIONS.............................. 38 TABLE 11: RESULTS REGARDING THE SHIFT, LOSS AND NOISE CALCULATIONS FOR THE 30R122 MATERIAL D-CORE

TCA............................................................................................................................................................. 41 TABLE 12: RESULTS REGARDING WINDING HOT SPOT CALCULATIONS FOR 30R122 MATERIAL D-CORE TCA ....... 41 TABLE 13: RESULTS OF FLUX SHIFT, LOSSES, LOAD AND NOISE CALCULATIONS FOR THE 30R122 MATERIAL EY-

CORE TRAFOSTAR......................................................................................................................................... 44 TABLE 14: RESULTS OF THE WINDING HOT SPOT TEMPERATURE CALCULATION FOR 30R122 MATERIAL EY-CORE

TRAFOSTAR. ................................................................................................................................................. 44 TABLE 15: RESULTS OF THE FLUX SHIFT, LOAD, LOSSES AND NOISE CALCULATIONS FOR 30R122 DY-CORE

TRAFOSTAR. ................................................................................................................................................. 47 TABLE 16: RESULTS OF THE WINDING HOT SPOT TEMPERATURE CALCULATIONS 2 FOR MATERIAL 30R122 DY-

CORE TRAFOSTAR......................................................................................................................................... 47 TABLE 17: RESULTS OF SHIFT, LOAD, LOSS AND NOISE CALCULATIONS FOR MATERIAL 30R122 DY-CORE TCA... 49 TABLE 18: RESULTS OF THE WINDING HOT SPOT TEMPERATURE CALCULATIONS FOR MATERIAL 30R122 DY-CORE

TCA............................................................................................................................................................. 50 TABLE 19: SELECTED POINT FOR THE LINEAR APPROXIMATION OF H=H(B) .......................................................... 53 TABLE 20: THE RESPECTIVE VALUES FOR PERMEABILITY AND INCREASE FACTORS CALCULATED FROM THE VALUES

OF THE MAGNETIZING FIELD AND INDUCTION FIELD. .................................................................................... 56

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TABLE OF CONTEST 1. INTRODUCTION............................................................................................................................................. 7

1.1 BACKGROUND ............................................................................................................................................... 7 1.2 PROBLEM STATEMENT................................................................................................................................... 7 1.3 GOAL ............................................................................................................................................................ 7

2. MAGNETIC EXAMPLE AND APPROXIMATIONS.................................................................................. 8 2.1 INTRODUCTION OF THE MAGNETIC CIRCUITS (SCHOOLBOOK EXAMPLE) ........................................................ 8 2.2 MAIN PREPARATIONS BEFORE THE BUILDING OF MODELS FOR DIFFERENT POWER TRANSFORMERS TYPES .. 10

3. CONSTRUCTION OF THE TRANSFORMERS CALCULATION MODELS....................................... 10 3.1 D-CORE ....................................................................................................................................................... 10 3.2 INDUCTION FIELD SHIFT DETERMINATION AS THE RESULT OF THE INFLUENCE OF DC CURRENT.................. 14 3.3 EY-CORE..................................................................................................................................................... 17 3.4 DY CORE..................................................................................................................................................... 20

3.4.1 DY-core Trafostar............................................................................................................................... 20 3.4.2 DY-core TCA ...................................................................................................................................... 22

3.5 TY-3-CORE ................................................................................................................................................. 25 3.5.1 TY-3 core Trafostar ............................................................................................................................ 25

3.6 TY-1 CORE .................................................................................................................................................. 29 3.6.1 TY-1 core Trafostar ............................................................................................................................ 29

4 MAIN CALCULATIONS ............................................................................................................................... 32 4.1 NO LOAD LOSSES INCREASE DURING INFLUENCE OF A DC CURRENT ........................................................... 32 4.2 HARMONICS DETERMINATION ..................................................................................................................... 33 4.3 CALCULATION OF THE INCREASE FACTORS OF THE WINDING LOSSES DUE THE INFLUENCE OF THE DC CURRENT........................................................................................................................................................... 33 4.4 PERMITTED LOAD TIME CALCULATION DUE TO CORE RESTRICTIONS (FLITCH PLATE) .................................. 35 4.5 DETERMINATION OF THE NOISE INCREASE DURING THE DC CURRENTS INFLUENCE..................................... 36 4.6 DETERMINATION OF THE WINDING HOT SPOTS TEMPERATURE VERSUS TIME DURING THE INFLUENCE OF DC CURRENT........................................................................................................................................................... 36

4.7 RESULTS ...................................................................................................................................................... 39 4.7.1 D-CORE .................................................................................................................................................... 39

4.7.1.1 D-core TCA ..................................................................................................................................... 39 4.7.2 EY-CORE.................................................................................................................................................. 42

4.7.2.1 EY-core Trafostar ............................................................................................................................ 42 4.7.3 DY-CORE ................................................................................................................................................. 45

4.7.3.1 DY-core Trafostar............................................................................................................................ 45 4.7.3.2 DY-core TCA ................................................................................................................................... 48

4.7.4 TY-3 CORE, T-CORE & TY-1 CORE TYPES ................................................................................................ 50 4.7.5 COMMENTS OF THE RESULTS CALCULATIONS........................................................................................... 50

5 CONCLUSIONS AND FUTURE WORK ..................................................................................................... 51 APPENDIX.......................................................................................................................................................... 52

A.1 APPROXIMATION OF THE MAGNETIZING FIELD WITH KNOWN INDUCTION FIELD )(BHH = . ................... 52 A.2 APPROXIMATION OF THE PERMEABILITY WITH KNOWN INDUCTION FIELD )(Bμμ = .............................. 56 A.3 INPUT DATA SHEETS FOR THE CREATED MODELS ........................................................................................ 59

A.3.1 D-core ................................................................................................................................................ 59 A.3.2 EY-core .............................................................................................................................................. 60

A.3.3 DY-CORE................................................................................................................................................. 61 A.3.3.1 DY-core Trafostar ........................................................................................................................... 61 A.3.3.2 DY-core TCA................................................................................................................................... 62

REFERENCE...................................................................................................................................................... 63

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1. Introduction

1.1 Background The influence of DC currents on the power transformers have been an important question for many years, however no simple calculation tool have been developed. E.g. currently ABB had no general tool for determination of permitted load time, or even a trustful knowledge about the DC current influence on the properties of the involved constructing materials. Some rough approximations have only been made in the earlier works to describe this DC currents influence on the transformers. The customers' demands for more detailed information about the influence of DC currents on power transformers have increased dramatically under the last couple of years. This makes this matter of highly priority. There are several aspects to be considered that limits the possible applied load on the transformers such as no load loss and winding loss increase factors, permitted load time and the hot spot temperature in winding.

1.2 Problem statement The problem with DC current influence on the transformers has been tackled before but only with some rough approximations and most of the calculations were performed manually for specific models. No general tool for simple calculations due to DC current influence exists at the moment. E.g. one of the approximations made in earlier calculations neglects the air flux in the winding. The reason why the problems haven't been handled properly earlier is lack of time (there was always something more urgent that had to be done). ABB has for a long time been aware of the problem, and has after customers' demands for more information about the matter, decided to work with this subject. This investigation report contributes to this work.

1.3 Goal The aim of this project regarding the influence of DC currents was the following issues.

• Theoretically create the reasonable magnetic flux models for the different types of power transformers such as D, EY, DY, T, TY-3 and TY-1 core types both for Trafostar and TCA transformer types.

• Create calculation tools in order to calculate the additional core losses, permitted load time, noise calculations, hot spot temperature and increase of the winding losses with the main focus on the Trafostar transformers

• Well written program documentation for the calculation tools. • Project report in English.

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2. Magnetic example and approximations

2.1 Introduction of the magnetic circuits (schoolbook example) In figure 1 a typical school book example can be seen (without an air gap). Note: In all the following figures only the magnetizing windings are shown

Figure 1: The basic circuit (school book example)

It is assumed that the following parameters are known.

:1l Magnetic path length 1(red) [ m ] :2l Magnetic path length 2(green) [ m ] :1A Core area region 1 [ 2m ] :2A Core area region 2 [ 2m ]

N: Number of turns in winding I Winding current [ A ]

:1μ Permeability region 1 (red) [ ( )( )mA

sV*

* ] :2μ Permeability region 2 (green) [ ( )

( )mAsV

** ]

Task: Calculate the instantaneous values of the magnetic flux densities (inductions fields) 1B and 2B . The equivalent magnetic circuit of the figure 1 can be seen in figure 2

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Figure 2: Magnetic circuit of the school book example

wφ Total flux in winding [Wb ]

1Fφ Total flux in the iron core [Wb ]

1R Reluctance region 1 [ ( )sVA

* ]

2R Reluctance region 2 [ ( )sVA

* ]

1R is the reluctance of the core leg and 2R is the reluctance of the region 2 which will be divided in two parts in the later calculations. Those parts are yokeR (reluctance corresponding to the sum of the upper and lower parts (yoke) in the region 2) and limbR (reluctance corresponding to the vertical part of the region 2(side limb)) they are combined in this example because of the same area in both parts. The winding flux is equal to the total flux in the core 1FW φφ = (no air flux is considered).

