Note: Descriptions are shown in the official language in which they were submitted.
-l- 51,489
TOROUE DETERMINATION FOR CONTROL
OF AN INDUCTION MOTOR APPARATUS
CROSS REFERENCE TO RE_ATED APPLICATION
The present application is related to the follow-
ing Canadian patent applications Serial No. 500.751 by D.J.
Shero et al. and entitiled "Induction Motor Synthesis
Control Apparatus And Method" Serial No. 500,752, by H.
Dadpey et al. and entitled "Induction Motor Regenerative
Brake Control Apparatus And Method" and Serial No. 500,750
by D. J. Shero et al. and entitled "Induction Motor Control
Apparatus And Method".
BACKGROUND OF THE INVENTION
-
Field of the Invention:
This invention relates in general to the determin-
ation of output torque delivered by an induction motor
drive driven from a dc power source with an inverter.
Description of the Prior Art:
It is known that the input power to an induction
motor drive apparatus, that is supplied power from a dc
source is ln accordance with the relation ship
Power in z voltage dc * current dc (1)
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where the voltage dc and current dc can be readily
measured.
The output power of the motor is the input power
minus all power losses in the motor drive system and the
output torque of the motor is
PI~-Power ~osses
Torque out Inverter Frequency rotor losses) ~2~
An output torque sensor can be coupled with a motor shaft
to measure the output torque and this permits an empirical
. determination of the motor drive apparatus power losses,
when the input power is known and using above equation 2.
For many applications of the motor, it might not
be desired to couple a toraue sensor with the motor shaft
to measure the output torque.
It is known to sense the ac voltage and A~
current of the motor. For a three phase induction motor
this has required sensing all three phase voltages and all
three phase currents, or sensing two of the phase voltages
and two of the phase currents an~ deriving the third phase
voltage and current in relation to the sensed parameters.
This can present a problem in relation to the variable
frequency operation of the motor.
SU~MARi OF T~ INVENTION
The present invention provides an output torgue
determination for an ac motor drive apparatus using already
known input dc voltage and input dc current parameters, and
either determining the motor output torque in relation to
predetermined lookup tables of torque versus speed or by
determining the motor power losses through operation of a
mathematical model of the motor apparatus as selected in
accordance with the operational speed of the motor, without
the actual sensing of the actual output torque, the motor
ac voltage or the motor ac current parameters.
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2a 67880-~1
In accordance with the present invention there is
provided in an apparatus for determining the output torque of an
AC motor in a transit vehicle eneryized by an inverter operative
with a DC power supply, the combination of
firs~ means connected with said power supply for establishing
the input DC power provided to the inverter by said power supply,
second means connected with said motor for establlshing the
motor speed,
third means for establishing the inverter losses as a
predetermined function of motor current with the motor current
being determined from an expression equal -to a motor current
versus ~onstant slip frequency relationship as a factor multiplied
by the ratio of the voltaye across the motor when divided by the
desired voltage across the motor for a constant volts per hertz
operation of the motor,
fourth means for establishing motor losses as a predetermined
function of said determined motor current, inverter frequency and
determined harmonic losses,
fifth means for establishing friction and windage losses as a
0 predetermined function of motor speed.
sixth means for establishing the output torque in relation to
the difference between said input DC power minus the sum of the
inverter losses, the motor losses and the friction and windage
losses when divided by said inverter frequency.
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ln accordance with the present invention -there is also
providecl in a method of determining the on-:Line output torque of
an AC motor energized by an inverter operative with a DC power
source, the steps of:
determining the input power to saicl motor in relation to the
power supplied by the power source through the inverter,
sensing the rotational speed of the motor with a tac~lometer
haviny an output frequency,
selecting a first determination of motor output toryue when
the tachometer frequency is greater than a first predeter1nined
~requency, with the firs-t determination of motor output torclue
being in relation to the sum of the respective power l~sses in the
inverter and in the motor anc1 with the ou-~put torque of said first
determination being determined in relation to the input power
minus the sum of the power losses, and
selecting a second determination of motor output torque when
the tachometer frequency is less ~han said first predetermined
frequency.
In accordance with the present invention there is also
provided an apparatus for determining the on-line output torque
~or any frequency in a broad array of operating frequencies of an
~C motor having a rated air gap flux and energized through an
inverter by a DC power source supplying a voltage and a current to
the inverter, said inverter and motor being adapted to operate at
a requested frequency within the array of frequencies while in
selected braking and synthesis modes of operation, the combination
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means connected wi-th sald power source for mea~1ring the
input DC power fLom said power source, a portion of which power i~
deliverable to the motor to produce output torque :Ln relation to
the product of said voltage and said current a-t said requested
frequency while in said selectec1 moc1e;
means connected with the motor for measuring the motor speed;
means providing a first determinakion of the output torque as
a function of the deliverable power ancl the motor speed when the
motor speed is ~reater than a predetermined motor speed, said
output torque first determination providing means including means
for establishing the motor current as a function of the motor slip
and the voltage across the motor and establishing the inverter
frequency, means ~or determining the inverter losses as a function
of motor current, and means for determining the motor losses as a
function of the inverter ~requency and the motor speed, and
means providing a second determination of the output torque
as a function of the deliverable power when the motor is operating
at rated air gap .f~lux and the motor speed is less than said
predetermined motor speed, said first and second output torque
c1eterminations belng provided in real time while operating the
motor.
In accordance with the present invention there is also
provided an apparatus for determining the on-line output torque
for any frequency in a broacl array of operating frequencies of an
AC motor having a rated air gap flux and energized through an
inverter by a DC power source supplying a voltage and a current to
the lnverter, said inverter and motor being adapted to operate at
a requested frequency within the array of frequencies while in
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2d 67880-41
selectecl braking and syn~hesls modes of operation, the combination
of:
means connected with said power source for measuring the
input DC power from said power source, a portion of which power is
deliverable ~o the motor to produce ou~pu-t torque in relation to
the produ~t of said voltage ancl said current at said requested
frequency while in said selected mode;
means connec-ted ~ith the motor for measurlng the motor speed;
means provicling a first determination of the outpu-t torque as a
1~ function of the deliverable power and the motor speed ~hen the
motor speed is greater than a predetermined motor speed, with the
first determination of the output torque including a calculation
of respective power losses in relation to the inverter operation,
and in relation to the motor operation, with the output torque of
said first determination means being determined as the difference
between the input power and ~he calculated power losses considered
in relation to the inverter frequen~y, and
means providing a second determination of the output torque
as a function of the deliverable power when the motor is operating
at rated air gap flux and the motor speed is less than said
predetermination motor speed, said first and second output torque
determinations being provided in real time while operating the
motor.
In accordance with the present i.nvention there is also
provided in a method of determining the on-line output torque of
an AC motor energized by an inverter operative with a nc power
source, the steps of:
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2e 678~0-41
determinlny -the input power to said motor ill relation -to the
power supplied by the power source to the motor through the
inverter,
sens:ing the rotati.orlal speed of the motor with a tachometer
having an output frequency,
selecting a first determination of motor output torque when
the tachometer frequency is greater than a first predetermined
frequency, including establishing the inverter losses ln
accordance with the motor current, establishing the motor losses
in accordance with the invertex frequency and establishing
friction and windage losses in accordance with the motor speed,
and,
selecting a second determination of motor output torque when
the tachometer frequency is less than said first predetermined
:Erequency.
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BRIEF DESCRIPTION OF THE DRAWINGS
Figure 1 shows a prior art dc power supply
operating through an inverter to energize a three phase ac
motor;
5Figure 2 shows a prior art transit vehicle
operative with a propulsion motor to determine the movement
of that vehicle along a roadway track;
Figure 3 shows schematically the torque feedback
determination apparatus of the present invention for
controlling a three phase ac induction motor;
Figures 4~, 4B, 4C and 4D show a program flow
chart for determining the torque feedback provided in
Figure 3 using a mathematical model of the motor drive
apparatus;
lSFigures 5A, 5B and 5C show a program flow chart
for establishing the determined torque feedback using
torque versus speed memory lookup tables;
Figure 6 schematically shows the inverter appara-
tus and braXing apparatus provided for the control of an
induction motor in accordance with the present invention;
Figure 7 shows illustrative prior art GTO turn-on
current information; and
Figure 8 shows illustrative prior art GTO turn-
off current information.
25DESCRIPTION OF THE PREFERRED EMBODIMENT
In Figure l there is shown a prior art torque
control apparatus for a three phase ac motor, and lncluding
: a dc power source lO energizing an inverter 12 for provid-
ing three phase energization of the ac motor 14 coupled
-39 with a load 16 which could be a transit vehicle. A torque
: sensor 18 is connected with the output shaft of the ac
motor 14 for sensing the output torque 37 delivered to the
load 16. A power controller 26 (also referred to as the
car control) receives as inputs an acceleration request 24
from the transit vehicle operator, the transit vehicle
weight 28, and the transit vehicle wheel diameters 31. The
: power controller 26 produces a tor~ue effort request 30
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which represents the torque to be achieved by the ac motor
14 in order to accelerate the transit vehicle, load 16, at
the rate defined by acceleration request 24. A jerk
limiter 24 takes the torque effort request 30 and jerk
S limits it to provide a jerk limited torque-effort request
27 to the motor controller 20. The motor contrsller 20
produces GT0 firing pulses 38 for the inverter 12, in order
to match the torque feedback 37 to the jerk limited torc~e
effort request 22.
