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Claims  |
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I claim:
1. A method to be used with a PWM inverter which receives a sinusoidal
command signal having a peak command value and a high frequency carrier
signal having a peak carrier value and generates a series of high
frequency voltage pulses therefrom the fundamental component of which over
the period of the sinusoidal command is a phase voltage, a modulation
index being the ratio of the peak command value to a peak carrier value,
the phase voltage being variable within a phase voltage range and being
substantially linearly related to the sinusoidal command signal when the
modulation index is less than unity, the method for extending the linear
relationship between the sinusoidal command signal and the phase voltage
to include the entire phase voltage range, the method comprising the steps
of:
(a) determining if the modulation index is greater than unity and where the
modulation index is greater than unity;
(b) identifying a modifier signal in phase with the sinusoidal command
signal;
(c) mathematically combining the modifier signal with the sinusoidal
command signal to provide a modified sinusoidal command signal; and
(d) providing the modified sinusoidal command signal to the inverter for
comparison with said carrier signal to generate a phase voltage;
wherein, the modifier signal is such that, when the modified sinusoidal
command signal is compared to the carrier signal, the inverter generates a
fundamental component phase voltage that substantially maintains the
linear relationship with the command signal.
2. The method of claim 1 wherein the step of identifying a modifier signal
includes the step of identifying a square wave.
3. The method of claim 2 wherein the step of identifying includes the steps
of solving the equation:
##EQU8##
for S, where V.sub.c is the peak value of the sinusoidal command signal,
and using S as the magnitude of the square wave.
4. The method of claim 1 wherein the step of mathematically combining
includes the step of adding the modifier signal to the sinusoidal command
signal.
5. The method of claim 1 further including the step of, prior to providing
the modified sinusoidal command signal, altering the modified sinusoidal
command signal so that no part of the sinusoidal command signal has a
magnitude which is greater than the peak carrier signal value.
6. The method of claim 3 further including the step of, prior to providing
the modified sinusoidal command signal, altering the modified sinusoidal
command signal so that no part of the sinusoidal command signal has a
magnitude which is greater than the peak carrier signal value.
7. The method of claim 2 wherein the step of identifying a square wave
includes the steps of solving the equation:
##EQU9##
for S, where K.sub.s, is a constant and M.sub.i is the modulation index,
and using S as the magnitude of the square wave.
8. The method of claim 1 wherein the inverter includes a look-up table
consisting of a plurality of modifier signals which have been generated
off line as a function of different sinusoidal command signals, and the
step of identifying includes the step of identifying a modifier signal in
the look-up table which corresponds to a specific sinusoidal command
signal.
9. The method of claim 2 wherein the inverter includes a look-up table
consisting of a plurality of modifier signals which have been generated
off line as a function of different sinusoidal command signals, each
modifier signal including square wave magnitude data generated by solving
the equation:
##EQU10##
for S, where S is the magnitude of the square wave and V.sub.c is the peak
sinusoidal command signal value.
10. An apparatus to be used with a PWM inverter which receives a sinusoidal
command signal having a peak command value and a high frequency carrier
signal having a peak carrier value and generates a series of high
frequency voltage pulses therefrom the fundamental component of which over
the period of the sinusoidal command is a phase voltage, a modulation
index being the ratio of the peak command value to a peak carrier value,
the phase voltage being variable within a phase voltage range and being
substantially linearly related to the sinusoidal command signal when the
modulation index is less than unity, the apparatus used for extending the
linear relationship between the sinusoidal command signal and the phase
voltage to include the entire phase voltage range, the apparatus
comprising:
(a) a comparator to determine if the modulation index is greater than unity
and where the modulation index is greater than unity;
(b) a first calculator to identify a modifier signal in phase with the
sinusoidal command signal;
(c) a second calculator to mathematically combine the modifier signal with
the sinusoidal command signal to provide a modified sinusoidal command
signal; the second calculator supplying the modified sinusoidal command
signal to the inverter for comparison with said carrier signal to generate
a phase voltage;
wherein, the modifier signal is such that, when the modified sinusoidal
command signal is compared to the carrier signal, the inverter generates a
fundamental component phase voltage that substantially maintains the
linear relationship with the command signal.
11. The apparatus of claim 10 wherein the first calculator identifies a
square wave.
12. The apparatus of claim 11 wherein the first calculator includes a
selector which identifies a square wave by solving the equation:
##EQU11##
for S, where S is the magnitude of the square wave and V.sub.c is the peak
sinusoidal command signal value.
13. The apparatus of claim 10 wherein the second calculator includes an
adder which mathematically combines by adding the modifier signal to the
sinusoidal command signal.
