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2010 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2010), October 3-5, 2010, Penang, Malaysia
Study into Effects of Stator Resistance Variation
on Direct Torque Controlled Surface Permanent
Magnet Synchronous Motor Drive
Ali Jafarian Abianeh, Hew Wooi Ping
Department of Electrical Engineering, University of Malaya, Kuala Lumpur, Malaysia
E-mail: [email protected]
Abstract— in the recent past decade, direct torque
controlled (DTC) permanent magnet synchronous motor
drives have been widely utilized in many relevant
applications, as they offer numerous advantages compared
to other control schemes. In fact, for the stator flux
estimation of such scheme the motor parameter dependency
is much less than the previous control schemes and it is just
concerned with stator resistance. In practice, the stator
resistance is prone to large variations caused by
temperature which indeed are very likely to occur. The
mismatch between actual and estimated resistance could
deteriorate the performance of the DTC drive by
introducing errors in the stator flux linkage estimation
especially at lower speeds. This paper analytically
investigates different effects of stator resistance variations
on the performance of the direct torque controlled SPMSM
drives while the simulation results are also provided.
In order to reduce the level of torque and flux ripples,
some researchers have proposed using multilevel inverters
[5], and some others have focused on SVM based inverter
switching technique [6]. In this regard, Intelligent based
SVM-DTC schemes [7] and adaptive based SVM-DTCs
[8], [9] have shown a superior performance as they have
the capability of dealing with uncertainties and
nonlinearities present in motor dynamics.
The rest of DTC problems could be referred as
estimation related issues. In fact, in a DTC scheme the
stator flux linkage is estimated through integration of the
difference between input voltage and resistive voltage
drop as given by (1).
Keywords- permanent magnet synchronous machine, stator
resistance, torque control.
Where φs0 is the initial stator flux linkage vector and Rs,
vs and is are the stator resistance, applied stator voltage
vector and stator current respectively.
Due to presence of integrator in equation (1) any dc
offset in the measured stator current and voltage could
lead to drift of the estimated stator flux linkage. In this
regard many researchers have proposed different
techniques which each one has its own pros and cons [4].
Furthermore, according to (1) for a desirable DTC drive
scheme, the initial rotor position must be known. In fact,
unkown initial rotor position even could lead to wrong
direction of motor rotation on the starting stage.
Numerous estimation methods are proposed which
managed to detect the initial rotor position in spite of their
pertaining limitations [4].
From equation (1) it could be clearly seen that the only
motor parameter employed for the flux estimation, is the
stator resistance. Despite less motor parameter
dependency of DTC drives compared to other schemes,
the error in flux estimation could also occur due to stator
resistance variation. Actually, such a resistance changes is
mainly due to temperature variation which is caused by
motor losses. In this regard, copper loss is more dominant
and it is proportional to stator current value. Therefore, the
larger the stator current is the higher resistance variation
could occur. Basically, the stator resistance could change
up to 1.5-1.7 times of its nominal value. As a result, a
mismatch between estimated and actual stator resistance
value happens, and therefore the estimated flux and torque
provides wrong values for the DTC scheme, which
deteriorates the performance of such motor drives. It
should be noted that in the case of lower speed and higher
ϕ s = ∫ (vs − Rs is )dt + ϕ s 0
I.
INTRODUCTION
In the past few decades, the field oriented control
(FOC) scheme has been widely used in different
applications of electric motor drives. However, such a
control scheme involves algorithm complexity as well as
sluggish response [1].
In the late 80’s Takahashi and Depenbreck proposed a
new control scheme that was based on direct control of
torque and stator flux linkage by employing the proper
voltage vectors which was referred as direct torque control
(DTC). The first proposed DTC schemes were designed
for the Induction motors. However, in the late 90’s the
PMSM drives based on DTC scheme appeared in the
literature [2], [3], which combined the advantages of DTC
and PMSMs so as to satisfy the requirements for high
performance drive applications.
