Offline inductance identification of a permanent magnet synchronous motor (PMSM) is essential for the design of the closedloop controller and position observer in sensorless vector controlled drives. On the base of the offline inductance identification method combining direct current (DC) offset and high frequency (HF) voltage injection which is fulfilled at standstill, this paper investigates the inverter nonlinearity effects on the inductance identification while considering harmonics in the induced HF current. The negative effects on dq axis inductance identifications using HF signal injection are analyzed after selflearning of the inverter nonlinearity characteristics. Then, both the voltage error and the harmonic current can be described. In addition, different cases of voltage error distribution with different injection conditions are classified. The effects of inverter nonlinearities on the offline inductance identification using HF injection are validated on a 2.2 kW interior PMSM drive.
I. INTRODUCTION
The permanent magnet synchronous motor (PMSM) has been widely used in industrial applications because of its high efficiency, high power density, good dynamic performance, etc.
[1]
,
[2]
. Parameter identification is important for performance improvement of the PMSM sensorless vector control which is sensitive to parameter variations, and there have been many studies on the PMSM online parameter identification methodology
[3]
,
[4]
. In order to meet the emerging demand for modern generalpurpose drive applications, more attention should be paid to the offline parameter identification of PMSMs at standstill using only a voltagesourceinverter (VSI) fed drive. The dq axis inductances of a PMSM obtained from offline identification using high frequency (HF) voltage injection are needed for the proper tuning of closedloop controllers and the modelbased sensorless control scheme
[5]

[7]
. These critical parameters should be acquired before motor startup. Otherwise the drive system would exhibit poor behavior or even fail to operate. Especially, parameter identification at standstill is essential for applications with load machinery connected, since it is not permissible to make the rotor deviate from its initial position during selfcommissioning.
The offline parameter identification methods investigated in
[8]
,
[9]
require a signal generator that is not suitable for generalpurpose drives. This is due to the fact that these signal generatorbased test methods need specific instruments. The practical identification strategy using a VSIfed drive without additional instruments to get accurate dq axis inductances is important for PMSM control
[10]
. HF injection is an effective method to estimate inductances by analyzing the relationship between the HF voltage and the induced HF current. However, inverter nonlinearities cause an error of the injected HF signal and influence the accuracy of identification results.
Generally, offline inductance identification strategies using HF injection can be divided into two classes. The first one focuses on HF voltage injection
[11]
,
[12]
. It is easy to implement and the deterioration of the induced current can reflect the influence of the inverter nonlinearities. The second identification method is executed by injecting a HF current signal as the reference of the currentloop using a PI controller. In
[13]
, a proportional resonant (PR) controller was adopted to improve the performance of HF current injection. In this paper, the HF voltage injection based method is adopted.
The main contribution of this paper is that the investigation of the VSI nonlinearities influencing offline inductance identification is proposed by analyzing HF current attenuation and the existence of harmonic components. Additionally, different cases of voltage error distribution with different DC offsets and HF amplitudes are classified. This can effectively describe the negative effects of inverter nonlinearities. All of the signal generation and parameter calculations are performed by a generalpurpose PMSM drive. In order to obtain the characteristics of inverter nonlinearities, a selflearning of the inverter nonlinearity parameters using a linear regression algorithm is adopted. The proposed method is validated on a 2.2 kW interior PMSM drive.
II. PROPOSED INDUCTANCE IDENTIFICATION METHOD
 A. Scheme of dq Axis Inductance Identifications
A VSIfed drive is adopted to implement offline inductance identification at standstill, and the proposed scheme is shown in
Fig. 1
. The dq axis inductance (
L_{d}
and
L_{q}
) identifications are executed by selecting terminal 1 and 2, respectively. Then, the HF sinusoidal voltage is superposed on the corresponding axis. In addition, a supplementary DC current is injected into the daxis by using current closedloop control. This can keep the rotor stay at its initial position and attenuate the influence of inverter nonlinearities.
Proposed scheme of dq axis inductance identifications.
The HF voltage equation in the dq axis synchronous rotating frame can be expressed as follows:
where
R_{s}
is the stator resistance,
p
is the differential operator,
ω_{e}
is the electrical angular velocity,
u_{d}
and
u_{q}
are the dq axis voltages,
i_{d}
and
i_{q}
are the dq axis currents, and
φ_{d}
and
φ_{q}
are the dq axis flux linkages, respectively. The rotor can be kept at standstill during the identification process. Therefore, (1) can be simplified into:
Thus,
L_{d}
and
L_{q}
can be estimated according to the HF model as follows:
where the symbol ‘^’ means the estimated value,
U_{inj}
and
ω_{h}
are the voltage amplitude and the frequency of the injected HF signal, respectively, and
I_{dh}
and
I_{qh}
are the amplitudes of the induced dq axis HF currents.