14. The apparatus of claim 10 wherein the second calculator further
includes a clipper that, prior to providing the modified sinusoidal
command signal, alters the modified sinusoidal command signal so that no
part of the sinusoidal command signal has a magnitude which is greater
than the peak carrier signal value.
15. The apparatus of claim 11 wherein the first calculator includes a
selector which identifies a square wave by solving the equation:
##EQU12##
for S, where S is the magnitude of the square wave, K.sub.s is a constant
and M.sub.i is the modulation index.
16. The apparatus of claim 10 wherein the inverter includes a look-up table
consisting of a plurality of modifier signals which have been generated
off line as a function of different sinusoidal command signals, and the
first calculator includes a searcher for searching the look-up table to
identify a modifier signal in the look-up table which corresponds to a
specific sinusoidal command signal.
17. An apparatus to be used with a PWM inverter which receives a sinusoidal
command signal having a peak command value and a high frequency carrier
signal having a peak carrier value and generates a series of high
frequency voltage pulses therefrom the fundamental component of which over
the period of the command signal is a phase voltage, a modulation index
being the ratio of the peak command value to a peak carrier value, the
phase voltage being variable within a phase voltage range and being
substantially linearly related to the sinusoidal command signal when the
modulation index is less than unity, the apparatus used for extending the
linear relationship between the sinusoidal command signal and the phase
voltage to include the entire phase voltage range, the apparatus
comprising:
(a) a comparator to determine if the modulation index is greater than unity
and where the modulation index is greater than unity;
(b) a first calculator to identify the magnitude of a square wave;
(c) a square wave generator for generating a square wave that is in phase
with the sinusoidal command signal and has the magnitude identified by the
first calculator; and
(d) a second calculator to mathematically combine the square wave with the
sinusoidal command signal to provide a modified sinusoidal command signal,
the second calculator supplying the modified sinusoidal command signal to
the inverter for comparison with said carrier signal to generate a
fundamental component phase voltage;
wherein, the square wave magnitude is such that, when the modified
sinusoidal command signal is compared to the carrier signal, the inverter
generates a fundamental component phase voltage that substantially
maintains a linear relationship with the command signal.
18. The apparatus as recited in claim 17 wherein the second calculator
includes a clipper that, prior to providing the modified sinusoidal
command signal, alters the modified sinusoidal command signal so that no
part of the sinusoidal command signal has a magnitude which is greater
than the peak carrier signal value. |
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Claims |
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Description |
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FIELD OF THE INVENTION
The field of the invention is electrical power conversion equipment and,
more particularly, PWM control methods used with DC to AC inverters and AC
to DC converters.
DESCRIPTION OF THE ART
Many AC motor applications require that a motor be driven at various
speeds. Motor speed can be adjusted with an adjustable speed drive (ASD)
which is placed between a DC voltage source and an associated motor that
can excite the motor at various AC frequencies. One commonly used type of
ASD uses a pulse width modulated (PWM) inverter and associated PWM
controller which can control both voltage and frequency of signals that
eventually reach motor stator windings.
Typical motors have three phases which are separately controlled via an
inverter and a controller. Referring to FIGS. 1(a)-1(c), although only a
single command voltage V.sub.c and signals generated therefrom are
illustrated, a three phase PWM inverter for controlling a three phase
motor is driven by three such sinusoidal reference voltages, a separate
reference voltage corresponding to each of the three motor phases (i.e.,
each of three motor stator windings). In the interest of providing a
simple explanation of a PWM inverter only one sinusoidal command voltage
V.sub.c and signals generated therefrom are explained here in detail.
Referring specifically to FIGS. 1(a) and 1(b), a PWM controller receives
three sinusoidal command signals V.sub.c and a carrier signal V.sub.T,
compares each sinusoidal command signal V.sub.c to the carrier signal
V.sub.T and generates a firing signal V.sub.f corresponding to each
sinusoidal command signal. When a sinusoidal command signal V.sub.c is
greater than the carrier signal V.sub.T, a corresponding firing signal
V.sub.f is high. When a sinusoidal command signal V.sub.c is less than the
carrier signal V.sub.T, a corresponding firing signal V.sub.f is low.
The firing signals V.sub.f are used to control an associated PWM inverter.
A PWM inverter consists of a plurality of switches that alternately
connect associated motor stator windings to positive or negative DC
voltage buses to produce a series of high frequency voltage pulses that
excite the stator windings.
