Download Current Control of VSI-PWM Inverters

562-
IEEE
TRANSACTIONS ON
INDUSTRY APPLICATIONS,
VOL. IA-21.
NO. 4,
MAY/WUNE 1985
Current Control of VSI-PWM Inverters
DAVID M. BROD,
MEMBER, IEEE, AND
DONALD W. NOVOTNY,
SENIOR MEMBER, IEEE
Abstract-The inherent limitations of commanding voltages and
currents in a three-phase load with an inverter are examined. An overview
of several current controllers described in the literature is presented, and
computer simulations are used to compare performance. A switching
diagram is developed which reveals some of the operating characteristics
of hysteresis controllers. For ramp comparison controllers, a frequency
transfer function analysis is used to predict the line currents and provide
some insight into the compensation required to reduce the current errors.
INTRODUCTION
C URRENT-CONTROLLED PWM inverters offer
substantial advantages in eliminating stator dynamics a in
high-performance ac drives and are widely applied in such
systems. A basic VSI-PWM system with current control is
shown in Fig. 1. Presently, current controllers can be
classified as hysteresis, ramp comparison, or predictive
controllers. Hysteresis controllers utilize some type of hysteresis in the comparison of the line currents to the current
references [1]-[4]. The ramp comparison controller compares
the current errors to a triangle waveform to generate the
inverter firing signals [5]. Predictive controllers calculate the
inverter voltages required to force the currents to follow the
current references [4], [6], [7].
This paper presents a general overview of the behavior and
inherent limitations of current controllers when commanding
currents in a three-phase load. Typical simulation results for
several current controllers are presented to illustrate important
performance characteristics. The hysteresis controller and the
ramp comparison controller are studied in greater depth
because of their simplicity and widespread use. A switching
diagram for a hysteresis controller is developed and utilized to
help explain the controller operation. For the ramp comparison controller, a frequency domain transfer function analysis
is presented, and its use in compensator design is illustrated.
GENERAL CURRENT CONTROLLER PROPERTIES
Before analyzing specific controllers, the general properties
of current controllers are examined. The concept of the
voltage (current) vector is utilized because it is a very
convenient representation of a set of three-phase voltages (or
currents). The voltage vector is defined by the following
Paper IPCSD 84-31, approved by the Industrial Drives Committee of the
IEEE Industry Applications Society for presentation at the 1984 Industrial
Applications Society Annual Meeting, Chicago, IL, April 3-6, 1984.
Manuscript released for publication August 9, 1984. This work was supported
in part by the Wisconsin Alumni Research Foundation and in part by the
Wisconsin Electric Machine and Power Electronics Consortium.
D. M. Brod is with the Borg-Warner Corporation, Wolf and Algonquin
Roads, Des Plaines, IL 60018.
D. W. Novotny is with the Department of Electrical and Computer
Engineering, University of Wisconsin, 1415 Johnson Drive, Madison, WI
53706.
b !H Cntroler
l(
6
Fig. 1. Basic system diagram of PWM current controller.
expression:
V
2
(Va+ dub+d2vc)
3
(1)
where
a = ej(2,x/3)
which defines a two-timensional vector (or complex number)
associated with the three-phase voltages. The actual voltages
can be recovered from v and the zero sequence component v0
using the equations
VaI=UI
cos 0+VO
Vb =
|V|I COS
0-
)+ O
Vc=
1171
0+
+)
Cos
(2)
where 0 is the angle between the voltage vector and the real
axis.
Fig. 2 shows the basic circuit of a three-phase voltage
source inverter. Notice that the dc bus midpoint is assumed to
be the ground reference. The inverter operates in one of eight
conduction modes to produce one of six nonzero voltage
vectors or a zero voltage vector as illustrated in Fig. 3. The
line-to-ground voltages: vag, Vbg, and vcg are uniquely specified
by the inverter with the line-to-neutral voltages equal to the
line-to-ground voltages if the neutral is connected to the dc bus
midpoint. Otherwise, the line-to-neutral voltages sum to zero,
and the inverter cannot apply a zero sequence voltage across
the load.