μAlR = (2.1-1)

Magnetic circuit law applied on the loop gives the following

NIlHlH =+ 2211 (2.1-2) Well known relationship between induction field B[T] and the magnetizing field H[A/m] is

[ ]THB μ= (2.1-3)

[ ]mABH /μ

=⇒

By using equation (2.1-2) and (2.1-3) one gets

NIlBlB=+ 2

2

21

1

1

μμ (2.1-4)

The same flux in both regions yields

[ ]WbBABA 2211 ==φ (2.1-5)

[ ]TBAAB 1

2

12 =⇒

10

Equation (2.1-5) inserted in equation (2.1-4) results in the new equation

NIlBAAlB

=+ 22

1

2

11

1

1

μμ (2.1-6)

Our desired results can now be obtained from the equations (2.1-7) and (2.1-8)

2

2

2

1

1

11

μμl

AAl

NIB+

= [T] (2.1-7)

12

12 B

AAB = [T] (2.1-8)

This was a typical school book example. Unfortunately in the real life it’s not always that simple but the main principle is the same For instance the permeability is not a constant and the known parameters are limited only to the dimensions of the particular circuit and feeding voltage source. The feeding voltage source determines the total winding flux, which can be approximated as long as the voltage, frequency and number of the winding turns are known. The problem becomes more complicated when the air flux and the influence of the dc current have to be taken into consideration. But still the main magnetic principle of the circuit remains the same as in example above.

2.2 Main preparations before the building of models for different power transformers types In order to create the effective and helpful tool that could easily be used by a single user who is less interested in all the details of the calculation background. It is necessary to “construct” the relationship between H and i.e. H=H(B) Unfortunately one approximation wasn’t enough for creation of all the models. And another approximation for the determination of the permeability μ with known induction field ][TB had to be created μ=μ(B). The reason for that was that Microsoft Excel could not find any solution for more advanced models with the first mentioned approximation, see Appendix A.1 and A.2.

3. Construction of the transformers calculation models

3.1 D-core One of the simplest power transformers is the D-core transformer, see figure 3 and table 1. It reminds a lot of the example in part 2.1 in this report "Introduction of the magnetic circuits". The main difference is that windings are placed on both limbs in the D-core otherwise they are completely identical. For this core type it can be assumed that

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Trafostar=TCA but only regarding the magnetic circuit principle (the same model can be used for the calculations). The mechanical construction is not identical for those two design types.

Figure 3: D-core transformer

Lh Limb height [m] Lp Limb pitch [m] Yh Yoke height [m] A Area for respective region [m2] Wh Winding height [m] l1 Magnetic path length 1

region 1 (red) [m]

l2 Magnetic path length 2 region 2 (green)

[m]

lair1 Magnetic path length of the air 1(in winding)

[m]

lair2 Magnetic path length of the air 2 (outside)

[m]

Table 1: Transformer resignations for D-core. Due to the symmetry only half the circuit may be considered. This makes the model less complicated and results in the following equivalent magnetic circuit (see figure 5). By considering only half of the circuit one can make this model resemble the example in 2.1, but this time including the air flux and the influence of the DC current. That's why the model will be more complicated. When it comes to the definition of the magnetic path length of the air flux, the Magnetic path length

2airl will be neglected (assumed to be =0) and the 1airl will be set equal to the winding

height in all following models, see figure 3. The reason for that is that the magnetizing field outside the winding is much smaller than the magnetizing field in the winding

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The equation (3.1-1) describes already an approximation, where the magnetizing field outside the winding is approximated by a constant. By using the earlier mentioned approximation

12 airair HH << the magnetic air path can be reduced to just 1airl in equation (3.1-2).

NIlHlH airairairair =+ 2211 (3.1-1) NIlHHH airairairair ≈⇒<< 1112 (3.1-2)

This approximation is absolutely not perfect but is accurate enough in order to give some reasonable results. The area approximation of the air flux is that it is assumed to be the area between the winding and the core limb, see figure 4, table 2 and equation (3.1-3)

Figure 4: Air area approximation in the winding.

][2

21limb

2

mAAAWA wd

Air −=−⎟⎠⎞

⎜⎝⎛= π (3.1-3)

Figure 5: Magnetic equivalent of the D-core power transformer

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N Winding turn number

I Winding current [A] Φw Total flux in winding [Wb] ΦAir Air flux [Wb] ΦF1 Total flux in the iron core [Wb] RAir Reluctance of the air [A/(V*s)] R1 Reluctance of region 1 [A/(V*s)] R2 Reluctance of region 2 [A/(V*s)]

Table 2: Magnetic resignations D-core In order to solve this system a solver, which is a mathematic tool already existing in Microsoft Excel, has to be used. The solver basically guesses some value. In this case it is the induction field value in the region 1 1B which has to fulfill the magnetic requirements of the circuit. Under the assumption that induction field 1B is known (correctly guessed) the model creation can be started. The magnetizing field 1H is directly approximated from the induction field 1B by using our approximation ])[( TBH . The total flux in iron 1Fφ can be calculated by multiplying the area and induction field of the region 1.

⎩⎨⎧

=⇒

][]/[

111

11 WbBA

mAHB

Fφ

21

21 AFFB φφ =⇒ Induction field in region 2 [T ]

22 HB ⇒ Magnetizing field in the region 2 [ mA / ] Our magnetizing current can be calculated from the first loop equation (see figure 5) which is exactly the same as equation (2.1-2)

][2211 AN

lHlHI +=

Now when the magnetizing current is known the second loop can be used to determine the magnetizing field in the air ]/[ mAH Air equation (3.1-4)

]/[ mAlNIHAir

Air = (3.1-4)

By using the magnetizing field of the air both the induction field ][TBAir and flux of the air

][WbAirφ can be determined by equations (3.1-5) and (3.1-6)

][00 THlNIB AirAir

Air μμ == (3.1-5)

][0 WbBAlNIA AirAirAir

AirAir == μφ (3.1-6)

The requirement that has to be fulfilled by the solver is that

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01 =−− AirFw φφφ (3.1-7)

, where ][Wbwφ is known (by voltage) and ][1 WbFφ , ][WbAirφ have to be calculated.

3.2 Induction field shift determination as the result of the influence of DC current At this moment our base model for the D-core is complete. The next step is to introduce the DC current ][AIDC into our model. This can easily be done by introducing an additional parameter resulting in a DC addition ][Wbshiftφ to the winding flux. This additional term is the result of the DC current influence on the transformer which leads to the change of the total winding flux: the total flux is no longer pure AC and now contains a DC component, see equation (3.2-1).

][WbDC

shift

AC

wwTot φφφ += (3.2-1) The addition of the DC part in the winding flux will result in that the original AC part will be shifted by the DC component, see figure 6.

The influence of DC current on the flux in the winding

-1

-0,5

0

0,5

1

1,5

-100 -50 0 50 100

Degrees

Flux

[Wb] FluxWindingAc[Wb]

FluxShift[Wb]FluxWinding[Wb]

Figure 6: The DC currents influence on the flux of the winding

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The DC shift in flux depends on magnitude of the DC current, transformer design, core type, and core material. The addition of the DC current results in another noticeable effect: the extreme value of the excited current can for that unsymmetrical (DC current) case only be reached once in a period but two times for the symmetrical (pure AC) case, see figure 7.

Current plot for the symmetrical and unsymmetrical cases

-300

-200

-100

0

100

200

300

400

500

600

700

0 50 100 150 200 250 300 350 400

Degrees

I[A] I (unsym)[A]

I (sym) [A]

Figure 7: Current plot for the symmetrical respective unsymmetrical case

In order to actually calculate the DC shift in the flux with the known DC current the solver has to fulfill another requirement; equation (3.2-2), see also figure 8

lowerupper AA = (3.2-2) , where:

upperA Current area above the DC current [ 2m ]

lowerA Current area under the DC current [ 2m ] The resulting current during a period is only the DC component (the resulting AC current is 0)

In order to implement the aforesaid in Microsoft Excel the following has to be done for a half of the period: • Subtract the DC current from the total magnetizing current ][AII DCw − , because

the DC current should be our new zero level. • Divide the values of ][AII DCw − at the start and end phase positions to reduce the error

in the area calculation.

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• Now by summation of the all the values over a half of period of ][AII DCw − the equal area criteria can be checked in equation (3.2-2). The summation of over a half period corresponds to an integration which yields 0=− lowerupper AA if equation (3.2-2) is fulfilled, see figure 8.

Figure 8: Winding and DC current plot.

After the DC currents are introduced in our model the induction field shift in the core limb

][1 TB shift and the nominal magnitude of the induction field [ ]TB nom1 (peak value of the induction field in the core limb for pure AC case) can be calculated by use of equation (3.2-3) and (3.2-4)

][2

minmax TBBBnom−

= (3.2 -3)

minBBB nomshift −= (3.2-4) Those parameters will be useful in the loss calculation later on. Note: The calculations of the above mentioned items were solved in Microsoft Excel in two steps, as Microsoft Excel could not solve it all at once. The first step was to determine the induction field ][1 TB to meet the circuit magnetic requirements and in the second step the DC current was introduced in order to calculate the influence on the transformer. Step 1 No DC current/flux shift considered With start guess induction field 1[ ]B T fulfill equation (3.2-5) in order to get the start values for step 2.

