Calculation of Losses in the Core Clamps of a Transformer Using 3 D Finite Element Method by fsm71579

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									                                           O.W. Andersen

             USER’S MANUAL, FLD12
 COMPLEX POTENTIAL TRANSFORMER LEAKAGE FLUX


                                     PROGRAM DESCRIPTION


FLD12 is a subprogram of the general purpose magnetic field program FLD8. It consists basically of
an input and an output routine for this program. Input and output can be in either metric or English
units.

FLD12 calculates vector potentials as complex numbers. That is of particular importance when sheet
windings are present, and when there are phase shift connections. The program calculates reactance,
losses and forces in core type transformers with only the most essential information as input. Design
program TRA1 generates input automatically for FLD12.

The earlier scalar potential program FLD5 with the same input and output has now been superseded by
FLD12, because the faster calculation time with FLD5 is no longer an issue.

FLD12 has been reprogrammed to produce bitmap picture files for processing by standard Windows
XP and Vista programs. Windows Command Prompt is still used to run FLD12, but compatibility with
earlier versions of Windows can no longer be guaranteed.



                                    PROGRAM INSTALLATION


FLD12 must be installed in directory (folder) \FLD12 on the same computer and in the same unit
(usually C) as FLD8 and GRAPHICS. The programs are supplied on the Internet as e-mail attachments
together with installation instructions. The input and output routines in FLD12 are completely
decoupled from FLD8. They are supplied in Fortran source code and can be changed using a suitable
compiler, such as Watcom.


                                   RUNNING THE DEMO INPUT


Here all the Command Prompt commands, directories (folders) and file names will be in capital letters.
However, they are case insensitive, and small letters can also be used.

An input file DEMO.INP is in directory FLD12. To run the program with this input, enter:

RUN DEMO.INP

After a few seconds, a flux plot with 25 flux lines appears on the screen. It has been drawn on a Visual
Basic Form. If the picture appears to be cropped, adjust the file \GRAPHICS\SIZESCR.FIL. At the
same time a bitmap picture file PLOTFILE.BMP has been produced in directory GRAPHICS. Close
the form and enter command:

PLOT
                                                    -2-


The flux plot now reappears in a standard Windows program. The tic-marks to the left and at the
bottom show the positions of horizontal and vertical finite element grid lines. The windings are red. In
other cases, winding segments which don’t carry current will be green.

If it is now desired to print the flux plot, crop the picture file first to remove empty space and save it.
Rather than printing it directly, it is recommended to transfer the picture file to Microsoft Word. Here
it can easily be resized and comments added before printing.

Output from FLD12 is stored in file OUTPUT in directory FLD12. To display it on the screen, enter:

FILE OUTPUT

Batch command FILE starts the standard Windows program NOTEPAD. It will be used here for
viewing, editing and printing text files. The first time it is invoked, it should be set to Courier New size
9, word wrap, and to no top and bottom extra text when printing. The window should always be
maximized.

The output file does not contain any detailed information about vector potentials at the nodes and flux
densities in the triangles. Such information can be provided in file OUTPUT (in directory FLD12) by
entering the command
DETAILS
and answering the prompts. As a suggestion, in response to the prompts, answer:
Horizontal lines 25 and 26
Vertical lines 2 and 30
File OUTPUT can now be displayed and printed, as before.
To display the finite element grid on the screen, enter:
GRID
After the form is closed, the grid also reappears with the command:
PLOT


                                                  INPUT


The demo input file can be viewed with the command:

FILE DEMO.INP

What the numbers mean can be found on the input sheets, pages 4 and 5. For an explanation of what
else can be done with the input file, copy it first to a new file with the command:

COPY DEMO.INP NEW.INP

Introduce headings with the command:

HEADINGS NEW.INP
                                                  -3-


To see how the file now has been modified, enter:

FILE NEW.INP

The abbreviated headings on the input file also explain the numbers. With a little experience, that
explanation suffices to enter new numbers and to make up new input files.

Old input as similar as possible is first copied to a new input file. Then headings are introduced and the
file changed. Numbers always start in columns 1, 11, 21 and so on. They can be entered with or
without decimal point.

Before the new file can be run, the headings must be removed. Do this first with:

CLEANUP NEW.INP

A file without headings can have headings introduced and be viewed at the same time with:

HEADFILE NEW.INP

Headings can also be removed and the file run at the same time with:

CLEANRUN NEW.INP

An input file in a different format is in

INPUT.FIL

Numbers start in column 60 and are preceded by more detailed explanations. It can be copied to a new
file INPNEW.FIL, where also more terminals, layers and segments can be added in the same format.
Then a standard input file NEW.INP can be created with the command:

INPUT INPNEW.FIL NEW.INP

Input must be entered very carefully, following explanations on the input sheets and instructions
elsewhere in this manual. Small mistakes like a comma instead of a decimal point or a number starting
in the wrong column are not tolerated. Some mistakes are caught by the program and are explained on
the output. Another way to catch mistakes is by giving a command such as:

CHECK NEW.INP

The input must here be in the standard format, without headings. A picture similar to a flux plot, but
without flux lines, will be displayed on the screen. Mistakes with the geometry can be caught this way.
COMPLEX POTENTIAL TRANSFORMER LEAKAGE FLUX                                    PROGRAM FLD12                   INPUT SHEET 1

Numerical data are entered with the first digit in columns 1,11,21 etc., as indicated. Decimal point is optional.

IDENTIFICATION (line 1)                                                                  Max. 80 characters, including blanks

                                                                                                      Col.       Data     Line
INPUT UNITS (mm=1, inches=2)                                                                       *1  1
NUMBER OF PHASES                                                                                      11
FREQUENCY                                                                                             21
NUMBER OF WOUND LIMBS (often 3)                                                                       31
                                                                                                                            2
FRACTION OF WINDOW HEIGHT CALCULATED (0.5 or 1)                                                       41
Z - LOWER BOUNDARY (often zero, can be negative)                                                   *2 51
Z - UPPER BOUNDARY (upper yoke or beyond)                                                             61
CORE DIAMETER                                                                                         71
DISTANCE WINDING - TANK (or to right field boundary)                                                   1
AL/CU SHIELD (no=0, yes=1)                                                                         *3 11
SYSTEM SHORT CIRCUIT GVA (zero if no external impedance)                                              21
OPTIONAL PER UNIT IMPEDANCE (usually zero)                                                         *4 31                    3
PEAK FACTOR (Often 1.8 and never more than 2.0. See explanation below)                             *5 41
NUMBER OF TERMINALS (6)                                                                              51
NUMBER OF LAYERS (or windings, 30)                                                                   61
IN LAYERS WITH NEGATIVE CURRENT:
   DISPLACEMENT/ELONGATION (0, 1 or 2)                                                                    1
  AMOUNT (mm or inches)                                                                                  11
LOSS FACTOR, TANK              )                                                                         21
             LEG               )     zero, if not known (see manual)                                                        4
                                                                                                         31
             YOKE              )                                                                         41
SCALE, FLUX PLOT (on printer or plotter, but not used on newer computers)                                51
NUMBER OF FLUX LINES                                                                                     61

For the terminal data, use only as many lines as there are terminals (6).
 TERMINAL NUMBER (usually 1 for layer 1)                             1
 CONNECTION CODE                                                                                                            5
  I,Y: 1 D: 2 Auto: 3 (see manual)                             *6 11                                                        6
 AT CALCULATED LOAD CONDITION:                                                                                              7
  MVA (usually positive, must give balanced ampereturns)            21                                                      8
  KV (never zero)                                                   31

For the layer (winding) data, use one pair of lines for each layer. The layers are entered in a sequence starting from the inside.
 LAYER NUMBER                                                         1                                                      9
 LAST SEGMENT NUMBER (200)                                          11                                                     11
 INNER RADIUS                                                        21                                                     13
 RADIAL WIDTH                                                        31                                                     15
 TERMINAL NUMBER                                                      1
 NUMBER OF PARALLEL GROUPS (usually 1)                          *7 11                                                       10
 DIRECTION OF CURRENT (-1 or 1)                                      21                                                     12
 COPPER/ALUMINUM (1 or 2)                                            31                                                     14
 SPACER BLOCKS, NUMBER (between disks)                               41                                                     16
                     WIDTH                                           51

