Two Flux Weakening Schemes for Surface-Mounted Permanent-Magnet
Synchronous Drives - Design and Transient Response Considerations
Dragan S. Marit*, Silva Hiti"', Constantin C. S t a n d * ,James M. Nagashima"*,and David B. Rutledge'
* California Institute of Technology,
Dept. of Electrical Engineering, M/S 136-93,
Pasadena, CA 91 125
Dragan@caltech.edu
** General Motors, ATV - Torrance,
3050 W. Lomita Blvd., PO Box 2923,
Torrance, CA 90509-2923
HitiS@pcssmtp.hac.com
- Two recently published flux weakening schemes
for surface-mounted permanent-magnet synchronous
(SMPMS) drives have offered estimable advantages over
previous solutions. The first scheme uses values of the voltage
reference to generate an adequate flux weakening component
of the stator current, whereas the second method detects an
increase of the tracking error in torque-producing current
component and uses the error signal for the flux weakening
control. The viability of the algorithms has been
demonstrated and the schemes have been compared and
analyzed in steady state operation. The transient behavior is
very important for electric vehicles and other highperformance applications. Here, we analyze the transient
response characteristics of both schemes when a sudden
change in the torque reference is received. The paper also
describes an approach to practical regulator design and
parameter selection for the flux weakening control section.
suggested in [I] to sudden torque changes is sluggish and
can even result in instability due to the loss of the current
control at high-torque operating points. The error detection
algorithm has much faster transient response to sudden
torque changes, but results in a constantly present steady
state error in the torque regulation.
In [l], the schemes have been compared and analyzed
in steady state operation. Here, the flux weakening
schemes described in [ l ] are compared and evaluated
during torque transients in the flux weakening region.
The transient behavior to sudden torque changes is
very important for high-performance applications such as
electric propulsion. The procedure for selection of the
parameters in the flux weakening section has not been
addressed thus far. This paper provides guidelines for the
regulator design and selection of the parameters in the flux
weakening for improved steady state and transient
characteristics of the drive system. Both schemes use space
vector modulation (SVM). If the over-modulation and sixstep PWM inverter modes are utilized, [3], the
performance degradation was observed due to the high
harmonic content in the stator currents. The problem was
thoroughly addressed and a remedy suggested in [4].
As a result of this work, we show that the voltage
reference scheme ought to be used only if a rapid transient
torque response is not required during the flux weakening
operation. Otherwise, the faster algorithm of the second
scheme should be employed. An approach to the practical
design of the flux weakening control sections is described
and the avenues for possible application of other results of
the control theory are opened. The simulation results are
provided to illustrate the transient characteristics of the
schemes.
Abstract
1. INTRODUCTION
Due to their distinct characteristics, but also because of
improvements in and reduced cost of permanent magnet
(PM) technologies, PM machines have been used in many
electrical drive applications. Some applications, like
electrical propulsion, require a wide operating range above
the motor base speed, i.e., a wide range of flux weakening
operation.
The straightforward approach to flux weakening
operation is to calculate the magnetizing current reference
from the surface-mounted permanent magnet synchronous
(SMPMS) machine equations, assuming that all machine
parameters are known. Limits for the magnetizing and
torque current in the flux weakening region are calculated
with a presumption that a SMPMS drive operates in the
voltage or in the voltage and current limits. However, the
method is very sensitive to uncertainties related to the
system parameters. Recently published flux weakening
schemes, [I], [2], [7], offered important advantages. The
schemes in [ l ] and [2] do not use any machine parameters
for calculations in the flux weakening region and DC bus
voltage measurement is not necessary for the flux control.