0093-9994/85/0500-0562$01.00 © 1985 IEEE
563
BROD AND NOVOTNY: CURRENT CONTROL OF VSI-PWM INVERTERS
Fig. 2. Power circuit configuration of VSI inverter.
Im
13
expected to experience interaction between the phases if the
load has no neutral connection.
Some current controllers may exacerbate this interaction
between the phases by adding offsets to the current errors. The
added offsets may cause unexpected and even incorrect lineto-neutral voltages to be produced. For example, if the current
in phase A is too low and the currents in phases B and C are
too high, then the controller probably should apply the voltage
vector v1, (A +, B -, C -), to reduce the current errors
quickly. If the current controller adds an offset to the phase A
current error, the controller might switch phase A low and
attempt to drive all three line currents lower by producing the
voltage vector v8, (A -, B-, C- ). Under this condition the
load is effectively allowed to coast, and the controller seems to
experience a lack of control during this time.
In this example, the controller attempted to command a zero
sequence current change. A current controller should not need
to command a zero sequence current change because the
current errors sum to zero if the three-phase current reference
sums to zero. Adding offsets to the line current errors may be
beneficial in reducing the inverter switching frequency.
However, if the offsets are added improperly, higher current
errors and a poorer dynamic response may result.
Effect of DC Voltage Limit
14
VI
Re
I
1
Fig. 3. Six
nonzero
v6
voltage vectors associated with VSI inverter.
For a current controller to operate properly, there must be
sufficient voltage to force the line currents in the desired
direction. For loads with low counter EMF the dc bus voltage
is not critical, but as the counter EMF is increased, a point is
reached where the current controller is not able to command
the desired current. This condition is reached when the line-toneutral voltages approach a six-step quasi-square wave. In the
following sections it is assumed that there is sufficient voltage
to command the line currents.
Inverter Switching Frequency
To determine the factors that influence the inverter switching frequency, let one phase of the load be described by the
following differential equation:
Effect of Unconnected Neutral
A current controller can exhibit an ambiguity when comv = Ri + Ldi/dt + e
(3)
manding the firing signals to an inverter that supplies a load
with an unconnected neutral. When an inverter leg switches
state, the resulting voltage vector is dependent on the state of where
the other two inverter legs. For example, if phase A switches
v line-to-neutral load voltage,
from high to low, the following inverter voltage vectors can
i line current,
result:
e counter EMF,
L leakage inductance.
tl(A+, B-, C-)-'i78(A-, B-, C-)
The time A t in which the line current will increase by A i can
be
found from (3) assuming that v and e do not change
92(A+, B+, C-)-4-(A-, B+, C-)
appreciately over the interval and that the resistance is
97(A+, B+, C+)--i4(A -, B+, C+)
negligible:
96(A +, B-, C+)-*i7(A-, B-, C+).
At = LAi/(v - e).
(4)
If the neutral is not connected, the individual line-to-neutral
voltages are not independent of each other, and the response of
each line current will depend not only on the state of the
corresponding inverter leg but also on the state of the other
two inverter legs. Therefore, a current controller can be
Equation (4) shows that the inverter switching frequency is
influenced by several factors: inductance and the counter EMF
of the load, dc bus voltage, and the current ripple. The
fundamental of the line-to-neutral voltage and the counter
EMF vary periodically. Therefore, either the inverter switch-
564
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS. VOL.
ing frequency or/and the current ripple will vary over a
fundamental inverter period.
4
DESCRIPTION OF CURRENT CONTROLLERS
Several of the current controllers described in the literature
are briefly reviewed to provide a basis for the subsequent
sections.
Hysteresis Controller: Three Indepenfent Controllers
One version of hysteresis control, described in [1], uses
three independent controllers, one for each phase. The control
for one inverter leg is shown in Fig. 4. When the line current
becomes greater (less) than the current reference by the
hysteresis band, the inverter leg is switched in the negative
(positive) direction, which provides an instantaneous current
limit if the neutral is connected to the dc bus midpoint.
