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Review of off-line synchronous inductance measurement method for

permanent magnet synchronous machines

Conference Paper · November 2014

DOI: 10.1109/ITEC-AP.2014.6940943

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 ITEC Asia-Pacific 2014 1569949753

Review of Off-Line Synchronous Inductance

Measurement Method for Permanent Magnet
Synchronous Machines
Yuting Gao, Ronghai Qu, Senior Member, IEEE, Yu Chen, Jian Li, Member, IEEE, Wei Xu, Senior Member, IEEE
State Key Laboratory of Advanced Electromagnetic Engineering and Technology,
School of Electrical and Electronic Engineering,
Huazhong University of Science and Technology, Wuhan 430074, China
E-mail: ronghaiqu@hust.edu.cn

Abstract—In permanent magnet synchronous machines PMSMs, as the absence of permanent magnets is not possible.
(PMSMs), the synchronous inductances have important Therefore, standard procedures for the inductance
influences on both steady state and dynamic performances, such measurement in PMSMs are demanded, which have not been
as designing control strategies, predicting torque and flux-
established yet. This paper presents a review of the various
weakening capabilities. However, standard procedures for the
inductance measurement have not been established yet. This off-line methods that have been used to measure the
paper presents a comprehensive analysis and comparative study inductances based on their accuracy, experimental setup and
of the various off-line methods that can be used to measure the complexity. Some online identifications of the inductances are
inductances based on their accuracy, experimental setup and very effective methods since the Ld and Lq vary with different
complexity. These approaches can be classified into three main operating conditions [13-17]. However the online
categories: load test, DC standstill test and AC standstill test.
measurements often lead to complex numerical algorithms
Moreover, with the employment of finite element analysis (FEA),
qualitative comparisons are done among these methods. such as the Recursive Least-Square Method. To simplify the
calculations, this paper introduces many off-line measurement
I. INTRODUCTION methods that are much easier to apply.
This paper is organized as follows. First, the saturation and
Permanent magnet synchronous machines (PMSMs) are
cross-saturation effect, which play significant roles in the
widely used in modern adjustable AC drives for their high
inductance determination, are investigated in Section II.
torque density, high efficiency and easy maintenance [1]. The
Second, the load test, which is commonly applied for line-start
three essential parameters of the PMSM are back
PMSMs and have the advantages of considering the effect of
electromotive force (EMF), d- and q-axis inductances [2]. The
saturation & cross-saturation on the variation of inductances in
back-EMF can be easily measured by the no-load voltage, that
different operating conditions, is described in Section III. For
has been demonstrated in many papers. However, the accurate
other types of PMSMs, the standstill tests including the DC
estimations of synchronous inductances are relatively complex
standstill method and AC standstill method which are also
because they are influenced by saturation and cross-saturation
introduced in Section III, are preferable due to their control
effect [3-4]. Various approaches have been presented to
limitations and complications at running conditions. In Section
estimate the inductances, including analytical methods, finite
IV, qualitative comparisons are done to show the accuracy of
element methods (FEMs) and experiments. The analytical
each method with the employment of finite element analysis
methods claim to obtain precise results based on some
(FEA). To sum up, conclusions are drawn in Section V.
linearization and the neglect of saturation [5-7]. The accuracy
of this method depends greatly on the saturation factor and
II. SATURATION AND CROSS-SATURATION EFFECT ON
form factor, which bring rough results but with a relatively
SYNCHRONOUS INDUCTANCES
fast computation speed. The FEM methods are the most
efficient approaches as well known. They can be classified A. Influence of the Saturation Effect on Synchronous
into the frozen permeability method (FPM) [8], vector control Inductances
method (VCM) [9], and differential flux leakage method Since the magnetic loading of PMSMs is designed to be
(DFM) [3]. In FEM, the saturation and cross-saturation effects relatively high to achieve large torque density, the main flux
can be taken into account, giving a reliable prediction of the paths are much easier to saturate than the conventional
inductances during the design process. However, both the synchronous machines. A 35 kW interior permanent magnet
analytical method and FEM require the knowledge of the synchronous machine (IPMSM), whose parameters are listed
specific size, configuration and material properties of the in Table I, is built to investigate the saturation effect on the
machine, which may not always be available. For this reason, inductances. From the measured d- and q-axis inductances in
it is essential to determine the inductances by experiments. Fig. 2, it can be found out that as the current increases, more
Unlike conventional field excited synchronous machines, the saturated the magnetic circuit is, thus lower the inductances
tests [10-12] that require zero rotor field are not suitable for are. Therefore, many literatures [18-20] take saturation effect
into consideration for the improvement of accuracy.