10In Figure 2 there is shown a plurality of transit
vehicles 16 operative with a roadway track 17. The power
source 10 is coupled through a third rail 19 and a power
pickup member 21 to the inverter 12 carried by each vehicle
16. The motor la is connected with the drive wheels to
propel the vehicle 16 alon~ the track 17~ The torque
sensor 18 is coupled with the motor 14 for providing a
torgue feedback signal 37 to the motor controller 20.
In Figure 3 there is shown a suitable motor
control apparatus for operation with the present invention
24 to control a three phase ac motor, such as the propulsion
motor of a mass translt passenger vehicle. The vehicle
operator can provide a power controller 26 with a vehicle
acceleration request which, by taking into account the
vehicle weight and vehicle wheel diameters, the pow~r
controller 26 translates into a torque effort rec~est
signal 30 which is input to a signal limiter 32 for pre-
venting unreasonable torque effort requests. A jerk
limiter 34 is provided in relation to a desired jerk rate
36 for establishing a jerk limited torque request 22 for
the comfort of the vehicle passengers. A torque feedback
determination apparatus 38 determines the torque feedback
40 by measuring the system input power in relation to the
dc voltage 43 and dc current 44 provided by a power supply
10 and in relation to the inverter frequency 48 and the
synthesis mode 50 and the tachometer speed 52 provided by a
tachometer 54 coupled with the propulsion motor 14 to
estimate the output torque of the motor 14. The torque
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feedback signal 40 is supplied to the negative input of a
summing junction 58 for comparison with the jerk limited
torque request signal 22 supplied to the positive input of
the summing junction 58. The resulting torque error signal
60 is supplied to a motor controller 62. A car control
enable signal 64 from the operator permits the propulsion
motor 14 to run or not. Other needed inputs by the motor
controller 62 consist of the dc line voltage 43 and the
synthesis mode 50 of the inverter. The motor controller 62
outputs the braking thyristors enable 68, the requested
brakin~ angle 70, the requested inverter frequency 48, and
the requested inver~er voltage percent 74 to the inverter
and braking synthesis apparatus 76, which in addition has
as an input and output the control state signal 78 and
provides the synthesis mode signal 50 to the motor control-
ler 62 and to the torque feedback determination apparatus
38. When the motor 14 is in brake operation with addition-
al voltage supplied by the transformer braking circuit 80,
the control state signal 78 operates to keep the synthesis
mode in six-step and prevent a change to quasi six-step or
PWM modes. The lnverter and braking synthesis apparatus 76
outputs the inverter GTO firing pulses 82 to the inverter
12 and the brak~ GTO firing pulses 86 to the braking
circuit 80. The inverter 12 drives the motor 14 in power
and in brake operation and the braking circuit 80 operates
with the motor 14 when additional braking torque is desired
above base speed operation.
In Figures 4A, 4B, 4C, and 4D there is shown a
program flow chart for determining the torque feedback in
relation to the deliverable power of the motor 14 and as a
function of the motor speed. If the tachometer frequency
52 is less than 12 Hz, then a lookup table is employed for
an empirical determination of the motor torque. Above a
tachometer frequency of 14.5 Hz, the motor torque is
determined as a function of a plurality of calculated power
losses including stator losses, friction and windage and
similar losses, where the input power minus the sum of
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these losses when divided by inverter frequency as shown by
above equation (2) establishes the output torque of the
motor. ~he rotor current losses are not included. Betw~en
12 Hz and 14.5 Hz, the same method is repeated as the last
time torque was calculated in order to provide a band of
hysteresis for control stability.
In relation to Figure 4A at block 100, the input
power is computed as the dc line voltage times the dc line
current, and the input power is plus or minus depending
upon whether the motor 14 is regenerating in brake or is
drawing current from the line in power. At block 102 a
decision is made to see whether the torque in relation to
speed tables were last used for the calculation of the
torque. These torque lookup tables, which have been
digitized and are empirically established as a function of
input power for selected increments of speed are stored in
the computer memory for the below 14.5 Hz low speed range
of operation, since for such low speed operation the torque
versus input power is not linear. The power loss calcula-
tion model is satisfactory for motor operation above 12 Hz.
For this reason block 102 checks to see if the program is
already using the lookup table method, and if the answer is
yes, at block 104 a check is made to see if the tach
frequency 52 is greater than 14.5 Hz. If the answer is no
at block 102 then at block 106 a check is made to see if
the tach frequency 52 is less than or equal to 12 Hz. If
the an~wer is no at block 104 or if the answer is yes at
block ~06, then the lookup table method routine is branched
to at block 108. If the answer is yes at block 104 or if
the answer is no at block 106, then the power loss calcula-
tion method at block 109 is started at block 110 where a
flag is set indicating that the power loss calculation
method is being used, so the next time through the program,
the appropriate path is taken to check if the ta~le method
should be used or not. At block 112 in order to calculate
the power losses in the system, RMS motor current is
needed, and motor current is determined as a function of
7 51,~89
the slip times the ratio of the voltage across the motor
over the desired voltage across the motor for constant
volts per hertz ratio operation. A typical example of a
constant volts per hertz ratio for a propulsion motor is
9.33. For example, at 100 Hz and a voltage to frequency
ratio of 9.33, the motor would re~uire about 933 volts
line-to-line for this operation. For a 600 volt dc power
supply which can only supply about 468 volts line-to-line
to the motor, the ratio of these two voltages is a multi-
plying factor to determine the motor current. The function
of slip is provided by a predetermined lookup table in this
regard, which lookup table can be established by a well
known motor model in relation to a sine wave voltage
applied to the motor, and for a given slip of the motor a
particular motor current is provided with the assumption
that a particular motor temperature remains constant. At
block 114 the conduction losses in the inverter switching
devices are calculated, where the same equation is used
regardless of the s~fnthesis switching mode. The conduction
losses are established as
conduction losses = 4.05~ motor current (3)
At block 116 the switching losses are caiculated, with a
first equation determining switching losses in PWM or quasi
six-step where the average switching frequency is 400 Hz
and the switching losses for a 400 Hz carrier frequency
would be
switching losses = 312 + 1.2~ motor current (4)
The switching losses in six-step are lower because the
average switching frequency is not 400 Hz but rather the
switching losses are determined in accordance with the
relati~nship
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switching losses in six-step = (5)
.009 * motor current * inverter frequency
At block 118 ~he snubber losses are determined,
which are the losses in the snubbers across the GT0 switch
devices and are the resistive and capacitive losses in
c~arging and discharging the snubber circuits. In PWM or
quasi six~step the snubber losses are determined as
snubber losses = (6)
[l.0 - 10 5 ~ ~72) + (1.35 * 10-5 * I2)] * F
where V is the dc line voltage, I is the motor current, and
F is the switching ~requency of which an average value of
400 Hz is used for PWI-i and quasi six step. For six-step
operation, the snubber l~sses are
snubbér losses =
l(6.0 ~ lO * V ) I (5.3 ~ 10 ~ I )] * E
At block 120, the inverter losses are determined
as the sum of the conduction losses, the switching losses
and the snubber losses.
Now it is necessary as shown in Figure 4B to
calculate the motor losses and beginning at block 122 the
stator resistive losses are determined as
stator resistive losses = (8)
3 x (the stator resistance) x (motor current2)
which is a straight forward I R loss for three stator
windings. At block 124 the core losses are determined for
the constant volts over frequency range, where the desired
motor voltage is still available and for the example of a
motor 14 having a base speed of 45 Hz and a 9.33 volts per
hertz constant which deter}nines the parameter in the first
core loss equation, using the equation
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core losses = 186~ x in~rerter freauency
A second equation is used for the six-step mode of opera-
tion above the constar.t volts over frequency range where
the desired voltage is not available,
core losses =
1.162 mtor vol.age 1.6 ~ inverter frequency (10)
lnverter frequency
For the particular motor being controlled the stray losses
in block 126 uses a state of the art equation well known by
people who design motors ~hich is
stray losses = 2.12 ~ motor torque (11)
where torque is the pre~ious torque.
At block 128 the harmonic losses are determined
using a series of lookup tables and equations. It is
extremely difficult to create a loss model for harmonic
losses that is precise. Therefore, an abbreviated approach
is taken with the harmonic losses to arrive at an approxi-
mate value. To form ~he lookup tables and equations to be
used by the microprocessor to calculate harmonic losses,
the ac motor is run at several operating speeds and loads
in a laboratory environment. At each operating point
inverter input power, ou.put motor torque and RMS motor
current are measured. From this data, all defined losses,
except harmonic losses, are calculated using their respec-
tive equations. The motor output power is derived by
multiplying the output torque times the inverter frequency.
This output power is subtracted from the input power to
arrive at the remaining losses. These remaining losses
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should approximate the harmonic losses providin~ that the
derived loss equations are reasonably accurate. The
estimated harmonic losses at all operatin~ points are then
correlated to arrive at a combination of lookup tables and
equations that the microprocessor can use to calculate
harmonic losses.