Referring to FIG. 1(c), an exemplary sequence of high frequency pulses
V.sub.h that an inverter might generate across a stator winding can be
observed along with an exemplary low frequency alternating phase voltage
V.sub.phf. The phase voltage V.sub.phf is the fundamental component of the
high frequency pulse sequence V.sub.h. The high frequency pulses V.sub.h
are positive when the firing signal V.sub.f is high and negative when the
firing signal V.sub.f is low. The maximum magnitude of each pulse V.sub.h
is half the DC potential between the positive and negative DC bus lines.
Thus, where the DC potential is V.sub.dc, the maximum magnitude is
+V.sub.dc /2 and the minimum magnitude is -V.sub.dc/ 2.
By firing the PWM switches according to the firing signals V.sub.f, the
widths of the positive portions 10 of each high frequency pulse relative
to the widths of the negative portions 12 over a series of high frequency
pulses V.sub.h varies. The varying widths over the period of the command
signal V.sub.c generate the low frequency fundamental component
alternating phase voltage V.sub.phf.
The low frequency phase voltage V.sub.phf in turn produces a low frequency
alternating phase current I.sub.ph that lags the voltage by a phase angle
.PHI.. The phase current I.sub.ph drives the motor which operates at the
frequency of the phase current I.sub.ph.
By changing the frequency of the sinusoidal command signal V.sub.c, the
frequency of the phase current I.sub.ph, and thus the motor speed, can be
altered. For example, by increasing the frequency of the sinusoidal
command signal V.sub.c, the frequency of the phase current I.sub.ph can be
increased and motor speed can in turn be increased. Motor speed can be
decreased by decreasing the sinusoidal command signal V.sub.c frequency.
In addition, by changing the peak-to-peak of the sinusoidal command signal
V.sub.c while maintaining a constant frequency, the amplitude of the
fundamental component phase voltage V.sub.phf can be altered. For example,
referring to FIG. 2(a), a carrier signal V.sub.T and a plurality of in
phase sinusoidal command signals V.sub.c0 -V.sub.c4 which are
characterized by different peak values are illustrated. FIG. 2(b)
illustrates the effective command voltages V.sub.c0e through V.sub.c4e
corresponding to the command voltages V.sub.c0 through V.sub.c4 in FIG.
2(a). The effective phase voltages V.sub.c0e -V.sub.c4e are the parts of
the command voltages V.sub.c0 -V.sub.c4 which are below the carrier peak
value V.sub.T. When a zero sinusoidal command signal V.sub.c0 is provided,
the effective command voltage V.sub.c0e is zero. On the other hand, where
a high sinusoidal command signal V.sub.c4 is provided (i.e. where the peak
sinusoidal command signal is much greater than the peak carrier signal),
the effective command voltage V.sub.c4e approximates a square wave having
a fundamental component 4/.pi. times the maximum DC voltage value. In the
present case, where the maximum DC voltage value is V.sub.dc /2, the
maximum fundamental phase voltage V.sub.ph4 approaches 2V.sub.dc /.pi..
Thus, the range of possible fundamental phase voltages is between 0 and
2V.sub.dc /.pi..
Ideally a linear relationship should exist between the sinusoidal command
signals V.sub.c and the fundamental component phase voltage V.sub.phf such
that any change in the sinusoidal command signal V.sub.c magnitude is
mirrored by a linear change in the fundamental component phase voltage
V.sub.phf magnitude assuming that the maximum phase voltage is not
surpassed. Unfortunately, typical PWM controllers can only provide a
linear relationship between the control voltage V.sub.c and the
fundamental component phase voltage V.sub.phf over a reduced range of
possible phase voltage.
When a PWM inverter is used to provide a phase voltage outside the reduced
linear range, the phase voltage gain is sharply reduced which in turn
restricts the range of accurate speed and torque regulation. In other
applications such as for utility interfacing as a voltage source
converter, reduced gain restricts the range of fluctuations in the utility
voltage which can be handled effectively to keep the desired bus voltage
and power factor with low harmonic distortion.
Referring again to FIG. 1(a), an amplitude modulation index M.sub.i is
defined as the ratio of the peak sinusoidal command signal value V.sub.c
and the peak carrier signal value V.sub.T. By increasing the modulation
index M.sub.i, the amplitude of the fundamental component phase voltage
V.sub.phf can be increased.
PWM inverter operation can be divided into three modulation index magnitude
dependent modes. Referring again to FIGS. 2(a), a first mode of operation
is referred to as the linear mode of PWM operation which occurs when the
modulation index M.sub.i is less than one. In FIG. 2(a), all three
sinusoidal command signals V.sub.c0, V.sub.c1, and V.sub.c2 drive an
inverter in this linear mode of operation. Referring also to FIG. 2(c), in
the linear mode, any increase in the command voltage magnitude is followed
by a linear increase in the fundamental component voltage V.sub.phf.
