Download Transient Peak Currents in Permanent Magnet Synchronous Motors

Transient Peak Currents in Permanent Magnet
Synchronous Motors
for Symmetrical Short Circuits
Michael Meyer, Joachim Böcker
Institute of Power Electronics and Electrical Drives, Paderborn University
33095 Paderborn, Germany
Abstract—To enable constant-power areas with
permanent magnet synchronous motors, flux weakening has
to be applied. In that mode, the inverter protection measure,
e.g. in case of overcurrent, is a crucial point, since the opencircuit induced terminal voltage may exceed the allowed
inverter limit. That is why a symmetrical short circuit (SSC)
is currently favoured as alternative protection measure.
Though the steady state short circuit current may be
acceptable, this contribution shows that the transient
current may seriously exceed the allowed limit. The worst
case SSC will not occur during flux weakening but in
regenerative operation at rated speed. Such high peak
currents include the risk of permanent demagnetization of
the magnets. As a result, SSC should applied only during
flux weakening as protection measure, while for the lower
speed range the convential converter shut down is proposed.
Index Terms-- Permanent magnet synchronous motor, short
circuit, protection measure, transient behavior
I. NOMENCLATURE
The following symbols are used in this paper:
iα , i β
α- and β-axis terminal voltage
components
α- and β-axis current components
id , iq
d- and q-axis current components
ψ α ,ψ β
α- and β-axis flux components
R
Ld , Lq
Stator resistance
d- and q-axis inductances
ω RS
Rotor electrical angular velocity
ε RS
Rotor electrical angular position
ε ψS
Angle of flux vector in stator-fixed
coordinates
Angle of flux vector in rotor-fixed
coordinate
Permanent magnet flux
uα , u β
ε ψR
ψp
I max
U DC
p
Maximum length of the stator current
space vector
DC link voltage
Number of pole pairs
Note that vectors are labeled with bold italic
characters, e.g. ψ is the stator flux vector.
II. INTRODUCTION
Requirements for traction motors, e.g. in the drive
train of hybrid electric vehicles (HEV) or for railway
vehicles are a high starting torque, a good torque per
volume and power per volume ratio as well as a high
efficiency. This is why permanent magnet synchronous
motors (PMSM) became more and more popular for these
applications. The performance of PMSM drives increased
significantly in recent years due to the development of
rare earth permanent magnet materials, e.g. NdFeB or
SmCo. The described traction motor applications require
a wide constant-power area. In this operation mode, it is
necessary to weaken the permanent flux of the motor by
applying a negative d-axis current, to ensure that the
motor terminal voltage will not exceed the allowed limits
(flux weakening operation). This bears the risk that in
case of an inverter protection shut down during the flux
weakening operation the induced open-circuit terminal
voltage may exceed the allowed inverter limit [4]. That is
why a symmetrical short circuit (SSC) is to be discussed
as alternative protection measure. To do so, an accurate
knowledge of the SSC current behavior is required. While
the steady state SSC currents of the motor are generally
well known, the behavior of the transient SSC currents
depends on the operating point of the motor at the instant,
the short circuit is triggered. As the transient SSC current
peaks in negative d-axis direction, not only the magnitude
of the short circuit current poses a threat to the inverter
but also the generated flux poses a serious threat to the
permanent magnets, which may lead to an irreversible
demagnetization. It has to be ensured that under all
conceivable operating conditions at the instant of the
SSC, the transient SSC currents do not exceed the short
term current limit, determined by the current limit of the
power semiconductors as well as by the properties of the
permanent magnet material. A SSC must not be triggered
at motor operating points, for which the SSC current
behavior will damage the PMSM drive.
An analysis of PMSM behavior with respect to various
faults was conducted in [1], [3]. The behavior of PMSM
drives in field weakening operation, in case of an inverter
shut-down was examined in [4]. In [5] the PMSM
transient SSC behavior was discussed, but an estimation
of the SSC peak currents depending on the PMSM
operating point at the instant of the short circuit was not
done.
In this paper, after a derivation of the SSC behavior,
where the ohmic voltage drop is neglected, a transient
SSC model is derived and validated. It is shown, that the
transient SSC behavior depends significantly on the
motor operating point at the SSC instant. The worst case
operating point within the feasible area of operation with
respect to the SSC behavior is identified and the worst
case transient peak current and flux are approximated.