*1: See page 19 about output in English units.
*2: Instead of having lower and upper boundaries at the yoke positions, it is recommended to have them at twice the yoke
   distances from the windings, as a weighted average of conditions inside and outside of the transformer window.
*3: If there is an AL/CU shield, flux lines will be parallel to the tank.
*4: Overrides the calculated total impedance in calculations of forces and stresses, if given different from zero.
*5: Only for dc component, and not including the factor 2 = peak/rms ac.
*6: See the manual (page 9) also about phase shift connections.
*7: 2 for a normal sheet winding with half the window height calculated, otherwise usually 1.
COMPLEX POTENTIAL TRANSFORMER LEAKAGE FLUX                                PROGRAM FLD12                 INPUT SHEET 2

Additional layer (winding) data                                 Col.                       Data                       Line
 LAYER NUMBER                                                    1
 LAST SEGMENT NUMBER (200)                                     11
 INNER RADIUS                                                   21
 RADIAL WIDTH                                                   31
 TERMINAL NUMBER                                                 1
 NUMBER OF PARALLEL GROUPS (usually 1)                       *1 11
 DIRECTION OF CURRENT (-1 or 1)                                 21
 COPPER/ALUMINUM (1 or 2)                                       31
 SPACER BLOCKS, NUMBER (between disks)                          41
                      WIDTH                                     51

For the segment data, use one pair of lines for each segment. Up to 200 segments can be entered, in a sequence starting with
the inner layer, and with increasing z-coordinates within each layer. Empty spaces are not considered as segments.
If because of symmetry only half the window height is calculated, give segment data for only that half.

SEGMENT NUMBER                                                     1
Z - COORDINATE, MINIMUM                                           11
                MAXIMUM                                           21
NUMBER OF TURNS, TOTAL                                            31
                  ACTIVE (always positive)                        41
NUMBER OF STRANDS, PER TURN (per group)                            1
RADIALLY ACROSS LAYER (for all turns)                        *2   11
STRAND DIMENSION, RADIALLY                                        21
                   AXIALLY                                   *3   31

SEGMENT NUMBER                                                     1
Z - COORDINATE, MINIMUM                                           11
                MAXIMUM                                           21
NUMBER OF TURNS, TOTAL                                            31
                  ACTIVE (always positive)                        41
NUMBER OF STRANDS, PER TURN (per group)                            1
RADIALLY ACROSS LAYER (for all turns)                        *2   11
STRAND DIMENSION, RADIALLY                                        21
                   AXIALLY                                   *3   31

SEGMENT NUMBER                                                     1
Z - COORDINATE, MINIMUM                                           11
                MAXIMUM                                           21
NUMBER OF TURNS, TOTAL                                            31
                  ACTIVE (always positive)                        41
NUMBER OF STRANDS, PER TURN (per group)                            1
RADIALLY ACROSS LAYER (for all turns)                        *2   11
STRAND DIMENSION, RADIALLY                                        21
                   AXIALLY                                   *3   31

*1: 2 for a normal sheet winding with half the window height calculated, otherwise usually 1.
*2: Used to calculate bearing surface for compressive stress in spacer blocks and in insulation, due to accumulated axial
    forces. With machine transposed cable, where the true number of strands radially across the layer varies between a
    minimum and a maximum, the minimum number should be entered.
*3: For a normal sheet winding with half the window height calculated, half the depth of the sheet.
    Sheet winding is assumed, if axial strand dimension  100 mm.
    Round wire is assumed, if strand dimensions radially and axially are given equal and  4 mm.
                                                   -6-


                                      SPECIFICATION OF INPUT


LINE 1. The identification can consist of up to 80 characters, including blanks. Most combinations of
letters, numbers and special symbols on the keyboard can be used.

LINE 2. The z-coordinate for the lower boundary is usually given as zero. It corresponds to the radial
centerline if only half the window height is calculated, and to the lower yoke or beyond if the full
window height is calculated. In some cases it is desired to make a second run with the lower boundary
moved down and the upper boundary moved up, in order to approximate the conditions outside of the
transformer window. The z-coordinate of the lower boundary can then be made negative, which
permits the winding segments to remain in the same positions. However, the output z-coordinates
always refer to the lower boundary.

LINE 3. The distance winding-tank is the radial distance from the outer radius of the outer layer. In
most cases, more than the true distance would be entered, to approximate average conditions around
the perimeter. However, it is not recommended to make the distance more than about half the depth of
the windings. The value is not very critical.

If an AL/CU shield is specified, the program puts in a flux line with vector potential zero along the
tank wall. Otherwise, flux lines will be forced perpendicular to the tank. A reference potential zero is
then put in by the program at the tank at the radial centerline.

If the optional per unit impedance is not specified (given as zero), the program calculates forces and
stresses for a symmetrical short circuit, using the system impedance (calculated from the system short
circuit GVA) in series with the calculated transformer impedance. Of course, this only makes sense if
windings belonging to only two terminals carry current, because it is only then that the calculated
transformer impedance has any real meaning. If it is desired to get the forces and stresses for a
different current, the optional per unit impedance should be specified as the inverse of "times normal"
ac current.

The peak factor is for the maximum dc current component at short circuit. It does not include the 2
factor to get from rms to peak ac current.

LINE 4. If 1 is entered in col. 1, all layers with negative current are displaced in the z-direction by the
amount entered in col. 11. If 2 is entered in col. 1, all layers with negative current are elongated by the
amount entered in col. 11. Z-min is then kept unchanged, and all segments and open spaces belonging
to the layer are elongated in the same ratio. A negative displacement or elongation can also be entered.

It is explained elsewhere how tank, leg and yoke losses are calculated proportional to the factors
specified as input. These factors must be established from tests. They will be different for different
sizes and types of transformers, and for different manufacturers. One difficulty is that the transformers
are not truly axi-symmetric, and there is no fixed ratio between the actual and the calculated flux
entering the different parts. No great accuracy is therefore to be expected from these calculations.
                                                    -7-


Loss tests with and without the tank provide a clue about the tank loss, although the other losses will
not be exactly the same in the two cases. A separation of leg and yoke loss can only be done from a
statistical analysis of tests and calculations for several transformers (method of least squares).
Such an analysis for a line of medium size transformers gave the following result:
Tank loss factor: 45
Leg loss factor: 7
Yoke loss factor: 20
In the absence of such a study, it is better to give the loss factors as zero, and not perform this part of
the calculation.
The scale of the flux plot is unimportant for the plot on the screen. It is used for the plot on the printer
or plotter, but if the value is given too large, it is reduced automatically by the program. The plot is
automatically tilted 90 degrees, if this permits a better utilization of the paper.

LINES 5-8. The terminals should be numbered consecutively, usually starting with no. 1 for layer 1.
"I" is the connection for single phase units. For three phase units, the MVA rating is for all the phases
and KV is the line kilovolts. The values must correspond to the calculated load condition and tap
position, and must give balanced ampereturns. MVA can be zero, but never KV. Auto and phase shift
connections require special consideration, and will be dealt with separately (pages 8 and 9).
For single phase units, connection code = 1, except for auto connection. If the windings on two single
phase limbs are in series, number of parallel groups = 1, in parallel = 2 (layer data).

LAYER DATA. A layer in this context is a winding or part of a winding belonging to a certain
terminal. Layers are usually concentric, but can also be above each other and belong to different
terminals. If so, enter the lower layer first. Concentric layers do not have to be separated radially, but
normally they are. If a winding has one or two axial cooling ducts, it can be specified with two or three
layers.
If the layer has two parallel paths with a lead connection in the middle, the number of parallel groups
is entered as 2, otherwise usually as 1.
To provide uniformity among the users, layers belonging to the inner main winding can be specified as
having a negative current, but this is really immaterial as far as the program is concerned.
The number and tangential width of spacer blocks are used to calculate compressive stress in the
spacers, and combined bending and tension or compression in the conductors between the spacers. If
there are no spacer blocks, number and width are given as zero.

SEGMENT DATA. A segment is defined as part of a layer which can be considered uniform in
conductor arrangements and current densities. The number of segments which is specified in the input
should be kept to a minimum, and entering individual disks as segments should be avoided. Narrow
gaps between segments should be eliminated by specifying z-max for one segment equal to z-min for
the one above.
If a segment comprises a whole layer with two parallel groups, the number of turns in the segment is
specified as the sum of the numbers in the two groups.
When because of symmetry only the upper half of the window height is calculated, the segment data
refer to only that half.
                                                   -8-


                                         AUTO CONNECTION


                          Terminal 1                                        Terminal 2
   N1 turns                                                            N1 turns

                          Terminal 2                                         Terminal 1

   N2 turns                                                          N2 turns



            Fig. 1                                          Fig. 2


Referring to the figures, in the input N1 is assigned to terminal 1, N2 to terminal 2. If there are also
other terminals which are not auto connected, these are assigned numbers 3 and 4.