1
 TABLE I However, with respect to Fig. 3(b) and (c), the flux paths are
DESCRIPTION OF THE PROTOTYPE
more saturated without d-axis current injection, since Id
Item Quantity demagnetizes d-axis flux paths. Therefore, in Fig. 4(a), if Id is
Power 35 kW fixed as a constant, the Ld gets smaller and smaller with the
No. of poles 6
increase of Iq. This is because the saturation of d-axis is
affected by q-axis flux linkage, which is the so-called “cross-
No. of slots 36 IPM
saturation” effect. So d-axis flux paths become more saturated
Stator outer diameter 240 mm Dynamometer
as a consequence of larger Iq. For Fig. 4(b), as Id increases, Lq
Rated current (rms) 97.5 A gets larger due to the demagnetization of d-axis current.
Torque
DC bus voltage 380 V sensor
Air gap length 1mm III. MEASUREMENT METHODS FOR SYNCHRONOUS
Core length 80mm Fig. 1. Experiment of the 35kW IPM INDUCTANCES
A. Load Test
1.3 3.5
The phasor diagram of the PMSM at running condition is
1.2
3 shown in Fig. 5. The left one is for generators, while the right
one is for motors [26]. From the diagram, the synchronous
Ld (mH)

1.1
Lq (mH)

2.5 inductance can be deduced as

1
a. for generators:
2
0.9 E  U cos   IR1 cos    
Ld  0 (1)
0.8
20 40 60 80 100 120
1.5
20 40 60 80 100 120
 I sin(   )
Current Id (A) Current Iq (A) U sin   IR1 sin    
(a) (b) Lq  (2)
Fig. 2. Saturation effect on d- and q-axis inductances: (a) d-axis inductance; (b)  I cos(   )
q-axis inductance. b. for motors:
E0  U cos   IR1 cos    
Ld  (3)
 I sin(   )
U sin   IR1 sin    
Lq  (4)
 I cos(   )
where E0 is back-EMF, U is phase voltage, I is stator current,
R1 is the stator resistance, ω is angular speed, φ is power factor
(a) (b) (c)
angle, and θ is load angle. E0 can be estimated by no load
Fig. 3. Magnetic field distributions with different currents: (a) only d-axis voltage, which will bring about deviations since the back-EMF
current; (b) only q-axis current; (c) d- and q-axis currents on load is not identical with the no load voltage. U, I, R1 and φ
can be easily measured by the load test. Meanwhile, θ can be
1.1
3.4
obtained either by means of a shaft sensor [26]-[28], by
1.5
1.05
4
3.2 applying another synchronous machine [29]-[30], or using
some analytical calculations [5], [31], [32].
Lq (mH)
Ld (mH)

3
1 3
1 2.8
0.5 2.6
100 200 0.95 2
100 2.4
50 100 0.9 200
50 2.2
100
Current Id (A) 0 0 Current Iq (A) Current Id (A) 0 Current Iq (A)
0

(a) (b)
Fig. 4. Cross-saturation effect on d- and q-axis inductances: (a) d-axis
inductance; (b) q-axis inductance.