Referring also to FIGS. 1(a)-1(c), as the command voltage V.sub.c
magnitude is increased in the linear mode, the ratio of time during which
the command voltage V.sub.c waveform is above the carrier voltage V.sub.T
to the time during which it is below the carrier voltage V.sub.T increases
linearly which is reflected in the firing signal V.sub.f, the high
frequency voltage pulse V.sub.h, and eventually in the magnitude of the
phase voltage V.sub.phf.
Referring again to FIGS. 2(a) and 2(c), the linear relationship between
command voltages V.sub.c0, V.sub.c1, V.sub.c2 and related phase voltages
V.sub.phf0, V.sub.phf1, V.sub.phf2 can be observed. Referring also to FIG.
3, the phase voltage gain G.sub.v as a function of the modulation index
M.sub.i is illustrated. The phase voltage gain G.sub.v is the ratio of the
peak fundamental component phase voltage V.sub.phf to peak command voltage
V.sub.c times the D.C. bus value
##EQU1##
Up to a modulation index M.sub.i of 1.0 the phase voltage gain G.sub.v is
constant.
A second mode is the non-linear or pulse dropping mode of operation which
occurs when the modulation index M.sub.i exceeds one. In FIG. 2(a),
sinusoidal command signals V.sub.c3 and V.sub.c4 drive an inverter in this
non-linear mode. Here a specific increase in the peak command voltage
V.sub.c does not linearly increase the peak fundamental component phase
voltage V.sub.phf. For example, referring to FIG. 2(c), where an initial
command voltage is V.sub.c2 and the modulation index is 1.0, a 30%
increase in the command voltage to V.sub.c3 where the modulation index is
1.3 may only result in a 13% increase in the resulting phase voltage
V.sub.ph3 (i.e. V.sub.ph3 =1.13 V.sub.ph2).
Referring still to FIG. 2(a), where the modulation index M.sub.i exceeds
one, only the portion of the command voltage V.sub.c3 which is below the
peak value of the carrier voltage V.sub.T is effective for modulation
purposes. Referring also to FIG. 3, the phase voltage gain G.sub.v reduces
sharply in a non-linear fashion, hence the reference "non-linear mode".
The third mode is often referred to as the six step mode which occurs when
the peak value of the command voltage V.sub.e is much greater than the
peak value of the carrier voltage (i.e. V.sub.c .apprxeq.5V.sub.T). In
FIG. 2(a), command voltage V.sub.c4 corresponds to a modulation index
M.sub.i of 5.0 and therefore approaches the six step mode of operation.
Here, the effective command voltage V.sub.c4e approaches a square wave and
the phase voltage gain G.sub.v approaches zero. In this mode, the AC phase
voltage V.sub.phf4 starts to saturate and reach its theoretical maximum of
2V.sub.dc /.pi..
One method which can be used to extend the linear region of PWM operation
is to have DC bus voltage maintained at a higher than needed value so that
PWM operation always remains within a desired linear region. Besides the
added cost of some form of control to maintain a higher bus voltage, this
solution results in a cost and size penalty for the DC bus capacitor banks
along with increased switching losses.
U.S. Pat. No. 5,329,217 entitled Compensated Feedforward Voltage for a PWM
AC Motor Drive which issued to Kerkman, et al. on Jul. 12, 1994, describes
another method which can be used to extend the linear region of PWM
operation to include the entire range of fundamental phase voltages (i.e.
zero to 2V.sub.dc /.pi.). In this method, in the non-linear mode of
operation, the command voltage V.sub.c is multiplied by a gain factor
which increases as an inverse function of the PWM gain. In other words,
when the phase voltage gain drops in the non-linear region, the magnitude
of the sinusoidal command signal V.sub.c is increased to maintain the
desired output voltage. The drawback of this method is that the sinusoidal
command signal V.sub.c, after being adjusted for the falling gain, results
in an extremely high modified command voltage value.
To approach within 0.5% of the sinusoidal command signal V.sub.c required
to provide the maximum phase voltage V.sub.phf the modulation index
M.sub.i typically needs to be pushed to a value of 5.0 or beyond. This
means that a PWM controller must be able to handle peak command voltages
V.sub.c at least as large as five times the peak carrier signal V.sub.T.