Occasionally, there is a variation of auto connection, as shown in Fig. 3.

                                                                      Terminal 1

   N1 turns            N3 turns                                         Terminal 3
                                                                      Terminal 2

   N2 turns



              Fig. 3                                        Fig. 4

Since part of the winding is not really auto connected in this case, it is necessary to introduce a third
terminal to handle the situation, as indicated in Fig. 4. For terminal 3 the connection should be
specified
with code 1, and the MVA should be the difference of the MVAs for terminals 1 and 2. Buck or boost
of terminal 3 is specified with the direction of the current. With buck connection, terminal 2 has the
highest MVA, with boost terminal 1. For auto connection, per unit impedances will be calculated on
the basis of the MVA for terminal 1.

With buck connection:
MVA3 = MVA2 * N3/N2                       MVA1 = MVA2 - MVA3

With boost connection:
MVA3 = MVA1 * N3/(N2 + N3)                MVA2 = MVA1 - MVA3
                                                   -9-

                                    PHASE SHIFT CONNECTIONS

With Z or zig-zag connection, the neutral connected winding is given code 5 and is assigned to one
terminal, and the terminal connected winding is given code 6 and is assigned to another terminal. With
P or polygon connection, codes 7 and 8 are used in a similar way. With ED or extended delta
connection, code 9 is used for the main and 10 for the terminal connected winding. In all cases, MVA
is given as the total for the two windings for both terminals. FLD12 will take into account differences
in phase angles, when calculating these cases. The two phase shift windings should never be put in
above each other, since that will result in excessive radial flux. Phase shift terminals are always
specified consecutively, with increasing codes in sequence.

The rated voltage for each terminal is given as volts per turn times no. of turns for P and ED
connections, and this times 3 for Z-connection.

                                   TANK, LEG AND YOKE LOSSES

For the tank, leg and yoke, losses are set equal to:
Loss = factor * area * (flux/m)2 watts
where for the tank and the core leg, the area is taken as:
area = 2 * radius * (axial depth) mm2
and the flux per meter circumferential depth:
(flux/m) = Amax - Amin weber/m

Amax and Amin are the maximum and the minimum vector potentials at the tank and the core leg,
respectively.

For each yoke:
area = 2 * Rmin (Rmax - Rmin) mm2

           Rmax Amax - Rmin Amin
(flux/m) = ───────────── weber/m
             (Rmin + Rmax)/2

Index min refers to values at the core leg, max to values at the tank.

                                LOCATIONS OF AXIAL GRID LINES

Grid density break lines are put in at the core leg, at all layer boundaries and at the tank. They are
assigned numbers so that the maximum grid line spacings do not exceed certain percentages of the
distance between the core leg and the tank.

1. Between the outer layer and the tank: 10%
2. Between the core leg and the inner layer, and also between layers: 5%
3. Within layers: 2.5%

In sheet windings, there are two axial grid lines per turn.
                                                 - 10 -


                            LOCATIONS OF RADIAL GRID LINES
                      WITH SHEET WINDINGS AND ONE SHEET AXIALLY

                              HALF WINDOW HEIGHT CALCULATED




                                             Equally spaced lines, with maximum spacing 5% of the
                                             section depth or E1/4, whichever is smaller.
                         E1




                                             The grid line spacings are determined by E4. Starting
                                             from the top, there are four spacings of 0.5% of E4, then
                                             four of 1%, four of 2%, four of 4% and seven of 10%.

                         E4




             Fig. 5



                              FULL WINDOW HEIGHT CALCULATED


As above for the upper half of the sheet winding, as a mirror image for the lower half.
                                                 - 11 -


                             LOCATIONS OF RADIAL GRID LINES
                      WITH SHEET WINDINGS AND TWO SHEETS AXIALLY

                               HALF WINDOW HEIGHT CALCULATED




                                             Equally spaced lines, with maximum spacing 5% of the
                                             section depth or E1/4, whichever is smaller.
                         E1




                                             The grid line spacings are determined by E4. Starting
                                             from the top, there are four spacings of 0.5% of E4, then
                                             four of 2%, four of 5%, four of 10%, four of 5%, four of
                                             2% and four of 0.5%.
                         E4




                          E2                 Equally spaced lines, with maximum spacing 2.5% of the
                                             section depth or E2/2, whichever is smaller.
             Fig. 6



                               FULL WINDOW HEIGHT CALCULATED


As above for the upper half of the sheet winding, as a mirror image for the lower half.
                              - 12 -


                LOCATIONS OF RADIAL GRID LINES
         IN CALCULATIONS WITH HALF THE WINDOW HEIGHT
                    AND NO SHEET WINDINGS



                          Equally spaced lines, with maximum spacing 5% of the
                          section depth or E1/4, whichever is smaller.
            E1




                          If E4 is equal to or greater than 1/3 of the section depth,
                          the lines will be unequally spaced, as shown. The
                          spacings are determined by E4. Starting from the top,
                          there are four spacings of 1% of E4, then two of 1.5%,
                          two of 2%, two of 2.5%, two of 3%, two of 4%, eight of
                          5% and four of 7.5%.

            E4




                          If E4 is less than 1/3 of the section depth, the lines within
                          E4 are equally spaced, and with a maximum spacing of
                          1.5% of the section depth.


Fig. 7
                         - 13 -


           LOCATIONS OF RADIAL GRID LINES
    IN CALCULATIONS WITH THE FULL WINDOW HEIGHT
               AND NO SHEET WINDINGS




                     Equally spaced lines, with maximum spacing 2.5% of the
                     section depth or E1/4, whichever is smaller.
          E1




                     If E4 is equal to or greater than 1/3 of the section depth,
                     the lines will be unequally spaced, as shown. The
                     spacings are determined by E4. Starting from the top,
                     there are two spacings of 0.5% of E4, then two of 1%,
                     two of 1.5%, two of 2%, four of 3%, and seven of 4%.
                     Then the spacings repeat in the reverse order.




                     The lines within E4 are symmetrical about the radial
                     centerline.
          E4




                     If E4 is less than 1/3 of the section depth, the lines within
                     E4 are equally spaced, and with a maximum spacing of
                     1% of the section depth.




          E2
                     The same rules apply as between the upper boundary and
                     the windings.

Fig. 8
                                            - 14 -

                                    LIST OF SYMBOLS


AC       Complex vector potential at the centroid of a triangle
ACOND    Area of one conductor corrected for rounded corners, sq.mm
AL       Complex vector potential at node L
AM       Complex vector potential at node M
AMAX     Maximum vector potential
AMIN     Minimum vector potential
AN       Complex vector potential at node N
ASEG     Area of winding segment, sq.mm
BH       Complex flux density, horizontal (radial) component
BHABS    Absolute value of BH
BHR      Real value of BH
BRLEG    Maximum radial flux density at core leg
BRTNK    Maximum radial flux density at tank
BZ       Complex flux density, vertical (axial) component
BZABS    Absolute value of BZ
BZR      Real value of BZ
CD       Complex current density
CDABS    Absolute value of CD
CDAVG    Average value of current density
CDENS    Array of complex current densities in triangles
CPOT     Array of complex vector potentials at nodes
CRIT     Critical stresses etc., and their locations
 1,1     N/sq.mm tension due to radial forces
 2,1     N/sq.mm compression due to radial forces
 3,1     Minimum number of spacer bars
 4,1     N/sq.mm compressive stress in spacer blocks or insulation due to axial forces
 5,1     N/sq.mm bending stress plus tension or compression
 1-5,2   Location (segment number)
D        Twice the area of a triangle, sq.m
DATLN    Array containing data for contour lines (applies more to FLD8)
 I,1     Relative permeability, contour line (segment) I
   2     Average conductivity, m/(ohms*sq.mm)
   3     Phase connection. If code 8, 9 or 10, total current.
   4     Direction of phase current (0.0, 1.0 or -1.0)
   5     Calculation code 0.0 to 10.0 (User's manual FLD8, page 6)
   6     Applied voltage number
   7     Number of parallel conductors (contour lines)
   8     Min. x or r-coordinate, mm
   9     Max. x or r-coordinate, mm
 10      Min. y or z-coordinate, mm
 11      Max. y or z-coordinate, mm
 12      Uniform current loss, kW/m or kW/circle
 13      Sum of D (twice area), sq.m
 14      Sum of D times radius to centroid, cub.m
 15      Actual loss, kW/m or kW/circle
DCORE    Core diameter
DCTOT    Total kW dc loss in the transformer section
DISPL    Angular displacement for periodicity condition (0.0 in FLD12)
                                            - 15 -