 B. SelfLearning of the Characteristics of Inverter Nonlinearities
Usually, a saturation function is adopted to formulate the equivalent voltage error of inverter nonlinearities without considering the HF injection
[14]
. Due to the effects of the inverter nonlinearities, the relationship between the phase voltage reference and the phase current is shown in
Fig. 2
. The phase voltage reference, which increases nonlinearly with a linear increase of the phase current considering the influence of the inverter nonlinearities is indicated as curve 1. Normally, the phase voltage reference is composed with the stator resistance voltage drop and phase voltage error caused by inverter nonlinearities. After subtracting the stator resistance voltage drop, which is proportional to the phase current, the phase voltage error can be illustrated by curve 2. Δ
U
and ΔI denote the saturation values of the voltage error and the induced current, respectively. They are relevant to the inverter nonlinearities.
Relationship between phase voltage and phase current.
As can be seen from
Fig. 2
, curve 2, which indicates that the voltage error increases linearly with the current in the linear region, performs a resistance behavior. When the current exceeds the linear region and enters the saturation region, the voltage error is kept constant. In this period, the gradient of curve 1 which indicates the phase voltage reference decreases. It contains the sum of the stator resistance and the device onresistance. Additionally, the gradient
K
_{2}
in the linear region includes the gradient
K
_{1}
in the saturation region. Thus, their relationships can be expressed as follows:
In
Fig.2
, the intercept Δ
U
of line 1 can be derived by linear regression. Then:
Therefore,
K
=
K
_{2}
–
K
_{1}
, and the inverter nonlinearity parameters are finally obtained. In order to analyze the harmonic contents in the induced current,
K
and Δ
U
should be obtained automatically before the inductance identification by using a selflearning method. The way to get
K
and Δ
U
requires keeping the rotor at a standstill.
The selflearning method for acquiring
K
and Δ
U
is shown in
Fig. 3
. Firstly, a linearly increasing current is injected into the daxis by the closedloop control. Then the reconstructed phase voltage obtained from the PWM signals
S_{a,b,c}
and the DC bus voltage
U_{dc}
contains nonlinearity information. The gradient and intercept of the phase voltage reference used to estimate
K
and Δ
U
can be calculated by using the linear regression algorithm. The related formulas used to calculate Δ
U
and
K
_{1}
are shown as follows:
Selflearning method for acquiring the inverter nonlinearity characteristics at standstill.
The calculation of the coefficient
K
_{2}
is similar to
K
_{1}
, and the saturation value of the phase current also can be obtained:
III. HARMONIC ANALYSIS OF THE INDUCED HF PHASE CURRENT
 A. Inverter Nonlinearity Model
Assuming that the HF phase current is
I
_{1}
sin(
ω_{h}t
), then the corresponding HF phase voltage can be described as
U_{m}
cos(
ω_{h}tφ
). The phase inductance is
L_{s}
(
θ_{r}
) depending on
θ_{r}
. The nonlinearity model of the arbitrary phase
x
(a, b or c) is shown in
Fig. 4
.
Nonlinearity model of arbitrary phase x.
The model of
Fig. 4
can be described as follows:
where
f
(
i_{x}
) is the voltage error caused by inverter nonlinearities. It is difficult to obtain a general solution of (8) due to the existence of the nonlinear function
f
(
i_{x}
). Therefore, a practical method is adopted to analyze the HF current harmonics. There are four cases considering the different relationships between the voltage error and the HF current.
 B. Partial HF Current Located in the Linear Region  Case 1
The voltage error induced by the harmonic components of the HF current is neglected since its value is very small. In addition, only the fundamental component of the HF current is used for the voltage error analysis. In order to simplify the harmonic analysis, the sigmoid function is approximately replaced by the saturation function.
The first case, which is the most complicated, is that the partial HF current is located in the linear region. The relationship between the voltage error and the HF current is shown in
Fig. 5
.
Relationship between the voltage error and the HF current (case 1).
From
Fig. 6
, the voltage error is a clipped sinewave which can be expressed as:
where Δ
U
_{1}
is the DC voltage,
U
_{0}
sin(
ω_{h}t
) is the HF sinusoidal voltage, and u(·) is the unit step function. Only the first two terms in (9) induce the HF harmonic current. The first term is the HF fundamental component, and the second term, defined as Δ
u
^{’}
, is solely analyzed into a Fourier series to describe the HF harmonics.