For the particular ac motor drive system used,
correlation of the harmonic loss data yielded a lookup
table for each waveform slnthesis technique. Each of these
lookup tables relates harmonic losses at no load to an
operating parame~er. ~or P~IM synthesis operation, the
lookup table relates losses as a function of inverter
frequency. For quasi si~.-step synthesis, losses are
represented as a function of requested voltage percent.
For six-step synthesis, losses are represented as a func
tion of inverter frequency. I~ the motor is providing
torque, the laboratory data revealed that the harmonic
losses increased in magnitude. The following equation was
found to approximate the harmonic losses over the full
motor load range:
harmonic losses =
rated slip frequency ~llA)
At block 128 the synthesis mode is checked.
Depending on the mode, one of three lookup tables is
accessed to obtain the harmonic losses the motor would
experience under similar operating conditions but at no
load. This lookup table value along with the slip frequen-
cy and the rated slip frequency are used in equation llA to
determine the harmonic losses in the ac motor.
At block 130 the motor losses are determined as
the sum of the stator losses, the core losses, the stray
losses and the harmonic losses.
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At block 132 the windage loss has two components,
the shaft fan loss and the rotor windage loss, and a well
known equation is used with a four pole motor for determin-
ing these losses which is
RPM 3
windage loss = S99 x 1800 (12)
where 1800 is the base frequency of the motor such that at
base frequenc~ the windage loss is 599 watts. At block 134
the friction losses of the rotor and fan for the motor 14
are determined again as a ratio of
friction losses = 14800RPM (13)
where 104 watts is known to be the loss due to fxiction at
the base speed of 1800 RPM and a linear ratio above and
below base speed is provided. At block 136 the total of
the friction and windage losses is determined as the sum o
lS the losses established in blocks 132 and 134.
At block 138 a check is made to see if the
transformer within the braking circuit 80 is not shorted,
which would be the case when the motor 14 is operating with
transformer braking in the six-step mode. When the motor
is operating without transformer braking, at block 140 the
braking losses are established for the thyristors
braking losses = 4.05 * motor current (14)
If the transformer braking is provided, then at block 142
the transformer braking losses are determined as the sum of
the snubber and ~T0 switching losses plus the GT0 and line
diode conduction losses plus the diode bridge losses plus
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the transformer resistive loss plus the transformer core
loss, where these respective losses are determined as set
forth in Figure 4C in respective blocks 133, 135, 137, 139
and 141 for the equation relationship used ~to determine
each of those losses. At block 144 the powar losses are
determined as the motor loss established at block 130 plus
the friction and windage loss established at block 136 plus
the brake loss established at block 140 or 142 plus the
inverter loss established at block 120. In block 146 the
corrected powe~, which is the deliverable power, is deter-
mined as the input power, which is the computed power of
block 100, minus the computed power losses of block 144.
At block 148 the torque feedback TEF is set equal to the
correeted power divided by the inverter frequency, which is
the deliverable po~er di~ided by the stator frequency.for
the motor 14 and this is the torque feedback 37 shown in
Figure 3. In bloc~ 150 since a division operation was
provided in block 1~8 that could result in an overflow
condition when the inverter frequency is small, the block
150 determines that there was not an overflow. If the
register did overflow, the result is erroneous because only
the lower portion of the result is in the register and the
most significan~ informatio~ is lost, so if there was an
overflow, at block 152 a check is made to see if the torque
was positive or negative. If negative torque is present,
the motor is in the brake mode and for a positive torque
the motor is delivering power in the power mode. If the
torque is positive, at block 154 it is clamped to maximum
positive torque. In block 156 if the torque is negative,
it is clamped to maY.imum negative torque to protect against
inaccuracies in the computation of a finite number of bits,
and a return is made.
The program shown in Figura 5A named Table Method
is called at block 108 of the program shown in Figure 4A.
At block 160 a flag is set to indicate that the operation
is using the lookup torqua tables. Block 162 provides a
check to see if the tach frequency 52 is less than zero to
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determine if the operation is ~oing ne~ative, and if so
several operations are by~assed. The tach frequency 52 may
be less than zero when the vehicle is in a rollback opera-
tion during a start of forward movement and upon the
friction brakes being released, so at this time open loop
power is applied to get the vehicle moving forward. If the
tach frequency 52 is negative, the program goes to block
164 where an open loop calculation of torque is made in
relation to the desired slip. If the tachometer frequency
is greater than zero at block 162, then at block 166 a
check is made to see if the jerk limited torque request 22
is less than zero which would be present for a braking mode
of operation, and if so at block 168 a check is made to see
if the tacho~eter frequency. 52 is less than a minimum
frequency where the brake torque can be calculated and the
Table Method is valid. For a transportation vehicle at
very low tachometer frequencies it is desired to apply the
friction brake.
If the tachometer frequency is above the minimum
frequer,cy at block 168 where calculations are proper, the
program goes to block 170 where a parameter A is set equal
to tach frequency, wilh one bit being 1/64 Hz. At block
172, a parameter 3 is set equal to the integer portion of
A, which is determined by dividing A by 64, such that
anything between 0 to 63 would result in a zero integer
value, anything between 64 and 127 would result in a one
integer value and so forth. In block 174, a parameter C is
set equal to the offset from the starting location of the
very first table to the starting location of the table
corresponding to the integer tach frequency. Since the
tables have 32 entries each, in block 174, by multiplying
the B integer value by 32, the desired table is selected in
this manner. If the integer value is zero this is the
first table, if the integer value is one multiplying it by
32 would give the address 32 which is the start address of
the second table, and so forth. In block 176 a parameter D
is determined by adding the start address of the very first
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table to the parameter C which is the offset from that
table. Parameter C is the start address of the table if
the very first table is started at location zero, but since
the first table may start scmeplace else, by adding the
offset to the start address of the first table, the pointer
D is provided to point to the start address of the table
that is desired. At block 178 the Calculate Table Torque
Routine is called, which uses the now selected table and
returns a value of torque from that table which in block
10 180 is set equal to E and called torque low. In block 182
the parameter F is set equal to D plus 32, which adds 32 to
the top address of tile table in readiness to pick up the
next table. At block 18~ the Calculate Table Torque
Routine is again called, and in block 186 the torque high
value is returned as G, such that the parameter E gives the
torque low value for a particular frequency equal to the
integer portion of the tach frequency and the parameter G
gives the torque high value corresponding to a higher
frequency equal to the integer portion of the tach fre-
quency l 1. Knowing these two torque values corresponding
to the dc input power at the integer tach frequency plus
one, it is possible to interpolate between these two torque
values to arrive at the torque corresponding to the actual
tach frequency. This interpolation is performed by blocks
25 188, 190, 192, 194, and 196. To find the slope of the line
between the two torques E and` G, in block 188 the
torque/tach frequency slope is set equal to the difference
between the parameters G and E. In block 190 the parameter
H is set equal to ~ times 64. In block 192 the fractional
part of the tachometer frequency or delta tach is estab-
lished as the tach frequency minus the parameter H, since
the integer portion B times 64 when subtracted from the
original number should give the remainder, which is the
tach di~ference or fractional portion. In block 194 the
parameter J is set equal to the delta total torque which is
the additional torque that will be added to the torque low
value E, and is the slope of the line calculated in block
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188 times the delta tach divided by 2 or 64. At block 196
this fractional torque is added to the torque Low to give
the calculated torque feedback.
In Figure 5C, the flow chart for the calculate
table torque routine is provided. The selected table has
32 entries, and the last entry in the table is the scale
factor used to compact the data and which is used to divide
the computed power value. At block 200 the computed power
scale factor is read as the last entry in the table. The
middle entry of the table corresponds to zero computed
power, while the first or top entry corresponds to regener-
ated power and- the second to last or bottom entry corre-
sponds to consumed power. So the table ranges from
negative power at the top through zero power to positive
power at the bottom. At block 202 the parameter M, which
is scaled input power, is set equal to the computed power
divided by (8 x SC~LE FACTOR L). At block 204 the parame-
ter N is set equal to the scaled input Power M times 2 to
properly scale the value, and this results in a number
between -15 and 1~ and represents the offset from the
middle of the table. A~ block 206 a check is made to see
if that number N is greater than 14. If the number is
greater than 1~, at block 208 since there are only .14
positive power entries to deal with, N is set equal to 14
and this means that entry 14 is desired. If the number is
less than 14, at block 210 a check is made to see if the
numb~r is less than -15 since there are -15 entries from
the zero power point. I the number is less than -15, this
means that it should be clamped to -15 at block 212. In
block 214 a table entry is selected from the table at the
address equal to the offset rom the middle of the table
plus the top address of the table plus 15 because there are
lS entries prior to the middle of the table. The offset is
a signed value either plus or minus depending upon which
part of the table is being used, and R is set equal to that
entry properly scaled by the number 26 to convert to the
desired torque units of measure. In block 216 the
~.'Z~
16 51,489
parameter P is set equal to the next higher entry from the
table multiplied by 2 , and this yives two consecutive
entries from the table. Each of these two torque valles
correspond to a different dc input power with 0 correspond-
ing to a lower dc input power than does P. The actual dcinput power is between these two dc input powers. To find
the torque corresponding to the actual dc input power, it
is necessary to interpolate between these two torque
values. This in~erpolation is performed by blocks 218,
220, and 222. In block 218 the slope Q is determined by
subtracting the two entries and since the entries in the
table are a known amount different from each other the
slope does not require any division. In block 220 the
fractional part of the torque is determined, by multiplying
the entry number N by 28 and subtracting the product from
the original computed power to result in the difference
between the actual dc input power and the dc input power
corresponding to the torque R. This power difference is
multiplied by the slope Q and then multiplied by 2 8 for
scaling purposes to arrive at the value delta torque.