DPEL    Displacement = 1.0, elongation or contraction = 2.0
DZ      Difference in z-coordinates
DZDE    Displacement or elongation in mm (can be negative)
E1-16   Temporary storage
EAL     Modulus of elasticity for aluminum, 7200*9.81 N/sq.mm
ECU     Modulus of elasticity for copper, 13000*9.81 N/sq.mm
EDAX    kW/cub.dm eddy current loss due to axial flux
EDLEG   kW eddy current loss due to flux entering the core legs
EDPU    Per unit eddy current loss
EDRAD   kW/cub.dm eddy current loss due to radial flux
EDTNK   kW eddy current loss due to flux entering the tank
EDTOT   Total kW eddy current loss
EDWND   kW eddy current loss in the windings
EDYOK   kW eddy current loss due to flux entering the yokes
EMAG    Magnetic energy, watts-seconds
FEDDY   Factor for eddy current losses (FEDDY*(mm*tesla)**2 = kW/cub.dm)
FLAXI   = 2.0 (axi-symmetric field) in FLD12
FLEG    Factor for eddy current losses in the core legs
FLXLN   Number of flux lines
FPEAK   Peak factor for short circuit current (usually 1.8)
FR      N/circle radial force on a triangle
FRACT   Fraction of window height used in the calculations (0.5 if sheet winding present)
FREQ    Frequency
FSC     Factor for short circuit forces and stresses
FSUPL   Axial short circuit force, lower support, newton
FSUPU   Axial short circuit force, upper support, newton
FTANK   Factor for eddy current losses in the tank
FYOKE   Factor for eddy current losses in the yokes
FZ      N/circle axial force on a triangle
HORF    Array containing first horizontal grid lines, detailed output
HORL    Array containing last horizontal grid lines, detailed output
I       Index
I1      Temporary storage
I5      Number of triangles
I6      Number of nodes
IBIT    Array containing triangle information
IDENT   Array containing run identification
IEXIT   Branch exit number
ILN     Array of last contour point numbers, numbered consecutively
ISHT    1 if sheet winding present, 0 if not
ITRI    Function giving node numbers in triangle, counter-clockwise
J       Index
J1-2    Temporary storage
K       Index
K1      Temporary storage
KEXIT   Branch exit number
KLAST   Last value of index K
L       Index
                                            - 16 -


LAYER    Array containing information about layers
 J,1     Layer number
   2     Last segment number
   3     Inner radius
   4     Radial width
   5     Terminal number
   6     Number of parallel groups
   7     Direction of current
   8     1.0 = copper, 2.0 = aluminum
   9     Number of spacer blocks
 10      Width of spacer blocks
LIMBS    Number of wound limbs
LINC     Array containing coordinates for all the contour points, mm
LINTO    Total number of contour lines (no. of segments + 1)
LNHOR    Number of horizontal grid lines
LNVER    Number of vertical grid lines
M        Index
MASS     Logical unit number for mass storage device
MVAMX    Maximum MVA referred to one terminal
N        Index
NCOD3    = 0 for codes 3.0 and 9.0, = 1 otherwise
NEWAX    N/cub.dm axial force at peak rated current
NEWRAD   N/cub.dm radial force at peak rated current
NEXIT    Branch exit number
NIO      Inverse permeability
NLAY     Number of layers
NOCOL    Number of columns in the grid (LNVER-1)
NSEG     Number of winding segments
NTERM    Number of terminals
NVOLT    Number of independent voltages U
OUT      Logical unit number for output device
PCHH     Array containing horizontal grid line numbers, where the grid density changes.
PCHV     As above, but for vertical grid lines
PER      = 0 in FLD12
PERMO    Permeability of free space, 4.0**1.0E-7
PHASE    Number of phases
PHCUR    Rms phase current
PI       3.1415927
POINT    Array containing node numbers for all the contour points
RC       Radius at triangle centroid
RESAL    Resistivity of aluminum at 75 deg.C, 3.44E-5 ohms*mm
RESCU    Resistivity of copper at 75 deg.C, 2.1E-5 ohms*mm
RL       Radius at point L
RM       radius at point M
RMIN     Minimum radius to left field boundary (core leg)
RN       Radius at point N
RPU      Per unit short circuit resistance
RTANK    Radius to tank
S1       Alphanumeric string
SAXMN    Minimum accumulated axial force, newton
                                          - 17 -


SAXMX   Maximum accumulated axial force, newton
SCALE   Scale of flux plot
SCGVA   Short circuit GVA
SCOND   Current density in conductor, A/sq.mm
SEG     Array containing information about winding segments
I,1     Segment number
    2   Minimum z-coordinate
    3   Maximum z-coordinate
    4   Total number of turns
    5   Number of active turns
    6   Number of strands per turn
    7   Number of strands radially across layer
    8   Radial strand dimension
    9   Axial strand dimension
  10    Layer number
  11    Minimum radius
  12    Maximum radius
  13    kVA (with sign, indicating direction of current)
  14    Total radial force, newton
  15    Total axial force, newton
  16    Min. N/axial mm, radially
  17    Max. N/axial mm, radially
  18    Max. accumulated axial force within the segment, newton
  19    Max. N/cub.cm, axially
  20    N/sq.mm tension or compression due to radial forces
  21    Minimum number of spacer bars
  22    N/sq.mm compressive stress in spacer blocks or insulation due to axial forces
  23    N/sq.mm bending stress plus tension or compression
  24    kW/cub.dm gross volume dc loss
  25    Total kW dc loss, not including leads and connections
  26    kW eddy current loss due to axial flux
  27    kW eddy current loss due to radial flux
  28    Total kW eddy current loss
  29    Average per unit eddy current loss
  30    Maximum per unit eddy current loss (max. pu current density if sheet winding)
  31    Fill factor (conductor area/segment area)
  32    Force from bottom of layer up to maximum in segment, newton
  33    Force from bottom of layer up to top of segment, newton
  34    Force from top of layer down to maximum in segment (negative)
  35    Force from top of layer down to bottom of segment (negative)
  36    R-min for max pu eddy current loss
  37    R-max for max pu eddy current loss
  38    Z-min for max pu eddy current loss
  39    Z-max for max pu eddy current loss
  40    Average current density in sheet winding
SHLD    1.0 if aluminum or copper shield, 0.0 if no shield
SQR2    2 = 1.414214
SQR3    3 = 1.732051
STRNG   Alphanumeric string
                                            - 18 -


TERM    Array containing information about terminals
  I,1   Terminal number
    2   Connection code, I,Y: 1, D: 2, Auto: 3, Z: 5-6, P: 7-8, ED: 9-10
    3   MVA
    4   kV
    5   Number of active turns in series (negative for negative current)
    6   Dimensioning kVA, including correction for auto connection
    7   Volts per turn (always positive)
TRI     Array containing triangle information
 I,1    Relative permeability
   2    Conductivity, m/(ohms*sq.mm)
TSTR    Array of alphanumeric strings
TWOPI   2.0*3.1415927
UNITS   1 for input in mm, 2 for input in inches
V       Array of input data
VERF    Array containing first vertical grid lines, detailed output
VERL    Array containing last vertical grid lines, detailed output
VERT    Array containing data for the nodes (vertices)
  I,1   x or r-coordinate, meters (initially mm)
    2   y or z-coordinate, meters (initially mm)
VOL     Cub.dm volume of triangular element
VTURN   Volts per turn
WARN    Warning code (refers to format number for message)
WNTNK   Radial distance between outer winding and tank
WS      Magnetic energy in triangle, watts-seconds
XPOS    Array containing x or r-positions of vertical grid lines, mm
XPU     Per unit short circuit reactance
YPOS    Array containing y or z-positions of horizontal grid lines, mm
Z1-4    z-coordinate
ZANG    Angular displacement, phase shift connection
ZB      z-coordinate, lower boundary (yoke or radial centerline)
ZC      z-coordinate of triangle centroid
ZKVN    kV for neutral connected winding, phase shift connection
ZKVT    kV for terminal connected winding, phase shift connection
ZKVR    kV for complete winding, phase shift connection
ZL      z-coordinate, point L
ZM      z-coordinate, point M
ZN      z-coordinate, point N
ZNEW    New z-coordinate
ZOPT    Optional per unit impedance
ZPU     Per unit short circuit impedance
ZSGMN   Minimum z-coordinate for all segments
ZSGMX   Maximum z-coordinate for all segments
ZSMN    Minimum segment z-coordinate, meters
ZSMX    Maximum segment z-coordinate, meters
ZSYST   Per unit system short circuit impedance
ZTOT    Per unit total impedance (system + transformer)
ZU      z-coordinate, upper boundary (yoke)
ZUSED   Per unit impedance used in calculations of forces and stresses
                                                   - 19 -


                                        USER PROGRAMMING


The input and output routines for FLD12 are completely decoupled from the main program, and can be
changed by the program user. The routines are supplied in source code in the files:

INP12.FOR  Input routine for both metric and English units.
OUTMET.FOR Output routine for metric units.
OUTENG.FOR Output routine for English units.