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Start guess:

1[ ]B T Equation to fulfill

0w F Airφ φ φ− − = (3.2-5) Step 2 Determination of the circuits magnetic with DC current included With DC current considered another guess parameter have to be introduced in solver the constant flux shift [ ]shift Wbφ which have to fulfill the equal area criteria equation (3.2-2) Start guesses:

1[ ]B T and [ ]shift Wbφ Equations to fulfill

0w F Airφ φ φ− − =

lowerupper AA =

3.3 EY-core The principal magnetically difference between Trafostar and TCA transformers models can be seen in figure 9 and 10 see also table 3. I.e. TCA has separated magnetic frame parts and differs from Trafostar.

Figure 9: EY-core Trafostar

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Figure 10: EY-core TCA case

Lh Limb height [m] Lp Limb pitch [m] Yh Yoke height [m] A Area for respective region [m2] Wh Winding height [m] l1 Magnetic path length 1

region 1 [m]

l2 Magnetic path length 2 region 2

[m]

l3 Magnetic path length 3 region 3

[m]

Table 3: Transformers resignations EY-core Due to symmetry only half of the circuits need to be considered for both the Trafostar and TCA cases. The magnetic principle for both design types are identical because of the symmetry assumption (Trafostar=TCA). The magnetic equivalent of half of that circuit can be seen in figure 11.

Figure 11: Magnetic equivalent of the half of the DY-core power transformer which is valid for both the

Trafostar and TCA designs

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N Winding turn number

I Winding current [A] Φw Total flux in winding [Wb] ΦAir Air flux [Wb] ΦF3 Total flux in the iron core [Wb] RAir Reluctance of the air [A/(V*s)] R1 Reluctance of region 1 [A/(V*s)] R2 Reluctance of region 2 [A/(V*s)] R3 Reluctance of region 3 [A/(V*s)]

Table 4: Magnetic resignations EY-core

Our start guesses for this transformer are the induction field ][3 TB and flux shift ][Wbshiftφ . This results in the following.

⎩⎨⎧

=⇒

][]/[

333

33 WbBA

mAHB

Fφ

⎩⎨⎧

==

⇒][][

11

223

3

3

TBTB

A

AF

F

F

φ

φ

φ

⎩⎨⎧

⇒⎭⎬⎫

]/[]/[

2

1

2

1

mAHmAH

BB

The magnetizing fields ]/[1 mAH , ]/[2 mAH and ]/[3 mAH are directly obtained from the inductions fields The magnetizing current can be calculated by using the equation (3.3-1) (see loop 1 in figure 11.)

NIlHlHlH =++ 112233 (3.3-1)

][112233 AN

lHlHlHI

++=⇒

Thereafter the magnetizing current is calculated by use of the equations (3.1-4), (3.1-5) and (3.1-6) that can be used in order to calculate the magnetizing field, induction field and flux of the air.

⎪⎩

⎪⎨

⎧⇒

][][

]/[

WBTB

mAH

Air

Air

Air

φ

The flux shift in the core leg can be determined by checking the equal area criteria in the equation (3.2-2) and by using the equations (3.2-3) and (3.2-4). The induction field shift in the core leg can be determined in a similar way as in D-core part.

20

3.4 DY core

3.4.1 DY-core Trafostar The main model for DY-core for the Trafostar case can be seen in figure 12 and table 5. Even for this case the symmetry assumption is used and it brings results, although only half of the circuit is considered.

Figure 12: DY-core Trafostar

Lh Limb height [m] Lp Limb pitch [m] Lps Limb pitch side [m] Yh Yoke height [m] A Area for respective region [m2] Wh Winding height [m] l1 Magnetic path length 1(red) [m] l2 Magnetic path length 2(blue) [m] l3 Magnetic path length

3(orange) [m]

l4 Magnetic path length 4(green)

[m]

lAir Magnetic path length of the air region(Only the length parallel to core leg considered)

[m]

Table 5: Transformer resignations DY-core Trafostar By observing our transformer model in figure 12 and by taking advantage of the symmetry criteria in the circuit, our magnetic equivalent got the following appearance, see figure 13.

21

Figure 13: The magnetic equivalent of the DY-core Trafostar

N Number of turns in winding

I Winding current [A] Φw Total flux in winding [Wb] ΦAir Air flux [Wb] ΦF3 Total flux in the iron core [Wb] RAir Reluctance of the air [A/(V*s)] R1 Reluctance of region 1 [A/(V*s)] R2 Reluctance of region 2 [A/(V*s)] R3 Reluctance of region 3 [A/(V*s)] R4 Reluctance of region 4 [A/(V*s)]

Table 6: Magnetic resignations DY-core TCA The start guesses for the solver in Microsoft Excel in this case are the inductions fields ][2 TB ,

][3 TB and the flux shift ][Wbshiftφ . These result in the following:

⎩⎨⎧ =

⇒3

3333

][][

μφ WbAB

TB F

⎩⎨⎧ =

⇒2

2222

][][

μφ WbAB

TB F

, where the permeability values 2μ and 3μ are directly obtained from μ=μ(B) approximation, see Appendix A2. After the values of the total flux in the iron core ][3 WbFφ and the flux ][2 WbFφ have been obtained the flux ][4 WbFφ could be determined by equation (3.4-1)

][234 WbFFF φφφ −= (3.4-1)

22

Then it follows

⎩⎨⎧

⇒

⎪⎪⎭

⎪⎪⎬

⎫

=

=

4

1

4

44

1

21

][

][

μμ

φ

φ

TA

B

TA

B

F

F

]/[

]/[

]/[

]/[

4

44

3

33

2

22

1

11

mABH

mABH

mABH

mABH

μ

μ

μ

μ

=

=

=

=

From the equation (3.4-2) obtained from a loop in the figure 13 the magnetizing current can be calculated.

332211 lHlHlHNI ++= (3.4-2)

][332211 AN

lHlHlHI

++=⇒

After the magnetizing current is known (by using the equations (3.1-3), (3.1-4) and (3.1-5)) the magnetizing field, the induction field and the flux in the air can be obtained. The solver requirements that have to be met are:

03 =−− AirFw φφφ (3.4-3) 0112244 =−− lHlHlH (3.4-4)

The flux shift and the induction field shift can be calculated in a similar way as in earlier models.

])[( TBμ approximation was used in this model for determination of the magnetic requirements of the system instead of the ])[( TBH used in the previous models as the solver didn't manage to find the solution with the ])[( TBH approximation in this model

3.4.2 DY-core TCA The figure 14 shows the drawing of the DY-core transformer TCA case. The symmetry of the circuit makes it possible to consider only the half of the circuit. It simplifies the calculations considerably. The magnetic equivalent of this model can be seen in figure 15, see also table 7 for transformer parameters.

23

Figure 14: DY-core TCA case

Lh Limb height [m] Lp Limb pitch [m] Lps Limb pitch side [m] Yh Yoke height [m] A Area for respective region [m2] Wh Winding height [m] l11 Magnetic path length

11(green) [m]

l12 Magnetic path length 12(red) [m] l13 Magnetic path length

13(blue) [m]

l21 Magnetic path length 21(orange)

[m]

l22 Magnetic path length 22(pink)

[m]

Table 7: Transformer resignations DY-core TCA

24

Figure 15: Magnetic equivalent of our circuit

Calculation procedure: Introduced solver guess parameters

13[ ]B T Induction field in the wound limb (outer frame)

21[ ]B T Induction field in the wound limb (middle frame) [ ]shift Wbφ Flux shift in the winding flux

From which it follows

⎩⎨⎧

⇒13

1313

][μ

φ WbB F

⎩⎨⎧

⇒21

2121

][μ

φ WbB F

⎪⎩

⎪⎨

⎧⇒

⎪⎩

⎪⎨

⎧

===

⇒⎭⎬⎫

22

11

12

22

11

12

21

13

][][][

2221

1113

1213

μμμ

φφ

φ

φ

φ

TBTBTB

A

A

A

F

F

F

F

F

, where the permeability values are directly approximated from the inductions fields.

]/[]/[]/[

,,,

2222

1111

1212

22

11

12

2222

1111

1212

mAHmAHmAH

BBB

B

B

B

μ

μ

μ

μμμ

===

⇒⎪⎭

⎪⎬

⎫

The loop equations which are taken from the magnetic equivalent in figure 15 results in the following relation for the magnetizing current.

NIlHlHlH =++ 131312121111 (3.4.2-1)

][131312121111 AN

lHlHlHI ++=

25

With the current being known the equations (3.1-4), (3.1-5) and (3.1-6) can be used to calculate the magnetizing field, induction field and flux of the air.

⎪⎩

⎪⎨

⎧

====

=⇒

][][

]/[

0

00

WbBAATHB

mAHI

AirAirlNI

AirAir

AirlNI

Air

lNI

Air

Air

Air

Air

μφμμ

Finally the magnetic requirements for solver to fulfill are

0=−− AirFw φφφ (3.4.2-2) 022222121131312121111 =−−++ lHlHlHlHlH (3.4.2-3)

Here ][Wbwφ is known and the ][WbFφ , ][WbAirφ are guessed by the solver. The equation (3.4.2-3) was obtained from loop 1 in figure 15. After that the magnetic requirements were met and the Microsoft Excel yielded the basic results (without the DC current). The influence of the DC current can now be included in our model. The calculation of the flux shift and induction field shift in the core limb can be performed as in the previous models.