Delta torque represents the difference in torque repre-
sented by the difference be~ween the actual dc input power
and the dc input power associated with the torque R. In
blocX 222, the torque corresponding to the actual dc input
power is calculated by adding delta torque to the torque R.
A return is then made from this calculate table torque
routine.
The motor output torque is determined in relation
to t~e dc input parameters and not by sensing the ac motor
voltaga and current and not by coupling a torque sensor to
the motor. For a tachometer frequency less than or equal
to 14.5 Hz, the table method of determination is utilized
on the assumption that with a given power going into the
motor there will be a particular torque out of the motor
depending on applying substantially the same voltage. The
determined motor torque may vary by as much as 5% due to
loss calculation error and ignored parameters such as
17 51,489
temperature, but that is close enough for transit motor
control applications. At slow motor speeds up to base
speed, the motor is in the constant volts per hertz opera-
tion, and with known voltage across the motor, there is
going to be a correspondence between the motor output
torque and the dc input power. The provided torque versus
power tables are calculated by making a mathematical model
of the motor and calculating what the corresponding power
relationships are. At the higher speed range above base
speed where the voltage may vary because the inverter runs
out of voltage or bra~.ing is added, the torque versus power
lookup tables are not satisfactory and it is better to
calculate the power losses in relation to motor operation
and then subtract the calculated power losses from the
lS input power and divide the diference by the inverter
frequency to determine the motor output torque. At higher
speeds where the input power is large, then some error in
the determined power losses by comparison is not signifi-
cant in the determination of motor output torque. At the
lower speeds where the inverter and motor losses are a
large portion of the total input power, then the same error
in the power loss calculation can result in an undesired
large error in the determined motor output torque. From
approximately 12 Hz operation to base speed, hoth methods
~5 of calculati~g motor torque yield comparable results.
However, the loss calculation method does not need exten-
sive memory consuming lookup tables as the lookup table
method needs, therefore, the loss calculation method is
preferred about 12 H~
If the motor is braking at tach freauencies below
10 Hz, an additional problem arises, the torque versus
input power relationship ceases to be a function. That is,
there is more than one value of torque which corresponds to
some input power. Therefore, it is not possible to use
either method of torque calculation to determine the output
tor~ue of a braking motor at very low operational frequen-
cies. On mass transit vehicles, it is desirable to perform
,.:
5~
18 51,489
friction braking at such low speeds; therefrom, this
drawback is not a proble~. In such a situation, an open
loop estimate is made of the torque of the motor by saying
that the torque is proportional to the slip of the motor.
This estimation is not used by the controller as the
controller will not attempt to perform closed loop motor
braking at such low frequencies, but this estimation is
included for completeness. Also if the tach frequency is
negative, meaning the mass transit vehicle is rolling bacX
in a direction opposite the desired direction of movements,
a correct calculation of torque cannot be performed due to
the same reason. ~ollback recovery of such a transit
vehicle is achieved by an open loop control operation where
again the calculated torque is not used by the controller.
lS For sake of completeness, the torque is estimated as a
function of slip.
For the torque calculation lookup table method, a
set of 16 different lookup torque tables is provided. Each
lookup table is for a different tach frequency from 0 to 15
integer hertz values of tach frequency, and the torque is a
function of input power plus a function of the speed or
tach frequency. In this way 16 two dimensional lookup
tables are used to provide the operation of a three dimen-
sional lookup table, with the speed dimension provided by
the plurality of tables. Each table is arran~ed into 31
different locations of power, with tor~ue corresponding to
power, so for each power value there is a torque value that
corresponds. The power is broken up into 31 different
points, with the middle entry in the table being zero
computed power, lS entries of negative power or braking
prior to the middle and 15 entries of positive power after
the middle. The 32nd entry is a scale factor used to
compast the data. For each location there is stored the
corresponding amount of torque. Each table is made big
enough to cover the maximum expected output torque values.
At the higher speed end of where the lookup table method is
used, there ~ay be 50 to 100 kilowatts of power to
.
19 51,489
represent the maxlmum output torque, and at the low end,
there may be 3 kilowatts o power to represent full output.
Such a large dynamic range for dc input power would require
very large lookup tables. To reduce the size of the tables
a scaling factor is used to indicate the difference in
watts between each table location. For example, at the
high speed end there may be a difference of 3.3 kilowatts,
while at the low speed end each location may represent a
jump of about two hundred watts. Using a model of the
motor apparatus, the losses are calculated by an offline
calculation of the po~er losses, and when added together
and added to the power due to the motor torque they should
be equal to the lnput power to the system. Using such
offline calculations, a lookup table at each integer tach
frequency and representing the relationship between torque
and input power at that tach frequency is calculated.
If the input power happens to fall between two
points which it normally does, the table is used to obtain
the torque value at the next lower power and the torque
value corresponding to the next higher power, and assuming
a straight line connec~ing those two torque values then
interpolation in relation to the actual input power will
determine the output torque. Also tach frequency is
assumed to be either 0, 1, 2,..., or 15 Hz. Since the tach
frequency is rarely equal to an exact integer value, some
additional interpolation is reqùired to determine the
torque at the actual tach frequency. This interpolation is
done by determining the torque values for the current input
power for the two integer tach frequencies that bound the
actual tach frequency. Each of these torque values is
arrived at by using the interpolative procedure previously
described to calculate torque from input power at the
particular tach frequency. Interpolation is then used to
arrive at a final torque feedback value somewhere between
the intermediate torque values calculated from each of the
two tach fre~ency tables. For example, for a tach fre
quency of 10~ Hz, the torque value at 10 Hz is obtained and
,~ ~
20 51,489
the torque value at 11 Hz is obtained and interpolation
between these two torque values is used to determine the
output torgue.
In relation to the scaling, the input power is
5 scaled so that 1 bit equals 7.6294 times 10 3 watts, and
the torque is scaled so that 1 bit equals .1146 pound feet.
The scaling byte in the table is set up so that 1 bit
equals 15.625 watts per table location. The table is a
byte lookup table so there is a torque value range of 0 to
255, and the table values cannot be scaled in the same
units as torque is scaled because torque can go up to 800
pound feet which would require more than 255 bits. Thusly
each table torque value is equal to .1146 times 26 pound
feet.
In Figure 6 there ~is schematically shown the
inverter and brake circuit apparatus coupled for control-
ling an induction motor. The inverter 12 includes GT0-1
and GT0-2 switches connected to the DC power source 10 to
energize pole A of the motor 14. The GT0-3 and GT0-4
switches are connected to the DC power source 10 to ener-
gize pole B of the motor 14. The GT0-5 and GT0-6 switches
are connected to the DC power source 10 to energize pole C
of the motor 14. The brake circuit apparatus 80 is shown
for phase A of the motor 14. Identical brake circuits are
provided but are not shown for each of phase B and phase C.
The voltage snubber circuit 250 is shown for the GT0-1, and
the current snubber circuit 252 is shown.
The thyristors THl and TH2 are shown for the
brake circuit 80, and are operative to short circuit the
30 transformer 254. The thyristor snubber circuit 256 is
shown. The GT0 switch GBl is provided to ~odulate the
voltage provided by the brake circuit 80 to the motor 14
when the thyristors THl and TH2 are not conducting. The
GT0 switch GBl is provided with a voltage snubber circuit
35 258 and a current snubber circuit 260. The line diode DBl
includes a snubber circuit 262. The diode bridge 264
operative with the primary winding of the transformer 254.
~,
,
i5~
21 51,489
In Figure 7 there is shown the energy absorbed by
a GTO switch when it is switched on.
In Figure 8 there is shown the energy absorbed by
a GTO switch when it is switched off.
A Toshiba SG800EX21 GTO was used for each pole of
the inverter. The gate current for this GTO is 5 amperes,
and the anode di/dt is determined by the pole inductor,
such as L1 for pole A. The inductor Ll is 7 microhenries
and with a 600 volt power source 46, this provides 85
amperes per microsecond di~dt, or .16 watt per pulse as
sh~wn in Figure 7. For each turn on of the switch GTO-l in
pole A of the inverter 12, there is a loss of .16 watt of
power. This loss is multiplied by the turn-on frequency
per second to determine the ~ower loss in watts per second.
The turn-off iosses are determined with the curve
shown in Figure 8. This curve is assumed to be a straight
line and using the slope of this line times the current
through the GTO'switch will establish the energy loss per
turn-off, which is then multiplied by the number of turn-
of switches per second to determine the resulting power
loss.