Therefore, the hysteresis band specifies the maximum current
ripple assuming neither controller nor inverter delays. The
inverter switching frequency will vary over a fundamental
inverter period since the current ripple is specified by the
hysteresis band. In a system without a neutral connection, the
actual current error can reach double the hysteresis band
assuming the three-phase current reference sums to zero. A
more complete discussion of this phenomenon is given later.
IA-21. NO. 4, MAYiJUNE 1985
Fig. 4. Hysteresis control for one phase.
i
a
Fig. 5. Ramp comparison controller for one phase.
frequency of the triangle wave and produces well-define
harmonics [5]. Multiple crossings of the ramp by the current
error may become a problem when the time rate of change of
the current error becomes greater than that of the ramp.
It will be shown later that there is an inherent magnitude and
phase error in the line currents. The errors can be reduced by
increasing the controller gain or adding compensation. The
gain of the controller can be adjusted by either adjusting the
triangle amplitude or amplifying the current error.
Hysteresis Controller: Three Dependent Controls
In many applications the inverter switching frequency can
be reduced if a zero voltage vector is applied at the appropriate
time. Also, the maximum current error can be limited within
the hysteresis band, rather than twice the band, with proper
control.
Predictive Controller: Constant Switching Frequency
A controller has been suggested which incorporates three
The constant inverter switching frequency predictive condependent hysteresis controls and the intelligent application of
the zero voltage vector to accomplish these improvements. troller calculates an inverter voltage vector, once every sample
The controller is described in 141 and requires knowledge of period, that will force the current to track the current
the load counter EMF for the proper application of the zero command. The controller is shown in Fig. 6 and is described
voltage vector. A new inverter state is commanded when the in [41, [6]. The following is a description of the controller
current error in any line exceeds the hysteresis band providing operation. Let the load be represented by the differential
an instantaneous current limit. First, a zero voltage vector is equation in (3) which can be converted to a difference equation
applied if the counter EMF is in the appropriate direction to and solved for i:
reduce the current error in all three lines. If a line current error
i (T,,, + 1) =f(i (Tn), &(Tn), &(Tn))
(5)
exceeds the hysteresis band after a time delay, then new
inverter firing signals are obtained from comparators without where the inverter voltage and counter EMF are assumed to be
hysteresis. If the line current error is positive (negative), then constant over one sample period. A current command i*(Tn+ I)
the corresponding inverter leg is fired in the positive (nega- can be substituted for i(Tn + 1), and the voltage vector j(T,)that
tive) direction, reducing the current error as quickly as changes the current from i( T,) to the commanded value
possible.
i*(Tn+) can be written as follows:
Comparison Controller: Asynchronous
Sine-Triangle PWM with Current Feedback
A ramp comparison control for one inverter leg is shown in
Fig. 5. The controller can be thought of as producing
asynchronous sine-triangle PWM with the current error
considered to be the modulating function. The current error is
compared to a triangle waveform and if the current error is
greater (less) than the triangle waveform, then the inverter leg
is switched in the positive (negative) direction.
With sine-triangle PWM, the inverter switches at the
Ramp
6(Tn) = g(i*'(Tn + 1,)
i (Tn),
If the load is described by (3), then
R[i*(Tn± I) - i(Tn)e
I
-
e- TRIL
(6)
j(Tn))
TR L]
v(Tn) is given by
j(T)
(7)
where T is the sample period. The voltage vector u(T,) has to
be transformed into a weighted average of three inverter
voltage vectors. Each inverter voltage vector is applied for a
portion of the sample period with each inverter leg switched
565
BROD AND NOVOTNY: CURRENT CONTROL OF VSI-PWM INVERTERS
jxt
r,
e
Es L8
VS
aB(Lm+LmLtr1 Xi
E
Xt
e
Xm Xt
X ode+ fori
m
Fig. 6. Predictive controller: constant inverter switching frequency.
once every sample period, producing a constant inverter
switching frequency of 1/(27). For faster dynamic response,
the closest inverter voltage vector to v(Tn) can be applied for
the entire sample period.
The actual current lags the current reference by as much as
one sample period. Longer lags probably will be realized due
to calculation delays. The current ripple cannot be explicity
specified as with a hysteresis controller but the inverter
switching frequency is well-defined. The controller does not
provide an inherent instantaneous current limit.