This paper is organized as follows:
Section III: PMSM model
Section IV: SSC model neglecting the ohmic voltage
drop
Section V: Transient SSC model
Section VI: Saturation effects
Section VII: Identification of the worst case SSC
currents
Section VIII: Conclusions
III. PMSM MODEL
In the stator-fixed α , β reference frame, the voltage
equation of a PMSM is given by:
⎡uα ⎤
⎡iα ⎤ ⎡ψα ⎤
⎢u ⎥ = R ⎢i ⎥ + ⎢ψ ⎥
⎣ β⎦
⎣β⎦ ⎣ β⎦
(1)
Neglecting saturation effects, the stator flux vector ψ
in stator-fixed α,β coordinates is given by
⎡ψ p + Ld id ⎤
⎡ψ α ⎤
⎥
⎢ψ ⎥ = Q (ε RS ) ⎢ L i
q q
⎣ β⎦
⎣
⎦
(2)
u q = Lq iq + Riq + ω RS Ld id + ω RSψ p
(3)
(4)
IV. SSC MODEL NEGLECTING THE OHMIC
VOLTAGE DROP
Ld id 0
ψ p (t )
εψS 0
ε RS 0
(5)
From Eqns (1) and (5) follows directly that in case of a
SSC, the change of the stator flux in stator-fixed
coordinates is determined by the ohmic voltage drop. In
this section a model of the SSC currents neglecting the
ohmic voltage drop is derived. The impact of the ohmic
voltage drop on the SSC is covered in the next section.
ψ p0
ψα
∆ψ max = Ld id max
ψ p max
Fig. 1 Flux behavior of a PMSM in case of a SSC
It is assumed that the SSC is triggered at time t = 0 . In
the following, all quantities related to this time instant are
subscripted with ‘0’. Thus
ψ (t = 0) = ψ 0 = ψ 0 ∠ε ψS 0
ψ p (t = 0) = ψ p 0 = ψ p ∠ε RS 0
(6)
Neglecting the ohmic voltage drop means that from the
instant of the SSC on, the stator flux ψ remains
unchanged in stator-fixed coordinates. Assuming that the
rotor retains its angular velocity during the relevant time
period for the electrical transients, the permanent flux
vector will keep on turning with the rotor electrical
frequency ω RS .
(7)
The evolution of the current i(t) can be obtained from
the difference vector between the frozen stator flux and
permanent flux vector,
∆ψ (t ) = ψ 0 − ψ p (t )
(8)
Let t max be the time when the magnitude of ∆ψ (t )
reaches its peak value. At this time ψ p max = ψ p (t max )
points to the opposite direction of ψ 0 (see Fig. 1). Thus
the rotation angle with that peak value is
ε RS max = ε RS (tmax ) = εψS 0 + π
The terminal voltages in case of a SSC are by
definition:
uα = u β = 0
Lq iq 0
ψ0
ψ p (t ≥ 0) = ψ p ∠(ε RS 0 + ω RS t )
The voltage equations in the rotor-fixed reference
frame are given by
ud = Ld id + Rid − ω RS Lq iq
∆ψ (t )
ψ (t ≥ 0) = ψ 0
where Q is the rotation matrix
⎡cos(ε RS ) − sin(ε RS )⎤
Q (ε RS ) = ⎢
⎥
⎣ sin(ε RS ) cos(ε RS ) ⎦
ψβ
(9)
The time t max can then be calculated as
t max =
εψS 0 + π − ε RS 0
ω RS
(10)
Note that the orientation of ∆ψ (t max ) coincides with
the negative d-direction, and thus bears a serious risk of
demagnetization of the permanent magnets. Since the
currents can be calculated by
id =
∆ψ d
, iq =
Ld
∆ψ q
Lq
The steady state SSC currents are resulting as
(11)
from ∆ψ d , ∆ψ q it can be seen that, at least for Ld ≤ Lq ,
the maximum current is reached simultaneously with the
maximum of ∆ψ and hence given by
ψ 0 +ψ p
imax = −
(12)
Ld
It should be noted that for interior permanent magnet
synchronous machines (IPMSM) L d is significantly
smaller than L q . For a given ∆ψ max , a small value of
L d will lead to an additional boost of the SSC current
magnitude. Summarizing the results, the behavior of the
SSC currents in rotor oriented coordinates, neglecting the
ohmic voltage drop can be described by the following
equations:
i d (t ) = −
ψp
Ld
ψ0
i q (t ) = −
Lq
+
ψ0
Ld
cos(ω RS t + ε RS 0 − ε ψS 0 )
(13)
i q (t ) = e
K d cos(ϕ d (t )) + i d , SS
t
τ
R 2 + ω RS 2 Ld Lq
ψP
Ld
(16)
≈0
The above approximations of the steady state currents
are valid for a sufficient high speed. It should be noted
that they match with the DC-components of Eqn (13).
