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O.W. Andersen
USER’S MANUAL, FLD16
INDUCTION MOTOR FIELD PROGRAM
INTRODUCTION
Squirrel cage induction motors always have different numbers of slots in stator and rotor in order to
suppress noise and vibration, and to provide a uniform generation of torque. A stator and rotor slot
pitch can look as shown in Fig. 1.
Fig. 1. One stator and rotor slot pitch of a squirrel cage induction motor.
The motor sets up a rotating magnetic field, which is essentially sinusoidal and with a constant peak
flux density in the air gap. For the purpose of magnetic field calculations, conditions repeat for each
slot pitch. Magnetic flux densities will be the same, only phase shifted by an angle determined by the
number of slots and the number of poles. In magnetic field calculations by the finite element method,
the phase shift can be taken care of by calculating vector potentials as complex numbers. However, the
geometry in Fig. 1 does not lend itself to such calculations limited to one slot pitch, because the angles
spanned by a stator and rotor slot are different.
FLD16 circumvents this difficulty by artificially making the stator slot pitch the same as that of the
rotor, as shown in Fig. 2. In the process, the stator slot width is adjusted in the same ratio as the slot
pitch, keeping flux densities both in yoke and teeth unchanged. Also, the peak current flowing in a
stator slot is adjusted in the same ratio, keeping stator MMF unchanged.
Fig. 2. Stator slot pitch adjusted to be the same as that of the rotor.
The boundary conditions are zero vector potentials at the inside and outside radii, and a periodicity
condition applied to the upper and lower boundaries of Fig. 2. Vector potentials have the same
absolute values in corresponding positions, but with angular displacements corresponding to the rotor
slot pitch.
-2-
The FLD16 user specifies a per unit voltage and slip, for which the calculations are to be made. An
initial per unit stator current is also specified, but this is adjusted automatically to give the right air gap
flux that corresponds to the specified voltage.
The flux will be less when the motor is loaded than it is at no load, due to the voltage drop through the
stator resistance and leakage reactance. These impedances are independent of the load, and are
calculated separately.
Stator current, powerfactor and torque are part of the program output. Fig. 3 shows a flux plot from a
calculation near rated conditions for a motor with rotor bars sometimes described as inverted T-bars.
Fig. 3. Flux plot at peak stator current near rated conditions for 500 kW motor.
Calculations are made here with a centered stator slot, but just by changing a code in the program
input, a stator tooth will be centered, as shown in Fig. 4.
Fig. 4. Flux plot near rated conditions with centered stator tooth.
Whether a stator slot or tooth is centered will affect the calculated results to some extent, so normally
both calculations should be made and average values should be used. In Fig. 4 the currents in the two
stator slot halves are phase shifted plus and minus an angle corresponding to half a rotor slot pitch
from the maximum value, so that zero MMF will occur at the center in the same way as in Fig. 3.
Fig. 5 shows a calculation at no load with zero slip. Tests are most easily performed and are most
reliable for this condition.
Fig. 5. Flux plot at peak stator current at no load.
-3-
Since everything is assumed to vary sinusoidally, the permeability at a given point is assumed to be
constant through the cycle. Therefore, it should not be calculated for the peak flux density, but for a
somewhat lower value. The reduction factor is determined in such a way that the calculated stator
currents agree most closely with tests at no load, and it turns out to be about 0.9. That also takes into
account that the stator and rotor cores have a stacking factor slightly less than one.
Flux densities appear to be zero in the teeth in Fig. 5, but that is only because the flux plot is drawn at
the instant when the current in the stator slot goes through its maximum value. That is the way flux
plots are normally drawn, but it is also possible to draw the plot at a different phase angle. In Fig. 6 the
stator slot current goes through zero.
Fig. 6. Flux plot at zero stator current at no load.
Locked rotor conditions are the most difficult ones to analyze analytically, and this is where the need
for more accurate finite element calculations is the greatest. Fig. 7 shows a flux plot for a locked rotor.
The outer parts of the rotor teeth saturate very heavily due to tangential flux, in particular adjacent to
the neck above the bar. A very fine finite element meshing is required in this region, which also
explains the advantages of a one slot pitch analysis in terms of speed and accuracy.
Fig. 7. Flux plot at locked rotor.
The program can handle a large variety of stator and rotor slots. Fig. 8 shows another combination.
Fig. 8. Random winding stator slot and trapezoidal rotor bar.
-4-
In Fig. 9 there is an iron bridge above the rotor bar which saturates very heavily even at rated
conditions. The performance of the motor actually depends on this saturation. This is also a case which
demonstrates very clearly the advantages of accurate numerical field calculations.