This method runs into implementation problems in both the analog and
digital domains. In the analog domain this method is difficult to
implement because amplifiers and the like tend to saturate where the
command voltage V.sub.c is excessive. In the digital domain, increased
command voltage V.sub.c levels require additional memory for storing large
digital words needed to identify and differentiate large sinusoidal
command signals. These analog and digital problems result in a reduction
in the dynamic range of associated control as the PWM controller must
handle a wide range of control voltages.
U.S. Pat. No. 5,153,821, entitled Pulse Width Modulation Method For
Providing Extended Linearity, Reduced Communication Losses And Increase In
Inverter/Converter Output Voltage, which issued to Blasko on Oct. 6, 1992,
describes another method for extending the linear range of PWM operation.
This method provides a modified non-sinusoidal command signal to the PWM
inverter. This method extends the linear range of inverter operation to
the point where the modulation index M.sub.i is equal to 1.1547. However,
after the modulation index M.sub.i exceeds 1.1547, this method faces the
same problems with linear operation as described above.
Therefore, it would be advantageous to have a method for controlling a PWM
controller which could extend the linear range of PWM operation to all
possible phase voltage values without requiring special hardware or
additional memory and without reducing the dynamic range of control by
using excessive sinusoidal command signals V.sub.c.
SUMMARY OF THE INVENTION
The present invention is a new method to maintain the fundamental component
gain of a PWM inverter after a modulation index M.sub.i exceeds unity and
the PWM inverter enters the typically non-linear pulse dropping region of
operation. The method adds a square wave to a sinusoidal command signal
V.sub.c in the non-linear region of operation as a function of the
modulation index M.sub.i. The addition of the square wave linearizes PWM
operation all the way to the six-step mode of operation. The modulation
index required by the method does not exceed 1.273 and therefore, the
dynamic range of associated control is increased. For practical
implementation, a simple function governing the addition of the square
wave is suggested.
One object of the present invention is to provide a method whereby a linear
relationship can be maintained between a sinusoidal command signal and the
fundamental component of output phase voltage. By adding a square wave to
a sinusoidal command signal in the non-linear region of PWM operation
wherein the square wave is calculated to increase in magnitude as an
inverse function of the PWM gain drop-off, linear operation of the PWM
inverter (or converter) can be extended so that it covers a range of
fundamental phase voltages from zero to 4/.pi. times the maximum value of
the DC voltage.
Another object of the present invention is to extend the range of linear
PWM operation without requiring additional hardware or memory and without
reducing the dynamic range of associate control. Because the modulation
index M.sub.i does not exceed 1.273, the command voltage remains in a
relatively minimal range which is not likely to saturate an analog
amplifier and which does require additional bits in memory to identify
voltage magnitude.
Other and further objects and aspects of the present invention will become
apparent during the course of the following description and by reference
to the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1(a) is a graph illustrating carrier and command voltages, FIG. 1(b)
is a graph illustrating resultant firing signals, and FIG. 1(c) is a graph
illustrating the high frequency pulses generated by a PWM inverter, a
resulting low frequency phase voltage, and an associated fundamental
component phase current;
FIG. 2(a) is a graph illustrating various command voltages, FIG. 2(b) is a
graph illustrating effective command voltages corresponding to the command
voltages of FIG. 2(a), and FIG. 2(c) is a graph illustrating the
relationship between command voltages and fundamental component phase
voltage of a typical prior art PWM.
FIG. 3 is a graph illustrating the fundamental component gain of a typical
prior art PWM inverter;
FIG. 4(a) is a graph illustrating the shapes of effective command voltages
at the boundaries of the non-linear region of operation, FIG. 4(b) is a
graph showing a command voltage where M.sub.i >1.0, FIG. 4(c) is a graph
showing a square wave according to the present invention, and FIG. 4(d)
shows a modified effective command voltage;
FIG. 5 is a schematic diagram of a motor control system according to the
present invention;
FIG. 6 is a schematic of a single phase of a PWM inverter; and
FIG. 7 is a schematic of the command voltage modifier of FIG. 5.
DESCRIPTION OF THE PREFERRED EMBODIMENT
In the following description, all "c" subscripts will refer to sinusoidal
command signals, all "f" subscripts will refer to fundamental components
of associated signals or voltages, all " " symbols will identify peak
values of corresponding wave forms and all "ph" subscripts will refer to
phase signals, unless the description indicates otherwise.
Referring to FIGS. 1(a) and 4(a), assuming a peak carrier voltage V.sub.T
of one, when increasing the command voltage V.sub.c past the point where
the modulation index M.sub.i is equal to one, the effective command
voltage goes from being a sine wave V.sub.sine of unit magnitude at
M.sub.i =1 to an approximate square wave V.sub.square of unit magnitude
for M.sub.i >5.0. Thus, in order to maintain a linear relationship between
the command voltage V.sub.c and a resulting fundamental component of phase
voltage where the modulation index M.sub.i is greater than unity, a method
must be provided to control the progression of the effective command
voltage to maintain the linear relationship.