3.5 TY-3-core

3.5.1 TY-3 core Trafostar In figure 16 the TY-3 a Trafostar power transformer is shown. This is a 3-phase transformer which makes the magnetic behavior of the circuit harder to describe and the earlier used symmetry assumptions can’t be used for this transformer type.

Figure 16: TY-3 core Trafostar

26

Resignations are shown in table 8. Lh Limb height [m] Lp Limb pitch [m] Lps Limb pitch side [m] Yh Yoke height [m] A Area for respective region [m2] Wh Winding height [m] l11 Magnetic path length

11(blue) [m]

l12 Magnetic path length 12(red) [m] l13 Magnetic path length

13(black) [m]

l22 Magnetic path length 22(green)

[m]

l23 Magnetic path length 23(grey)

[m]

l33 Magnetic path length 32(pink)

[m]

l42 Magnetic path length 42(orange)

[m]

l43 Magnetic path length 43(yellow)

[m]

Table 8: Transformer resignations TY-3 core Trafostar The magnetic equivalent for this transformer can be seen in figure 17.

Figure 17: Magnetic equivalent of the TY-3 transformer Trafostar type

27

Rij Reluctances for respective region

[A/(V*s)]

ΦA Flux in phase a [Wb] ΦB Flux in phase b [Wb] ΦC Flux in phase c [Wb] Ia Magnetizing current in phase

a [A]

Ib Magnetizing current in phase b

[A]

Ic Magnetizing current in phase c

[A]

N Number winding turns for respective phase

ΦAF Flux in iron phase a [Wb] ΦBF Flux in iron phase b [Wb] ΦCF Flux in iron phase c [Wb] ΦAAir Flux in air phase a [Wb] ΦBAir Flux in air phase b [Wb] ΦCAir Flux in air phase c [Wb]

Table 9: Magnetic resignations DY-core Trafostar The Magnetic equivalent in figure 17 results in the following equation system. The flux of each phase can be directly determined from the applied voltage source, number of the winding turns, frequency and the introduced flux shift equations (3.5.1-1), (3.5.1-2) and (3.5.1-3)

][)cos( WbtNV

shiftshiftAC Aa

AAA φωω

φφφ +=+= (3.5.1-1)

][)3

2cos( WbtNV

shiftshiftAC Bb

BBB φπωω

φφφ +−=+= (3.5.1-2)

][)3

4cos( WbtNV

shiftshiftAC Cc

CCC φπωω

φφφ +−=+= (3.5.1-3)

The fluxes 2φ , 3φ and 4φ can be expressed by 1φ and equations (3.5.1-4), (3.5.1-5) and (3.5.1-6)

][12 WbAFφφφ += (3.5.1-4) ][123 WbBFAFBF φφφφφφ ++=+= (3.5.1-5)

][134 WbCFBFAFCF φφφφφφφ +++=+= (3.5.1-6) The inner and outer loop equations corresponding to the equivalent circuit in figure 17 are:

aa INlHlHlH −=−+ 131312121111 (3.5.1-7)

bbaa ININlHlHlH −=−+ 232322221313 (3.5.1-8)

ccbb ININlHlHlH −=−+ 333332322323 (3.5.1-9)

cc INlHlHlH =++ 434342423333 (3.5.1-10)

28

0434342423232222212121111 =+++++ lHlHlHlHlHlH (3.5.1-11) After that the main equation system is defined the creations of the model can be started. As in the earlier models start guesses will be introduced. This model is much larger than the previous ones and will require four start guess parameters, namely the inductions fields ][11 TB , ][13 TB , ][23 TB and ][33 TB . We then get.

⎪⎪⎩

⎪⎪⎨

⎧

====

⇒

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

====

⇒

⎪⎪⎭

⎪⎪⎬

⎫

]/[]/[]/[]/[

][][][

][

3333

2323

1313

1111

33

23

13

11

333333

232323

131313

1111111

33

23

13

11

mAHmAHmAHmAH

WbBAWbBAWbBA

WbBA

BBBB

B

B

B

B

CF

BF

AF

μ

μ

μ

μ

μφμφμφμφ

⎪⎪

⎩

⎪⎪

⎨

⎧

⎪⎪⎩

⎪⎪⎨

⎧

⎪⎪⎭

⎪⎪⎬

⎫

⇒

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⇒

⎪⎪⎭

⎪⎪⎬

⎫

⇒

⎪⎪⎩

⎪⎪⎨

⎧

⎪⎪⎭

⎪⎪⎬

⎫

⇒

⎪⎪⎭

⎪⎪⎬

⎫

]/[,]/[]/[]/[

,][,][][

][][][][

4342

32

22

12

4342

32

22

12

4342

32

22

4

3

2

121

mAHHmAHmAHmAH

TBBTBTB

WbWbWb

TB

CF

BF

AF

μμμμμ

φφφ

φφφφ

When the magnetizing field is known for all the regions, loop equations (3.5.1-7)-(3.5.1-11) can be used in order to determine the phase currents and make sure that magnetic behavior of the circuit is satisfied. Equation (3.5.1-7) gives

][121211111313 AN

lHlHlHI

aa

−−= (3.5.1-12)

Equations (3.5.1-7) and (3.5.1-8) give

][2222121211112323 AN

lHlHlHlHI

bb

−−−= (3.5.1-13)

Equation (3.5.1-11) gives

][434342423333 AN

lHlHlHI

cc

++= (3.5.1-14)


][4343424232322323 AN

lHlHlHlHI

bb

+++= (3.5.1-15)

Equations (3.5.1-12)-(3.5.1-15) describe the magnetic relations of the phase currents in the transformer. It should also be noted that two expressions for the phase current bI equations exist as well, see equations (3.5.1-13) and (3.5.1-15) The solver requirements to be met for this transformer in order to satisfy the magnetic behavior of the circuit are equations (3.5.1-11), (3.5.1-16)-(3.5.1-18) and the comparison of those two expressions of the phase current bI is done by the equations (3.5.1-13) and (3.5.1-15).

29

][0 WBAirAAFA =−− φφφ (3.5.1-16) ][0 WBAirBBFB =−− φφφ (3.5.1-17) ][0 WBAirCCFC =−− φφφ (3.5.1-18)

The Microsoft solver can’t provide a direct solution to this complex calculation situation. Therefore the user has to find by trial and error various [ ]shift Wbφ in the different phases, until the same desired DC current are reached for all the phases. By fulfilling the equal area criteria of the current equation (3.2-2) for each phase, the DC current can be obtained by the solver for respective winding independently.

3.6 TY-1 core

3.6.1 TY-1 core Trafostar The singe phased TY-1 transformer has a lot of the similarities with TY-3 transformer Trafostar case. The current in the middle limb is directed in such way that the winding flux will be pointing in opposite direction compared with fluxes in the outer windings. This is done in order to make the fluxes work together instead of "against" each other see figure 18.

Figure 18: TY-1 core transformer Trafostar case

The symmetry assumption has not been used in this case, because similarity to the TY-3 core was used instead. The magnetic equivalent of the circuit can be seen in figure 19.

30

Figure 19: Magnetic equivalent of the TY-1 transformer core Trafostar

Almost all the parameters are the same as defined in "3.4.1 TY-3 core Trafostar. The flux densities in respective winding A, B and C are

][)cos( WbtNV

shiftshiftAC Aa

AAA φωω

φφφ +=+= (3.6.1 -1)

( ) ][)cos( WbtNV

shiftshiftAC Bb

BBB ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=+−= φω

ωφφφ (3.6.1-2)

][)cos( WbtNV

shiftshiftAC Cc

CCC φωω

φφφ +=+= (3.6.1-3)

The flux densities 2φ , 3φ and 4φ can be expressed by equations (3.6.1-4), (3.6.1-5) and (3.6.1-6).

][12 WbAFφφφ += (3.6.1-4) ][123 WbBFAFBF φφφφφφ ++=+= (3.6.1-5)

][134 WbCFBFAFCF φφφφφφφ +++=+= (3.6.1-6) Equations (3.6.1-7)-(3.6.1-11) constitute the inner and outer loop equations corresponding to the equivalent circuit in figure 19.

aa INlHlHlH −=−+ 131312121111 (3.6.1-7)

bbaa ININlHlHlH +=−+ 232322221313 (3.6.1-8)

ccbb ININlHlHlH −−=−+ 333332322323 (3.6.1-9)

cc INlHlHlH =++ 434342423333 (3.6.1-10) 0434342423232222212121111 =+++++ lHlHlHlHlHlH (3.6.1-11)

As for the TY-3 Trafostar model, the 4 start guesses will be made and the flux shift density components will be assumed to be known for all the windings.