The decay time constant τ is given by.
τ=
2 Ld Lq
(17)
R ( L d + Lq )
The exact analytical solutions of ϕ d (t ) , ϕ q (t ) are
rather complicated and not suitable for further
investigations.
Therefore
some
simplifying
approximations shall be introduced. For sufficient high
ω RS , i.e.
2
ω RS
>> R 2 / Ld Lq
(18)
ε ψS 0 = ε ψS (t ), t ≥ 0
(14)
t
−
Rω RSψ p
≈−
V. TRANSIENT SSC MODEL
The ohmic voltage drop causes the decay of ψ to its
steady state value, by which the transient process is
characterized. Again it is assumed that during the
transient process, the motor retains constant speed.
Neglecting saturation effects, the behavior of the SSC
currents is described by the solution of the system of two
coupled first-order differential equations given by Eqns
(4) and (5). The solution obtained from a computer
algebra program is given as follows
τ
R + ω RS Ld Lq
2
2
it can be assumed that the angle of the stator flux
vector remains unchanged while its magnitude is
decaying with the time constant τ , i.e.
⎡ψ α ⎤
⎡iα ⎤
⎢ψ ⎥ = − R ⎢i ⎥
⎣ β⎦
⎣β⎦
−
iq, SS = −
Lqω RS 2ψ p
sin(ω RS t + ε RS 0 − ε ψS 0 )
A. Derivation of the SSC model
Taking now the ohmic voltage drop into consideration,
in case of a SSC, the stator flux vector will not stay
frozen in stator oriented coordinates any more. It is
forced to move in the opposite direction of the stator
current vector.
i d (t ) = e
id , SS = −
(15)
K q cos(ϕ q (t )) + i q , SS
Here, i d , SS and i q , SS are the steady state SSC currents
while exp(−t / τ ) K i describe the enveloping curves of the
decaying magnitudes of the transient current components.
(19)
The approximate values ϕ d (t ) , ϕ q (t ) , K d , K q can
then be obtained from Eqn (13), i.e. from the results of
the SSC currents, neglecting the ohmic voltage drop, as
follows
Kd =
ψ0
Ld
, Kq =
ψ0
Lq
(20)
ϕ d (t ) = ω RS t + ε RS 0 − εψS 0
ϕ q (t ) = ω RS t + ε RS 0 − εψS 0 +
π
2
If the assumptions in Eqn (18) and (19) hold, Eqns
(15) to (17) in combination with Eqn (20) provide an
analytical approximation of the transient SSC current
response. The accuracy of these approximations will be
shown in the following section.
B. Simulation Results
Simulations were carried out using Matlab/Simulink to
compare the approximations of the preceding section with
the original model given by Eqns (1) to (5). Figures 2 to 5
show the results for both SSC current components for
two different speeds. The curves of the original model
(“sim”) are plotted in solid green, the curves of the
approximated model (“calc”) in dashed red. The
simulations were conducted with a PMSM model with
the following parameters:
400
TABLE I
PMSM parameters used in SSC simulation model
R = 0,02Ω
L d = 0,201 mH
Stator resistance
d-axis inductance
300
Lq = 0,445 mH
q-axis inductance
100
ψ p = 53,41 mVs
Permanent flux
p =8
Number of pole pairs
-100
nrated = 2200 rpm
Rated speed
-200
I rated = 380 A
Rated magnitude of current vector
-300
U DC = 565 V
DC-link voltage
-400
iq,sim
iq,calc
q-current (A)
200
0
0
0.01
0.02
0.03
0.04
0.05
time (s)
Fig. 5 q-axis SSC currents n=0.15nrated , id0=0A, iq0=Imax
800
id,sim
id,calc
600
The figures show that the approximated SSC currents
match closely, although not completely with the original
model. The approximation bases on the assumptions of a
sufficient high speed (Eqn (18)) and that the stator flux
vector retains its orientation in stator-fixed coordinates
during the process of decay (Eqn (19)). The accuracy of
that assumption can be seen from Fig. 6 , which depicts
the transient flux trajectories.