Fig. 9. Cast aluminum rotor winding with closed rotor slot.
Figures 10, 11, 12 and 13 just show various additional rotor slot geometries, which the program is
capable of handling.
Fig. 10. Cast aluminum rounded trapezoidal bar.
Fig. 11. Double squirrel cage with top rectangular bar.
Fig. 12. Double squirrel cage with top round bar.
Fig. 13. Cast aluminum triple squirrel cage.
-5-
SOLUTION
The current in the stator slot is determined first from the initial per unit current specified in the input.
With this current, the finite element solution must be repeated a sufficient number of times to get the
right permeabilities which correspond to the flux densities in the various parts of the iron. The
procedure is explained in the FLD8 manual. When convergence has been achieved or the number of
iterations reaches a maximum usually set at 50, the program calculates the air gap flux and converts
that to an air gap voltage, induced by this flux. The voltage drop through the stator resistance and
leakage reactance is added, to get the terminal voltage.
The initial stator current is probably not right, and the current must be corrected for a second
calculation, where the terminal voltage will come closer to the specified value. If necessary, normally
up to four calculations like this are performed, or whatever maximum number is specified in the input.
If convergence is achieved earlier, the number is less.
Due to the double set of iterations, on permeabilities and stator current, the total number of finite
element solutions will often be in the order of 100. Even so, with a modern personal computer, the
solution time is only a few seconds.
Since the calculation is two dimensional, the resistance of the rotor bars per meter length must be
corrected to take into account ventilating ducts, bar extensions and endrings. In order to avoid
unnecessary complications in the layout of the finite element grid, bar clearances in the slots are
eliminated. This increases the bar cross section to fill out the slot, and the calculated bar resistance
must also account for that.
PROGRAM DESCRIPTION
FLD16 is a subprogram of the general purpose magnetic field program FLD8. It consists basically of
an input and an output routine for this program, but everything can now be installed in one directory
(folder). The routines are supplied in Fortran source code, and can be modified by the user. Input and
output can be either in metric or English units.
Design programs INDOP and INDAN generate input automatically for FLD16.
PROGRAM INSTALLATION
FLD16 is transmitted as a zip-file. It is extracted and installed in any directory (folder). The program
can also be installed on a memory stick and run from there.
-6-
RUNNING THE DEMO INPUT
Here all the Command Prompt commands and file names will be in capital letters. However, they are
case insensitive, and small letters can also be used.
To run the program with an input file DEMO.INP, enter:
RUN DEMO.INP
After a few seconds, a flux plot with 20 flux lines appears on the screen. It has been drawn on a Visual
Basic Form. If the picture appears to be cropped or too small, adjust the file SIZESCR.FIL. At the same time
a bitmap picture file PLOTFILE.BMP has been produced. Close the form and enter command:
PLOT
The flux plot now reappears in a standard Windows program. The conductors are red.
If it is now desired to print the flux plot, crop the picture file first to remove empty space and save it.
Microsoft Office Picture Manager or Microsoft Paint can be used for that. 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 FLD16 is stored in file OUTPUT. It is shown automatically on the screen when a run has
been made, and it can be brought back with command:
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.
To display the finite element grid on the screen, enter:
GRID
After the form is closed, the grid also reappears with the command:
PLOT
-7-
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. 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
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
New 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 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.
INDUCTION MOTOR FIELD PROGRAM
PROGRAM FLD16
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
INPUT UNITS (mm=1, inches=2)
STATOR SLOT OR TOOTH CENTERED (1 or 2)
KW
BASE KVA
FREQUENCY
NUMBER OF POLES
STATOR PUNCHING MATERIAL (code 1 or 2)
OUTSIDE STATOR DIAMETER
INSIDE STATOR DIAMETER
STACK LENGTH (gross length, including ventilating ducts)
NUMBER OF VENTILATING DUCTS (for radial ventilation)
WIDTH OF VENTILATING DUCT
NUMBER OF STATOR SLOTS
NUMBER OF VENTILATING HOLES (for axial ventilation)
HOLE DIAMETER
STATOR SLOT WIDTH, MIN.