31

⎪⎪⎩

⎪⎪⎨

⎧

====

⇒

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

====

⇒

⎪⎪⎭

⎪⎪⎬

⎫

]/[]/[]/[]/[

][][][

][

3333

2323

1313

1111

33

23

13

11

333333

232323

131313

1111111

33

23

13

11

mAHmAHmAHmAH

WbBAWbBAWbBA

WbBA

BBBB

B

B

B

B

CF

BF

AF

μ

μ

μ

μ

μφμφμφμφ

⎪⎪

⎩

⎪⎪

⎨

⎧

⎪⎪⎩

⎪⎪⎨

⎧

⎪⎪⎭

⎪⎪⎬

⎫

⇒

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⇒

⎪⎪⎭

⎪⎪⎬

⎫

⇒

⎪⎪⎩

⎪⎪⎨

⎧

⎪⎪⎭

⎪⎪⎬

⎫

⇒

⎪⎪⎭

⎪⎪⎬

⎫

]/[,]/[]/[]/[

,][,][][

][][][][

4342

32

22

12

4342

32

22

12

4342

32

22

4

3

2

121

mAHHmAHmAHmAH

TBBTBTB

WbWbWb

TB

CF

BF

AF

μμμμμ

φφφ

φφφφ

After all the important magnetic parameters are determined the equations (3.6.1-7)-(3.6.1-10) can be used in order to calculate the winding currents Equation (3.6.1-7) gives

][121211111313 AN

lHlHlHI

aa

−−= (3.6.1-12)


][2323222212121111 AN

lHlHlHlHI

bb

−++= (3.6.1-13)

Equation (3.6.1-9) gives

][434342423333 AN

lHlHlHI

cc

++= (3.6.1-14)

And the equations (3.6.1-9) and (3.6.1-10) gives the second expression for the current bI

][)( 4343424232322323 AN

lHlHlHlHIb

b−−−−

= (3.6.1-15)

In order to fulfill the magnetic requirement of this the circuit the solver should fulfill the following. The outer loop equation (3.6.1-11), the difference between both expressions for the magnetizing current bI should be zero (equations (3.6.1-13) and (3.6.1-15)) and the difference between the calculated total flux and the known total flux in respective winding should be zero as well equations (3.6.1-16)-(3.6.1-18).

][0 WBAirAAFA =−− φφφ (3.6.1-16) ][0 WBAirBBFB =−− φφφ (3.6.1-17) ][0 WBAirCCFC =−− φφφ (3.6.1-18)

The solving procedure for the determination of the DC current is the same as for TY-3 model i.e. by changing the [ ]shift Wbφ for respective winding guess and aiming for the correct DC current by trial and error.

32

4 Main calculations The aspects considered here are depending on the basic magnetic DC calculation results from the previous section i.e. flux shift [ ]shift Wbφ and the magnetizing current [ ]I A over a period as well as the nominal induction field [ ]nomB T . There are several aspects to be considered when a transformer is biased by a DC current. All important aspects should be investigated. The most serious conditions are found and that will be decisive for the transformer load. These aspects are no load losses increase, increase of the winding losses, permitted load time, noise increase and the winding hot spot temperature determination due DC current influence. Below are these aspects described in detail.

4.1 No load losses increase during influence of a DC current The losses that develops at no load condition at the energized transformer are the so called no load losses. The no load losses develop by definition in the core of the transformer and the contribution from the dielectric phenomena in the insulating material as well as the joule losses in the winding from the no load current is negligible. The no load losses in the iron core have three different components: Hysteresis losses hP , classic eddy current losses eP and excess (ac Hysteresis) losses xP . An already existing iron loss calculation model is used.

]/)[,(),(),(),( kgWfBPfBPfBPfBP nomxnomenomhnomTotnom++= (4.1-1)

, where

),( fBP nomTotnom Total nominal iron loss [ kgW / ]

(No influence of DC current pure AC) f Operating frequency 25, 50 or 60 [ Hz ]

nomB Magnitude of the nominal induction [T ] field in the core (peak value of the

pure AC induction field in the core) The total losses with the influence of a DC current can be obtained from the equation (4.1-2).

1max max2

1min min2

( , ) ( , ) ( ( , ) ( , ))

( ( , ) ( , ))[ / ]AC DCTot e nom h x

h x

P B f P B f P B f P B f

P B f P B f W kg+

= + +

+ + (4.1-2)

33

, where shiftnom BBB −=min Peak AC induction field and the [T ]

subtraction of the DC currents influence influence

shiftnom BBB +=max Peak AC induction field and the [T ] addition from the influence of the DC current

The main interest of these calculations is to calculate the increase of the no load losses due the influence of the DC current compared to the original losses (pure AC case). This can be done by dividing the equation (4.1-2) with (4.1-1)) and it will result in the increase factor in percentage, see equation (4.1-6).

[%]nom

DCAC

Tot

Totfactor P

PIncrease += (4.1-6)

4.2 Harmonics determination In order to calculate hot spot temperature and the winding losses, the harmonics of the magnetizing current have to be determined. ABB did has already an internal tool for the harmonics determination which is created in Microsoft Excel “Harmonics Calculator - OKG_PULS.xls”. It should be noted that for the harmonics determination this tool require the input over one whole period with equal intervals and in most of the created models calculated the magnetizing current only over half of the period. Another important thing that should be taken into consideration is that this tool only determines the first eleven harmonics and can deviate from the correct value up to 23% for some cases. Those eleven harmonics are directly related to the nominal frequency which is either 50 or 60 Hz in this report. The results of the harmonics calculations will be used later on in the calculation of the hot spot temperature and the winding losses

4.3 Calculation of the increase factors of the winding losses due the influence of the DC current. There are three different types of the winding losses:

• the resistance losses (ohmic losses) • eddy current losses (caused by the leakage flux crossing the conductor elements

stands)

34

• and structural part losses (other eddy current losses are the losses in the structural parts of the transformer caused by the leakage flux in cables, busbars, entrance leads, tank and so on).

Calculation of the increase factors for resistance 1rF , eddy current 1wF respective structural parts 1sF during the influence of DC current (relation between the total losses in respective

part with influence of DC current and without ACloss

DCACloss

PP

factorIncrease += ) can be done by use of

equations (4.3-1)-(4.3-3) Resistance increase loss factor:

229

11

n

n rated

IFrI=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ (4.3-1)

Eddy current increase loss factor:

129 2ln 2ln

11

n n

rated

I fI f

n

Fw e e⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

=

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

∑ (4.3-2)

Structural parts loss increase factor:

129 2ln 0.8ln

11

n n

rated

I fI f

n

Fs e e⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

=

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

∑ (4.3-3)

, where

ratedI Rated current for the respective transformer winding [ A ]

1f Nominal frequency of the transformer [ Hz ]

1nffn = Frequency related to respective harmonics [ Hz ] n Index of the respective harmonics (=1,2,3…29)

nI Current for n: th harmonic [ A ] One of those parameters that need to be given a closer look at is the current. In order to obtain the current spectra the results of the magnetizing current harmonics calculations which are peak values have to be converted to rms values. The current spectra will have 29 values - one for each index and will be directly related to the rms value of the magnetizing current harmonics calculations (se equation below). Note that the harmonics calculation tool only generates the first eleven harmonics so the assumption that current spectra will be zero for the higher harmonics will be made in our calculations. The first eleven values of current spectra will be defined as follows:

35

rms nn rated harm DCI I I I= + + for 1=n [ A ]

rms nn harmI I= for 112 ≤≤ n [ A ]

The rest of the values will be set to zero

0nI = for 2911 ≤< n [ A ]

4.4 Permitted load time calculation due to core restrictions (flitch plate)

ABB has a specific instruction that describes the permitted load time versus, the induction field in the iron core. In order to make the calculating process smother the approximation of this instruction was made in the similar way as in section A.1 and A.2. The final partial linear approximation compared with extracted point from the instruction sheet can be seen in figure 20.

time(B(T)) approximation

0,01

0,1

1

10

100

1000

1,8 2,05 2,3 2,55

B[T]

time[

min

]

pointsFinalApp

Figure 20: Approximation of the permitted load time compared with line approximation Note that this approximation can only be used between 1,90 and 2,6 T. For the values below the 1,90 , the permitted load time can be assumed infinite and for the values above the 2,6 T,

36

the permitted load time can be assumed to be 0. The precision of this approximation model corresponds to deviation by at most 17,8% within the induction field range. The reason for that is the limitation to 8 intervals that can be used in the approximation of our model in the Microsoft Excel.

4.5 Determination of the noise increase during the DC currents influence ABB has a simple calculation rule for the determining noise due to DC. This was included in the aspects to consider. According to the equations the noise depends on two main parameters: the maximum induction field in the core and the type of the overlap at the core joint in the transformer. bI The noise increase during the influence of DC current have been calculated by equation (4.5-1)

])[(

)(

max

max

dBBNoise

BNoiseNoiseNoiseNoise shiftACDCACincrease

−

=−= ++ (4.5-1)

, where

maxB Nominal peak value of induction field (no DC current) [T]

shiftB +max Nominal induction field added with the induction field shift [T] factor which is a contribution from the DC current

4.6 Determination of the winding hot spots temperature versus time during the influence of DC current Hot spot is the hottest temperature in the winding where the temperature deviates significantly from the rest of the winding. For the creation of the tool for the determination of the hot spots temperature "International standards IEC 60076-7 Loading guide for oil-immersed power transformers" was used. In chapter 8.2 the following equation for the hot spot determination was given: Hot-spot temperature versus time

( ) ])[(

)(1

1)(

2

1

2

CtfKHg

tfR

RKt

hiy

r

hioi

x

oroiah

θ

θθθθθθ

Δ−+

Δ+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛Δ−⎥

⎦

⎤⎢⎣

⎡+

+Δ+Δ+=

(4.6-1)

The equation (4.6-2) describes the relative increase of the top-oil according to the steady state value and the equation (4.6-3) describes the relative increase of the hot-spot-to-top-oil gradient according to the steady state value.