In DTC scheme, the desired torque value could be
obtained through controlling the amplitude and rotational
speed of stator flux. As a result, fast torque response with
a simpler control algorithm is attainable. However, there
are a few shortcomings to DTC schemes which in order to
provide a high performance drive, it has to be addressed
properly. In this regard, excessive torque and flux ripples,
drift in stator flux linkage estimation due to measurement
errors, position sensor requirement for locating the initial
rotor position as well as error in stator flux linkage
estimation due to resistance variations, could be pointed
out [4].
978-1-4244-7647-3/10/$26.00 ©2010 IEEE
361
(1)
load torque values, this effect could be more severe as
even it can makes the drive to become unstable.
In the recent decades, many researchers have studied
different effects of stator resistance variations on DTC
drives. Some has just focused on Induction motor DTC
drives [10], [11] and some others evaluated such a
behavior for PMSMs [12], [13]. However, in terms of
PMSMs DTC drives, most of the reported papers are
restricted to interior mounted permanent magnet
synchronous motor (IPMSM) and lack of enough research
for surface mounted permanent magnet synchronous
motors (SPMSM) is noticeable. Although, the response of
DTC SPMSM drive to stator resistance variations might
be similar to the field weakening-based DTC IPMSM
drives, still there are some considerable differences
between them especially in terms stator current changes.
In this paper, different effects of stator resistance
variation on performance of DTC SPMSM drives are
analytically investigated. Moreover, the expected
responses to such a variation in terms of electromagnetic
torque, stator flux and stator current are confirmed
through simulation results. It is also demonstrated that the
lower values of actual stator resistance could lead to
unstable drive conditions in SPMSM DTC drives as
previously was reported for IMs and IPMSMs DTC drives
[11], [12]. In simulation stage, two different types of stator
resistance variations are employed. First, it has been
changed by a single step so as to provide a perfect
comprehension of expected response changes. Then, a
linear stator resistance variation, which represents more
realistic resistance changes, is applied so as to display the
problem caused in practical DTC SPMSMs drives.
II. PRINCIPLES OF DIRECT TORQUE CONTROL
The equations of surface mounted permanent magnet
synchronous machine (SPMSM) in the rotor reference
frame are as follows:
ϕ sd = Lsd isd + ϕ m
(2)
ϕ sq = Lsq i sq
(3)
ϕ s = ϕ sd 2 + ϕ sq 2
(4)
result, it is preferred for the flux and torque estimation to
be carried out in the stationary reference frame which
enables elimination of coordinate transformation and
quicker dynamic response. The SPMSM equations in the
α-β reference frame could be outlined as below:
ϕ sα = ∫ (v sα − Rs i sα )dt + ϕ sα 0
(6)
ϕ sβ = ∫ (v sβ − R s i sβ )dt + ϕ sβ 0
(7)
ϕ s = ϕ sα 2 + ϕ sβ 2
T=
3
p(ϕ sα i sβ − ϕ sβ i sα )
2
(8)
(9)
Where Rs is the stator resistance, φsα0 and φsβ0 are the
initial components of stator flux, φsα and φsβ are the two
components of stator flux, isα , isβ are the two components
of stator current, vsα , vsβ are the two components of stator
voltage in the stationary reference frame.
Block diagram of a classical DTC SPMSM drive is
shown in fig 1. It could be clearly seen that such a control
scheme has some main parts including hysteresis
controllers, switching table and Flux-torque estimation
unit. Also, it should be noted that besides the initial flux
vector detector (encoder), just two current sensors and
one dc bus voltage sensor are employed for measurement
purposes.
Figure 1. Block diagram of classical DTC SPMSM drive
T=
3
pϕ m i sq
2
(5)
Where φsd and φsq are the direct and quadrature axis
components of stator flux, Lsd , Lsq are the direct and
quadrature axis components of stator inductance, isd , isq
are the direct and quadrature axis components of stator
current in the rotor flux reference frame. Also, φm is the
magnetic rotor flux, p is the pair of motor poles and T is
the produced electromagnetic torque.