400
d-current (A)
200
0
-200
-400
-600
-800
0.2
-1000
0.18
0
0.01
0.02
0.03
0.04
β flux component (Vs)
400
iq,sim
iq,calc
300
200
100
ψ(t=0)
0.16
0.16
Fig. 2 d-axis SSC currents n=nrated , id0=0A, iq0=Imax
q-current (A)
ψ(t=0)
0.05
time (s)
(b)
0.18
β flux component (Vs)
-1200
0.2
(a)
0.14
0.12
0.1
0.08
0.06
0.14
0.12
0.1
0.08
0.06
0.04
0.04
0.02
0
0.02
ε ψ S0
-100
0
0.02 0.04 0.06
α flux component (Vs)
-200
εψ S0
0
-0.02
0
0.02 0.04 0.06
α flux component (Vs)
Fig. 6 Stator flux trajectories for n= nrated (a) n=0.15 nrated (b)
-300
-400
0
0.01
0.02
0.03
0.04
0.05
time (s)
Fig. 3 q-axis SSC currents n=nrated , id0=0A, iq0=Imax
800
id,sim
id,calc
600
400
VI. SATURATION EFFECTS
d-current (A)
200
0
-200
-400
-600
-800
-1000
-1200
The accuracy of both assumptions Eqn (18) and Eqn (19)
improves with the rising τω RS . The lower accuracy for
small τω RS seems to be acceptable, as the importance of
the consideration of the transient SSC behavior decreases
with lower values of τω RS .
0
0.01
0.02
0.03
time (s)
Fig. 4 d-axis SSC currents n=0.15nrated , id0=0A, iq0=Imax
0.04
0.05
In Fig. 6, the trajectory of the stator flux in stator-fixed
coordinates is given. During the transient process, in spite
of the high magnitude of the SSC currents, the stator flux
decreases, following the depicted trajectory (green) to its
steady state. Thus, the flux generated by the transient
SSC stator currents with their high magnitudes is bound
to compensate for the difference between the relatively
slowly decaying stator flux and the permanent flux,
rotating with ω RS . During the SSC, the flux magnitude
will not exceed the level of normal operating condition.
Thus no unusual saturation effects in the stator yoke will
be expected. However, the decaying stator flux is now
rotating with a speed of −ω RS in rotor-fixed coordinates.
This may lead to saturation effects in the rotor when the
stator flux vector ψ is aligning with the d-direction. A
finite element analysis is needed for an accurate
estimation of the saturation effects inside the rotor.
VII. IDENTIFICATION OF THE WORST CASE SSC
CURRENTS
The simulation results proofed that the accuracy of the
derived approximated SSC current model is sufficient to
estimate the time developing of the SSC currents. The
peak SSC current occurs at time t max , when for the ϕ d
of Eqn (20) cos(ϕ d (t max )) = −1 holds for the first time
after the SSC was triggered. t max is calculated according
to Eqn (10). Summarizing the results of Section V, the
maximum magnitude of the transient SSC currents is
given by
imax = id max = −
ψp
Ld
−
t max
−e τ
ψ0
ε ψR is positive in motor operation mode and negative in
regenerative operation mode. So, for a given speed, t max
adopts its minimum value, while at the same time the
stator flux magnitude is at its maximum, when the
machine is operating at rated torque in regenerative
mode. The maximum ω RS , with which this operating
point can still be realized is determined by the constant
flux ellipses, which are depicted in green in Fig. 7. They
form the voltage limit at different speeds ω RS . For any
given ω RS the values of i d , i q have to be located inside
the flux ellipse determined by Eqn (24).