B. Influence of the Cross-Saturation Effect on Synchronous

Inductances
Regarding cross-saturation, numerous papers have analyzed
(a) (b)
or accounted for its effect on equivalent circuit parameters, Fig. 5. Phasor diagram of typical PMSMs: (a) generators; (b) motors.
especially d- and q-axis inductances [21-25]. Because the
cross-coupling effect is particularly significant in PMSMs, this In [26]-[28], a notched collar illuminated with optocoupler is
paper will give a brief investigation on its influence. The clamped on the rotor shaft, and the sensor received light
cross-saturation effect on magnetic field distributions is signals to detect the rotor position, i.e. the phase of E0. Then
analyzed in Fig. 3. In Fig. 3(a) and (b), only rated d- and q- the load angle θ, which is the phase difference of E0 and U,
axis currents are applied to armature windings respectively, can be determined from the phase difference between the zero
while in Fig. 3(c), rated current (including d- and q-axis crossing of the supply voltage waveform and reference pulse
components) is injected into stator windings. Comparing (a) from the optocoupler respectively. The circuit layout is
and (c), the absence of q-axis current desaturates the iron core. depicted in Fig. 6.

2
 B. DC Standstill Test
In Fig. 8(a), after disconnecting the switch, the time
constant of the system can be observed when the current i
reaches 63.2% of its maximum value in Fig. 8(b) [34]-[35].
Hence the inductance in the equivalent circuit can be deduced.
With the employment of this principle, in DC standstill tests
one can also determine the synchronous inductance values by
applying a step DC voltage and observing the transient
reponse of the system, when the d- and q-axis of rotor is
aligned with stator magnetic motive force (MMF) respectively.
Fig. 6. Circuit layout of load angle measurement through a shaft sensor i

In [29]-[30], an additional PMSM (ASM), of which the pole Switch

I0
pair is the same with the test motor (TSM), is needed. The DC voltage Req
load angle can be estimated as the phase difference of the two source
line voltage waveforms. From Fig. 7, it can be found that the ‒ u +
O t
TSM should have two shaft ends to connect the prime motor Leq i
and the ASM respectively, which is not a usual case in (a) (b)
conventional PMSMs. Fig. 8. Basic principles of DC standstill tests: (a) equivalent circuit model of
the PMSM; (b) transient current response.
iA iA
+ PMSM PMSM
Switch A Switch + A
q
d
DC voltage R q DC voltage d
R
source source
uAB uAB
B C B C
‒ ‒