Referring also to FIGS. 4(b) through 4(d), in the present method, the
command voltage V.sub.c is added to an in phase square wave S to produce a
modified command or modified effective voltage V.sub.mc. The modified
effective voltage V.sub.mc is then compared with the carrier voltage
V.sub.T to provide the firing signals to a PWM inverter. The magnitude of
the square wave S is a function of the modulation index M.sub.i. The
modified effective voltage V.sub.mc is the summation of a clipped sine
wave V.sub.cc and the square wave S, the summation never having magnitude
greater than unity. To progress into the region where the modulation index
M.sub.i greater than one, the square wave S pushes out the clipped sine
wave V.sub.cc beyond the effective unit boundary, eventually eliminating
the sine wave completely to provide a square wave S of unit magnitude at
the outer bound.
Analytical Development
Referring to FIG. 5, the fundamental component V.sub.mcf of the effective
voltage is linearly related to the fundamental component of the phase
voltage V.sub.phf. Thus, by maintaining a linear relationship between the
fundamental component V.sub.mcf of the modified effective voltage and the
magnitude of the command voltage V.sub.c, the linear relationship between
the peak command voltage V.sub.c and the peak fundamental phase voltage
V.sub.phf can be extended.
As well known, where two signals are in phase, the fundamental components
of the two signals add to produce a composite fundamental signal. Thus,
referring to FIG. 4(d), the fundamental component of the effective command
voltage can be determined by adding the fundamental components of the
clipped sine wave V.sub.cc and the square wave S. Thus:
V.sub.mcf =S.sub.f +V.sub.ccf. Eq. 1
Because the fundamental component V.sub.mcf of the effective command
voltage must linearly track the peak command voltage V.sub.c, the peak
command voltage V.sub.c can be substituted for the modified effective
voltage V.sub.mcf in Equation 1 so that:
V.sub.c =S.sub.f +V.sub.ccf Eq. 2
Given Equation 2, if both the fundamental component S.sub.f of the square
wave and the fundamental component V.sub.ccf of the clipped wave can be
expressed as functions of the magnitude S of the square wave, the square
wave required to maintain the linear relationship between the command
voltage V.sub.c and the resulting fundamental phase voltage can be
determined.
It is well known in the art that the fundamental component of a square wave
can be expressed as:
##EQU2##
where S is the magnitude of the square wave.
Deriving an S dependent function for the fundamental component V.sub.ccf of
the clipped control voltage is more complex. To derive the fundamental
component V.sub.ccf of the clipped voltage, we start with a gain function
for a PWM inverter in the typically non-linear region of operation (i.e.
where M.sub.i >1). As well known in the industry, the fundamental
component gain G.sub.v of a PWM inverter where the modulation index
M.sub.i is greater than one and the command voltage V.sub.c is purely
sinusoidal, can be expressed as:
##EQU3##
Assuming the carrier voltage V.sub.T has a peak magnitude of one:
m.sub.i V.sub.c Eq. 5
Normalizing Equation 4 for maximum DC voltage (V.sub.dc/ 2, combining
Equations 4 and 5, and solving for the fundamental component of the phase
voltage:
##EQU4##
Referring again to FIG. 4(d), when a square wave of magnitude S is added to
the command voltage, the portion of the clipped waveform V.sub.cc
effecting the fundamental component V.sub.phf of phase voltage is reduced
so that:
##EQU5##
In other words, Equation 7 represents the fundamental component V.sub.ccf
of the clipped sinusoidal command voltage which contributes to the
fundamental phase voltage V.sub.phf.
Combining Equations 2, 3, and 7 yields the Equation:
##EQU6##
Equation 8 includes only two variables, the command voltage V.sub.c and
the square wave magnitude S. The command voltage V.sub.c is known and
therefore, the square wave magnitude S required to maintain the linear
relationship between the command voltage V.sub.c and the fundamental
component of phase voltage can be determined.
For each value of the command voltage V.sub.c, Equation 8 provides a value
S indicating the magnitude of the square wave that is required to be added
to the command voltage V.sub.c to linearize the relationship between the
fundamental component of the phase voltage V.sub.phf and the command
voltage V.sub.c.