DETERMI~I.L.TIO~l OF POWER LOSS RELATIONSHIPS
To estimate ~he torque in the ac motor, the dc
input voltage and dc input current are measured by the
microprocessor via an A/D converter. These two parameters
are multiplied together to arrive at the input power to the
inverter an~ braking circuit. By subtracting the losses of
all components in the system, except rotor resistive losses
in the motor, the micro can determine the useful portion of
power that contributes to the torque force generated by the
motor. The torque of the motor can then be determined by
dividing the useful power by the inverter frequency. If
rotor resistive losses were included in the system losses,
rotor mechanical frequency would be used instead of invert-
er frequency to calculate torque.
5~
22 51,489
S'fSTEM LOSSES
The determined system losses can be broken downinto four basic groups. The first of these groups consists
of losses in the inverter, which are further broken down to
GTO and diode conduc~lon losses, GTO switching losses, and
losses in the snubber circuits. The second group of system
losses consists of electrical losses in the ac motor, which
are stator resistive losses, magnetic core losses, harmonic
losses, and stray losses. The third group of system losses
consists of mechanical losses in the ac motor, whlch are
composed of friction and windage losses in the motor. The
fourth group of system losses is present only if the
optional braking circui' is included in the motor control
operation and consists of transformer braking losses, ~hich
are different depending upon whether the transformer is not
presently being used and is shorted by the thyristors or
whether the transformer lS presently being used and the
thyristors are kept off. If the transformer is shorted by
the thyristors, the transformer braking losses consist of
the conduction losses of the thyristors only. If the
transformer is not shorted by the thyristors, the trans-
former braking losses consist of snubber and switching
losses in all semiconductors in ~he brake circuit, conduc-
tion losses in all semiconductors except the bri`dge diodes,
conduction losses in the bridge diodes, resistive losses in
the transformer, and magne'ic core losses in the trans-
former. The total system losses are determine~ by adding
up the losses in each ~roup and then adding all of the
groups together.
CALCUL.~TING RMS MOTOR CURRENT
In order to calculate each group of these system
losses, it is necessary to derive relationships between the
loss to be calculated and various known parameters in the
ystem. Many of the losses are a function of motor cur-
rent. The microprocessor could read in the RMS value of
motor current directly; however, due to the high cost of
components necessary to perform such a function (isolated
23 51,489
current to voltage transducer, wire to connect transducer
to control logic module, and RMS volta~e detector circuit,
a relationship between RI~S motor current and the known
variables slip frequency, motor voltage, and constant V/F
voltage was developed. This relationship is based on the
principle that as long as the air gap flux is kept con-
stant, current will be a known function of slip frequency,
neglecting temperature variations. This known relationship
. between motor current and slip frequency at rated air gap
flux is stored in a lookup table for the micro to access,
such that the micro can find the value of motor current
knowing the slip frequency if the motor is at rated air gap
flux. It is also known that the motor current is directly
proportional to the applied motor voltage given that slip
frequency is held constant. ~nowing this relationship, the
motor current is calculated, even if the motor is not at
rated air gap flux by using the following equation:
motor current = F(sli~) ~ motor vol g (16)
DFS motor voltage
where F(slip) is the motor current vs. slip frequency
relationship and desired motor voltage is the voltage
necessary to produce rated air gap flux. By performing the
lookup table function and the above calculations, the micro
provides a good estimate of the present motor current.
This motor current is then used in several loss
~5 calculations.
INVERTER LOSSES
Conduction losses in the GTOs and frae-wheeling
diodes in the inverter can be approximated as a function of
motor current. The instantaneous conduction loss in a GTO
~ or dioda is equal to the current flowing through the device
multiplied by the voltage drop across the device. It is
unnecessary to calculate the instantaneous conduction loss,
so an average conduction loss in the semiconductor devices
..
.,~.
~L2~
24 51,489
is determined. The average voltage drop of the GTOs
conducting an average amount of current is approximately
1.8 volts and the averaqe voltage drop of the diodes
conducting an average amount of current is approximately
1.2 volts. The voltage drop for both of these devices
varies slightly with the amount of conduc~ing current,
however, the constant values of 1.8 and 1.2 volts are
assumed. Assuming a constant voltage drop of 1.8 volts,
the conduction losses in all of the inverter GTOs is
defined as follows:
GTO C d ti L 1 8 ~ 9 ~ Motor Current * 6 (17)
where loss is in ~atts, motor current is in amps RMS, .9 is
used to convert RMS current tO average current, 4 is due to
each GTO conducting on average only ~ of the time, and 6 is
the total number of GTOs in .he inverter. This equation is
simplified to the followincJ:
GTO Conduction Loss = 2.43 ~ Motor Current (18)
Similarly, assuming a constant voltage drop of 1.2 volts,
the conduction losses in all of the inverter free-wheeling
diodes is defined as follows:
Diode Conduction Loss = 1.2 * 9 ~ ~ 6 (19)
where loss is in watts, motor current is in amps RMS, .9 is
used to convert RMS current to average current, 4 is due to
each diode conducting on average only 1~ of the-time, and 6
is the total number of free-wheeling diodes in the invert-
er. This equation is further simplified to the following:
~65~5~
.
2S 51,489
Diode Conduction Loss = 1.62 ~ Motor Current (20)
These equations are actually only partially valid because
it is assumed that each diode and each GTO conducts only
of the time. Actuall~, the GTOs will conduct more than
of the time and ~he diodes will conduct less than ~ of the
time during ~otoring operation. This subtle shift in
conduction time will actually increase the GTO losses and
decrease the diode losses, but ~he difference is neglected
for simplicity. In braking~ the opposite effect will
occur. Neglecting these shifts in conduction times, the
final inverter conduction losses are equal to the GTO plus
the diode conduction losses or:
Inverter Conduction Losses = 4.05 ~ Motor Current (3)
This equation is used regardless of synthesis mode.
15 Inverter GTO switching losses are dependent upon
the synthesis mode because the~ are dependent on switching
frequency. In P~lM and quasi six step, the switching
frequency hovers around 400 Hz, while in six-step, the
swikching requency is equal to the inverter frequency;
therefore~ a need exists for an equation for PWM and quasi
six-step and an equation for six-step. Switching losses in
a~GTO are calculated using data from the GTO manufacturer.
Switching losses are composed of two components, turn-on
losses and turn-off losses. GTO manufacturers supply
curves relating turn-on losses to anode di/dt and curves
relating turn-off losses to anode current. Curves for the
GTOs used in the present inverter (SG800EX21) are shown in
Figures 7 and 8. From these curves the following relation-
ships are derived for turn on and turn off losses for the
six GTOs in the inverter:
GTO Turn On Losses = .26 ~ (400/2) ~ 6 = 312 (21)
, . ~
p~:~
26 51,489
GTO Turn Off Losses = .0011 (.9 * IM) *
(~00/2) * 6 = 1.2 ~ IM (22)
Inver~er GTO Switching Losses = 312 + (1.2 ~ IM) (4)
In the above equations, the .26 number is read from the
turn on loss cur~e for an IGM of 5 amps and an anode di/dt
of 85 amps/microsecond; the ~00 represents the average
swi~ching frequency during PWM and quasi six-step; the 2
reflects that current will be flowing through the GTO only
~2 of the time that the GTO is being switched, effectively
reducing the switching frequency by a factor of 2; the 6
represents the number of GTOs in the inverter; the .0011 i5
the estimated slope of the curve in the turn-off loss
curve; and IM is RMS motor current in amps. During si~-
step synthesis, the switching losses are reduced because
the switching frequency is equal to the fundamental invert-
er frequency i~nstead of 400 Hz; therefore, a different
equation is used if the synthesis mode used is six-step.
In six-step, all GTO turn-offs occur while current is
flowing through the GTO, so that the effective switching
frequency is not divided by two as was the case with the
PWM and quasi six-step s~nthesis. Also, in six-step,
turn-on losses are negligibly small because each GTO is
switched on initially when the motor current is flowing in
the opposite direction. Eventually the motor current
swltches direction and the GTO begins to conduct; however,
under such circumstances the turn-on losses will be quite
small. Also, during six-step, the turn-off losses per
switching cycle will be higher because the current being
switched off will normally be higher than the RMS value of
the motor current. The exact magnitude of the current
depends upon the phase angle between voltage and current.
The current value where the GTOs are turned off is estimat-
ed to be about 1.4 times the RMS motor current, due to
harmonicc, etc. The equations that define the switching
losses durin~ six-step synthesis are as follows:
~, .