                                          PROGRAM OUTPUT


If output in English units is desired, change OUTMET to OUTENG in file RUN.BAT. Commands
OUTMET and OUTENG can also be given separately, after a run has been made.

The first part of the output is simply a repetition of the input, except that now "z - lower boundary" is
always zero, and all z-coordinates are with respect to the lower boundary.

Positions of vertical and horizontal grid lines are needed if it desired to run program DETAILS (see
page 2), to get detailed information about vector potentials at the nodes and flux densities in the
triangles.

The information for each segment includes forces and stresses at peak short circuit current, and dc and
eddy current losses at normal current.

Losses are at 75 deg. C, and do not include losses in leads and connections. Strands are assumed to
have rounded corners, with an area 0.5 mm2 subtracted if width times depth exceeds 15 mm2,
otherwise 0.25 mm2 subtracted. If width = depth and not more than 4 mm, round wire is assumed.

KVA is kilovolts times amperes within the segment. For a negative current, the KVA also comes out
negative. If the KVA does not add up to zero for all the segments, the program aborts and prints an
error message.

Force per unit axial length is the total across the width of the layer.

Maximum accumulated axial forces are given both for each segment and for each layer. For a segment,
it is the force accumulated only within that segment. Forces on the supports from the layers are for one
phase.

Most of the forces can come out negative or positive. The signs then refer to negative and positive
directions along the r and z-axes. If a stress comes out negative, it is due to a negative force.
                                                  - 20 -


                                         POST PROCESSING


After the main program has been run, the run identification and all the essential calculated and input
information are in file \GRAPHICS\FOR.FIL, and can be retrieved for further processing. This
includes all the vector potentials, current densities and node and contour line coordinates.

Post processing can also be done in directory FLD8, according to the FLD8 manual.

A particularly useful post processor determines ideal locations of crossover points. If the calculations
involve a disk or helical winding with N parallel conductors, there will be N-1 crossover points in the
winding, where the conductors change positions. Some manufacturers make these crossover points
evenly spaced, but since the axial flux density is not uniform, this can lead to quite high losses due to
circulating currents. Ideally, spacings should be inversely proportional to average axial flux density,
which again is proportional to differences between vector potential times radius at the outer and inner
radius of the winding. Before ideal locations can be determined, the main program must have been run
with the upper boundary approximately twice the yoke distance from the winding ends, to make the
leakage field a weighted average of conditions under and outside of the yoke. The winding must
belong to only one layer in this case. Ideal locations can be calculated simply by giving the command:

LOCATIONS

This should be done after the output from the main program has been printed, because the file
OUTPUT is also used by the post processor. Some questions about the winding will appear on the
screen, and it is useful to have the output from the main program available, in order to answer the
questions.

Another post processor is for drawing a graph of current density distribution in individual turns in
sheet windings. It is started with the command:

GRAPH

Here also, it is advisable to have the output from the main program available, in order to answer the
questions on the screen. Followed by the command PLOT, the graph can be printed in the same way as
explained earlier for flux plots and grids.

FLD12 also generates an output file SEGMENT.FIL, which contains all the 40 input and output items
for each segment, listed on page 17. They are in metric units if written by Fortran subprogram
OUTMET, in English units if written by OUTENG. The format specification is FORMAT(40(F14.4)).
                                                     - 21 -


                              DETAILS, REACTANCE CALCULATION


After running DEMO.INP, from the end of file \FLD8\OUTPUT, the magnetic energy is:

W = 1203.5 Ws

This is for half of one phase at peak rated current. For the low voltage winding, the rms value of the
rated current is:

IN = 25000/(6.43) = 2255.3 A

Short circuit reactance, referred to this winding:

x = (41203.5)/(2255.32)2 = 0.1487 ohms/phase

Base impedance:

zN = 6400/(2255.33) = 1.6384 ohms

Per unit reactance, as from \FLD12\OUTPUT:

xpu = 0.1487/1.6384 = 0.0908
                                                 - 22 -


                              DETAILS, SHORT CIRCUIT STRESSES


Before reading this page, repeat the command in directory FLD12:

RUN DEMO.INP

From \FLD12\OUTPUT, for segment no. 2, both occuring at the winding end:

Across the width of the layer, for the whole circumference:
Min. newton per axial mm, radially = 7306.0
Axial force per cub.cm gross volume of a triangular element, including insulation and spacers (if any):
Max newton per cub.cm, axially = 28.90

(Average spacer pitch) - (half spacer width) = ((763+92)/20) - 25 = 109.3 mm

Gross area of segment = 92  550 = 50600 mm2
Net area of segment = 194  5  1.8  12.2 = 21301 mm2

Axial force on one strand = 28.90  109.3  1.8  12.2  50600 / (21301  1000) = 164.78 N

Bending stress = 164.78 109.3 / ((12  1.8  12.22)/6) = 33.612 N/mm2

Tension = 7306.0  550 / (2  21301) = 30.024 N/mm2

Since both stresses are tangential, they add directly:
Combined bending + tension = 33.612 + 30.024 = 63.64 N/mm2

as given in \FLD12\OUTPUT.

Axial bending between the spacers must be considered for each strand individually, since the strands
don’t support each other. This is different for tension and compression, which is taken up by the disk
acting as a whole, so that the stress evens out radially.

The factor 1/(2) in the equation for tension can be explained with reference to Fig. 9.




          Fig. 9

Taking only the vertical component of the force on the upper half introduces a factor 2/. Multiplying
that with (total force around the circumference)/2 makes the factor 1/. Since the force is taken up by
two cross sections of the cylindrical coil, the factor for the stress becomes 1/(2).
                                                  - 23 -

                                              BUCKLING

A slender column subject to compression may fail long before the stress limit is reached, due to the
unstable deflection known as buckling. It is described in “S. Timoshenko, Strength of Materials”, third
edition 1955, and in other textbooks on the subject. The description here is based on Timoshenko’s
theory of columns, pages 245-277 in his book. The theory is applied in FLD12 to determine the
number of axial spacer bars required to prevent buckling in windings subject to compression.

                                      y       P

                               

                            Fig. 8

Fig. 8 shows a slender column with one built-in and one open end subjected to a compressive force P.
As long as it is straight, the bending moment along the column is zero. For a small deviation from the
straight line, indicated as a dashed line in the figure and unavoidable for a slender column, the bending
moment is Py along the column. If P is not too large compared with the stiffness of the column, Py
will be balanced by bending stresses with the deflection at an equilibrium. However, when P is
increased, a point may be reached where equilibrium is no longer possible. The deflection increases
uncontrollably accompanied by increasing bending moments, and buckling occurs. Timoshenko’s
equation 144 relates this critical force Pcr with the modulus of elasticity E, moment of inertia Iz and
length :

Pcr = 2 E Iz / (4 2)

For copper of varying strengths, the modulus of elasticity is nearly constant. To prevent buckling in
windings, it is therefore immaterial what grade of copper is used.

For a column which is built-in at both ends, as shown in Fig. 9, the deflection pattern in Fig. 8 is
repeated four times. Timoshenko’s equation 146 for a column with two built-in ends can therefore be
derived from his equation 144 above with /4 substituted for :

Pcr = 42 E Iz / 2

    P                                                                                                   P
          A                                          b                                          C
                                                         B
                                                  Fig. 9

The maximum bending moment Pb occurs in the middle and at both ends (points A, B and C).
In order to apply the theory to the buckling of windings, reference is made to Fig. 10, which shows one
strand in a winding subject to compression, located between two adjacent axial spacer bars.
                                                   - 24 -

Only one strand is considered, because all the strands have the same compressive stress and are
assumed to flex individually, sliding against each other (without bonding). As long as the strand is
perfectly round, the bending moment at point B and all other points is zero. The bending moment from
the compressive force P is balanced exactly by the bending moment from the distributed inward force
acting along the circumference.