The relationship of the Fourier series in
Fig. 6
can be expressed as:
where the term
is deduced as follows:
Define:
HF voltage error component in case 1.
Then (11) can be simplified into:
Thus,
M
can be obtained:
For
A
and
B
, there are two situations:
i ) when
n
is even, there is:
ii ) when
n
is odd, there is:
As a result,
This conclusion demonstrates that when
n
≠1, there is
Then, it can be expressed as follows:
 C. Partial HF Current Located in the Linear Region – Case 2
When the amplitude of the HF current is high enough when compared with the DC component, the maximum and minimum values of the induced current exceed the positive and negative saturation values, respectively. The voltage error is shown in
Fig. 7
.
Relationship between the voltage error and the HF current (case 2).
From
Fig. 7
, the voltage error is a clipped sinewave which can be expressed as:
According to the same analytical method mentioned in case 1, just analyze the second term of the voltage error which causes the distortion of the induced current in (16). Thus, the clipped sinwave removing the DC component is shown in
Fig.8
, where
t
_{1}
,
t
_{2}
,
t
_{3}
, and
t
_{4}
are the clipped times, and
U
_{1}
, and
U
_{2}
are the saturation values.
The resolution of the Fourier series of
Fig. 8
is:
HF voltage error component in case 2.
According to the same analysis method introduced in case 1, the result is;
i ) when
n
≠1, (17) can be expressed into:
where A, B, C, and D can be expressed as:
ii ) when
n
=1, (17) can be expressed into:
 D. Whole HF Current Located in the Linear Region – Case 3
Similarly, in
Fig. 8
, when the whole HF current is located in the linear region, the induced voltage error has the same phase as the HF current shown in
Fig. 9
. In this case, the voltage error only contains the sinusoidal component without other harmonics and can be expressed as Δ
u
=
U
_{0}
sin
(
ω_{h}t
)+Δ
U
_{1}
. The equivalent HF resistance reaches its maximum that is equal to the gradient of the linear region.
Relationship between the voltage error and the HF current (case 3).
 E. Whole HF Current Located in the Saturation Region – Case 4
When the DC current component is high enough, the HF current will be far away from the linear region and the voltage error is constant as shown in
Fig. 10
. The equivalent HF resistance reaches its minimum and is equal to zero. Thus, the voltage error does not affect the identification result in this case.
Relationship between the voltage error and the HF current (case 4).
According to the inverter nonlinearity parameters, which are obtained by the selflearning method introduced in Part II, a twodimensional region can be established based on the HF amplitude and DC offset of the induced current.
Fig. 11
shows the regions of the four cases in the first quadrant, where the current saturation value Δ
I
is 0.1pu (the current base value is selected as 7.9A), and the voltage error saturation value Δ
U
is 0.044pu (the voltage base value is selected as 311V).
Different effects of the inverter nonlinearities described by twodimensional region with four cases.
Define the HF amplitude and DC offset of the induced current as
X
and
Y
, respectively. When
X
and
Y
meet 0.1<
X
+
Y
<0.1 and 0.1<
X
+
Y
<0.1, they belong to case 3. When
X
and
Y
meet –
X
+
Y
>0.1, they belong to case 4. In these two cases, the voltage error only contains the sinusoidal component without other harmonics. When
X
and
Y
meet the conditions 0.1<
X
+
Y
<0.1 and
X
+
Y
>0.1, they belong to case 1. Thus,
U
_{0}
=0.4
X
and
U
_{1}
=0.040.4
Y
. When
X
and
Y
meet the conditions 
X
+
Y
<0.1 and
X
+
Y
>0.1, they belongs to case 2. Thus,
U
_{0}
=0.4
X
,
U
_{1}
=0.040.4
Y
, and
U
_{2}
=0.040.4Y.
After analyzing the distribution of the four cases in the twodimensional region, the 2
^{nd}
and 3
^{rd}
harmonics are selected to be analyzed because the amplitudes of the highorder harmonics are very small and can be neglected. According to the harmonic analysis above, the amplitudes of the 2
^{nd}
and 3
^{rd}
harmonics, which are represented by
A
_{2}
and
A
_{3}
, can be obtained:
 F. Inverter Nonlinearity Effects on Offline Inductance Identification
According to the above analysis, the inverter nonlinearities cause voltage errors during the offline inductance identification process. The actual voltage drop of the dq inductances and stator resistance is
U_{inj}
sin(
ω_{h}
t)Δ
u
. Therefore, the induced current
I
_{d(q)}
can be expressed as follows:
According to (3), the estimated values of the dq inductances are larger than the theoretical values due to the effects of inverter nonlinearities. In addition, the relationship between the estimated and theoretical values can be obtained:
where 
U_{inj}
sin(
ω_{h}
t)Δ
u
 is the fundamental component amplitude of
U_{inj}
sin(
ω_{h}
t)Δ
u
, and
and
L_{d(q)}
are the estimated and theoretical values of the dq inductances, respectively.