While a real time analog or digital implementation of Equation 8 would be
possible given an extremely complex and high speed controller, where a
controller is not capable of extremely high speed calculations, it would
be extremely difficult to implement real time control using Equation 8
without additional hardware. Where a real time implementation of Equation
8 is not possible, a more iterative approach to finding a square wave
magnitude S should be employed. Where a controller is digital and includes
sufficient memory, Equation 8 can be solved prior to PWM inverter
operation for various command voltages V.sub.c and the data can be stored
in a look-up table accessible by the controller during operation to
determine the magnitude S of the square wave required to maintain the
linear relationship desired.
Where the platform of implementation is analog, a look-up table would not
be possible. Similarly, where a digital controller and cannot compute the
magnitude S of the square wave in the required time, some other solution
must be found. In these cases, a simplified equation could be used instead
of Equation 8 for either the analog or digital implementations. One
simplified Equation that could be used is:
##EQU7##
where K.sub.s is a square wave constant. Experiments have shown that
linear tracking using Equation 9 matches tracking using Equation 8 quite
well.
Hardware Implementation
Referring now to FIG. 5, the present invention will be described in the
context of an exemplary 3 phase motor control system 14 including a PWM
controller 16 and a PWM inverter 18.
The PWM controller 16 includes a carrier wave generator 28 and a comparator
module 30. Referring also to FIG. 1(a), the carrier wave generator 28
produces the carrier voltage signal V.sub.T which is provided to the
comparator module 30 along line 32. In addition, the comparator module 30
receives three modified effective sinusoidal command signals V.sub.mc,
V.sub.mc ', V.sub.mc ". The comparator module 30 compares each of the
modified sinusoidal command signals V.sub.mc, V.sub.mc ', V.sub.mc " to
the carrier signal V.sub.T and produces three firing signals V.sub.f,
V.sub.f ', V.sub.f ". In FIGS. 1(a) through 1(c) only a single sinusoidal
command signal V.sub.c and signals related thereto are shown in order to
simplify this description.
Where the sinusoidal command signal V.sub.mc is greater than the carrier
signal V.sub.T, the comparator module 30 produces a corresponding firing
signal V.sub.f which is "high." Where a sinusoidal command signal V.sub.mc
is less than the carrier signal V.sub.T, the comparator module 30 produces
a corresponding firing signal V.sub.f which is "low." Thus, three
pulsating firing signals V.sub.f, V.sub.f ', V.sub.f " are produced that
vary in width according to the amplitude of an associated command voltage.
The firing signals V.sub.f, V.sub.f ', V.sub.f " are provided to the PWM
inverter 18 which in turn provides phase voltages V.sub.h, V.sub.h ',
V.sub.h " with their respective fundamental components V.sub.phf,
V.sub.phf ', V.sub.phf " to the motor 20.
Referring now to FIG. 6, while the inverter and controller described
operate to control three separate phases of a three-phase motor, only
operation of a single phase will be explained in detail. It should be
understood that component corresponding to the two phases which are not
described are duplicative in configuration and operation.
For each phase of the three-phases of the motor 20, the inverter 18
includes a pair of switches S1 and S2 (BJT, GTO, IGBT or other transistor
technology may be used). Each pair of switches includes an upper switch S1
and a lower switch S2 and each connects to positive or negative DC buses
44, 46 respectively. Each switch S1 and S2 is coupled with an inverse
parallel connected diode D1, D2 respectively. Such diodes and their
function are well known in the art. A separate one of the three phase
winding 26 is electrically connected between the switches S1 and S2.
The firing signal V.sub.f is provided to the upper switch S1. In addition,
the firing signal V.sub.f is inverted by inverter 40 producing inverted
firing signals V.sub.f. The inverted firing signal V.sub.f is provided to
the lower switch S2.
When the inverter switches S1 and S2 are controlled by the firing and
inverted firing signals, as an upper switch S1 opens, the corresponding
lower switch S2 closes. When an upper switch closes, a corresponding lower
switch opens.
A DC voltage source connects the positive and negative DC busses 44, 46.
For the purpose of this description, the DC source can be thought of as
consisting of both positive and negative series arranged DC sources 48, 50
respectively, that connect the positive and negative DC buses 44, 46. The
positive terminal of the positive source 48 is connected to the positive
DC bus 44 and its negative terminal is connected at a node n to the
positive terminal of the negative DC source 50. The negative terminal of
the negative DC source 50 is connected to the negative DC bus 46. Both DC
voltage sources 48, 50 produce potentials of identical magnitude but of
opposite signs with respect to central point n on the DC voltage source.