The compression will shorten the strand, so that it moves inward at point B. The deflection will be as
shown by the dashed line in Fig. 11.




In practice, due to a fairly large number of axial spacer bars, the arcs in figures 10 and 11 will be close
to straight lines. The bending moment from the distributed inward force will be practically the same
after the deflection, and the bending moment at point B will change from zero to very nearly Pb. At
points A, B and C the angular displacements are zero. Both the flexing and the change of bending
moment along the strand will be very nearly equivalent to that of the column with built-in ends in
Fig. 9. FLD12 therefore assumes that the same critical length (or load) for buckling applies in both
cases and uses Timoshenko’s buckling formula 146 to determine the required number of axial spacer
bars. However, the recommended number is double the theoretical minimum, for several reasons.

The flexing may not be exactly as shown in Fig. 11. Axial spacer bars provide more or less rigid
support radially, partly because the core is not perfectly round. Flexing between two spacer bars may
affect flexing between the other spacer bars to some extent. For both reasons, the pattern of deflection
will not repeat exactly between spacer bars before buckling occurs. Also, there is obviously some
approximation involved in the use of the equation for a straight column with built-in ends. Fortunately,
a large safety factor applied to buckling is usually not difficult and expensive to achieve.

Sometimes the use of epoxy bonded CTC will make a winding stiff enough to withstand buckling,
even though the FLD12 calculation based on individual flexing of strands may show otherwise.

How buckling deforms a winding is shown in Fig. 12. It usually bulges outward at one point.




In Fig. 11, the bending moments in A, B and C will be nearly the same, Pb. When the critical stress is
approached, the winding will tend to give way sooner outward at a point A or C than inward at a point
B, because of the curvature of the winding. The buckling occurs at the weakest point around the
circumference, or where the bending moment happens to be highest.
                                                  - 25 -


                           TRANSFORMERS WITH PARALLEL CIRCUITS


                           A three winding rectifier transformer is used as an example. It has a primary
                           high voltage winding consisting of two parallel connected parts H1 and H2.
                           There are two secondary windings above each other, one wye connected and
                           one delta connected. Both are designed for the same rated MVA and kV.
      LY     H1
                           One problem here is to calculate short circuit reactance and forces for a
                           short circuit in one of the secondary windings, such as LY, not knowing
                           ahead of time what the current distribution is between H1 and H2.

      LD     H2            Four layers and four terminals should be specified in FLD12, one for each of
                           the four parts. The MVA should be given as zero for LD and 100% for LY,
                           but it is uncertain initially what the MVAs should be for H1 and H2, except
                           that the sum should be the same as the MVA for LY. Two methods for
                           finding the current distribution between H1 and H2 will now be explained.


Minimizing magnetic energy

Current distributions and circulating currents always adjust themselves to give minimum magnetic
energy. This is often the easiest way of finding the correct currents.

100% MVA can be specified initially for H1, zero for H2. Then gradually MVA is decreased for H1
and increased for H2 until minimum calculated magnetic energy is reached within a certain tolerance.
The current distribution will then be correct. Magnetic energy is the last item in \FLD8\OUTPUT.


Equalizing flux linkages

Since H1 and H2 are in parallel, for the correct current distribution they should have the same flux
linkages. Flux linkages are linear functions of currents.

Again, 100% MVA can be specified initially for H1, zero for H2. The difference in flux linkages is
recorded. Then current distribution can be changed by say 1%, to 99% in H1, 1% in H2. That will
probably make the difference in flux linkages closer to, but not quite zero. Linear extrapolation down
to zero establishes the correct current distribution for a third calculation. If desired, that can now be
checked by observing how the calculated magnetic energy is changed with small deviations from the
calculated currents. The changes should always be positive.
                                                   - 26 -


                                THREE WINDING TRANSFORMERS


Only one reactance is calculated each time FLD12 is run, based on the magnetic energy. This is not
sufficient to determine short circuit currents and forces in three winding transformers when all the
windings carry current.

A three winding transformer has an equivalent circuit with three reactances, which can be determined
from three FLD12 calculations, each time with currents in only two windings. In per unit, they must all
relate to the same base MVA. The theory behind this is presumed to be known to the FLD12 user and
will not be gone into here.

From this equivalent circuit, currents can be calculated for different loads or short circuit conditions.
Having done that, a final calculation with FLD12 can have the "optional per unit impedance" on input
line 3 specified different from zero, as the inverse of "times normal" ac current, if short circuit forces
are required.

The short circuited winding is assigned 100% MVA. For the other two, MVA is assigned in proportion
to the per unit current flowing through the winding.



         REGULATING WINDING CONNECTED THROUGH SERIES TRANSFORMER


Sometimes a regulating winding is connected through a series transformer to reduce the current in the on
load tap changer. The regulating winding can be connected in boost or buck position.

The main winding and the regulating winding are assigned to two separate terminals. In boost
connection both MVAs are positive and add up to the base MVA for the transformer. In buck
connection the main winding has positive MVA and the regulating winding negative MVA. Again, the
sum is equal to the base MVA. The specified kV always agrees with the number of active turns.

The impedance in the series transformer should be taken into account in the calculation of forces and
stresses. This can be done by specifying the “optional impedance” in the input different from zero.
                                                  - 27 -


                   IMPEDANCE BETWEEN WINDINGS ABOVE EACH OTHER


Below, in the flux plot to the left, the calculated leakage flux flows radially and enters the outer
boundary at right angles. In reality, this outer boundary usually consists of a tank around at least part
of the perimeter, where the radial leakage flux is counteracted by strong eddy currents, which are here
not taken into account in the calculations and would cause excessive losses in a normal transformer. A
normal transformer can therefore simply not be built like that. Nevertheless, the arrangement is
sometimes used in transformers for short intermittent duty. Due to the neglect of eddy currents in the
tank, the calculated impedance will be much too high, if it is done this way.




If a code=1 is put in for the AL/CU SHIELD in the FLD12 input, it changes the boundary condition
for the outer boundary into something which is probably more realistic in this case, as shown in the
flux plot to the right. However, the calculated short circuit impedance will be strongly dependent upon
the specified distance between the winding and the outer boundary and will be impossible to estimate
accurately without access to tests and calculations for similar transformers.



                                     SEQUENCE IMPEDANCES


Sequence impedances will first be discussed with reference to rotating machines.

Positive sequence current produces an MMF, which rotates in the same direction as the rotor. In a
synchronous machine, the resulting flux is dc with respect to the rotor. In an induction machine, the
frequency is very low. Negative sequence current sets up an MMF, which rotates opposite to the rotor
and produces large opposing induced currents. Zero sequence current sets up essentially zero MMF in
the air gap. The differences are profound and produce very different positive, negative and zero
sequence reactances.
                                                  - 28 -


In a transformer, there is no difference in reactance for positive and negative sequence. However, zero
sequence reactance is usually different.

Zero sequence current flows simultaneously without the usual 120 degree phase shift in the different
phases. It can only flow from the outside into a Y, Z or auto connected winding with the neutral
connected, so that the current has a return path. If the transformer also has a delta connected winding,
it will always act as if short circuited. Induced zero sequence current flows in a closed loop within the
delta.

If zero sequence current flows into a winding and there is no possibility of ampereturn balance with
induced current in another winding, the zero sequence reactance depends on the type of core. In a five
legged core and in single phase units, zero sequence flux has a return path through the core, and zero
sequence reactance will assume the very high value of a magnetizing reactance. In a three legged core,
the flux has no return path through the core and must find its way elsewhere, usually through oil,
structural parts and tank. In the tank and core clamps there will be induced currents, which lower the
reactance. The reactance will be much lower than a magnetizing reactance, but still very much higher
than a short circuit reactance. Without ampereturn balance, the reactance can not be calculated with
FLD12.

In the discussion which follows, zero sequence current is assumed to flow into a winding, where
ampereturn balance results from induced currents in other windings.

In a two winding transformer with Y-Y or Y-D connection, the zero sequence reactance will tend to be
the same as the positive and negative sequence reactance, also in a transformer with auto connection
without tertiary winding. With a three legged core, induced currents in tank and core clamps will lower
the reactance slightly.

Since a delta connected tertiary winding acts as if short circuited, its presence always influences the
reactance between the main windings.