5~
27 51,489
GT0 Turn 0~ Losses = 0 (23)
GT0 Turn Off Losses = .0011 * 1.4 * IM * Inverter
Freq ~ 6 (24)
= .009 ~ IM * Inverter Freq
5 Inverter GT0 Switching Losses = .009 ~ IM* Inverter ~5)
Freq
Inverter snubber losses include the losses in the
six voltage snubber circuits and three current snubber
circuits in the inverter. Losses occur in the voltage
snubber circuits due to the capacitor fully charging and
discharging during GT0 turn-on and turn-off times. Losses
occur in the current snubber circuits due to current
building and falling in the inductor duriny GT0 turn-on and
turn-off times. There are basically four different condi-
15- tions that exist in each inverter pole that cause energy to
be dissipated in the snubber circuits. Case number one is
when the motor currer.t is negative with the direction of
current flow out of the motor, and GT0-2 is off and is then
switched on. In this case, the motor current is initially
flowing through diode D1 and Ll, but after GT0-2 is turned
on the current will flow through GT0-2. In the meantime,
the yoltage snubber capacitor C2 for GT0-2 must discharge a
voltage of 600 volts where ~he energy dissipated equals .5
* C * V2; the voltage snubber capacitor Cl for GT0-1 must
charge up to 600 volts; the current through Ll must stop
flowing; and the diode D1 must turn off. As soon as GT0-2
is tur~ed on, the current through L1 starts to decrease and
the voltage across the GT0-2 snubber capacitor begins to
decrease. As soon as the current through L1 reaches zero,
it reverses direction and begins ~o charge up the GT0-1
`snubber capacitor. Also diode Dl takes about 2.5 microsec-
onds to turn o~f so it will conduct current in the reverse
direction for this small period of time. The GT0-1 snubber
capacitor will charge beyond the DC line voltage because of
the presence of the snubber induc~or and stray circuit
5~
28 51,489
inductance. As soon as the GT0-1 snubber capacltor voltage
passes the line voltage, the current in L1 and the stray
inductance begins to decrease. Some of this energy is
temporariIy transferred to the capacitor and the rest is
dissipated in Rl. The energy transferred to the capacitor
accounts for the overshoot voltage in the snubber capaci-
tor. Much of this energy is soon quickly dissipated in R4
and R1 with the remainder of the energy fed back to the DC
power source. The amount of losses in the pole's snubber
resistors Rl, R4, and R5 can be shown to equal:
Case #1 Snubber Loss = (.5 * C * V2) + (.5 ~ L ~ (25)
(IL2 + ID~))
where the first term represents losses due to the GT0-2
snubber capacitor discharging and the second term repre-
sents losses due to the GT0-1 snubber capacitor charging
and snubber inductor dissipating energy. The energy stored
in the GT0-1 capacitor at the end of the switching cycle is
not considered as a loss because it is stored energy at
this time. In the equation, C is 2 ~icrofarads and repre-
sents the snubber capacltance; V is the DC line voltage; Lis 9 microhenries and represents the combination of snubber
inductance (7 microhenries) and stray inductance (2 ~.icro-
henries); IL is the peak current through L1 attained while
charging the GT0-l snubber capacitor and not counting diode
current; and ID is the peak reverse diode current through
Dl. IL and ID are further defined ~y the following
equations:
IL = V ~ (C/L) (26)
ID = V/L ~ Trr (27)
where V, C, and L are defined earlier and Trr is the
reverse recovery of the diode and is equal to about 2.5
microseconds. Substituting these equations into the above
- ~
~!.f~/~ . J1 ~l
29 51,489
snubber loss eguation (25) results in the following equ~-
tion for the losses in one pole:
Case #1 Snubber Loss = C * v2 + .5 ~ (V2 / L) (28)
* Trr2
Case number two is where the motor current is
positive and flowlng through GTO-l when GTO-l is turned
off. In this case the motor current i5 initially flowing
in Ll. ~hen GTO-l is turned off, the GTO-l snubber capaci-
tor begins to charge up with the motor current. Also the
GTO-2 snubber capacitor begins to discharge, dissipating
the energy in tne R5 resistor (energy dissipated equals .5
* C ~ V2). Once, the ~ioltage across the GTO-l snubber
capacitor reaches the dc line voltage, the current in the
snubber inductor Ll and the stray inductance begins to
decrease. The energ~ ir. these inductors at this time is
equal to 5 r L * I2 where I is the motor current. All of
this energy is either dissipated in resistor Rl or tempo-
rarily transferred .o .he GTO-l snubber capacitor in the
- form of an overshoot voltage. This temporary overcharge of
the capacitor is soon dissipated in resistor R4. As one
can see, the diode and re~erse Ll current are not a factor
in case number two and the losses for case number two are
defined as follows:
Case #2 Snubber Loss = (.5 C ~ V2) + (.5 ~ L ~ (29)
I2)
:
where C, V, and L have been previously defined and I is the
RMS motor current.
Case number three occurs when the motor current
is positive and GTO-l is switched from off to on. Initial-
ly the motor current is flowing in diode D2, but aftar
GTO-l is turned off, the motor current flows through GTO-l
and inductor Ll. This case parallels case number one
~; descrlbed previously and the eguation for snubber lossas
'
.
6~ ~
51,489
for this case can be shown to be identical to the snubber
losses for case number one.
Case number four occurs when the motor current is
negative and flowing throuqh GTO-2 which is on. Then GTO-2
is turned off, and the current ends up flowing through
diode D1 and inductor L1. This case parallels case number
two described previously and the equation for snubber
losses for this case can be shown to be identical to the
snubber losses for case number two.
We have so far described the eneryy losses in the
snubbers of ona pole for all possible switching cycles. To
arrive at a power loss, we need to multiply these energy
losses by the number of times each case occurs per second
and by the number of poles in the inverter. During PWM and
quasi six-step synthesis, each of the four cases occurs at
a frequency equal to half the switching frequency. There-
fore, in PWM and quasi six-step, the following equation
applies: '
Inverter Snubber Losses = (30)
(2 ~ Case #i Loss + 2 ~ Case #2 Loss) * F/2 * 3
OR
Inverter Snubber Losses = (31)
((2 ~ ((C ~ V2) + (.5 * (V2 / L) ~ Trr2))) +
(2 * ((.5 ~ C * V2) ~ (.5 ~ L * I2))) ~ F/2 * 3
2S where F is the switching frequency.
In six-step synthesis, GTO-l is never turned on
when motor current is positive and GTO-2 is never turned on
when motor current is negative; therefore, cases number one
and three do not occur in six-step. However, cases two and
four occur at a frequency equal to the switching frequency
which is equal to the fundamental inverter frequency in
six-step. Also, when a switch occurs in six step the motor
current is usually higher than the RMS motor current. As
mentioned in previous loss calculations, the motor current
is approximated to be 1.4 times the ~MS motor current at
~ '~6~
31 51,489
this switch point. From this information the following
snub~er loss equation applies when in six-step synthesis:
Inverter Snubber Losses = (32)
(2 ~ Case #2 Loss) ~ F * 3
OR
Inverter Snubber Losses = (33
(2 ~ (( 5 ~ C * V2) ~ (.5 * L * (1.4 ~ I)2))) ~ F * 3
Using an avera~e switching frequency of 400 Hz
for PWr~ and ~uasi six-step, a 2 microfarad capacitor, 9
microhenries of total inductance, and a diode reverse
recovery time of 2.5 microseconds, the snubber loss equa-
tions reduce to the following:
Inverter Snubber Losses (PWM and Quasi Six-Step) ~ (34
((1.0 ~ 10~ 5) * V2) + ~1.35 ~ 10~-5) ~ I2)) ~ E
AND
Inverter Snubber Losses (Six-Step) = (35)
(~6.0 ~ 10~ 6) * v2) + (5.3 * 10(-5) * I2) ~ F
where V is DC line voltage, I is RMS motor current, and F
is the inverter switching frequency which for six-step is
equal to the fundamental inverter frequency.
MOTOR ELECTRICAL LOSSES
The first motor loss to be defined is the stator
resistive loss. This loss is caused by the motor stator
ha~ing resistance which dissipates power as the fundamental
motor current flows through the stator. The following
equation defines this loss:
; Motor Stator Resistive Loss = 3 ~ R ~ I2 (36)
where the R is the stator resistance and I is the fundamen-
tal motor current in RMS amperes. The factor of 3 is added
to account for the fact that in a three-phase motor there
are three stator windings. The stator resistance actually
c~
32 51,489
increases with temperature, but the resistance variation is
not great enough to require measuriny the motor temperat.ure
in order to compensate for this variation. A constant
value is assumed for the stator resistance of the motor of
.0204 ohms. The above equation for this application
becomes as follows:
Motor Stator Resistive Loss - .0612 * I2 (37)
Core losses in the motor are magnetizing losses.
The flux in the motor follows the sine wave motor current;
however, energy is lost as the flux changes polarity, due
to hysteresis properties of the motor. When operating in
the constant volts per hertz mode, the motor flux is kept
at a constant amplitude, and core losses are, therefore,
proportional to the fundamental inverter frequency. Using
core loss data from the motor manufacturer for the particu-
lar motor enables deriving the following equation for core
losses while operating i~ the constant volts per hertz mode
(the motor data consists of saying that there are 1864
watts of core losses in the motor, given that the motor is
at the rated speed of 45 Hz and rated line-to-line voltage
of 4Z0 volts):
Core Loss = 1864 ~ (Inverter Freq/45) = (38)
41.4 ~ Inverter Freq
When not operating in the constant volts per hertz mode,
the above formula does not apply because flux is no longer
being kept at a constant amplitude; therefore, a different
equation which allows for variation in the motor flux must
be developed. The ollowing equation is used in such a
case: .