Fig. 7. Transient model for induction motor.
TRANSIENT MODEL OF INDUCTION MOTOR
Ln
4-
9
TORQUE-O.N-m, VDC=300V, FREQ=3OHz, BAND=1. 5A
-
m
I
Ln
1.
14'
Predicitve Controller: Minimum Switching Frequency
9
0. ; 0
The minimum inverter switching frequency controller is
I
similar to the previous predicitve controller in that the proper
E-4
I
z
I
II
Iinverter voltage vector is found with knowledge of the load
1
8
and operating conditions. When the current error vector
magnitude exceeds a specified value, the controller predicts
9.
the current trajectory for each possible inverter state and
determines the length of time that the current error vector will
I
-1
-1
I
i
i
i
4I ti-n
remain within the specified value. The inverter switching
0-71
1.43
2.14
2.86
3.57
4.29
5.00
that
state
inverter
the
selecting
by
frequency is minimized
TIME (Secs ) * 10maximizes the following expression:
Fig. 8. Simulated current error waveform of typical 10-hp induction motor
itn
I
.
I
I
cn
I
tn
I
,
t(k)/nc(k)
(8)
Vv.
v. /
XL . 4.
TIME tSc
.
U0l
J..,
't. .,
I.. v1
operating at no-load and one-half rated speed with hysteresis controller
incorporating three independent controls.
where
predicted time before next inverter switching,
nc number of commutations required to reach the new
inverter state,
k new inverter state.
The controller is discussed in detail in [7]. This controller
provides an instantaneous current limit, although there will be
a calculation delay in determining the next inverter state. The
controller response might be slower than any of the hysteresis
controllers due to the necessary calculations, but this was not
investigated.
SIMULATION OF THE CURRENT CONTROLLERS
Simulations were carried out on a VAX 11/780 using the
Advanced Continuous Simulation Language (ACSL). Since
the inverter switching period is very short, the constant flux
linkage model for the transient behavior of the induction
machine is employed in the simulations. The machine is
assumed to be in a high-performance field-oriented drive
operating at constant speed over the period of the simulation.
The magnitude and phase of the counter EMF is calculated
assuming that the actual currents track the current references
t
within a reasonable tolerance. This assumption will create an
error for the ramp comparison controller which can produce
relatively large current errors if the controller gain is low.
The transient model for the induction motor is shown in Fig.
7 with the parameters for a typical 220-V three-phase 60-Hz
four-pole 10-hp induction motor listed in the Appendix. The
simulations only investigated the steady-state operation of the
current controllers using the transient model with an operating
frequency of 30 Hz and were run for one and one-half cycles
with the load currents initialized to zero. The simulations did
not incorporate controller or inverter delays and did not
account for variation of motor parameters, i.e., saturation.
More detail on the simulation technique and additional
simulation results can be found in [8].
Hysteresis Controller: Three Independent Controls
Simulation results for the hysteresis controller with three
independent controls illustrate the properties of the controller.
Fig. 8 shows the current error when the induction motor
operates at no-load and one-half rated speed with the hysteresis band set at 1.5 A. The average inverter switching frequency
IEEE TRANSAC7'C)\ O
566
-t W (- v -,, -CA ?
-.I
NOO>CI 'A!
U{. JCN IOT3.rR
1_
T1kJJ$4]r
I U`
)POULE=rE.N-M, JID=30:,J FRE>-)OHz, 3ANM-)
TRANSIENT MODEL OF INDUCTION MOTOR
:
c
T`RilRGLE FPEQ=2000Hz
TRIANGLE PEAKC i).
urrent
E
Reference
~Current
- IAREF
IA
j~~
s._|.s
40
-;
0
0
1.43
'0.0
0.83
1.67
TIME
2.50
(Secs)
3.33
4.17
*102
Fig. 9. Simulated waveforms of stalled typical 10-hp induction motor with
hysteresis controller incorporating three independent controls.
is approximately 2600 Hz. The figure shows that the maximum line current error can be double the hysteresis band.