STATOR SLOT WIDTH, MAX. (same as min. if open slots)
NECK WIDTH (above wedge, can be given as zero for open slots if neck = slot width)
NECK DEPTH
SLOT DEPTH (total from air gap to bottom)
TAPER DEPTH (zero if open slots)
WEDGE DEPTH
RADIUS AT BOTTOM (zero if open slots)
AIR GAP
ROTOR PUNCHING MATERIAL (code 1 or 2)
BORE DIAMETER (towards shaft)
NUMBER OF VENTILATING HOLES (for axial ventilation)
HOLE DIAMETER
NUMBER OF ROTOR SLOTS
ROTOR SLOT DEPTH (total from air gap to bottom, also for closed slots)
NECK WIDTH (zero for closed slots)
NECK DEPTH
TAPER DEPTH (zero for round and rounded trapezoidal bars)
REFERENCE ROTOR TEMPERATURE (degrees C, often 180)
ROTOR WINDING CODE
*1: 0.35 mm H-10, code=1 0.5 mm 1.3 W, code=2
*2: 1 = Trapezoidal or rectangular bars (see input sheet 2)
2 = Inverted T-bars
3 = Double or triple cage, rectangular top bar
4 = Double or triple cage, round top bar
5 = Cast aluminum rounded trapezoidal bars
Semiclosed stator slot (for random winding)
Col.
1
11
21
31
41
51
*1 61
1
11
21
31
41
51
61
71
1
11
21
31
41
51
61
71
1
*1 11
21
31
41
1
11
21
31
41
51
*2 61
Data
Line
2
3
4
5
6
INDUCTION MOTOR FIELD PROGRAM
Trapezoidal or rectangular bars
ASSEMBLED / CAST AL BARS (1 or 2)
W1 = SLOT WIDTH, TOP
W2 = SLOT WIDTH, BOTTOM
BAR DEPTH
CLEARANCE, TWO SIDES
PU BAR RESISTIVITY (Cu = 1, Al  1.6)
BAR EXTENSION, ONE SIDE
PU BAR AREA, EXTENSION (1)
CU / CAST AL ENDRING
CROSS SECTIONAL AREA
AVERAGE DIAMETER
Inverted T-bars
ASSEMBLED / CAST AL BARS (1 or 2)
W1 = SLOT WIDTH, TOP
W2 = SLOT WIDTH, CENTER
W3 = SLOT WIDTH, BOTTOM
D2 = SLOT DEPTH, TOP
D3 = BAR DEPTH, TOP
D4 = BAR DEPTH, BOTTOM
CLEARANCE, TWO SIDES, TOP
BOTTOM
PU BAR RESISTIVITY (Cu = 1, Al  1.6)
BAR EXTENSION, ONE SIDE
PU BAR AREA, EXTENSION (1)
CU / CAST AL ENDRING
CROSS SECTIONAL AREA
AVERAGE DIAMETER
PROGRAM FLD16
Col.
1
11
21
31
41
51
61
*1 71
Line
W1
7
W2
1
11
1
11
21
31
41
51
61
1
11
21
31
*1 41
8
W1
7
D3
1
11
21
31
41
51
D2
W2
D4
8
51
61
Assembled double cage or cast aluminum triple cage
ASSEMBLED / CAST AL BARS (1 or 2)
1
W1 = SLOT WIDTH, TOP
11
W2 = SLOT WIDTH, LEAKAGE
21
W3 = SLOT WIDTH, CENTER
31
W4 = SLOT WIDTH, BOTTOM
41
D2 = SLOT DEPTH, TOP
51
D3 = SLOT DEPTH, LEAKAGE
61
D4 = BAR DEPTH, TOP
71
D5 = BAR DEPTH, BOTTOM
1
CLEARANCE, TWO SIDES, TOP
11
BOTTOM
21
PU BAR RESISTIVITY, TOP
31
BOTTOM (Cu = 1, Al  1.6) 41
BAR EXTENSION, ONE SIDE, TOP
51
BOTTOM
61
CU ENDRING, TOP / CAST AL ENDRING
CROSS SECTIONAL AREA
1
AVERAGE DIAMETER
11
CU ENDRING, BOTTOM (zeros, triple cage)
CROSS SECTIONAL AREA
21
AVERAGE DIAMETER
31
Cast aluminum rounded trapezoidal bars
W1 = SLOT WIDTH, TOP
W2 = SLOT WIDTH, BOTTOM
PU BAR RESISTIVITY (Cu = 1, Al  1.6)
BAR EXTENSION, ONE SIDE
CAST AL ENDRING
CROSS SECTIONAL AREA
AVERAGE DIAMETER
Data
INPUT SHEET 2
W3
7
W1
Can be round
D2
W2
D3
W3
8
9
W4
W1
7
W2
*1: The extended parts of the bars are sometimes machined off to make them shallower and increase their flexibility.