In the case of DTC scheme, the stator voltage could be
obtained through DC bus voltage and inverter switching
states. Thus, for measurement purposes just two current
sensors and a dc bus voltage sensor are sufficient. As a
362
At first stage of such scheme, the command torque
and stator flux values are compared with their
corresponding estimated values. Then, the obtained
torque and flux error values are applied to three-band and
two-band hysteresis controllers respectively. The outputs
of controllers along with stator flux position sector are
employed to index a look up table of optimum voltage
vectors. In fact, the selection of the voltage vectors are
made so as to restrict the torque and flux error values in
their allowable range of hysteresis bands. Consequently, a
proper motor drive control with quick torque response
could be attained.
III.
ϕ se = ∫ (v s − Rse is )dt
ANALYSIS OF THE STATOR RESISTANCE VARIATION
EFFECTS ON DTC SPMSM DRIVE PERFORMANCE
As it is mentioned earlier, one of the main problems
associated with DTC drive schemes is the stator winding
resistance variation. In practice, the stator resistance
changes as a result of temperature variation caused by
machine losses. In this regard, stator current could
substantially affect the losses amount as the produced
copper loss is proportional to it. The resistance variation
caused by machine losses could be represented by
equation (10) as follows:
Rs = Rs 0 [1 + α (T − T0 )]
(10)
Where Rs and Rs0 are the respective actual and initial
stator resistance, α is the temperature coefficient of
resistance and T and T0 are the actual and initial stator
winding temperature respectively.
As the resistance value varies by temperature, a
mismatch between actual and estimated stator resistance
occurs which can deteriorate the performance of DTC
drives to some different extent. For instance, in the case of
higher actual resistance values (caused by higher
temperature), the produced effects might just be restricted
to worsened stator flux response and higher copper losses
resulted from higher stator current values. However, it
should be noted that these effects could become more
severe depend on torque limit assigned by speed
controller, as even an inappropriate choice of these values
could make the drive to become unstable for such a
resistance variations. On the other hand, for the actual
resistance smaller than the estimated one (cold
environments), the unstable condition usually occurs as
also reported for IMs and IPMSMs DTC drives [11], [12].
In fact, this instability is caused by large voltage vectors
chosen based on wrong stator flux and torque error values.
In this section, different effects of stator resistance
variations on the performance of DTC SPMSM drives will
be analytically explained in details by means of motor
equations and stator current d-q plane.
Basically, as the actual stator resistance of PMSMs
( ) increases, from (11) it is understood that due to
constant applied voltage vector (vs) on that specific
moment, the stator current (is) momentarily decreases.
Thus, according to (12) since the estimated resistance
( e) is kept at its initial value, the resistive voltage drop
on the estimator is also reduced. Therefore, the smaller
voltage vectors (vs) are chosen by drive control scheme so
as to compensate for the lower voltage drop caused and
ensure providing estimated flux equal to desired value.
Consequently, the smaller chosen voltage vectors results
in smaller actual flux (φsa). The stator flux response to
lower actual stator resistance values could also be
similarly justified. In fact, for smaller actual resistance
( ) values, higher voltage vectors (vs) are chosen which
results in higher actual stator flux linkage. In this case, the
wrongly chosen large voltage vectors are known as the
source of DTC drive instability for such a resistance
variation, which later will be clearly demonstrated by
means of simulation results.
ϕ sa = ∫ (v s − Rsa is )dt
(11)
363
(12)
The effects of stator resistance variations on the stator
flux linkage could be summarized as follows:
⎧ Rsa > Rse ⇒ ϕ a < ϕ e
⎨
⎩ Rsa < Rse ⇒ ϕ a > ϕ e
(13)
In terms of the electromagnetic torque, the produced
effects are supposed to be similar with stator flux linkage.
However, since the motor drive is working as speed
closed loop with constant load torque, the actual output
torque would not be affected much. It is due to the fact
that the applied load forces the actual torque to be constant
and provides equilibrium with the applied load torque. As
a result, just some negligible variation of actual torque
might occur. On the other hand, the estimated torque will
be pushed to the higher values for larger actual resistance
values so as to enable the actual torque to be constant.