A load consisting of a stalled induction motor was simulated
to observe the operating characteristics of the hysteresis
controller when the load counter EMF is zero. The line current
and current reference are shown in Fig. 9. The inverter has
periods of high switching frequency which are interrupted
occasionally when the inverter applies a zero voltage vector
across the load. The periods of high switching frequency are
referred to as limit cycles in this paper. The switching
frequency during the limit cycles is approximately 6600 Hz.
Hysteresis Controller: Three Dependent Controls
Simulation results for the hysteresis controller with three
dependent controls and the programmed application of the
zero voltage vector indicate that the magnitude and phase error
in the line currents are small. Also, the current errors remain
within the hysteresis band. The average inverter switching
frequency is approximately 5350 Hz when the induction motor
operates at no-load and one-half rated speed with a hysteresis
band of 1.5 A. The switching frequency is much higher than
for the previous hysteresis controller. There are two reasons
for this: 1) the maximum current error is smaller, and 2)
when any one hysteresis band is exceeded, more than one
inverter leg may switch.
A load consisting of a stalled induction motor was also
simulated. The simulation showed that the limit cycles that
occur with the previous controller are eliminated, and the
average inverter switching frequency is much lower (approximately 550 Hz) than that of the hysteresis controller with three
independent controls.
Ramp Comparison Controller
Fig. 10 shows the line current, current reference, and
counter EMF that result with the ramp comparison controller
2.14
2.86
(Secs)
TIME
5.0
*1Q2
Fig. 10. Simulated waveforms of typical 10-hp induction motor operating at
no-load and one-half rated speed with ramp comparison controller.
MOTOR
TORQUE-O.N-m, VDC=300V, FREQ=30Hz
INVERTER SWITCHING FREQ=2000Hz
TRANSIENT MODEL OF INDUCTION
0
40
D
_
u)
cn
vi
--
w;
_
r0
40
D
0
l4
0.0
_ I
0.71
I-
1.43
I*
*-
2.14
TIME
2.86
(Secs)
3.57
*102
Fig. 11. Simulated waveforms of typical 10-hp induction motor operating at
no-load and one-half rated speed with constant inverter switching
frequency predictive controller.
without any additional compensation. Some hysteresis was
added to the controller to prevent multiple crossings of the
triangle ramp. There is a magnitude and phase error in the line
current which results in a sinusoidal current error. The
inverter switching frequency is approximately equal to the
triangle frequency of 2000 Hz. The current ripple amplitude is
reduced when the triangle frequency is increased.
Predictive Controllers
The predictive controllers performed as expected with a
typical simulation for the constant inverter switching frequency controller shown in Fig. 11. Additional simulation
results are given in [8].
567
BROD AND NOVOTNY: CURRENT CONTROL OF VSI-PWM INVERTERS
HYSTERESIS CONTROLLERS-THE SWITCHING
DIAGRAM
A switching diagram can be used to explain some of the
characteristics of hysteresis controllers. The diagram indicates
when and how a hysteresis current controller switches the
inverter given the current references and the load currents.
The derivation of the switching diagram for the hysteresis
controller with three independent controls is presented as
follows. Referring to Fig. 12, the current reference vector, the
actual current vector, and the current error vector along with
the A, B, and C axes of a three-phase set of coordinates are
drawn in the complex plane. The line current errors Aia, Aib,
and A i are the projections of the current error vector A i on
the corresponding A, B, and C axes. The hysteresis controller
switches the A inverter leg when Ai, exceeds the hysteresis
band as represented in Fig. 13 by two switching lines drawn
perpendicular to the A axis. The switching lines are located
from the current reference vector by a distance equal to the
hysteresis band. Similarly, the switching lints for phases B
and C can be drawn. Fig. 14 shows the switching diagram that
results when the switching lines for each phase are combined.
The switching diagram will move with the current reference
vector since the current reference vector locates the center of
the switching diagram in the complex plane. A somewhat
similar development is contained in [9].