The voltage errors caused by inverter nonlinearities vary with the signal injection condition. There are inevitable estimation errors in case 1, case 2 and case 3 due to the fundamental voltage error. In case 4, the fundamental voltage error is zero and the DC voltage error does not influence the inductance estimation.
IV. EXPERIMENTAL RESULTS
The analysis of the inductance identification algorithm has been validated in a 2.2kW interior PMSM (IPMSM) drive as shown in
Fig. 12
. The rated parameters of the IPMSM are listed as follows: 380V, 5.6A, 50Hz, 21Nm, and 1000r/min. An intelligent power module FP25R12KT4 is used. A STM32F103 ARM is adopted to execute the whole identification algorithm. The PWM frequency is 6kHz, and the dead time is 3.2μs. The frequency of the injected HF voltage signal is 300Hz. The current reference increases linearly with 0.03pu/s during the estimation of the inverter nonlinearity parameters. A 12bit absolute encoder is installed to obtain the actual position. This is used solely for showing the electrical angle of the rotor during the identification process.
Experimental platform of 2.2kW IPMSM.
The waveform of the whole identification process is shown in
Fig. 13
. The aphase current and the estimated
L_{d}
and
L_{q}
are given. The rotor position is obtained by the initial position identification method. This method can realize the identification process at standstill. The supplementary daxis DC current is 0.1pu. The injected HF voltage increases gradually to avoid overcurrent and to guarantee signal intensity. The dq axis inductances converge to stable values, 32.4mH (the theoretical value of
L_{d}
is 31.6mH) and 65.2mH (the theoretical value of
L_{q}
is 62.8mH), respectively.
Experimental waveforms of the dq axis inductance identification.
Fig.14
shows the actual and estimated rotor position waveforms during the inductance identification process with different initial rotor positions. Firstly, the HF voltage signal is injected to obtain the initial position. Then, a pulse signal is injected to detect the polarity. After acquiring the initial rotor position, the inductance identification process can be started. The actual rotor positions in
Fig. 14
(a), (b), (c), and (d) are 46º, 200º, 257º, and 328º, respectively, while the estimated rotor positions are 50º, 203º, 262º and 334º, respectively. From
Fig. 14
, the position stays constant during the identification process. The results indicate that the proposed inductance identification method can be operated at a standstill no matter what the initial rotor position is.
Experimental waveforms of the actual and estimated rotor position during the inductance identification process. (a) 46º, (b) 200 º, (c) 257 º, and (d) 328 º.
Fig. 15
shows the experimental waveforms of the selflearning process for the inverter nonlinearity parameters. From top to bottom, the daxis voltage reference of the current regulator output, the estimated inverter nonlinearity gradient parameter, and the estimated saturation value of the voltage are given. According to (4), the inverter nonlinearity gradient
K
is equal to 15.8Ω. The saturation value of the voltage error is 13.66V.
Experimental waveforms of the selflearning process.
Fig. 16
shows the harmonic analysis of the induced current with different HF amplitudes when the DC offset is 0.12pu. From these results, the amplitude of the 2
^{nd}
harmonic component increases as the HF amplitude becomes larger.
Harmonic analysis of the induced current considering the effects of the inverter nonlinearities with different HF amplitudes, (a) 0.15pu, (b) 0.2pu, (c) 0.25pu, and (d) 0.3pu.
Fig. 17
shows the harmonic analysis of the induced current with different DC offsets when the HF amplitude is 0.1pu. It can be seen that the amplitude of the 2
^{nd}
harmonic component decreases as the DC offset becomes larger.
Harmonic analysis of the induced current considering theeffects of the inverter nonlinearities with different DC offsets, (a) 0.05pu, (b) 0.15pu, (c) 0.2pu, and (d) 0.3pu.
Fig. 18
and
Fig. 19
show the theoretical and experimental results of the 2
^{nd}
and 3
^{rd}
harmonics of the induced current caused by voltage errors with different DC offsets and HF amplitudes. The experimental results closely match the theoretical results.