Similarly, for smaller actual resistance, the estimated
torque value will reduce. Thus:
⎧ Rsa > Rse ⇒ Ta < Te
⎨
⎩ Rsa < Rse ⇒ Ta > Te
(14)
In order to investigate effects of stator resistance
variation on the stator current, the constant torque and flux
curves for the SPMSM has to be drawn on the stator
current d-q plane. In fact these curves could be drawn
based on the stator flux and torque equations of SPMSMs
which is represented by equations (15) and (16) as
follows:
T=
ϕ=
∫ (L i
d d
3
pϕ m iq
2
(15)
+ ϕ m ) 2 + ( Lq i q ) 2
(16)
The constant torque and flux curves in stator current d-q
plane are drawn as shown in fig 2. Thus, by means of
drawn constant flux and torque curves, and previously
explained expected actual and estimated torque and flux
responses, the expected changes on stator current could be
easily understood.
Figure 2. Constant torque and flux curves for SPMSMs
As it is mentioned earlier, for any stator resistance
variation in DTC drives, the actual flux changes while the
actual torque is kept constant equal to load torque.
Consequently, the new intersection point of constant flux
and torque curves determines the new operating point of
stator current components and the resultant stator current
value respectively.
In the case of SPMSMs, since the direct axis
component of stator current does not need to build up flux,
it could be set to about zero. Thus, the stator current is just
including quadrature axis component (torque producing
component) of stator current. It also has the advantage of
providing maximum torque per ampere (MTPA) for the
SPMSMs drive, since the quadrature components is the
only torque producing current component in the case of
SPMSMs. Thus, it could be concluded that by choosing
reference stator flux equal to magnet flux, the intersection
point of constant flux and torque curves is almost placed
on q axis (MTPA path) so as to ensure maximum torque
per ampere which is desirable for many relevant
applications.
If the stator resistance increases, the actual flux is
lessened while the actual torque is unchanged. Thus,
according to fig 2 the operating point of stator current in
d-q plane move to the left hand side which implies that
some negative direct axis current component is produced
while the quadrature component is maintained at the
initial value. Consequently, the stator current increases for
the higher stator resistance values. On the other hand, for
the reduced stator resistance values, the actual stator flux
is increased while the actual torque is still not varied.
Thus, the stator current operating point moves to the right
hand side which produce positive direct axis current
component with the same quadrature component value.
Therefore, the stator current for the lower stator resistance
is also supposed to be risen up. However, due to unstable
conditions caused by large chosen voltage vectors, the
operating point will not be stabilized and just fluctuate on
different positive and negative values of direct axis
component of stator current as it would be clearly
illustrated by simulation results.
The expected displacement of the stator current
operating point for different stator resistance variations is
shown in fig 3. It should be noted that the initial operating
point is set to MTPA path which is a case for many
applications of DTC SPMSM drives.
Figure 3. Movements of stator current operating point in d-q
plane for certain stator resistance variations
364
Therefore, the expected stator current changes for
different variation of stator resistance could be
summarized as below:
⎧ Rsa > Rse ⇒ I a ↑
⎨
⎩ Rsa < Rse ⇒ I a ↑
(17)
According to the presented effects of stator resistance
variation on DTC SPMSM drives performance, it is
understood that stator resistance changes affect the stator
current in some different way compared to DTC IPMSM
drives, which in fact is due to unsimilar operating point of
stator current components in d-q plane. Basically, in most
of DTC IPMSM drives applications, the operating point is
set to be on MTPA curves. However, for IPMSMs such a
path is placed on the negative side of stator current d-q
plane. Therefore, for the increased stator resistance values,
the similar behavior with DTC SPMSM drives is expected
(stator current rise up). However, for lower stator
resistance the stator current decreases. In both DTC
drives, lower actual resistance values lead to unstable
drive conditions. It should be also mentioned that the
stator resistance variation has same effects on IPMSMs as
SPMSMs in terms of the actual torque and stator flux.
Besides the considered effects on stator current, stator
flux and produced torque values, a resistance mismatch
would result in wrong voltage vector selection which can
substantially affect the DTC drive performance. In fact, on
the moment of region transition a wrong voltage vector
would be chosen which does not satisfy the present stator
flux and torque error. Therefore, each transition will be a
source of further torque and flux ripples which is not
desirable.