The switching diagram confirms that the maximum line
current can be double the hysteresis band, 2h, and the
maximum spatial current error (magnitude of the current error
vector) is also double the hysteresis band. Fig. 15 shows a
current trajectory which results in the maximum error in a line
current. This trajectory occurs when the initial voltage vector
vI, (A +, B-, C- ), forces the line current vector to hit the
-A switching line which results in the zero voltage vector v8,
(A-, B-, C-). The line current error in phase A can
increase further because of the load resistance, load counter
EMF, or movement of the switching diagram due to variation
of the current references. The voltage vector will not change
until the actual current vector crosses another switching line.
The maximum current error occurs if the actual current vector
hits one of the outside corners of the switching diagram.
The switching diagram can also be used to show that limit
cycles, which are interrupted occasionally, can occur when the
load counter EMF is low. Fig. 16 shows a current trajectory,
indicated by the solid line, that may occur during a limit cycle.
The initial voltage vector VI, (A +, B-, C-), forces the
current vector to travel in the same direction as the voltage
vector since the counter EMF and resistance are assumed to be
zero. The current vector hits the + C switching line, causing
inverter leg C to switch and produce the inverter voltage
vector v2, (A +, B -, C+ ). Next, the current vector will hit
the -A switching line producing the voltage vector V3, (A -,
B-, C+ ). Continuing with the same line of reasoning, the six
nonzero voltage vectors are applied repeatedly, and a high
switching frequency results if there is a low leakage inductance and a small hysteresis band. Notice that the magnitude of
the current error vector is not reduced to zero during the limit
cycle. The dashed line in Fig. 16 represents a current
trajectory when there is a nonzero counter EMF.
Im
b
IAT
a
Re
Fig. 12. Current vectors in complex plane.
tm
Re
Fig. 13. Switching lines for phase A.
Im
Fig. 14. Switching diagram for hysteresis controller with three independent
controls located in complex plane.
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. IA-21 NO. 4, MAY/JUNE 1Q"
568
Fig. 15. Current trajectory which results in maximum line current error.
-B
IA
+C
va E_I_, _ _ _g
-A
+3,
Fig. 16. Current trajectory for two limit cycles. Solid line: zero load counter
EMF. Dashed line: nonzero counter EMF.
The frequency of the limit cycle can be found by dividing
the velocity of the current trajectory by the distance traveled in
one complete inverter switching period. The velocity is given
by (for zero counter EMF):
2
di 3 Vdc
(9)
vel=-=dt L
and the distance traveled by a limit cycle is approximately
-
=
(10)
d= 6h.
Therefore, the inverter switching frequency can be written as
(11)
vel Vdc
Sd 9hL
consider the limit cycle in the previous example. A zero
voltage vector occurs when one of the switching lines in the
sequence is skipped due to the load counter EMF, load
resistance, or the movement of the switching diagram. If the
switching diagram moves, the voltage vector VI, (A +, B -.
C - ), in Fig. 16 may cause the current vector to cross the - A
switching line instead of the + C switching line which results
in the zero voltage vector v8, (A -, B-, C- ). The
application of a zero voltage vector will significantly reduce
the inverter switching frequency when the counter EMF is low
since the velocity along a trajectory with a zero voltage vector
is much lower (zero if the counter EMF is zero) than with a
nonzero voltage vector.
RAMP-COMPARISON CONTROLLERS:
FREQUENCY-DOMAIN ANALYSIS
The following analysis assumes that the ramp comparison
controller produces asynchronous sine-triangle pulsewidth
modulation. Sine-triangle PWM produces fundamental line-toneutral voltages which are proportional to the ratio of the sine
wave peak and triangle peak. The block diagram of the
frequency domain model for the ramp comparison controller is
shown in Fig. 17. The line current f can be found from the
following quantities:
I* current reference,
E load counter EMF,
Z load impedance,
K system gain.
The system gain is given by the following expression [5], [10]:
K=KsG
where
=-V
2At
300
=s 9(1.5)(0.00336) =6614
HIz
(12)
which is close to the value estimated from the simulation. The
highest inverter switching frequency occurs during the limit
cycles when the counter EMF is zero (since the counter EMF
tends to reduce the switching frequency). Therefore, the
inverter has to be designed to handle the switching frequency
that occurs with zero counter EMF. This is an important
limitation of this type of controller.