ψ lim =
U dc
3ω RS
(24)
Note that the constant flux ellipse corresponding to the
voltage limit at rated speed intersects with the circle of
maximum current in the operating points (A) and (B).
Thus ω RS , rated is the maximum angular speed with
which the operating points (A) and (B) can be realized.
(21)
Ld
and the minimum (most negative) stator flux value in ddirection is given by
ψ d min = − ψ 0 e
−
tmax
τ
(22)
ψ p , L d and τ are determined by the motor parameters
and are independent of the operating point. However,
ψ 0 and t max depend on the operating point, particularly
on the speed ω RS , when the SSC happens. The higher
the stator flux magnitude ψ 0 and the smaller the time
t max till the SSC peak values occur, the worse the SSC
behavior. The time t max is given by Eqn (10). Note that
ε ψR 0 = ε ψS 0 − ε RS 0 is the orientation of the stator flux
vector ψ in rotor-fixed coordinates. t max decreases with
decreasing angle ε ψR 0 and with rising speed ω RS .
The higher the stator flux magnitude ψ0 , the smaller
the stator flux angle in rotor-fixed coordinates ε ψR 0 and
the higher the speed ω RS at the instant of the SSC, the
worse the SSC behavior.
Neglecting the reluctance torque, the reasonable
operating points of a PMSM in the i d , i q -plane are given
by the shaded area in Fig 7. It comprises only operating
points with negative d-axis currents within the circle of
maximum allowed current, which is depicted in red. The
ellipses of constant flux are depicted in green. The
operating points of maximum possible flux within the
shaded area are labeled with (A) and (B), respectively.
The corresponding currents are given by:
id = 0
i q = + / − I max
(23)
(A) and (B) are the operating points, where the
machine is producing rated torque in motor and in
regenerative operation mode, respectively. In both points
(A) and (B), ψ0 is of equal value. However, the sign of
Fig. 7 Operating points of a PMSM in current coordinates
The operating point with the worst case SSC behavior
is identified, as the point where the machine is operating
at rated speed and torque in regenerative operation mode.
The SSC current and flux magnitude in d-direction
were calculated with the initial conditions at the short
circuit instant according to the operating point
i d = 0, i q = − I rated , n = n rated . The calculated values
were validated by simulation results shown in Figures 8
and 9.
The maximum transient current is reached at
t max = 1 ms after the SSC was triggered and the peak
value is 2.84 times higher than the rated current. The
minimum flux in negative d-direction is reached at the
same time and is 3.05 times the permanent flux in
negative d-direction that will cause most probably a
permanent demagnetization of the magnets. So the worst
case SSC behavior for the given motor is not acceptable
because it would lead to severe damages of the motor
drive.
REFERENCES
1200
itotal,sim
itotal,calc
1000
total current (A)
800
600
400
200
0
0
0.01
0.02
0.03
0.04
0.05
time (s)
Fig. 8 total SSC currents n=nrated, id0=0, iq0=-Irated
0.2
Ψd,sim
Ψd,calc
0.15
flux in d-direction (Vs)
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.01
0.02
0.03
0.04
0.05
time (s)
Fig. 9 total SSC currents n=nrated, id0=0A, iq0=-Irated
VIII. CONCLUSIONS
In this contribution a method to analytically estimate
the transient SSC current developing is derived. The
transient SSC behavior depends significantly on the
operating point at the instant of the SSC. This operating
point is characterized by the stator flux vector and the
motor speed ω RS . The worst case operating point with
respect to the transient SSC behavior is reached, when the
machine is operating at rated speed and torque in
regenerative operation mode. Counterintuitive, flux
weakening operation is not the crucial point concerning
SSC. It is shown that for PMSM, where the electrical
transients do not decay with a time constant very small
compared to its rated speed, the transient SSC behavior
must not be neglected. Further on, it is shown that the
worst case transient SSC peak current may lead to an
irreversible demagnetization of the permanent magnets.
As a result, a solution for that problem may be to
distinguish the suitable protection measure depending on
the motor speed. During constant torque operation, a
conventional converter shut-down is reasonable, while
the discussed SSC should be applied only in the flux
weakening region.
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