The scheme described in [l] uses closed-loop control of
the phase voltage magnitude to generate a magnetizing
current reference for the flux weakening operation. The
approach described in [2] detects the steady-state error in
the torque current and, then, uses the error to generate the
magnetizing current reference. This principle was modified
and further developed in [ 11 for use with current control in
the synchronous reference frame. The voltage reference
control method proved to be robust, without steady state
error present in the second scheme, but computationally
more complex. Also, the response of the first algorithm
0-7803-5662-4199/$10.00 01999 IEEE
11. SMPMS MODELING AND FLUX CONTROL
As a review and to clarify the notation used
henceforward, the SMPMS machine equations in the
synchronous reference frame, [5], are given:
di
= R i + L - - dw L i
v
d
673
d
dt
e 4
(1)
ISIE'99 - Bled, Slovenia
where R, L are the stator resistance and inductance, we is
the electrical speed of magnetic field, P i s the number of
poles, ;1,is the flux linkage of permanent magnets, id is
the flux generating component of the stator current, and iq
is the torque generating component of the stator current.
Maximal allowable motor phase voltage and phase
current are determined by the inverter and machine ratings
and by the available DC link voltage vdc. The following
inequalities must be satisfied at any time:
2
' d
2
id
+
2
' q "ma,
+ i,2
_<
2
2
I,,,
.(4-a)
Fig. 1: SMPMS voltage and current limit circles.
(4-b)
In (4-a,b), V,, is the maximal available phase voltage
amplitude at the fundamental frequency and I,, is the
maximal phase current.
Typically, SMPMS control strategy at low speeds is
chosen to maximize the torque per ampere ratio, i.e., id is
set to zero if back EMF is sufficiently smaller than V,,
(U,&<< V,,).
Operation above the base speed,
Vm,//zm, is enabled by allowing negative id current to
flow and decrease the total flux in the machine air gap. It
should be noted that, above the base speed, the drive
always operates in the voltage limit. However, maximal
torque in the flux weakening region is obtained when
SMPMS drive operates in the voltage and current limits. In
Fig. 1, this is a trajectory from point A to B when the
speed increases from ul to u 2 . If the effect of the stator
resistance is neglected, RzO in equations (1) and (2), and
if the equality sign is used in (4-a,b), the mrfximal torque
achievable and the corresponding reference i d are obtained
as in [ 13 and [7]:
.
v
(9)
where T" is the value of the torque reference.
111. DESCRIPTION OF FLUX WEAKENING CONTROL
SCHEMES
Block-diagrams for the two flux weakening schemes
are shown in Figs. 2 and 3. In both schemes, two antiwindup proportional-integral (PI) controllers, implemented
in the synchronous reference frame, are used for current
control, [l]. A space vector modulator (SVM), [3], is
employed to generate IGBT gate signals. Transition into
the full six-step operating mode during the flux weakening
is enabled.
The first flux weakening method was described in [11.
It uses closed loop co~trolof the phase voltage magnitude
Vphuseto generate i d , Fig.2. Flux weakening mode is
entered when VPh,, approaches V,,, where v,,=o.577vdc
for sine wave mode and V,,,,=0.637Vdc for full six-step
operation.
Transformed phase voltages in the synchronous
reference frame, vd and vq, can be written as follows:
iy
i, = J I : ~- ~
where dd and dq are the outputs of current
Further, we have:
Solutions for id. and iq*for torque smaller than T,,,
also be obtained from (1) - (4):
ii = T * / K t o r q u e
, can
(8)
674
d
2
2
2
= d d +d,.
_
. .,
..... Flux weakening
...........
i Processor
Fig. 2. Flux weakening control scheme with closed loop voltage control.
Fig. 3. Flux weakening control scheme with current error detection.
It follows from (12) and (13), that the boundary of the
sine wave region is reached when d=0.867, whereas the
six-step mode is reached at d=O.956. The d2 value is used
as a feedback variable in the flux weakering loop, and
compared to the onset modulation index &-,[I]. The error
d'-d,,,' is regulated by an anti-windup PI controller, whose
output is limited (to avoid irreversible demagnetization)
between 0 and minimal allowable id*= -&,,,U .
675
To achieve a rapid transient response during the step
changes in the torque command, the scheme given in Fig.