Core Loss = 1.162 ~ (V/F)1-6 * F (39)
... .
33 51,489
where V is line-to-line motor voltage and F is fundamental
frequency of the applied ;nverter voltage waveforms.
Stray losses are a collection of extraneous
electrical motor losses which are not included in any of
the other motor losses. A rough estimate of these stray
losses indicates that they are pr~oportional to the motor
torque value. ~ stray loss of 1630 watts at a torque of
768 LB-FT is read from motor data for the motor used in the
AC drive setup. From this data the constant is calculated
which relates stray losses to motor torque as follows:
Stray Loss = C * Motor Torque (40)
C = Stray Loss / Motor Torque
C = 1630 ~ 768 = 2.12
Stray Loss = 2.12 * Motor Torque (11)
Where core loss is in watts and motor torque is the last
calculated motor torque in units of LB-FT.
Harmonic losses are caused by the harmonic
currents flowing in the motor which produce resistive
heating of the motor. These losses are functions of
several parameters such as synthesis mode used, fundamental
inverter frequency, and torque output of the motor. It is
extremely difficult to derive equations for the harmonic
losses, correlate the data, and form a combination of
looXup tables and equations which allow for the calculation
of these harmonic losses. An accurate but very time
consuming way is to use a spectrum analyzer and measure the
RMS voltage and current values for all major harmonics.
The spectrum analyzer can also be used to then measure the
phase angle between the voltage and current for each
harmonic. Knowing these three parameters the losses are
calculated due to each harmonic using the following
eguation:
34 51,489
Harmonic Power = V * I * Cos(Phase Angle) (41)
After calculating ~he power loss due to each harmonic, an
addition of power losses from each harmonic provides the
total loss due to all of the harmonics. Data must be taken
at several operating frequencies and motor torques. By
gathering enough information, the data is correlated to
arrive at a co~bination of lookup tables and equations
which enable the processor to calculate the harmonic losses
under all operating circumstances. Another much easier
method is to measure the input power to the entire AC drive
system (input power to inverter) and measure the output
torque of the motor at various operating frequencies and
motor loads (alonc3 with other helpful parameters such as
motor current, motor temperature, etc.). The following
formula calculates remaining losses in the system. These
remaining losses should be approximately equal to the
harmonic losses pro~iding that the derived loss equations
are reasonably accurate:
Remaining Loss = P - ((T ~ F~ + Loss) (42)
Where P is input power, T is motor torque, F is fundamental
inverter frequency, and loss is total defined losses in the
system (not includin~ rotor losses). After calculating
harmonic losses for several operating frequencies and motor
torques, the data can be correlated to arrive at a combina-
tion of lookup tables and equations that define harmonic
losses over all operating conditions. This latter method
was used to formulate equations and lookup tables used by
the micro to calculate the harmonic losses in the present
AC drive system~
MOTO~ MECHANICAL LOSSES
Windage losses in the motor are the losses due to
both the rotor and fan displacing air as the rotor turns.
It is well known that these losses are proportional to the
rotor speed cubed by some proportionality constant. This
~2~
51,489
constant is determined from data supplied by the motordesigner. Eor the motor used in the present system, the
equation for windage losses is as follows:
Windage Losses = 599 ~ (RPM/1800) (12)
where RPM is the speed of the rotating shaft in RPMs.
Frictlon losses in the motor are the losses
caused by the friction between the motor shaft and housing
as the rotor turns. It is well Xnown that these losses are
directly propor~ional to the rotor speed by some propor
tionality constant. Like windage losses, this constant is
determined from data supplied by the motor designer. For
the motor used in the present system, the equation for
friction losses is as follows:
Friction Losses = 104 ~ (RPM~1800) (l3?
where RPM is the speed of the rotating shaft in RPMs.
~R~ IG TRA~lS~ORMER LOSSES
If the bra~.ing transformer is included the
resulting losses must be calculated and added to the losses
calculated so far. If the braXing transformer is not being
used, such that the thyristors on the motor or primary side
o the transormer are gated on and thus are shorting out
the primary side of the transformer, the transformer and
devices on the primary side of the transformer are not
conducting any current and thus have no power losses.
However, the thyristors have a voltage drop associated with
them and are conducting the motor current; therefore, the
thyristors have conduction power losses. Assuming an
average voltage drop of 1.5 volts for the thyristors and
realizing that each of the six thyristors conducts current
only half of the time, the equation for brake thyristor
conduction losses is:
36 51,~89
Brake Thyristor Conduction Loss = 1.5 ~ ((.9 ~ I) (14)
~ 2) * 6
= 4.05 ~ I
Where 1.5 is the thyristor voltage drop, I is RMS motor
current in amps, .9 converts RMS motor current to average
motor current, 2 is due to each thyristor conducting only
of the time, and 6 is the total number of braking circuit
thyristors.
If the braking thyristors are not being gated on,
they ~lill each turn off the next time their respective
sinusoidal motor current crosses zero. When all thyristors
are off, the braking transformer and the components on the
primary side of the transformer will conduct current and,
therefore, have losses associated with them. In the
following equations, a DC line voltage of 700 volts is
assumed instead of the 603 volt nominal line assumed
previously. Using this higher voltage is justified because
during times when transformer braking is used, a consider-
able amount of current is being regenerated to the DC line.
In most cases, the line will not take all of this current;
therefore, the voltage will rise above the nominal 600 volt
line to a value of around 700 ~olts.
3raking circuit snubber and switching losses
consist o losses in the thyristor snubber circuits due to 25 the subsequent charging ar.d discharging of the capacitor in
the circuits; GT0 turn-on losses; GT0 turn-off losses; GT0
voltage snubber losses due to the subse~uent charging and
discharging of the capacitor in the circuits; GT0 current
snubbex losses due to the building up and building down of
current in the inductor in the circuits; and line diode
~oltage snubber losses due to the charging and discharging
of the capacitor in the circuits.
Thyristor snubber circuit losses occur because
each time the GT0 on the primary side of the transformer is
switched off, the capacitor in this snubber circuit charges
to the value 700 * .9, plus or minus depending on the
C~L
37 51,489
direction of current through the transformer, assuming a DC
line of 700 volts and a transformer turns ratio of .9.
Each time the CTO discharges, this capacitor must discharge
all of this voltaye. Each GTO is turned on and off at
S twice the fundamental inverter frequency; therefore, each
snubber circult charges twice and discharges twice each
fundamental inverter period. Each charge cycle and dis-
charge cycle is through the snubber resistor. The energy
losses in watt-seconds in this resistor for either a charge
or discharge cycle are equal to ~2 ~ C ~ v2 where C is the
value of the snubber capacitor in farads and V is the
voltage change in the capacitor of 700 ~ 1.1. To compute
the power losses in all three thyristor snubber circuits
the following equation is used:
Braking Th~ristor Snubber Loss =
~ ~ C * (700 ~ 9)2 * 4 ~ F * 3
where F is the fundamental inverter frequency in hertz; 4
is the total number of charge and discharge cycles per
fundamental inverter pèriod; and 3 is the number of thyris-
tor snubber circuits. ~ith a capacitor value of 1 micro-
farad, this equation reduces to the following:
Braking Thyristor Snub~er Loss = 2.4 ~ F
Brakiny GTO turn-on losses are determined the
same way as the inverter GTO turn-on losses were. From the
2S GTO manufacturer s curves, using an anode current di/dt of
85 amps per microsecond, and an IGM of 5 amps, the value of
.16 watt-sec/pulse is determined rom the curve. Using
this value, the following equation is derived for braking
GTO turn-on losses:
Braking GTO Turn On Loss = .16 * 2 ~ F ~ 3 =
.96 ~ F
38 51,489
Where F is the fundamental inverter frequency; 2 * F is the
number of times each braking GTO is turned on each second;
and 3 is the number of braking GTOs in the system.
Braking GTO tur~-off losses are determined in a
manner similar to tAe way the inverter GTO turn-off losses
were; however, to simplify matters a constant average GTO
tur~-off anode current of 350 amps is assumed. From the
GTO manufacturer's ~urves, using this 350 amp figure, the
. value of about .5 watt-sec/pulse is determined. Using this
value, the following equation is derived for braking GTO
turn-off losses:
Braking GTO Turn Off Loss = .5 * 2 * F * 3
= 3 * F
Where F is the fundamental inverter frequency; 2 * F is the
number of times each braking GTO is turned off each second;
and 3 is the number of braking GTOs in the system.
Braking GTO voltage snubber losses are similar to
the inverter Gl'O voltage snubber losses. When turning a
braking GTO off, the ca?acitor in the snubber circuit
20 charges up to a voltage of 700 volts, assuming a 700 volt
operating voltage, via a diode and since the charging is
via a diode, there are fe~ losses associated with charging
the capacitor. When turning the braking GTO on, the
capacitor must discharge this voltage through a resistor.