The limit cycle may be occasionally interrupted by the
intermittent occurrence of a zero voltage vector. For example,
(14)
A, triangle peak,
G additional gain and/or compensation.
Equation (11) can be used to estimate the inverter switching The line-to-neutral voltage phasor V is given by
frequency of the limit cycles observed in the simulation in Fig.
V=(I* - f),
9:
(13)
(15)
and the line-to-neutral voltage and load current are related as
follows:
P.
V=ILZ+E
(16)
Equations (15) and (16) can be equated and written as
(17)
K(I* - 1) = IZ +E.
Equation (17) can be rearranged to give an expression for the
load current
I KI*-E
(18)
569
BROD AND NOVOTNY: CURRENT CONTROL OF VSI-PWM INVERTERS
t*
I +
Fig. 17. Frequency domain transfer function model for ramp comparison
controller.
which shows that the counter EMF can have a significant
effect on the line currents and current error especially at
higher motor speeds. The magnitude and phase errors of the
line currents are reduced by increasing the controller gain or
including some type of compensation.
This frequency-domain analysis is substantiated by the
previous simulations. For example, substituting the following
information, obtained from Figs. 7 and 10,
k= 15z00
-= 76.6 z900
* = 12.4 0°
V
1+
(I9)
This expression can be useful in evaluating system response at
specific operating points (i.e., evaluating the effect of load or
changes in machine parameters).
As an illustration of the use of (18) for compensation
design, consider a proportional-integral compensator to improve the low-frequency response. The transfer function of a
proportional-integral controller is
G=Kc
I
+j-
-
(20)
where Tc is the time constant of the compensation network.
Using the transient model of the induction motor and
substituting (20) into (18), the following expression for the
current results:
KsKc(I + jwTC)] P*_I
jw(L+ ')K
+
rS
sK O(+jCOTC)
'21'
(22)
JX
A
into (18) results in a line current of 13.2z - 24.8° A which
corresponds to that shown in Fig. 10.
The analysis can be extended to incorporate the steady-state
equivalent circuit for the induction motor. This is useful since
the counter EMF is not normally known explicity. The
equivalent circuit can be reduced to an equivalent impedance
Zq which provides the following expression for the line
current
K+ Zeq
TC = Ts' = L,/r
which results in the following expression:
I
Z=0.66z72.9o
KI*
Using the concept of cancellation compensation, the time
equal to the motor
constant of the controller TC can be set
stator transient time constant
JrS
KsKc
.,
,jKsKc
E.
(23)
(1 + jws ')( 1 + KgK
The compensation gain KC would be set to as large a value as
permitted by the parasitic poles (and/or the delays associated
with the inverter switching and sampling effects).
For systems in which the closed loop phase error is a major
concern (i.e., field orientation), the best choice of a compensator will depend on the speed range of the system. If small
phase errors at high speed are required, the compensator
should be chosen to minimize the phase error near the cutoff
frequency. This design problem was not considered in this
project.
SUMMARY
Of the controllers studied, the hysteresis controller with
three independent controls is the simplest to implement. The
predictive controllers are the most complex and require
knowledge of the load and extensive hardware which may
limit the dynamic response of the controller. The ramp
comparison controller has the advantage of limiting the
maximum inverter switching frequency and producing welldefined harmonics, but the controller requires a large gain and
compensation to reduce the current error and generally has
lower bandwidth than hysteresis controllers.
The hysteresis controller with three independent controls
works very well except the inverter switching frequency is
higher than required when there is low counter EMF due to
limit cycles. The switching frequency can be reduced by
introducing zero voltages at the appropriate times (when the
counter EMF is low). A combination of a ramp comparison
controller for low-speed operation and a simple hysteresis
controller for high-speed operation may provide a good overall
solution.
570
IEBEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL, IA-21, NO. 4, MAY/JUNE 1985
APPENDIX
TYPICAL 10-hp INDUCTION MOTOR PARAMETERS
220 V, FOUR POLE, 60 Hz
r, Stator resistance, 0.195 Q.
rr Rotor resistance, 0.195 Q.
x15 Stator leakage reactance, 0.649 O.
x,, Rotor leakage reactance, 0.649 O.
xm Magnetizing reactance, 12.98 U.