Fig. 7. Circuit layout of load angle measurement through an additional PMSM (a) (b)
Fig. 9. Experimental setup of DC standstill tests: (a) d-axis inductance; (b) q-
As what has been mentioned above, the load angle is known axis inductance.
unless shaft sensors or additional PMSMs are available. In
In the DC test, d-axis inductance occurs when the stator’s
addition, the accuracy of θ is limited when the harmonic
MMF is aligned with d-axis, as Fig. 9(a) shows; while q-axis
contents of the two voltage signals are rich. However, in some
inductance corresponds to the alignment when the stator’s
analytical calculations [5, 31, 32], θ can be determined
MMF is aligned with q-axis of rotor, as Fig. 9(b) presents. So
without special equipment. First, a separate measurement for
Ld can be derived as:
Ld is needed. When the test PMSM is connected with an
inductor, so power factor φ is equal to 0, then Ld can be given Ld  1 Leq ( a )  1 Req eq ( a ) (6)
2 2
as: where Leq(a), Req, τeq(a) is the equivalent inductance, resistance
E U and time constant of the circuit in Fig. 9(a) respectively. For
Ld  0 (5)
I Lq measurement in Fig. 9(b), the inductance can be given as:
Once Ld is known, from (1) or (3), the load angle can be Lq  1 Leq (b )  1 Req eq (b) (7)
derived. After that, Lq can be worked out using (2). But this 2 2
measurement will bring about deviations as Ld is measured where Leq(b), Req, τeq(b) is the equivalent inductance, resistance
when load angle is zero, which is not the case in operating and time constant in Fig. 9(b) respectively. Similarly, many
conditions, especially for IPMSMs. literatures also take advantage of the voltage response of the
Since E0 is estimated by no-load voltage, the inherent errors inductances, e.g. the voltage waveforms are integrated to
of the above methods are the disregard of d-axis armature obtain flux linkage [36] or magnetic field energy [37] to
reaction on the back-EMF. The authors of [33] have proposed determine the inductance values. In [36], using the set-up
an improved method where a look-up table is used to correct illustrated in Fig. 9 and measuring the transient terminal
the error. For different stator currents, the armature reaction voltage and current, the magnetic flux and consequently the
varies, thus having different influences on E0. Therefore, a equivalent inductance of the circuit can be determined by:
t
table of E0 with I is referred to when the inductances are
calculated.  eq  [u AB (t )  Req iA (t )]dt 
In brief, the load tests are commonly applied for line-start Leq 
 0
(8)
i iA (t )
PMSMs and have the advantage of considering the effect of
saturation and cross-saturation effects on the variation of where iA is phase A current and uAB is line voltage of terminal
inductances in different operating conditions. Meanwhile, no A&B. In [37], the equivalent inductance is estimated by the
complex control strategy is required in the experiments. magnetic field energy:
t
However, in case of error accumulation from the large amount
of parameters, high-precision measuring instruments are 2Weq  [u AB (t )  Req iA (t )]iA (t )dt 
needed to improve accuracy. Leq   0
(9)
iA2 (t ) iA2 (t )

3
But during the decay process of the current in Fig. 8, the Laa  Ll  L0  L2 cos 2 (12)
synchronous inductance dramatically increases due to the
M ab  M 0  M 2 cos 2(  120 ) (13) 
saturation effect. In order to maintain the same saturation level
with the operating condition, the DC current which determines where Ll is the additional leakage flux component, L0 and M0
the saturation level of the magnetic field decays to a relatively are the components of self- and mutual-inductances due to the
large value instead of zero in Fig. 8(b). Despite that the space-fundamental air-gap flux, L 2 and M 2 are the second
saturation effect can be included in the above measurements harmonics components of self- and mutual-inductances
and derivations, the cross-saturation effect has not been taken resulting from saliency of rotor. The parameters L0, L2, M0 and
into account with the employment of the circuit connections M2 can then be determined from (12) and (13). Therefore, the
decipted in Fig. 9. Some improved schematics that consider synchronous inductance Ld and Lq can be derived when (14)
both saturation and cross-saturation are introduced in [26], and (15) are employed.
shown in Fig. 10. Similarly, the inductance can be obtained by Ld   L0  M 0    L2 2  M 2  (14)
measuring iA and uAB, calculating the equivalent time constant, Lq   L0  M 0    L2 2  M 2  (15)
flux leakage or field energy, and finally deriving the
Switch
inductances with the help of Eqs. 6 to 9. iA