As an example, say from the inside the transformer has a delta connected tertiary, a secondary and a
primary winding, where zero sequence current is fed into the outer primary. The secondary is shorted,
and the inner tertiary acts as if shorted. The current sharing between secondary and tertiary can be
found as explained on page 26. Say 100 MVA is specified for the primary, then perhaps 120 MVA for
the secondary and –20 MVA for the tertiary will be about right.

Phase shift connections also require special treatment. In extended delta connection, the main delta
connected winding acts as if short circuited, whereas the series winding is open. In zig-zag (Z) and
polygon (P) connections, the two winding parts on one core leg carry currents of the same phase in
opposite directions. Ampereturns are balanced within the same winding if the two parts have the same
number of turns. In calculations with FLD12, the two parts can be considered belonging to two separate
terminals. The Z-connection can be replaced by Y and the P by D.
                                                      - 29 -


                                UNSYMMETRICAL SHORT CIRCUITS


Short circuit currents are found by the method of symmetrical components, which is presumed to be
known.


LINE TO LINE

The transformer is presumed to be unloaded when the fault occurs. The figure shows a wye connected
winding, but the result is the same also for other connections.


                                                                    Ib1
                                                                    <

                     a                     Ia = 0                         Positive sequence

                                                                                              ^ Ib1 = Ib2
                                                               Z1                               = Ib/2

                                                                    Ib2
                                                                    <
         c                         b
                                           Ib = -Ic            Z2         Negative sequence



Since the sum of the three phase currents is zero, there is no zero sequence current in this case. The
sequence equivalent network is drawn for phase b. Positive sequence current Ib1 is equal to negative
sequence current Ib2, both of them equal to half the phase current Ib. As noted earlier, positive and
negative sequence impedances Z1 and Z2 are the same as the three phase short circuit impedance.
A driving voltage is only present for the positive sequence.

The resultant short circuit current will be the same as for a symmetrical three phase short circuit.
                                                    - 30 -


LINE TO NEUTRAL

                                                               Ia1
                                                                <

                      a
                                                                      Positive sequence

                             Ia                         Z1
                                                                                          ^ Ia1 = Ia2
                                                                                            = Ia0 = Ia/3
                                                                <
          c                          b                          Ia2
                                                         Z2           Negative sequence

                                                                <
                                                                Ia0 Zero sequence
                                                         Z0


Again, the transformer is presumed to be unloaded before the fault, so that Ib=Ic=0. Positive, negative and
zero sequence currents are all the same in this case, equal to one third of the phase current Ia. Z1 and Z2
are equal to the three phase short circuit impedance, as before, but now zero sequence impedance Z0 also
enters into the sequence network. If Z0 is also the same as Z1 and Z2, the short circuit current will again be
the same as for a three phase symmetrical short circuit. That is approximately true for a two winding
transformer with the other winding either delta connected, or wye connected with the neutral grounded.
Another case will now be discussed.

A three winding transformer has a primary high voltage winding with isolated neutral, a delta connected
stabilizing (tertiary) winding and a secondary wye connected winding with a line to neutral fault through
an external impedance Ze.

The figure shows the three windings. Phase windings which are drawn above each other are on the same
core leg. For a one per unit fault current, the principle of balanced ampereturns on each core leg gives
other per unit currents, as shown.
                                                  - 31 -




                I=2/3          1/3                    H




                I=1/3                                 T




             I=1     Ze                               L


                           a          b           c


The critical phase is phase a, where the fault occurs. With ILa=1, IHa=2/3 and ITa=1/3, MVAs for the three
windings must be specified accordingly, and an optional per unit impedance as the inverse of the per unit
short circuit current, calculated from the sequence network. Zero sequence impedance Z0 is
approximately equal to the short circuit impedance ZLT between secondary and tertiary windings. With
the short circuit through an external impedance Ze, Z0 in the sequence network on the previous page must
be replaced by Z0+3Ze.
                                                - 32 -


                           THE COMMAND PROMPT ENVIRONMENT


The Command Prompt window should be maximized and the size adjusted to fill the screen after right
clicking the top title bar. Cursor size small and letter size 12x16 pixels are recommended. If Command
Prompt goes into full screen mode by an application, it can be brought back with Alt-Enter.

Since many PC users are not familiar with Command Prompt, here are some hints and frequently used
commands. The commands are examples and may be modified in obvious manners. Large and small
letters are interchangeable.

Commands given once on startup, perhaps in a STARTUP.BAT file:
SET COPYCMD=/Y Deactivates warning on overwriting existing files.
PATH=C:\SYSTEM;C:\QBASIC Specifies search paths for executable files.
SUBST P: C:\DRIVEP Substitutes drive P for directory (or folder) C:\DRIVEP making P a virtual
drive (or unit).

Other commands:
C: Moves to unit C or another unit.
CD\ Changes to base directory.
MD GRAPHICS Makes directory GRAPHICS.
CD\GRAPHICS Changes directory to GRAPHICS, just below the base directory.
COPY OLD.INP NEW.INP Copies old file OLD.INP to a new file NEW.INP.
COPY /? Explains options available for command COPY.
REN OLD.INP NEW.INP Renames OLD.INP as NEW.INP.
DEL OLD.INP Deletes OLD.INP.
DIR *.INP Lists all files in the directory with extension INP.
DIR *.I?? Lists all files in the directory with three letter extension starting with I.
START NOTEPAD OUTPUT Invokes Windows program NOTEPAD with file OUTPUT.
START PLOTFILE.BMP Starts a standard Windows program to process the bitmap file.
PROGRAM FLD12          COMPLEX POTENTIAL TRANSFORMER LEAKAGE FLUX

SAMPLE CALCULATION


NUMBER OF PHASES 3.0                          SYSTEM SHORT CIRCUIT GVA   0.000
FREQUENCY 50.00                               OPTIONAL PU IMPEDANCE 0.0000
NUMBER OF WOUND LIMBS 3.0                     PEAK FACTOR 1.800
FRACTION OF WINDOW HEIGHT CALC. 0.5           LOSS FACTORS
Z - LOWER BOUNDARY     0.0                      TANK   0.00
Z - UPPER BOUNDARY   670.0                      LEG    0.00
CORE DIAMETER 567.0                             YOKE   0.00
DISTANCE WINDING-TANK 100.0                   SCALE, FLUX PLOT 0.350
AL/CU SHIELD 0.0                              NO. OF FLUX LINES 25.0


TERMINAL NUMBER      1.0          2.0
CONNECTION            D            Y
MVA                 25.000       25.000
KV                   6.400       50.000

LAYER    LAST     INNER      RADIAL   TERM.   PAR.     DIR.     CU/AL      SPACER BLOCKS
NO.      SEGM.    RAD.       WIDTH    NO.     GROUPS   CUR.                NUMBER WIDTH
 1.0      1.0     301.5       52.0     1.0     1.0     -1.0        CU       20.0    40.0
 2.0      2.0     381.5       92.0     2.0     1.0      1.0        CU       20.0    50.0

SEGM.   LAYER    Z-COORDINATE     NO. OF TURNS    NUMBER OF STRANDS     STRAND DIMENS.
NO.     NO.      MIN.    MAX.     TOTAL ACTIVE    PER TURN RADIALLY     RAD.   AXIALLY
  1.0    1.0       0.0   550.0     43.0   43.0      16.0      16.0       2.90    8.30
  2.0    2.0       0.0   550.0    194.0 194.0        5.0      35.0       1.80   12.20

RADIAL POSITIONS, VERTICAL GRID LINES
  1-10    283.5 292.5 301.5 308.0 314.5            321.0   327.5   334.0    340.5   347.0
 11-20    353.5 367.5 381.5 388.6 395.7            402.7   409.8   416.9    424.0   431.0
 21-30    438.1 445.2 452.3 459.3 466.4            473.5   498.5   523.5    548.5   573.5

AXIAL POSITIONS, HORIZONTAL GRID LINES
  1-10      0.0   41.2   82.5 123.8 165.0         192.5    220.0   247.5   275.0    302.5
 11-20    330.0 357.5 385.0 407.0 429.0           445.5    462.0   475.8   489.5    500.5
 21-30    511.5 519.8 528.0 533.5 539.0           544.5    550.0   580.0   610.0    640.0
 31       670.0


'MIN. NUMBER OF SPACER BARS' IS TWICE THE THEORETICAL MINIMUM CALCULATED FROM
TIMOSHENKO'S BUCKLING FORMULA FOR COLUMNS WITH BUILT-IN ENDS, EQUATION 146.
THIS IS THE RECOMMENDED MINIMUM NUMBER. IF THE WINDING IS MADE OF BONDED CTC,
THE NUMBER IS CALCULATED TOO HIGH, SINCE NO BONDING IS ASSUMED IN THE
CALCULATIONS.