37

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

0111)(1τkt

etf (4.6-2)

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−−−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ −

22022 /21212 1)1(1)( k

tk

t

ekektf w ττ (4.6-3)

aθ Ambient temperature [ C ]

oiθΔ Top oil (in tank) temperature rise at start [ K ]

orθΔ Top-oil (in tank) temperature rise in steady state [ K ] At rated losses (no load losses+load losses) R Ratio of load losses at rated current to no-load losses K Load factor (load current/rated current)

hiθΔ Hot-spot-to-oil (in tank) gradient at start [ K ] (= Hot spot temp initial - Top oil temp), see figure 21

H Hot-spot factor (=(Hot spot temp initial- Top oil temp)/gr) ), see figure 21

rg Average-winding-to average-oil (in tank) temperature [ K ] gradient at rated current. x Exponential power of the total losses versus top-oil (in tank) temperature rise (oil component). y Exponential power of the current versus winding

temperature rise (winding exponent). 0τ Average oil-time constant [min]

The constants 11k , 21k , 22k and the time constants 0τ and wτ are transformer specific and can and can be found in the table 1 in “International standards IEC 60076-7 Loading guide for oil-immersed power transformers”. The constants in our calculations are taken from this table which is shown below table 10.

38

Distribution transformers Medium and large power transformers

ON

AN

ON

AN

re

stri

cted

(s

e no

te)

ON

AN

ON

AF

rest

rict

ed

(see

Not

e)

ON

AF

OF

rest

rict

ed

(see

Not

e)

OF

OD

Oil exponent x 0,8 0,8 0,8 0,8 0,8 1,0 1,0 1,0Winding exponent y 1,6 1,3 1,3 1,3 1,3 1,3 1,3 2,0

Constant k11 1,0 0,5 0,5 0,5 0,5 1,0 1,0 1,0Constant k21 1,0 3,0 2,0 3,0 2,0 1,45 1,3 1,0Constant k22 2,0 2,0 2,0 2,0 2,0 1,0 1,0 1,0

Time constant τ0 180 210 210 150 150 90 90 90 Time constant τw 4 10 10 7 7 7 7 7

Note: If a winding of an ON or OF-cooled transformer is zigzag-cooled, a radial spacer thickness of less than 3 mm might cause a restricted oil circulation, i.e. higher maximum value of the function

f2(t) than obtained by spacers>=3mm Table 10: The Recommended thermal characteristics for exponential equations

ON, OF and OD describe different types of cooling of the transformers with natural, forced and directed oil flow respectively. The values which will be used in this report are for the Medium and large transformers with an OF cooling system see Table 10. In order to get a better insight of the relations between the temperatures parameters used for the determination of the hot spot temperature, see figure 21

Figure 21: Temperature relation of the main parameters in the hot spot temperature determination The influence that DC current will have on the hot spot temperature in this model calculation is the increase of the load factor K which is directly related to the increase of the effective load current. The hottest temperature of the winding should never exceed 140 0C , because gassing can start and lead to fatal electrical errors.

39

4.7 Results As a documentation of the performed work some results from the calculation with the created Microsoft Excel tool are presented in this section. Examples for several core types are given. Results of the calculation can be obtained from the created tool in Microsoft Excel. The user has to insert certain transformer data into the input data sheet and then pressing the button “Click to calculate”. The result will then appear after a while. The input data sheets of the created tools for main calculations can be seen in appendix A.3.

4.7.1 D-core The D-core type of the transformer is magnetically equivalent for the both mechanical design types: Trafostar and TCA. That’s why the result of only one of those designs needs to be presented.

4.7.1.1 D-core TCA This specific transformer (presented as an example here) has the operation frequency of 60 Hz and the an OF (oil forced) type cooling system. In the figures below the flux, induction field B[T], magnetizing current and winding hot spot temperature can be seen both for pure AC (no DC current) and the DC current cases, figures 22 and 23, respectively.

Figure 22: Flux, induction field (B), current and the winding hot spot temperature plots for pure AC case (no DC current case)

40

Figure 23: Flux, induction field (B), current and the winding hot spot temperature plots for DC current case at DC current of 20A. The result of calculations for various DC currents influences on the D-core TCA can be seen in tables 11 (shift, loss and noise) and table 12 (Hot spot, time).

41

Loss enhancement factors for winding & structural steel loss

DC current[A]

FluxShift in winding [Wb]

B1shift in the core limb [T]

B1nominal in the core limb [T]

Permitted load time[min]

Noise increase during the Bshift in the core limb [dB]

NLL increase factor

resistance losses Fr1=

eddy current losses Fw1

structural parts losses Fs1

0 0,00 0,00 1,77 inf 0 1,00 1,00 1,00 1,001 0,08 0,20 1,77 686,49 17 1,13 1,03 1,03 1,035 0,13 0,34 1,77 0,80 39 1,35 1,17 1,19 1,17

10 0,15 0,39 1,77 0,44 52 1,48 1,35 1,42 1,3620 0,18 0,47 1,77 0,33 75 1,71 1,74 1,91 1,7830 0,20 0,53 1,77 0,28 100 1,94 2,17 2,42 2,2240 0,22 0,59 1,77 0,23 126 2,17 2,62 2,92 2,6850 0,24 0,64 1,77 0,19 155 2,41 3,09 3,44 3,1660 0,26 0,69 1,77 0,18 187 2,67 3,58 3,99 3,6770 0,28 0,73 1,77 0,16 222 2,93 4,09 4,52 4,1880 0,29 0,77 1,77 0,15 260 3,21 4,62 5,08 4,7290 0,31 0,81 1,77 0,13 303 3,51 5,17 5,66 5,27

100 0,32 0,85 1,77 0,12 347 3,81 5,72 6,20 5,82150 0,39 1,03 1,77 0,05 637 5,53 8,68 9,20 8,78

Table 11: Results regarding the shift, loss and noise calculations for the 30R122 material D-core TCA

DC current[A]

Max hot spot temp after 300 min [0C] Load factor K

time limit hot spot (temp<140 C) [min]




0 86 1,00 inf inf inf inf 1 87 1,02 inf inf inf inf 5 93 1,08 inf inf inf inf

10 100 1,16 inf inf inf inf 20 115 1,32 inf inf inf 9430 132 1,47 inf inf 70 2140 148 1,62 124 59 24 1050 165 1,76 53 27 13 760 183 1,89 28 15 9 570 201 2,02 17 11 7 480 220 2,15 13 8 5 390 239 2,27 10 7 4 3

100 257 2,39 8 6 4 2150 357 2,95 4 3 2 1

Table 12: Results regarding winding hot spot calculations for 30R122 material D-core TCA

42

Load factor K vs time to reach a winding hot spot temperature of 140, 130, 120 and 100 0C can be seen in figure 24 for D-core TCA and OF cooling system.

Loading factor K vs time to reach a winding hot spot temperature of 140, 130, 120 & 100 C

0

0,5

1

1,5

2

2,5

3

3,5

0 50 100 150 200

time[min]

Load

ing

fact

or

time limit hot spot(temp<110 C)time limit hot spot(temp<120 C)time limit hot spot(temp<130 C)time limit hot spot(temp<140 C)

Figure 24: Load factor K vs time to reach a winding hot spot temperature of 140, 130, 120 & 100 0C Note: The thermal calculations are done with ambient temperature of 30 0C for all the models. The hottest temperature of the winding should never exceed140 0C , because gassing can start and lead to electrical fatal errors.

4.7.2 EY-core Because of the magnetic similarity of Trafostar and TCA mechanical designs it will be enough to present one of those designs. The results of EY-core Trafostar with operating frequency of 50Hz and OF cooling system will be presented below.

4.7.2.1 EY-core Trafostar In the figures below the flux, induction field B[T], magnetizing current and winding hot spot temperature can be seen both for pure AC (no DC current) and the DC current cases, see figures 25 and 26.

43

Figure 25: Flux, induction field, current and the winding hot spot temperature plots for the pure AC case (no DC current case) for EY-core Trafostar.

Figure 26: Flux, induction field, current and the winding hot spot temperature plots for DC the current case for EY-core Trafostar at DC current of 20 A. Result calculations due DC currents influence for EY-core Trafostar can be seen below in table 13 and 14.

44


DC current[A]




Permitted load time[min]


NLL increase factor




0 0,00 0,00 1,73 inf 0 1,00 1,00 1,00 1,001 0,07 0,13 1,73 inf 8 1,05 1,02 1,02 1,025 0,13 0,22 1,73 1077,19 17 1,15 1,08 1,08 1,08

10 0,17 0,28 1,73 61,59 24 1,23 1,16 1,17 1,1620 0,22 0,33 1,73 3,28 32 1,33 1,32 1,34 1,3330 0,25 0,34 1,73 2,06 33 1,35 1,50 1,54 1,5140 0,28 0,35 1,73 1,04 35 1,37 1,68 1,74 1,7050 0,30 0,35 1,73 0,83 36 1,39 1,87 1,93 1,8860 0,32 0,36 1,73 0,72 38 1,40 2,06 2,14 2,0870 0,34 0,37 1,73 0,64 39 1,42 2,25 2,34 2,2780 0,36 0,37 1,73 0,58 40 1,43 2,45 2,54 2,4790 0,38 0,38 1,73 0,52 42 1,45 2,65 2,75 2,67

100 0,40 0,39 1,73 0,48 43 1,46 2,85 2,96 2,88150 0,48 0,41 1,73 0,38 49 1,53 3,89 4,00 3,91

Table 13: Results of flux shift, losses, load and noise calculations for the 30R122 material EY-core Trafostar.