In this section different effects of stator resistance
variation on DTC SPMSM drives performance have been
analytically investigated. In order to have more
comprehensive impression of the produced changes by
such a resistance mismatch, the resistance variation effects
should be studied through simulation results. The next
section is devoted to simulation studies of DTC SPMSM
drive performance as the stator resistance is varied by step
and linear function.
IV. SIMULATION RESULTS
This section deals with simulation studies of stator
resistance variation effects on the performance of DTC
SPMSM drive. In this regard, drive response to some
different type of resistance mismatches (produced by step
and linear variation of actual resistance) is evaluated in
terms of stator flux linkage, electromagnetic torque and
stator current. It could be clearly seen that the drive
responses follows the earlier justified effects as it was
expected. In order to change the stator resistance of the
pertaining PMSM, the powersim model of this motor is
manipulated in a manner so as to provide the access to its
stator resistance by different desired function. The
employed PMSM parameters are shown in table 1. The
operating condition of the drive is set to 300 rpm (low
speed) and 4 n.m.
TABLE I.
PMSM MOTOR PARAMETERS
5.25 mH
0.1827 wb
Rotor flux linkage
Stator Resistance
(Ohms)
Stator Resistance
(Ohms)
Times(s)
Times(s)
Times(s)
Times(s)
Times(s)
Times(s)
Times(s)
Times(s)
Stator Current (A)
Stator Current (A)
Times(s)
Electromagnetic Torque
(n.m)
Times(s)
Electromagnetic Torque
(n.m)
Times(s)
Stator Flux (wb)
Stator Flux (wb)
Times(s)
Times(s)
Stator Current (A)
The simulation results for the higher actual stator
resistance values are shown in fig 4. At first, the stator
resistance is increased by 50% with a step function which
is applied on 0.5 sec. then, the stator resistance is raised
linearly up to 50% of its nominal value and the produced
effects are observed. It could be clearly seen that with
such a resistance variation, as justified earlier, Actual
stator flux reduces and the estimated torque increases
while the actual torque and estimated flux are remained
unaffected. Actually, there is some overshoot and
oscillation for step responses of the varied signals which is
mainly due to delay for selection of the appropriate
voltage vectors after the applied stator resistance variation.
Stator Resistance
(Ohms)
5.25 mH
q-axis stator inductance Lq
Stator Flux (wb)
0.9585 Ω
Electromagnetic Torque
(n.m)
Stator resistance Rs
d-axis stator inductance Ld
Stator Current (A)
4
Stator Resistance
(Ohms)
300 v
Magnetic pole pairs p
Stator Flux (wb)
1.675 kw
Electromagnetic Torque
(n.m)
Rated power PN
Rated voltage UN
Additionally, the stator resistance is decreased by 30%
of its nominal value with a step and linear function and the
produced effects are shown in fig 5. As it was mentioned
earlier, for lower actual resistance values drive becomes
unstable which in fact is due to large voltage vectors that
is wrongly chosen by drive control scheme.
Times(s)
Stator Flux Trajectories
Figure 5. Simulation results for DTC SPMSM drive responses to step
and linear resistance reduction (30% of nominal value)
Times(s)
Blue-colored: actual value, red colored: estimated value
The effects of stator resistance variations on the stator
current could be more clarified through simulating the
operating point of stator current on d-q plane. As it is
shown in fig 6, with unchanged stator resistance the
operating point is placed on q axis which implies almost
zero direct axis current components. However, with any
types of resistance changes it will be deviated to either left
(for higher actual resistance) or right (for lower actual
resistance) as it is clearly demonstrated in fig 6. In both
cases, some direct axis component of stator current is
produced which results in larger stator current values. In
the case of reduced stator resistance, since the drive has
become unstable, the stator current operating point starts
to fluctuate for different values of direct axis component
of stator current while maintains the same value of
quadrature axis component.