INDUCTION MOTOR FIELD PROGRAM
PROGRAM FLD16
WINDING FACTOR (pitch times distribution factor, less than one)
STATOR LOADING, A/CM or A/IN (total current in slot divided by slot pitch)
PER UNIT STATOR LEAKAGE REACTANCE
PER UNIT STATOR LEAKAGE RESISTANCE
KW WINDAGE AND FRICTION LOSS (at rated speed)
PERCENT SLIP (100 for locked rotor, zero at no load)
PER UNIT APPLIED VOLTAGE
INITIAL PER UNIT CURRENT (as close as possible to final value)
NUMBER OF FLUX LINES (about 20)
MAXIMUM NUMBER OF ITERATIONS (to get the right current, about 4)
ACCELERATION FACTOR (to speed up convergence, about 1.1)
INPUT SHEET 3
Col.
1
11
21
31
41
1
11
21
31
41
51
Data
Line
- 11 -
FORMULAS
The symbols used here are the same as those in the input and output subroutines for FLD16, which are
supplied in source code. The formulas are presented without derivation, only with some explanations.
All units are metric with dimensions in millimeters.
Base current:
CBASE = 0.1*ACM*PI*ID*KWND/NR
The base current is the current in one adjusted stator slot (as shown earlier on the flux plots) which
corresponds to rated (base) kVA and rated voltage.
where
ACM = Stator loading in amps per cm (total rated current in actual stator slot divided by slot pitch.
PI = 
ID = Inside stator diameter.
KWND = Winding factor (pitch times distribution factor, less than one).
NR = Number of rotor slots.
The winding factor enters the formula to make the stator MMF the same as in the actual machine.
Air gap flux in Weber which induces rated voltage in the stator winding, calculated as a scalar:
FLUX = SQR2*10000*KVA/(PI**2*KWND*F*ACM*ID)
where
SQR2 = 2
KVA = rated (base) kVA
F = frequency
Actual air gap flux at the inside stator diameter in Weber, calculated as a complex number:
ACFLX = (SQR2*2/PI)*(A2-A1)*0.001*LNET*NR/POLES
where
A2 and A1 = rms complex vector potentials at upper and lower boundaries at the air gap (Fig. 2).
LNET = Net stack length with ventilating ducts subtracted, but with the stacking factor disregarded.
POLES = Number of poles.
Per unit terminal voltage, calculated as a complex number as the air gap voltage plus the voltage drop
through the stator resistance and leakage reactance:
CPUV = (ACFLX/FLUX) + PUI*CMPLX(R1,XL1)
where
PUI = Per unit stator current (taken as real).
R1 = Per unit stator resistance.
XL1 = Per unit stator leakage reactance.
- 12 Powerfactor:
PF = REAL(CPUV/PUV)
where
PUV = Absolute value of CPUV
Per unit rotor current, calculated as a scalar:
PUIR = ABS(CROT)/CBASE
where
CROT = Induced current in rotor slot, calculated as a complex number (opposes current in stator slot).
Per unit magnetizing current, calculated as a scalar:
PUIM = ABS(CMAG)/CBASE
where
CMAG = CSTAT + CROT = Sum of currents in stator and rotor slot.
Per unit air gap flux or induced air gap voltage, calculated as a scalar:
PUFLX = ABS(ACFLX)/FLUX
Per unit magnetizing reactance:
XM = PUFLX/PUIM
Per unit rotor impedance, calculated as a scalar:
Z2 = PUFLX/PUIR
The rotor impedance is really air gap voltage divided by rotor current. The program knows the phase
angles of both, and can calculate the components of Z2:
R2 = Per unit rotor resistance divided by per unit slip.
X2 = Per unit rotor leakage reactance.
Per unit torque:
PUT = PUIR**2*R2*KVA/(KW + WFR)
where
KW = Rated kilowatts.
WFR = kW windage and friction loss.
This makes the per unit torque equal to one at rated conditions.
- 13 -
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.
- 14 -
OUTPUT
PROGRAM FLD16
INDUCTION MOTOR FIELD PROGRAM
6 POLE - 500 KW MOTOR
STATOR SLOT CENTERED
PERCENT SLIP 0.625
PU VOLTAGE
1. ITERATION 1.0486
2. ITERATION 0.9982
3. ITERATION 0.9998
PU TORQUE 0.9950
PU STATOR CURRENT 0.9509
PU ROTOR CURRENT 0.8717
PU MAGNETIZING CURRENT 0.3018
POWERFACTOR 0.8787
PU MAGNETIZING REACTANCE XM 3.1416
PU ROTOR REACTANCE X2 0.11571
PU ROTOR RESISTANCE R2 0.00676
PU ROTOR RESISTANCE R2/SLIP 1.08136