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Calculation of Losses in the Core Clamps of a Transformer Using 3 D Finite Element Method document sample
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 The earlier scalar potential program FLD5 with the same input and output has now been superseded by FLD12, because the faster calculation time with FLD5 is no longer an issue.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/(6.43) = 2255.3 A Short circuit reactance, referred to this winding: x = (41203.5)/(2255.32)2 = 0.1487 ohms/phase Base impedance: zN = 6400/(2255.33) = 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 = 42 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