Times(s)
Stator Flux Trajectories
Figure 4. Simulation results for DTC SPMSM drive responses to step
and linear resistance increase (50% of nominal value)
Blue-colored: actual value, red colored: estimated value
365
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Figure 6. Simulation results for displacement of stator
current operating point of DTC SPMSM drive for different
types of stator resistance variations
[10]
V. CONCLUSION
In this paper, different effects of stator resistance
variation on the performance of SPMSM DTC drive have
been studied in details. In this regard, the expected effects
are analytically investigated at the first stage and then,
they have been confirmed by simulation results obtained
through applied step and linear resistance variations. It is
understood that the performance of drive could be highly
influenced on the low operating speeds with high load
torque values. In order to address the problems caused by
stator resistance mismatch, some mitigation methods
(compensators, estimators) could be employed which
demands a comprehensive understanding of the produced
effects of stator resistance variation on the drive
performance which is carried out in this paper.
366
[11]
[12]
[13]
M. Meyer and J. Böcker, “Optimum Control for Interior
Permanent Magnet Synchronous Motors (IPMSM) in Constant
Torque and Flux Weakening Range”, 12th International Power
Electronics and Motion Control Conference (EPE-PEMC),
Portoroz, Slowenia, August 2006.
L. Zhong, M. F. Rahman, W. Y. Hu, and K. W. Lim, “Analysis Of
Direct Torque Control In Permanent Magnet Synchronous Motor
Drives”, IEEE Transactions On Power Electronics, vol. 12, no. 3,
may 1997.
L. Zhong, M. F. Rahman, W. Y. Hu And K. W. Lim, M. A.
Rahman, “A Direct Torque Controller For Permanent Magnet
Synchronous Motor Drives”, IEEE Transactions On Energy
Conversion, Vol. 14, No. 3, September 1999.
M.F.Rahman, E. Haque, L.Tang, L.Zhong, “ Problems associated
with the direct torque control of an interior permanent- magnet
synchronous motor drive and their remedies”, IEEE Transactions
on Industrial Electronics, vol.41, no.4, p.799-809, august 2004.
L.Lianbing, W.Xiaojun, S.Hexu, “A Variable Voltage Direct
Torque Control Based On DSP In PMSMs”, Proceeding Of IEEE
TENCON, 2002.
Swierczynski, D. Kazmierkowski, M.P, “Direct Torque Control of
Permanent Magnet Synchronous Motor (PMSM) Using Space
Vector Modulation (DTC-SVM) - Simulation and Experimental
Results”, Presented At Industrial Electronics Society, IEEE 2002
28th Annual, 2002.
E. Al-Radadi, " Direct Torque Neuro Fuzzy Speed Control Of An
Induction Machine Drive Based On A New Variable Gain
PIController", Journal of Electrical Engineering, VOL. 59, NO. 4,
2008.
Lin, C.-K, Liu, T.-H, Yang, S.-H, “Nonlinear position controller
design with input-output linearisation technique for an interior
permanent magnet synchronous motor control system”, IET
Journals on Power Electronics, Vol.1, No.1, 2008.
Foo, G, Rahman, M.F, “Direct torque and flux control of an IPM
synchronous motor drive using a backstepping approach”, IET
Journal on Electric Power Applications, Vol.3, No.5, 2009.
S. Mir, M.E. Elbuluk and D.S. Zinger, “PI and Fuzzy Estimators
for Tuning the Stator Resistance in Direct Torque Control of
Induction Machines”, IEEE Transactions On Power Electronics,
VOL. 13, NO. 2, March 1998
B.S. Lee and R. Krishnan, “Adaptive Stator Resistance
Compensator For High Performance Direct Torque Controlled
Induction Motor Drives”, Presented At IEEE Industry
Applications Conference, Thirty-Third IAS Annual Meeting., 1998,
pp. 423-430 vol.1.
M. E. Haque and M. F. Rahman, “A PI Stator Resistance
Compensator For A Direct Torque Controlled Interior Permanent
Magnet Synchronous Motor Drive,” Presented At Power
Electronics And Motion Control Conference, 2000. Proceedings.
PIEMC 2000. The Third International, 2000, pp. 175-179 vol.1.
L. Tang and M.F. Rahman, “A Novel PI Stator Resistance
Estimator For Direct Torque Controlled Permanent Magnet
Synchronous Machine Drive”, Unpublished.