THE COMPRESSIVE STRESS IN THE SPACER BLOCKS IS CALCULATED DUE TO ACCUMULATED
AXIAL FORCES WITHIN AND OUTSIDE OF THE WINDING SEGMENT. IF THERE ARE NO
SPACERS, THE COMPRESSIVE STRESS IN THE INSULATION IS CALCULATED IN THE SAME
WAY. IN BOTH CASES, THE PROGRAM ASSUMES A RADIAL LENGTH OF CONTACT EQUAL TO
THE SUM OF THE RADIAL STRAND DIMENSIONS.

STRESSES DUE TO COMBINED FORCES ARE BENDING STRESSES DUE TO 'MAX. N/CUB.DM,
AXIALLY' COMBINED WITH TENSION OR COMPRESSION DUE TO 'MIN. N/AXIAL MM,
RADIALLY'. BENDING STRESS IS CALCULATED FOR A BUILT-IN BEAM WITH A LENGTH
EQUAL TO THE AVERAGE SPACER PITCH MINUS HALF THE SPACER WIDTH. IT IS CALCULATED
TOO HIGH FOR BONDED CTC, AGAIN BECAUSE NO BONDING IS ASSUMED IN THE
CALCULATIONS.
SEGMENT NUMBER    1.0
  AMPERETURNS    -55989.6          FORCES AT PEAK SHORT CIRCUIT CURRENT
  KVA -4166.67                       TOTAL RADIALLY, NEWTON -5545547.0
DC LOSS                              TOTAL AXIALLY, NEWTON    -580281.4
  KW/CUB.DM GROSS VOLUME 0.1419      MIN. N/AXIAL MM, RADIALLY -5750.9
  KW TOTAL    8.354                  MAX. N/AXIAL MM, RADIALLY -10954.3
EDDY CURRENT LOSS                    MAX. ACCUM. AXIALLY, NEWTON   -580618.9
  KW DUE TO AXIAL FLUX    0.555      MAX. N/CUB.CM, AXIALLY -88.31
  KW DUE TO RADIAL FLUX 0.141      DUE TO RADIAL FORCES
  KW TOTAL 0.696                     N/SQ.MM TENSION/COMPRESSION -57.90
  PER UNIT, AVERAGE 0.0833           MIN. NUMBER OF SPACER BARS 16.7
  PER UNIT, MAXIMUM 0.3615         DUE TO AXIAL FORCES
  OCCURS BETWEEN                     N/SQ.MM IN SPACER BLOCKS 15.64
 R-MIN   R-MAX    Z-MIN   Z-MAX    DUE TO COMBINED FORCES
 314.5   321.0    544.5   550.0      N/SQ.MM BENDING+TENS./COMPR.    93.52

SEGMENT NUMBER    2.0
  AMPERETURNS     55989.6           FORCES AT PEAK SHORT CIRCUIT CURRENT
  KVA   4166.67                       TOTAL RADIALLY, NEWTON    6686240.5
DC LOSS                               TOTAL AXIALLY, NEWTON     -304434.6
  KW/CUB.DM GROSS VOLUME 0.0625       MIN. N/AXIAL MM, RADIALLY    7306.0
  KW TOTAL    8.495                   MAX. N/AXIAL MM, RADIALLY 13165.8
EDDY CURRENT LOSS                     MAX. ACCUM. AXIALLY, NEWTON    -304434.6
  KW DUE TO AXIAL FLUX    0.320       MAX. N/CUB.CM, AXIALLY -28.90
  KW DUE TO RADIAL FLUX 0.131       DUE TO RADIAL FORCES
  KW TOTAL 0.451                      N/SQ.MM TENSION/COMPRESSION    54.10
  PER UNIT, AVERAGE 0.0531            MIN. NUMBER OF SPACER BARS 0.0
  PER UNIT, MAXIMUM 0.4298          DUE TO AXIAL FORCES
  OCCURS BETWEEN                      N/SQ.MM IN SPACER BLOCKS    4.83
 R-MIN   R-MAX    Z-MIN   Z-MAX     DUE TO COMBINED FORCES
 438.1   445.2    544.5   550.0       N/SQ.MM BENDING+TENS./COMPR.     63.64


CRITICAL STRESSES ETC. AT PEAK SHORT CIRCUIT CURRENT:
DUE TO RADIAL FORCES                            SEGMENT NO.
  N/SQ.MM TENSION       54.10                       2.0
  N/SQ.MM COMPRESSION -57.90                        1.0
  MIN. NUMBER OF SPACER BARS 16.7                   1.0
DUE TO AXIAL FORCES
  N/SQ.MM IN SPACER BLOCKS OR INSULATION 15.64      1.0
DUE TO COMBINED FORCES
  N/SQ.MM BENDING+TENS./COMPR.    93.52             1.0

VOLTS PER TURN 74.419
MAX. RADIAL FLUX DENSITY AT TANK (PEAK, TESLA) 0.0132
MAX. RADIAL FLUX DENSITY AT CORE LEG 0.0696
BASED ON MAGNETIC ENERGY AND TOTAL LOSSES, WITH BASE MVA 25.000
PU TRANSFORMER SHORT CIRCUIT REACTANCE 0.0907411
                             RESISTANCE 0.0043
                             IMPEDANCE 0.0908
PU SYSTEM IMPEDANCE 0.0000
PU TOTAL IMPEDANCE 0.0908
PU IMPEDANCE USED IN CALCULATIONS OF FORCES AND STRESSES 0.0908

FOR THE WHOLE TRANSFORMER
EDDY CURRENT LOSS
  KW WINDINGS   6.883
  KW TANK       0.000
  KW LEG        0.000
  KW YOKE       0.000
  KW TOTAL      6.883
  PER UNIT, TOTAL 0.0681
DC LOSS, KW TOTAL 101.090
LOSSES REFERRED TO LAYERS ARE FOR ONE LIMB, REFERRED TO TERMINALS FOR THE
WHOLE TRANSFORMER.


LAYER NUMBER 1
   DC LOSS, KW    16.707
  EDDY CURRENT LOSS
   KW DUE TO AXIAL FLUX    1.111
   KW DUE TO RADIAL FLUX   0.282
   KW TOTAL    1.392
   PER UNIT, AVERAGE 0.0833
   PER UNIT, MAXIMUM 0.3615
   OCCURS BETWEEN
  R-MIN   R-MAX    Z-MIN  Z-MAX
  314.5   321.0    544.5  550.0
  FORCES AT PEAK SHORT CIRCUIT CURRENT, NEWTON
   MAX. ACCUMULATED AXIALLY 580619.0
   OCCURS IN SEGMENT NO.   1
   ON UPPER SUPPORT       0.0
  FLUX LINKAGE, REFERRED TO ONE TURN -0.03828893

TERMINAL NUMBER 1
  DC LOSS, KW   50.122
  EDDY CURRENT LOSS, KW   4.176
                     PER UNIT 0.0833
  PERCENT DEVIATION, VOLTS/TURN 0.000

LAYER NUMBER 2
   DC LOSS, KW    16.989
  EDDY CURRENT LOSS
   KW DUE TO AXIAL FLUX    0.640
   KW DUE TO RADIAL FLUX   0.263
   KW TOTAL    0.902
   PER UNIT, AVERAGE 0.0531
   PER UNIT, MAXIMUM 0.4298
   OCCURS BETWEEN
  R-MIN   R-MAX    Z-MIN  Z-MAX
  438.1   445.2    544.5  550.0
  FORCES AT PEAK SHORT CIRCUIT CURRENT, NEWTON
   MAX. ACCUMULATED AXIALLY 304434.5
   OCCURS IN SEGMENT NO.   2
   ON UPPER SUPPORT       0.0
  FLUX LINKAGE, REFERRED TO ONE TURN -0.00789063

TERMINAL NUMBER 2
  DC LOSS, KW   50.968
  EDDY CURRENT LOSS, KW   2.706
                     PER UNIT 0.0531
  PERCENT DEVIATION, VOLTS/TURN -0.024


TOTAL SHORT CIRCUIT FORCE FROM ALL THE LAYERS, UPPER SUPPORT       0.0

								
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