The losses in the resistor are equal to the total energy
stored in the capacitor which is defined as ~ ~ C * V2. By
multiplying this energy by the number of times each second
that the capacitor is discharged, equal to the brake GTO
switching frequency which is equal to 2 times the fundamen-
tal inverter frequency, and the number of braking GTO
voltage snubber circuits (3) the power losses are
calculated:
Braking GTO Voltage Snubber Loss =
~ * C ~ 7002 ~ 2 * F ~ 3
,
5~
39 51,489
Using a snubber capacitor having 2 microfarads of capaci-
tance, this equation reduces to the following:
Braking GT0 Voltage Snubber Loss ~ 2.9~ ~ F
Just like the intrerter voltage snubber capacitors, due to
current snubber inductance and stray inductance, the
snubber capacitor actually charges to more than 700 volts;
however, these extra losses are included in the current
snubber e~uation.
The braking circuit current snubber losses are
similar to the inverter current snubber losses. When
turning the braking GTO on, the current in the snubber
inductor builds up. ~lhen the bra~ing GTO is subsequently
turned off, the current builds down to zero by dissipating
the energy in the snubber resistor, ignoring the energy
that gets transferred to the voltage snubber capacitor.
During current build-up, the energy level reached in the
inductor is equal to ~. * L * (.9 ~ I)2 where L is the
inductance, .9 is the transformer turns ratio, and I is t~e
motor current at the time the GTO is turned off, therefore
1.1 ~ I is the current in the inductor at time of turn-off.
By multiplying this energy by the number of times each
second that the inductor current is built down, equal to
the brake GTO switching frequency which is equal to 2 times
the fundamental inverte- frequency, and the number of
braking GTO current snubber circuits (3) the power losses
are calculated:
Braking GTO Current Snubber Loss =
~ r L ~ ( 9 ~ I)2 ~ 2 * F ~ 3
This equation is further simplified by using an average
motor current at time of GTO turn-off of 400 amps, since
transformer braking generally produces motor currents of
around rated current magnitude. If greater accuracy of
loss estimation is required, a current proportional to RMS
5~
51,489
motor current can be used instead of a constant value of
400 amps. Using a current snubber inductance of 7 milli-
henry plus a stray ind~ctance of 3 millihenry, this equa-
tion becomes:~
Braking GT0 Current Snubber Loss = 3.9 ~ F
The line diode snubber loss is due to the charg-
ing up of the snubber capacitor to the 700 volt line
volta~e everytime the braking GT0 is turned on and dis-
charging to zero volts every~ime the braking GT0 is turned
of. Both cha~ging and discharging paths are through the
snubber resistor so losses equal to l,, ~ C ~ 7002 occur at
every GT0 turn-on or turn-off transition. There is a total
number of 4 ~ F transitions per second. The power equation
for the line diode snubber losses in all three phases of
the braking .rans~ormer c rcuit is as f~llows:
Braking Llne Diode Snubber Loss =
~ ~ C ~ 7002 * ~ ~ F ~ 3
Using a snubber capacitance of .5 microfarads, this equa--
tion becomes:
Braking Line Diode Snubber Loss = 1.47 * F
Since the equations are established for all of
the snubber and switching losses in the braking transformer
circuit as a function of a constant multiplied by fundamen-
tal inverter frequency, these losses are lumped together
into one equation in order to save the micro time in the
calculations. This equation is as follows:
Braking Switching Losses = 14.7 ~ F
The line diode in each phase of the braking
oircuit conducts current whenever the GT0 is of and the
~ 6.~ ~
41 51,489
GTO conducts current whenever tlle GTO is on. Instantaneous
line diode conduction losses are equal to the product of
the instantaneous voltage drop of the device and ~he
current being conducted b~ the device. Since average power
loss is needed, average diode voltage drop is selected of
1.2 volts for the diode. Also, the average current through
the diode is used. The combined conduction losses of all
three line diodes usiny these simplifications are deter-
mined using the following equation:
10BraXing Line Diode Conduction Loss ~
3 ~ ((ljl.l) * (.9 * I)) ~ 1.2 ~ Angle/180
Where 1.1 is the transformer turns ratio, I is the RMS
motor current, .9 converts RMS motor current to average
motor current, 1/1.1 converts average motor current to
average diode current, 1.2 is the diode voltage drop, angle
is in degrees a~d represents the portion out of each 180
degrees that the braking GTO is off, and angle/180 is the
percentage of time that the diode is conducting.
Braking GTO conduction losses are similarly
determined using an average GTO voltage drop of 1.8 volts:
Braking GTO Conduction Loss =
3 ~ ((1,'1.1) ~ (.9 ~ 1.8 * (180-Angle)/180
The term angle/180 is replaced by (180-Angle~/180 to
indicate the percentage of time that the GTO is conducting.
25By looking at the line diode and GTO conduction
loss equations, if the voltage drops in each device were
egual then the total conduction losses of all GTOs and line
diodes in the bra':ing circuit would reduce to the following
equation:
30Braking Conduction Loss =
3 ~ ((ljl.1) ~ (.9 ~ I)) * Voltage Drop
This single equation greatly simplifies the two separate
. eguations. Voltage drop i5 set equal to 1.6 volts by
42 51,489
averaging the diode drop of 1.2 and the GTO drop of 1.8
volts. 1.6 is chosen instead of the actual average of 1.5
because the GTOs are normall~ conducting a greater percent-
age of time than are the diodes. Including this voltage
drop of 1.6 volts the equation becomes:
Braking Conduction Loss = 3.9 i I
Th~ diodes in the full wave rectifier bridge also
have conduction losses associated with them. Two of the
diodes are always conducting no matter what the GTO is
doing. Assuminy a diode voltaye drop of 1.2 volts in these
diodes, the equation for diode bridge conduction losses in
all three phases is derived:
Braking Bridge Loss =
3 ~ ((1/1.1) - (.9 * I)) ~ 1.2 * 2
Where (1/l.1) ~ (~9 ~ I) is the average current through tha
diodes, I is the RMS motor current, 1.2 is the voltage drop
in a diode, and 2 indicates that two diodes are conducting
at all times. Simplifyi~g, this equation becomes:
Braking Bridge Loss = 5.9 * I
Transformer resistive losses are I2 ~ R losses
where I is the moto- current or secondary current and R is
the total resistance of both the primary and secondary
windings of the transformer. The primary winding resis-
tance is referred to the secondary side by multiplying the
actual resistance in the primary side by the turns ratio of
.9 squared. Assuming a transformer temperature of 130
degrees centigrade, the resistance of the transformer is
about .08 ohms. This value will vary with temperature, but
: in an effort to keep things simple, this resistance varia-
tion is ignored. The equation used to calculate trans-
former resistive losses is as follows:
~2~6~
43 51,489
Braking Transformer Resistive Loss = .08 ~ I
Where I is the RMS motor current~
The core loss in the transformer is caused by the
variation of the flux in the iron core and depends upon the
frequency, the maximu~ value of the flux density as deter-
mined by the excitation voltage, the shape of the excita~
. tion waveform, and the construction of the transformer. A
core loss equation which defines the losses in our trans-
former is as follows:
Braking Transformer Core Loss =
307 * lo( 13~5 ~ V/F - 1.272)
Where V is line-to-line RMS voltage across the transformer
and F is the fundamental invertçr frequency.
~OSS C..LCUL~.TI9N CO~ICLUSIONS
In the preceding text, the numerous equations
used by the microprocessor to calculate losses in the AC
drive system have been presented. Some of the more complex
equations which in-~Jolve non-inte~er powers are actually
performed with the assistance of lookup tables.
TORQUE CALCUL~.TIONS USING LOO~UP TABLES
At very low frequencies where losses become an
appreciable percentage, more than half of the overall
system power, a very accurate loss mode} would be required
to achieve any type of accurate torque calculation using
the (input power - losses)/frequency equation (l) to
calculate torque. To avoid this problem, lookup tables
relating torque to input power and frequency were devel-
oped. A different look~lp table exists for each hertz of
tach frequency from zero to 15 Hz. Each table represents
the relationship between torque and input power. Since at
each speed and load condition the same voltage is always
applied to the motor, true at low frequencies of below the
base speed of about 45 Hz, each time that condition exists,
44 51,489
there exists a definite relationship between the motor
torque and the input power, such that as motor torque
increases, so does the input power. Therefore, it is
practical to calculate the torque by simply calculating the
input power and using this torque/power relationship to
arrive at an answer. The loss model is used to calculate
these tor~ue/power relationships at each speed off-line.
The results are gathered into lookup tables which the micro
uses on-line for the torque calculation.
At higher frequencies, the on-line loss calcula-
tion method is preferred because it allows for much more
flexibility such as not requiring operation at constant
volts per hertz all of the time. Above base speed, where
such constant volts per hertz operation is not feasible,
the table method would be ~ery difficult to use because
somehow the micro would ha~e to compensate for voltage
differences. Such a compensation is very difficult.
Even though the lookup tables allow the calcula-
tion of torque at lower frequencies than does the on-line
loss model calcula~ior.s, calculating braking torques at
very low frequencies below about 10 Hz is not practical
using any method that is based upon reading only input
power. At these ver~ low frequencies, the relationship
between torque and input power ceases to be a function for
brake torques. If braking operation of the motor is
desired at such low frequencies, an open loop type control
is used because of this difficulty to measure torque.