Referring to FIGS. 1(b), 1(c), and 6, when the firing signal V.sub.f is
received by the inverter 18, the signal V.sub.f is directed to the first
switch S1 whereas the corresponding inverted signal V.sub.f is directed to
the second switch S2. When firing signal V.sub.f is high and V.sub.f is
low, the first switch S1 is closed and the second switch S2 is opened. In
this state, stator winding 26 is connected through line 27 and the first
switch S1 to the positive DC bus 44. This produces a positive DC voltage
pulse 31 across stator winding 26. This positive pulse 31 has an amplitude
equal to the magnitude of the positive DC voltage source 48 (i.e.
+V.sub.dc/ 2/) and a width equal to the width of the firing signal
V.sub.f.
When the signal V.sub.f goes low, the inverted firing signal V.sub.f goes
high. During this time, firing signal V.sub.f opens the first switch S1
and the inverted firing signal V.sub.f closes the second switch S2. This
disconnects stator winding 26 from the positive DC bus 44 and shortly
thereafter connects the stator winding 26, through line 27 and the second
switch S2, to the negative DC bus 46. When so connected, a negative DC
pulse 32 is generated across winding 26 having an amplitude equal to the
magnitude of the negative DC voltage source 50 (i.e. -V.sub.dc/ 2/) and a
width equal to inverted firing signal V.sub.f.
By changing the widths of the positive DC pulses 31 with respect to the
widths of the negative DC pulses 32 rapidly over time, a changing
fundamental component phase voltage V.sub.phf which follows the command
voltage V.sub.c can be provided across the stator winding 26. This phase
voltage V.sub.phf gives rise to a phase current I.sub.ph which lags the
voltage by a phase angle .PHI..
Referring again to FIG. 5, a command voltage modifier 52 provides the
modified command voltages V.sub.mc, V.sub.mc ', V.sub.mc " to the
comparator module 30. The command voltage modifier 52 receives initial
command voltages V.sub.c, V.sub.c ', V.sub.c ", and, where those initial
voltages have magnitudes which exceed the peak value of the carrier signal
provided by the carrier wave generator 28, the command voltage modifier 52
modifies the initial command voltages V.sub.c, V.sub.c ', V.sub.c ", thus
providing the modified command voltages V.sub.mc, V.sub.mc ', V.sub.mc "
in order to maintain the linear relationship between the initial
sinusoidal command voltages V.sub.c, V.sub.c ', V.sub.c " and the
fundamental components of the phase voltages applied to the motor.
Again, to simplify this explanation, while the command voltage modifier 52
includes components which modify each of the initial command voltages
V.sub.c, V.sub.c ', V.sub.c ", components required to modify only initial
command voltage V.sub.c are explained here. It should be understood that
identically configured components are provided for each of the three
initial command voltages C.sub.c, V.sub.c ', V.sub.c " and each grouping
of components operates in the same manner.
Referring now to FIG. 7, the command voltage modifier 52 includes a
modulation index calculator 60, a square wave calculator 62, a square wave
generator 68, a clipping circuit 70, an adder 72, a comparator 74, and a
double pole switch 76. The modulation index calculator 60 receives both
the initial command voltage V.sub.c and the carrier signal V.sub.T and
divides the peak value V.sub.c of the initial command voltage by the peak
value of the carrier signal V.sub.T to produce the modulation index
M.sub.i.
The modulation index M.sub.i is provided to the square wave calculator 62
which determines the magnitude S of the square wave required to maintain
the linear relationship between the sinusoidal command voltage V.sub.c and
the fundamental component of the phase voltage V.sub.phf.
Depending on the platform of implementation, the square wave calculator 62
will generate the magnitude S by solving Equation 8 or some other suitable
equation for magnitudes. In the alternative, the calculator 62 will search
a look-up table to identify a desired magnitude S given the modulation
index M.sub.i. The magnitude S is provided to both the square wave
generator 68 and the clipper circuit 70. The square wave generator 68
generates a square wave which is in phase with the initial command voltage
V.sub.c and supplies the square wave to the adder 72. In addition to
receiving the magnitude signal S, the clipper circuit 70 receives the
initial command voltage V.sub.c and clips any portion of the initial
command voltage V.sub.c which is greater than the magnitude 1-S and less
than the magnitude S-1 producing a clipped command voltage V.sub.cc which
is also provided to the adder 72. The clipped command voltage V.sub.cc is
added to the square wave by adder 72 to produce a modified command voltage
V.sub.mc on line 78. The modified command voltage V.sub.mc is provided to
one pole of the double pole switch 76. The other pole of the double pole
switch 76 is connected by line 80 to the initial command voltage V.sub.c.
The modulation index calculator 60 also provides the modulation index
M.sub.i to the comparator 74 w | | |