REFERENCES
David M. Brod (S'79-M'83) received the B.S.
degree from Northwestern University, Evanston,
IL, and the M.S. degree from the University of
Wisconsin, Madison, in 1982 and 1984, respectively, both in electrical engineering.
From 1979 to 1982, he was a co-op student at the
Borg-Warner Research Center, Des Plaines, IL. He
is currently a Research Engineer at the BorgWarner Research Center. His interests include
electric machines, variable frequency drives, and
power electronics.
Mr. Brod is a member of Eta Kappa Nu and Tau Beta Pi.
t1] A. B. Plunkett, "A current-controUed PWM transistor inverter drive,"
in Conf. Rec. 1979 14th Annu. Meet. IEEE Ind. Appl. Soc., pp.
785-792.
[21 S. C. Peak and A. B. Plunkett, "Transistorized PWM inverterinduction motor drive system," in Conf, Rec. 1982 17th Annu. Meet.
IEEE Ind. Appl. Soc., pp. 892-898.
[3] W. McMurray, "Modulation of the chopping frequency in dc choppers
and PWM inverters having current-hysteresis controllers," in Conf.
Rec. 1983 IEEE PESC, pp. 295-299.
[41 G. Pfaff, A. Weschta, and A. Wick, "Design and experimental results
of a brushless ac servo-drive," in Conf. Rec. 1982 17th Annu. Meet.
IEEE Ind. Appi. Soc., pp. 692-697.
t5] A. Schonung and H. Stemmler, "Static frequency changers with
'subharmonic' control in conjunction with reversible variable-speed
a.c. drives," Brown Boveri Rev., pp. 555-577, Aug./Sept. 1964.
[6] I. Takahashi, "A flywheel energy storage system having harmonic
power compensation," Univ. of Wisconsin, Madison, WEMPEC Res.
Rep. 82-3, June 1982.
[7] J. Holtz and S. Stadtfeld, "A predictive controller for the stator current
vector of ac machines fed from a switched voltage source," in Conf.
Rec. 1983 Annu. Meet. Int. Power Electronics Conf., pp. 16651675.
[8] D. Brod, "Current control in VSI-PWM inverters," M.S. thesis, Univ.
of Wisconsin, Madison, 1984.
[9] G. Pfaff and A. Wick, "Direct current control of ac drives with pulsed
frequency converters," Process Automat., vol. 2, pp. 83-88, 1983.
[10] P. Wood, Switching Power Converters. New York: Van Nostrand
Reinhold, 1981, ch. 4, pp. 152-153.
Donald W. Novotny (M'62-SM'77) received the
B.S. and M.S. degrees in electrical engineering
from the Illinois Institute of Technology, Chicago,
and the Ph.D. degree from the University of
Wisconsin, Madison, in 1956, 1957, and 1961
respectively.
Since 1961 he has been a member of the faculty at
the University of Wisconsin-Madison where he is
currently Professor and Director of the Wisconsin
Electric Machines and Power Electronics Consortium (WEMPEC). He served as Chairman of the
Electrical and Computer Engineering Department from 1976 to 1980 and as
an Associate Director of the University-Industry Research Program from 1972
to 1974 and from 1980 to the present. He has been active as a consultant to
many organizations including Marathon Electric Company, Borg Warner
Corporation, Barber Coleman Company, Otis Elevator Corporation, Allen
Bradley Company, Eaton Corporation, and the Wisconsin Department of
Natural Resources. He has also been a Visiting Professor at Montana State
University and the Technical University of Eindhoven, Eindhoven, Netherlands, and a Fulbright Lecturer at the University of Ghent, Ghent, Belgium.
His teaching and research interests include electric machines, variable
frequency drive systems and power electronic control of industrial systems.
Dr. Novotny is a member of ASEE, Sigma Xi, Eta Kappa Nu, and Tau Beta
Pi, and he is a Registered Professional Engineer in the State of Wisconsin.