3 is used, [I]. The error signal for the q axis current is
compared with the predetermined threshold value. When
the error increases above the threshold value, an
appropriate id* is generated. id* is proportional to the
filtered absolute value of the q axis current error,
and it is limited between 0 and &, . The sum of squares,
\ir\,
I>
*')
i d - f i , -, is compared to the maximal squared current
value, z : ~ from (4-b). If the current limit constraint is
satisfied, then i,' and id' are unchanged and used as
references for t p current regulators. Otherwise, (6) is used
to calculate i, . The low-pass filter is used to prevent
unnecessary current injection in the d axis due to highfrequency noise.
IV. ANALYSIS AND DESIGN SUGGESTIONS
It is known, [ 5 ] , that feed-fonvard compensation of
the cross-coupling terms in (1) and (2) is necessary for
good current regulation. The compensation is presented in
Fig. 4 for the case of the voltage control scheme, [l].
Analogously, it is implemented in the second algorithm
too. The compensation also offers an opportunity to design
controller parameters using results from the linear systems
control theory and to accomplish excellent overall system
performance.
The design task for the voltage control flux weakening
scheme requires that the proportional and integral gains of
two PI current controllers, as well as the gains for the flux
weakening section, are determined. When feed-forward
terms are used, the voltage equations of the SMPMS
machine reduce to (14) and (15).
di
vd = R i d + L d
dt
In the Laplace domain, these equations can be transformed
into the following form:
llmlter
where, Ks=2Vdc/3KA.Now, any method from control
systems theory can be applied to design the current
control. However, a relatively simple but effective method
is used here. The parameters of PI controllers are selected
using compensation by left-half-plane (stable) pole-zero
cancellation. The block-diagram of the current control is
shown in Fig. 5.
The compensation implies that, ideally, the following
relationship is satisfied:
After the compensation, the closed-loop current transfer
function becomes, according to Fig. 5 and using equations
(17) and(18):
(14)
di
v, = Ri, iL4
dt
;1
If the gain of the current sensors is KA and if we use
equations (10) and (1 l), we can write this as:
where wc=KpKs/Lis the closed-loop bandwidth. Now, we
first choose the desired bandwidth. This sets forth the
value for Kp. Then, Kiis calculated from (18). In practice,
the compensation is achieved only approximately, but it
still results in very good performance, when the
aforementioned feed-forward terms are employed.
We notice that the controller gains are simple
functions of v d c . As a result, the current controI can be
further improved by including, if necessary, a simple
algorithm for the adaptation of the PI coefficients when the
DC bus voltage varies.
Design of the PI controller in the flux weakening
section is more convoluted because of an additional nonlinearity that comes from the modulation index function,
(13). We are interested in designing the parameters in such
PI
Fig. 4. The feed-forwardcompensationof the cross-couplingterms in
equations (1) and (2), the voltage scheme.
Fig. 5. Equivalentblock-diagramfor decoupled current control using
proportional-integrallaw.
676
a way that the transfer function from the iq* to the output of
the flux weakening section, id*, provides an adequate
response at elevated speeds. From Fig. 5 , we can derive
that the transfer hnction between the current command
input and the duty cycle in the corresponding axis is:
Fig. 6 Simplified block-diagram for the design of the flux weakening PI
controller.
We can now reduce the block-diagram shown in Fig. 4 to
the structure given in Fig. 6. Bearing in mind that in the
flux weakening operation the square of the reference duty
cycle stays close to d,', the d2 function can be linearized
around the d,,, by retaining only the fist order terms in the
Taylor expansion for d'=f(dd,dJ. The linearization is
performed around the midpoint of the possible range for dd
and d, , when:
The feed-forward decoupling method is used in the
second scheme too. By analogy, the current regulation
satisfies the equations (14) - (17). The design follows
equations (18) and (19). Then, the iq sensitivity transfer
function is derived from (19) and Fig. 5 :
1+-
S
Therefore, the trayfer function for filtered
used to generate id in Fig. 3, is now:
iF,which is
S
Now, if we use Ko=(ddol,we obtain:
S
1+-
GFwo
(s)= K o K , K ,
. -a
S
1+
@M
Kfi/KP!