Analysis of the 2^{nd} harmonic caused by voltage error with different DC offsets and HF amplitudes, (a) the theoretical result, (b) the experimental result.
Analysis of the 3^{rd} harmonic caused by voltage error with different DC offsets and HF amplitudes, (a) the theoretical result, (b) the experimental result.
The experimental result of the
L_{d}
identification is given to verify the effectiveness of the proposed analysis.
Fig. 20
shows the estimated value of
L_{d}
under different injection conditions. It can be seen from these results that with the increasing of the DC current component, the estimated value of
L_{d}
is more accurate. When the injected DC component and the HF component are relatively small as case 3 shows, the maximum estimation error reaches 44.9%, while the maximum estimation error is reduced to 4.6% in case 4 when the DC component is high enough.
The experimental results of L_{d} estimation under different injection conditions.
V. CONCLUSION
The harmonic analysis method of inverter nonlinearity effects on offline inductance identification with a DC offset and high frequency voltage injection for PMSMs at standstill using a VSIfed drive was proposed. The parameters of the inverter nonlinearities can be obtained by the selflearning method. Then the current harmonic induced by inverter nonlinearities was analyzed in four different cases with different injection conditions. The rotor can be kept at the initial position during the inductance identification and inverter nonlinearity estimation processes. The proposed offline inductance identification can be applied in generalpurpose drives for PMSM sensorless control. According to the characteristics of the inverter nonlinearities, the harmonic analysis can be accomplished. The experimental results match the theoretical results closely.
Acknowledgements
The research work was supported by the Research Fund for the National Science Foundation of China (51207030, 51522701), grants from the Power Electronics Science and Education Development Program of Delta Environmental & Educational Foundation (DREK2015002), and the Fundamental Research Funds for the Central Universities (HIT.BRETIII.201407).
BIO
Gaolin Wang (M’13) received his B.S., M.S. and Ph.D. degrees in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2002, 2004 and 2008, respectively. In 2009, he became a Lecturer in the Department of Electrical Engineering, Harbin Institute of Technology, where he has been a Full Professor of Electrical Engineering, since 2014. From 2009 to 2012, he was a Postdoctoral Fellow in the Shanghai Step Electric Corporation, Shanghai, China, where he was involved in traction machine control for directdrive elevators. He has authored more than 50 technical papers published in journals and conference proceedings. He is the holder of 10 Chinese patents. His current research interests include permanent magnet synchronous motor drives, high performance directdrives for traction systems, position sensorless control of AC motors, and digital control of power converters.
Ying Wang received her B.S. degree in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2014, where he is presently working toward her M.S. degree in Power Electronics and Electrical Drives in the School of Electrical Engineering and Automation. Her current research interests include the position sensorless control of IPMSMs.
Li Ding received his B.S. degree in Electrical Engineering from Shanghai University, Shanghai, China, in 2013. He is presently working toward his M.S. degree in Power Electronics and Electrical Drives in the School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin, China. His current research interests include permanent magnet synchronous motor drives and position sensorless control.
Lei Yang received his B.S. degree in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2015, where he is presently working toward his M.S. degree in Power Electronics and Electrical Drives in the School of Electrical Engineering and Automation. His current research interest include permanent magnet synchronous motor drives.
Ronggang Ni received his B.S. and M.S. degrees in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2010 and 2012, respectively. He is presently working toward his Ph.D. degree in Power Electronics and Electrical Drives in the School of Electrical Engineering and Automation, Harbin Institute of Technology. His current research interests include the optimal design and control of permanent magnet synchronous motors.
Dianguo Xu (M’97, SM’12) received his B.S. degree in Control Engineering from the Harbin Engineering University, Harbin, China, in 1982, and his M.S. and Ph.D. degrees in Electrical Engineering from the Harbin Institute of Technology (HIT), Harbin, China, in 1984 and 1989 respectively. In 1984, he became an Assistant Professor in the Department of Electrical Engineering, HIT. Since 1994, he has been a Professor in the Department of Electrical Engineering, HIT. From 2000 to 2010, he was the Dean of the School of Electrical Engineering and Automation, HIT, and from 2010 to 2014, he was the Assistant President of HIT. He is presently the Vice President of HIT. He has published over 600 technical papers. His current research interests include renewable energy generation technology, sensorless vector controlled motor drives, and high performance PMSM servo systems. Dr. Xu is an Associate Editor of the IEEE Transactions on Industrial Electronics and the IEEE Journal of Emerging and Selected Topics in Power Electronics.
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