DC current[A]

Max hot spot temp after 300 min [0C]

Load factor K





0 84 1,00 inf inf inf inf 1 85 1,01 inf inf inf inf 5 87 1,04 inf inf inf inf

10 90 1,08 inf inf inf inf 20 96 1,15 inf inf inf inf 30 103 1,22 inf inf inf inf 40 109 1,30 inf inf inf inf 50 116 1,37 inf inf inf 62 60 123 1,44 inf inf 145 22 70 129 1,50 inf inf 57 13 80 136 1,57 inf 106 27 10 90 143 1,63 176 54 17 8

100 149 1,69 90 30 12 6 150 183 1,97 14 9 6 3

Table 14: Results of the winding hot spot temperature calculation for 30R122 material EY-core Trafostar.

Load factor K vs time to reach a winding hot spot temperature of 140, 130, 120 and 100 0C for the EY-core Trafostar can be seen below in figure 27.

45

Loading factor K vs time to reach a winding hot spot temperature of 140, 130, 120 & 100 C

0

0,5

1

1,5

2

2,5

0 20 40 60 80 100 120 140

time[min]

Load

ing

fact

or

time limit hot spot(temp<110 C)time limit hot spot(temp<120 C)time limit hot spot(temp<130 C)time limit hot spot(temp<140 C)

Figure 27: Load factor K vs time to reach a winding hot spot temperature of 140, 130, 120 & 100 0C for EY-core Trafostar.

4.7.3 DY-core The Trafostar and TCA designs are no longer equal magnetically as in previous models. That’s why result for both design need to be presented.

4.7.3.1 DY-core Trafostar In the figures below the flux, induction field B, magnetizing current and winding hot spot temperature can be seen both for pure AC (no DC current) and the DC current cases, see figures 28 and 29 for the DY-core Trafostar with frequency of 50 Hz and ONAF cooling system.

46

Figure 28: Flux, induction field (B), current and the winding hot spot temperature plots for pure AC case (no DC current case) for DY-core Trafostar.

Figure 29: Flux, induction field (B), current and the winding hot spot temperature plots for DC current case for DY-core Trafostar for DC current of 20A. The results due to DC currents influence on the DY-core Trafostar can be seen below in tables 15 and 16.

47


DC current[A]




Permitted load time [min]


NLL increase factor




0 0,00 0,00 1,73inf 0 1,00 1,00 1,00 1,001 0,14 0,18 1,73 2159,53 12 1,11 1,01 1,01 1,015 0,19 0,25 1,73 871,83 19 1,22 1,03 1,03 1,03

10 0,21 0,27 1,73 611,77 23 1,26 1,07 1,08 1,0720 0,24 0,30 1,73 444,50 27 1,32 1,14 1,17 1,1430 0,24 0,30 1,73 442,51 27 1,32 1,16 1,19 1,1640 0,27 0,34 1,73 328,83 34 1,41 1,28 1,37 1,3050 0,28 0,36 1,73 298,59 37 1,45 1,36 1,47 1,3860 0,29 0,37 1,73 275,81 40 1,49 1,44 1,57 1,4670 0,30 0,38 1,73 257,68 43 1,53 1,52 1,67 1,5480 0,31 0,40 1,73 242,73 46 1,57 1,60 1,78 1,6390 0,32 0,41 1,73 230,10 49 1,60 1,68 1,88 1,72

100 0,33 0,42 1,73 219,22 51 1,64 1,76 1,99 1,81150 0,38 0,47 1,73 180,23 66 1,81 2,20 2,53 2,27

Table 15: Results of the flux shift, load, losses and noise calculations for 30R122 DY-core Trafostar.

DC current[A]


Load factor K





0 75 1,00inf inf inf inf 1 75 1,00inf inf inf inf 5 76 1,02inf inf inf inf

10 77 1,03inf inf inf inf 20 79 1,07inf inf inf inf 30 80 1,08inf inf inf inf 40 83 1,13inf inf inf inf 50 86 1,17inf inf inf inf 60 88 1,20inf inf inf inf 70 90 1,23inf inf inf inf 80 92 1,26inf inf inf inf 90 95 1,30inf inf inf inf

100 97 1,33inf inf inf inf 150 109 1,48inf inf inf inf

Table 16: Results of the winding hot spot temperature calculations 2 for material 30R122 DY-core Trafostar.

As it can be seen in table 7 the winding hot spot temperature is not a problem for this specific transformer for DC current up to 150 A for respective winding. This because the DC biased exciting current is low compared with the rated winding current of 1132A, which results in a lower load factor compared with that in the earlier models. The DC current of 150 A corresponds to the total current in the neutral of 300 A.

48

4.7.3.2 DY-core TCA In the figures below the flux, induction field B, magnetizing current and winding hot spot temperature can be seen both for pure AC (no DC current) and the DC current cases, see figures 30 and 31 for DY-core TCA with frequency of 50 Hz and OF type of cooling system.

Figure 30: Flux, induction field (B), current and the winding hot spot temperature plots for pure AC case (no DC current case) for DY-core TCA.

49

Figure 31: Flux, induction field (B), current and the winding hot spot temperature plots for the DC current case for DY-core TCA for a DC current of 150 A. Results calculations for various DC current influences on the DY-core TCA.


DC current[A]




Permitted load time [min]


NLL increase factor




0 0,00 0,00 1,75inf 0 1,00 1,00 1,00 1,001 0,19 0,06 1,75inf 3 1,01 1,00 1,00 1,005 0,44 0,17 1,75 1527,01 12 1,08 1,02 1,02 1,02

10 0,51 0,20 1,75 1002,01 15 1,12 1,03 1,03 1,0320 0,51 0,20 1,75 1002,01 15 1,12 1,03 1,03 1,0330 0,59 0,23 1,75 484,18 19 1,16 1,10 1,15 1,1140 0,60 0,24 1,75 395,13 20 1,17 1,14 1,26 1,1550 0,61 0,24 1,75 335,76 21 1,18 1,17 1,40 1,2060 0,62 0,24 1,75 292,66 21 1,18 1,21 1,59 1,2570 0,62 0,25 1,75 259,57 22 1,19 1,26 1,82 1,3180 0,63 0,25 1,75 233,11 22 1,19 1,30 2,11 1,3790 0,63 0,25 1,75 211,30 22 1,19 1,35 2,45 1,45

100 0,63 0,25 1,75 192,89 22 1,19 1,40 2,85 1,53150 0,64 0,26 1,75 129,92 23 1,20 1,67 5,82 2,02

Table 17: Results of shift, load, loss and noise calculations for material 30R122 DY-core TCA.

50

DC current [A]


Load factor K





0 100 1,00inf inf inf inf 1 100 1,00inf inf inf inf 5 101 1,01inf inf inf inf

10 102 1,02inf inf inf inf 20 102 1,02inf inf inf inf 30 105 1,05inf inf inf inf 40 107 1,07inf inf inf inf 50 109 1,08inf inf inf inf 60 111 1,10inf inf inf 141 70 113 1,12inf inf inf 74 80 116 1,14inf inf inf 44 90 118 1,16inf inf inf 28

100 120 1,18inf inf 248 19 150 134 1,29inf 135 35 7

Table 18: Results of the winding hot spot temperature calculations for material 30R122 DY-core TCA.

4.7.4 TY-3 core, T-core & TY-1 core types The above core types are only mentioned for completeness and could not be finalized within the time available for this work. This applies for both the Trafostar and TCA designs.

4.7.5 Comments of the calculated results The calculation models were applied to some particular designs from the current production. The results are presented below. The various transformers differed in many aspects (e.g winding, core type, dimensions, type of cooling system and more) DC current has huge impact on the transformers. The biggest impact was noticed on the D-core and EY-core transformers, where the losses increased dramatically even with a small addition of DC current. This can be expected due to the core geometry which is in principle a closed loop. The winding hot spot temperature was not a problem for most of the transformers types. Dangerous temperatures were reached for two transformer types: the D-core TCA and EY-core Trafostar transformers. The D-core power transformer reached its limit already at a DC current around 20 A. The limiting factor is the permitted noise restriction of 150 dB. Under the consideration that the nominal noise of the transformers is 70 dB. The limiting factor for EY-core is permitted load time up to DC currents of 150 A. The dangerous winding hot spot temperature will be reached as well for EY-core Trafostar at DC current of 90 A but, will allow this DC current for 176 minutes which is much longer compared to the limitation of 0,53 minutes provided by the permitted load time calculation, see table 13 and 14.

51

According to the result from the calculations regarding DY-core transformers for both design types Trafostar and TCA the limiting factor is the permitted load time of 3 hours for the Trafostar and 2hours 10 min for TCA cases. No other restrictions were even exceeded for DC current in the region of 150 A. In the addition to above it can be mentioned, that DY-core transformers for both design types Trafostar and TCA are less sensitive to the addition of DC current compared to the earlier mentioned Transformers types. Note that T-cores are considered as less susceptible to DC, however this core type could not be included in this work, due to time limit.