S
S
1+-
(25)
O C
If we apply the compensation again, so that K,dKpfw=wc,
and if we use KFW'KdCMKifi, the equation (25) becomes:
S
1+G F w o = KFW
O M
S
The next step is to choose KFw,i.e., K+v so that the
flux weakening response is optimized. It is important to
see that an exessively high increase o f K F w is restricted by
presence of the limiter, Figs. 2 and 4. Once K,& is
determined, the proportional gain is K&= &/uC.
Having in mind the final value theorem, [6], we see
from (28) that the error would exist only during transients
and would go to zero as the steady state is approached.
However, the threshold condition, Fig. 3, results in a
steady state error in q axis current. The error is necessary
to provide i; during flux weakening and it is minimized by
increasing K , On the other hand, the K , constant cannot
be too high because of possible oscillations related to the
presence of the Bmiter, Fig. 3.
The calculation of i; begins in the flux weakening
section of Fig. 3 quickly after the torque command is
received, (28). Conversely, the voltage control scheme is
expected to provide a more sluggish response, whose
characteristics are dependent upon the additional
processing in the closed loop regulation of .;i
Consequently, although the first scheme provides better
steady state, [ 11, much better transient behavior is expected
from the second algorithm. The analysis presented here
was used to choose the starting values for the control
parameters in the simulation models.
V. SIMULATION RESULTS
The simulations were run at the constant speed of 0.7
p.u, i.e., deep in the flux weakening region. The torque
current reference changed from 0 to 0.2 p.u, with a slew
677
rate of 50 p d s e c . The transition into six-step was
disabled, i.e., d,,, set to 0.867, in order to observe i, and id
responses in a clear manner. Namely, in the overmodulation and six-step modes, these current components
contain the sixth harmonic, as discussed in [4].
The time domain response (P.u.) for the voltage
control scheme is shown in Fig. 7. A certain delay in id
response can be observed and an overshoot in i, is
noticeable. An attempt to decrease the i, overshoot (by
adjusting the control parameters results) in a larger id
delay, i.e., a trade-off between these two characteristics
must be made in a real drive system. We notice that the
settling time is relatively long compared to the time
duration of the reference transient and it is close to
approximately five times the rise time. This response is
also illustrated in Fig. 8, where iq versus id trajectory is
presented in the d-q plane. There, we can clearly see that
the iq overshoot is almost 20%. This imposes some
qualitative limitations in electrical drives where this
scheme is utilized and when a rapid transient response of
the torque control loop is demanded. A similar simulation
test was repeated for the second technique as well. The
transient problems associated with the id delay and the
overshoot in iq are not present, Figs.9 and 10. The
corresponding settling time is much shorter than in Fig. 7.
*
0.a
o.ai
o.ao5
0.3‘5
0.11
Time (sec)
0.315
o.3a
Fig. 7. iq and id transient responses, the voltage control scheme.
0.25-
0.2 .
0.15.
’,
0.1
.
0.05-
-0.3
-0.28
-0.26
jd
-0.24
-0.22
-0.2
Fig. 10. Transient i,, versus id trajectory, the error detection scheme.
The simulation results confirmed that the second
scheme offers advantageous transient performance.
However, the steady state error of the second scheme can
be observed in Figs. 9 and 10.
VII. CONCLUSION
Two flux weakening schemes for SMPMS machine
drives are discussed in terms of their transient response
characteristics. Guidelines for the selection of the control
parameters are suggested.
Using the theoretical and simulation results presented
here and experimental results given in [l], we conclude
that if a fast transient response is not required, the voltage
reference control scheme should be employed. It comes
with high efficiency because id automatically adapts to the
exact value needed during the flux weakening operation.
There is no steady state error in the q axis. On the other
hand, the second scheme offers a faster transient response
and the steady state error is not critical for applications
where the current control loop is an inner loop, e.g., in
velocity control. The optimum efficiency of the voltage
scheme is sacrificed in the error scheme and has to be
addressed separately by considering some additional
optimization routines as a part of the control software.
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] ] : O
n.
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-0.3
-0.28
.0.26
id
-0.22
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Fig. 8. Transient iq versus id trajectory, the voltage control scheme.
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