Switch Switch +
iA iA AC voltage PMSM
uA A
+ +
A PMSM A q PMSM source
d ‒
R1 uAB R1 uAB Rotor
q d
‒ position
DC DC Sensor θ
voltage voltage uB
B C ‒ B C B C
source I ‒ source I
iB iB +
DC voltage DC voltage
source II source II Fig. 11. Circuit connection of the AC standstill method
R2
R2
For the second method, when measuring Ld, the d-axis of
R3 R3
rotor is aligned with the axis of phase A winding. For Lq
(a) (b)
Fig. 10. Circuit layout of DC standstill test considering cross-saturation: (a) d- measurement, the q-axis is aligned with phase A axis. The d-
axis inductance; (b) q-axis inductance. and q-axis inductance can be estimated by [38]
Ld ( q )  2 3*U AB I A (16)
As mentioned above, the DC standstill tests have superiority
iA iA
in the elimination of high power supply and complex control
Switch + A PMSM Switch + A q PMSM
algorithms. However, the measurements which utilize d
AC AC
transient response of the circuit require the equivalent voltage q
voltage d
uAB uAB
resistance determination of 0.01Ω, as well as the use of inrush source source
C
current detectors with small sampling time. In general, the DC B B C
‒ ‒
standstill tests would fail to provide reliable results unless
advanced measuring instruments are available. (a) (b)
Fig. 12. Circuit connections of the second AC standstill method: (a) d-axis
C. AC Standstill Test inductance; (b) q-axis inductance.
In AC standstill tests, an AC voltage source (sinusoildal or Because the saturation effect will cause the distortion of
stepwise) is applied to a standstill PMSM and the inductances voltage waveform, [3] and [39] integrate the measured voltage
are determined from the terminal voltage and current. There to obtain the corresponding flux linkage, which is more
are two main methodological classifications to measure the sinusoidal than the voltage waveform. With the employment
inductances. The first one is through measuring the stator self- of this technique, the above two methods can take the
and mutual-inductance against rotor position, then the saturation effect into account, whereas they cannot consider
synchronous inductance can be obtained using Park cross-saturation effect on the inductances. Some improved
transformation. The other one is to measure the input approaches are proposed to consider the effect which are
inductance of the terminal when the d- and q-axis of rotor is introduced in [3], [40], [41] respectively.
aligned with stator magnetic motive force (MMF) respectively. In [3], the conventional two-axis machine model is
For the first method, [2] presents the basic principles. In Fig. modified in order to include the influence of saturation and
11, the rotor is locked at different positions and an AC voltage cross-saturation effects on the variation of self- and cross-
source is applied to phase A winding. Then the stator self- and coupling inductances in the d- and q-axis. Thus the
mutual-inductance, Laa and Mab , can be determined by synchronous inductances can be rewritten as:
measuring uA , iA and uB at each rotor position: Ld  Ldd  Ldq , Lq  Lqq  Lqd (17)
2
U  where the self- and cross-coupling inductances can be
Laa   A   R 2  2 f  (10)
 IA  calculated by:
M ab  U B (2 fI A ) (11) Ldd   d id i const . , Lqq   q iq (18)
q id  const .

Then the calculated stator self- and mutual-inductance can be Ldq   d iq , Lqd   q id (19)
id  const . iq  const .
plotted against rotor position. By applying the least square
curve fitting technique, the inductances are expressed in In (18) and (19), the current-dependent flux linkages can be
functions of rotor position α as: determined from the experimental tests performed at locked-

4
rotor position when the PMSM is supplied from the controlled measurement using transient response of the circuit requires
voltage source inverter (VSI). For example, for Ldd, the current the equivalent resistance determination of 0.01Ω, as well as
iq is controlled to be a constant value, while the voltage ud is the use of inrush current detectors with small sampling time.
changed in a stepwise manner. Then the time dependent flux The AC standstill tests do not need complex control
linkage Ψd can be determined from the recorded voltage ud and algorithms as well. Meanwhile, the AC standstill tests could
current id. provide accurate results without advanced measuring
In [40], the Ld and Lq are given as: instruments, hence they are frequently applied in the lab.
Ld  [ 0 sin(   )  a ] [i sin(   )] (20)
Lq  [ 0 cos(   )] [i cos(   )] (21) IV. FEA SIMULATION AND COMPARISON
where Ψ0 and Ψa are flux linkages produced by PM and stator As aforementioned, the principles and experimental setups
currents respectively. The above two flux linkages can be of different measurement methods have been carefully
estimated through measuring terminal voltages and currents at analyzed. In this section the accuracy of each method is
no load and load test respectively, which is similar to the qualitatively compared with the employment of FEA. The
method introduced in Fig. 12. parameters of the prototype are shown in Table I.
In [41], the two AC currents, i1 and i2, passing through the 1.4

test circuit are proportional to d- and q-axis currents at the 1.2

operating condition, respectively. Hence the saturation and
1
cross-saturation effect on synchronous inductances are
included. The synchronous inductances can be determined by:

Ld (mH)
0.8

2 2 Load test
1  U1( a )  1  U1(b ) 
0.6
AC standstill test (the first catagory)
Ld     R1 , Lq 
2
   R1
2
(22) AC standstill test (the second catagory, only consider saturation)
  2 I1( a )    2 I1(b )  0.4
AC standstill test (the second catagory,
consider saturation and cross-saturation)
0.2
where the subscript (a) and (b) means Fig. 13(a) and (b) DC standstill test (only consider saturation)
DC standstill test (consider saturation and cross-saturation)
respectively. Only the two AC current sources, oscilloscope 0
20 30 40 50 60 70 80 90 100
and wattmeter are needed in this method. However it requires Current (A)

that the neutral point can be connected with the current source. (a)
i2 i2 3.5
AC + A q PMSM AC + A PMSM
current u2 u2 d
current 3
source II ‒ d source II q
‒
2.5
Lq (mH)

B C B C 2

Load test
i1 + u1 ‒ i1 + u1 ‒ 1.5
AC standstill test (the first catagory)
AC standstill test (the second catagory, only consider saturation)
1
AC current AC current AC standstill test (the second catagory,
source I source I consider saturation and cross-saturation)
0.5
(a) (b) DC standstill test (only consider saturation)
Fig. 13. Circuit layout of AC standstill test considering cross-saturation: (a) d- 0
DC standstill test (consider saturation and cross-saturation)
axis inductance; (b) q-axis inductance. 20 30 40 50 60 70 80 90 100
Current (A)
As mentioned above, the AC standstill tests, which are (b)
preferable for the PMSMs supplied with VSI, do not need Fig. 14. Inductance comparison of different measurement methods: (a) d-axis
inductance; (b) q-axis inductance.
complex control algorithms as well. Futhermore, the AC
standstill tests could provide accurate results without advanced From Fig. 14, it can be found that as the stator current
measuring instruments, and are frequently applied in the increases, the inductances get lower due to the saturation
laboratory. effect. Moreover, the results of the tests that only consider
saturation are much higher than that of the tests taking both
D. Summary
saturation and cross-saturation into account. It can also be
The load tests are commonly applied for line-start PMSMs observed that the influence of saturation on Ld is more
and have the advantages of considering the saturation and significant than Lq, since the magnetic resistance of q-axis flux
cross-saturation effects on the variation of inductances in paths is much smaller than that of d-axis.
different operating conditions. However, the inherent error of
the method is the neglect of d-axis armature reaction on the V. CONCLUSION
back-EMF. In addition, in case of error accumulation of the
large amount of parameters, high-precision measuring Accurate estimation of synchronous inductances is quite
instruments are needed to improve accuracy. While for other important for predicting the performances of PMSMs. The
types of PMSMs, the standstill tests including DC standstill calculation methods including analytical and FEMs, can not be
method and AC standstill method are preferable due to their employed when the machine parameters are not available. So
control limitations and complications at running conditions. it is essential to obtain the inductances by experiments. This
The DC standstill tests have superiority in the elimination of paper presents a detailed analysis and comparative study of the
high power supply and complex control algorithms. But the various off-line methods for determining the inductances

5
including load tests, DC standstill tests and AC standstill tests. [17] W. T. Su, C. M. Liaw, “Adaptive positioning control for a LPMSM
drive based on adapted inverse model and robust disturbance observer,”
The load tests are applicable for line-start PMSMs, and are
IEEE Trans. Power Electron., vol. 21, no. 2, pp. 505-517, Mar. 2006.
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1988.
addition, many conventional load tests necessitate application
[19] J. Faucher, L. M. Michel, A. Chayegani, “Characterization of a closed-
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142, Mar. 1989.
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