5 Conclusions and future work The used approximations can be improved quite a bit. The limitation e.g. in the approximation of the μ=μ(B) or H=H(B) to 8 points have to be removed. This can be obtained by splitting up the approximation in a number of regions and create some transitions between the approximations. Improve approximation of air contribution and implement the created models in more trustful software than Microsoft Excel, such as Mat Lab and MathCAD. The results of this calculation tool depends partly on the mathematical model approximations and partly on the Microsoft Excels “Solver” performance. The later was pushed over its reliable performance level.

52

Appendix

A.1 Approximation of the magnetizing field with known induction field )(BHH = . In order to create this approximation a core steel specific datasheet from an dielectric steel manufactor was used. By selecting some carefully chosen points from that sheet the following partial linear plot could now be obtained see figure 32.

Partial linear approximation of the magnetizing force

0

2000

4000

6000

8000

10000

12000

0 0,25 0,5 0,75 1 1,25 1,5 1,75 2 2,25

B[T]

H[A

/m]

Figure 32: Partial linear approximation of the magnetizing field of the selected electric steel

The material that was used for this approximation was 27D (internal ABB definition) and the selected points for the linear approximation are found in Table 19.

53

B[T] H[A/m] Increase

point:8 2,02 10000 121428,6 point:7 1,95 1500 20400 point:6 1,9 480 3550 point:5 1,8 125 441,1765 point:4 1,63 50 86,95652 point:3 1,4 30 26,66667 point:2 1,1 22 20 point:1 0 0

Table 19: Selected point for the linear approximation of H=H(B)

In order to improve the linear approximation in figure 32 a circle approximation was introduced for some section in order to make our approximation little smoother. The linear parts of the approximation can be described by the equation (A.1-1) and the circular part by equation (A.1-2) The equation (A.1-1) is valid for ii BBB <<−1 (the linear part)

iB

iii BBkHmAHΔ

−− −+= )(]/[ 11 (A.1-1) And equation (A.1-2) is valid for rxB > (circular part)

22 ))((]/[factorHrr BxBrymAH −−−= (A.1-2)

Where:

i

ii B

HkΔΔ

= Linear increase factor [ TmA /)/( ]

B Known induction field [T ] ry Y-coordinate of the circle [ mA / ]

rx X-coordinate of the circle [T ] r Circle radius [ mA / ]

factorHB Relation between the B and H axis (scale factor) (can easily be seen in figure 22)

Note: Scale factor e.g. factorHB =8000 ( 4 units in B direction (=1T) corresponds to 4 unit in H

direction (=8000A/m) The tricky part with introducing the circular approximation is to find the connection points between the linear and circular approximated parts which have to be done with high precision. The parameters that have to be adjusted in order to succeed in doing that are the radius r and x and y-coordinates of the circle. The figure 32 gives enough information to make some reasonable start guesses for the circle parameters and after some minor modifications of these parameters the connecting points

54

were obtained (see figure 33 and 34). In the figures Hcircle is the circular approximation and the Hline is the linear approximation.

Figure 33: Determination of the starting point of the circular approximation

Figure 34: Determination of the ending point of the circular approximation of the H(B)

55

The circular approximation will be used between the connecting point ][84390133,1 TBstart = and [T],Bend 9521037981= and the linear approximation will be used outside this range, see figures 23 and 24. Now our approximation is almost complete. The only problem is the limitation around the saturation level ][05,2 TB ≈ . The well known fact is that the permeability of magnetic materials becomes almost equal to the permeability of the air when the transformers material reaches the saturation level.

00 μμμμ ≈= r When the saturation is reached ( ][05,2 TB ≈ ) (A.1-3) This leads to our final step in the approximation. The addition of another line approximation with the slope

01μ=ik .The final result of the approximation can be seen in figure 35.

Final approximation of the H(B) compared with line approximation

0

2000

4000

6000

8000

0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25

B[T]

H[A

/m]

Hfinal[A/m]Hline[A/M]

Figure 35: Final and the linear approximations of H(B) compared to each other

The final circular parameters for this approximation became:

]/[1948][7,1

]/[2026

MAyTx

mAr

r

r

===

56

A.2 Approximation of the permeability with known induction field )(Bμμ =

The necessity of this second approximation lies in the fact that Microsoft Excel failed to find a feasible solution for all the transformer models. This was actually the main approximation that was already finished in Microsoft Excel with the file name "3 - permeability curve - param determination.xls" and was a base for the previous mentioned approximation model ("Approximation of the magnetizing field with known induction field ][TB ”). Only some minor inconveniences were corrected before it could be used, such as the x and y - coordinates are now fixed and didn't change while the limits of the approximation were modified. Another important difference is that this approximation contains two different circular approximation parts, compared with the previous described one. The selected points of this linear approximation are taken from Table 19 and they are used in order to achieve the respective permeability values. This result with following permeability and increase factors, see table 20.

μ=B / H incr(μ) point:8 0,000202 -0,01569point:7 0,0013 -0,05317point:6 0,003958 -0,10442point:5 0,0144 -0,10706point:4 0,0326 -0,06116point:3 0,046667 -0,01111point:2 0,05 0point:1 0,05

Table 20: The respective values for permeability and increase factors calculated from the values of the magnetizing field and induction field.

These values can now be used in order to plot the linear approximation of the curve, see figure 36 The position of the first circle approximation should be somewhere between the limits 1,1 and 1,6 [T] and the second circle approximation should be somewhere in the range1,85 and 2,05 [T].. The equations used in this approximation are:

21

2110 )))(((

factorfactorBxBry rBrr μμμμμ −−+== (A.2-1)

22

2220 )))(((

factorfactorBBxry rBrr μμμμμ −−−== (A.2-2)

factor

i

BBBcB

iiir μμμμμΔ

−− −+== )( 110 (A.2-3)

57

There equation (A.2-1) describes approximation of the first circle, equation (A.2-2) second circle and equation (A.2-3) the linear approximation

1ry Y-coordinate of the first circle related to theμ axis

1rx X-coordinate of the first circle related to theμ axis

1r Radius of the first circle related to theμ axis

2ry Y-coordinate of the second circle related to the μ axis

2rx X-coordinate of the second circle related to theμ axis

2r Radius of the second circle related to theμ axis

factorBμ Relationship between B and theμ axis (scale factor)

(can easily be seen in figure 26)

BBxy

cfactor

i

i

ii Δ

Δ=

ΔΔ

=)( μ

μ Linear increase factor

Partial linear approximation permeability

0

0,01

0,02

0,03

0,04

0,05

0,06

0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 2,1 2,2 2,3 2,4

B[T]

mu

Figure 36: Partial linear approximation of the permeability dependent of the induction field

The circle parameters and connecting points can be determined in a similar way as in previous approximation. The starting and the ending points of the circles became

⎭⎬⎫

==

][59,1][15,1

1

1

TBTB

end

start The starting and ending connections points of the first circle

58

⎭⎬⎫

==

][04,2][9,1

2

2

TBTB

end

start The starting and ending connections points of the second circle

The last part of this approximation was to remove the limitation of this approximation (the original curve is only valid up to the saturation level around the 2,05 [T], but in this report we need to further then the saturation level). This was done by allowing the permeability μ to reach the final value of 0,0000013=μ (this value is almost equal to the permeability of the air 0μ ) at the saturation level ( ][05,2 TB ≈ ) and there after to keep the same value of permeability with increasing induction field. The final approximation can be seen in figure 27, where a comparison is made with the line approximation.

Final approximation of the permeability compared with line approximation

0

0,01

0,02

0,03

0,04

0,05

0,06

0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 1,9 2 2,1 2,2 2,3 2,4

B[T]

mu line

final

Figure 37: The final approximation compared with the line approximation

Both approximations are quite close to the original values but still they don't coincide completely. This will cause some deviations from the correct values, but the results will still be reasonable.

59

A.3 Input data sheets for the created models In order to calculate the permitted load time, no load losses, winding losses, winding hot spot temperature all that is needed to be done by the user is to fill in the yellow part of the sheet ( transformer data) and hit the button “Click to calculate” and the result will appear.

A.3.1 D-core

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A.3.2 EY-core

61

A.3.3 DY-core

A.3.3.1 DY-core Trafostar

62

A.3.3.2 DY-core TCA

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Reference [1] Calculations of DC current impact (GIC) on large 3-phase 3-limb core, Unit-#: 8329881, OKG order. Kurt Gramm. No: SETFO 2006-44 [2] Estimation of DC currents impact (GIC) on 4-limb single phase Transformer, Unit-#: 8720435-38, Puls, OKG. Kurt Gramm. No: SETFO 2006-228 [3] Study on DC bias for converter Transformer. Kurt Gramm. No: Document number PTTFO 2004-172. [4] Additional calculations on GIC impact on two 3-phase 3-limb transformers, OKG, serial numbers 8329881 and 8329 882. Kurt Gramm. No: SETFO 2006-118. [5] International standards: Power transformers-part 7: Load guide for oil immersed power transformers. No: IEC 60076-7:2005(E). [6] Likströmmagnetisering av transformatorer. Sören Peterson 2004-04-30. [7] No load loss calculation. No: 1ZBA 45 21-101 Note: Most of the references are ABB’s internal documentation and some of information was provided by my supervisor.

XR-EE-ETK_2007_0111

Documents

pure ac case

hot spot temperature

magnetic path

typical school

dc current

partial linear

power transformer

permitted