Generally, internal parameters of the motors and generators can be divided to the resistance and inductance components. They can become a cause of the changing internal parameters because they have sensitive characteristics due to external conditions. The changed parameters can generate the outputs which include error values from the speed and current controllers. Also, it can bring the temperature increase and mechanical damage to the system. Therefore, internal parameters of the motors and generators need to obtain their values according to the external conditions because it can prevent the mechanical damage caused by the changed parameters. In this paper, the offline parameter identification method is verified using the Goertzel algorithm. The motor used in the simulation and experiments is an interior permanent magnet synchronous motor (IPMSM), and the proposed algorithm is verified by the simulation and experimental results.
1. Introduction
Motors and generators are used in many areas such as robots, automobiles and ships which require reliability, efficiency, and precision. Motors and generators use controllers according to the purpose of the usage, and current and speed controllers are the most common controller among them. On the other hand, internal parameters of the motor and generator can be divided into the resistance and inductance components, and the current and speed controllers generate outputs according to each of the gain by internal parameters. However, these internal parameters have sensitive characteristics due to external conditions. Generally, internal temperature of motors and generators increases by the operation time, and it can change values of the internal parameters. Therefore, there is an error between the specification and actual parameters of the system, and it can bring the negative influence to the system. Also, temperature of the system can further be increased by error values of the controller, and it can generate the unbalanced power, mechanical vibration and a shortcircuit between wires. Therefore, the motor control needs the information of changed internal parameters by the increased temperature for the highefficiency control
[16
,
19]
.
The Goertzel algorithm is a method that performs the discrete Fourier transform (DFT) from the continuous signals. Generally, the fast Fourier transform (FFT) method has shorter calculation time than the DFT method. Hence, it is used often in the signal analysis method. However, the Goertzel algorithm has an advantage in the case of detecting numbers of smaller signals because it is faster than the FFT method. Also, it is possible to implement through the micro controller unit (MCU) of the low performance specification. Therefore, the Goertzel algorithm detects the highfrequency components of the voltage and current which can also be divided into real and imaginary components and can be expressed into the magnitude and phase of the highfrequency components
[1
,
3
,
5]
.
In this paper, parameters of the interior permanent magnet synchronous motor (IPMSM) are estimated using an offline identification method. Also, the proposed algorithm is divided into the two modes. First, the stator resistance value is estimated. Generally, the stator resistance estimation mode is a method using the relationship expression of the voltage, current and resistance. The offline parameter identification method is the estimation method in standstill, and it is applied to the daxis current of reactive component. Second, values of the d and qaxes inductance are estimated. The inductance estimation mode used the Goertzel algorithm, and it is method that injects a highfrequency voltage. The highfrequency voltages are generated, and are injected into the d and qaxes. Also, the injecting highfrequency voltages are analyzed to the Goertzel algorithm
[5
,
12]
.
Hence, the total impedance is calculated by magnitude and phase of d and qaxis from the component of highfrequency and each of the d and qaxis inductances is estimated by considering the ironloss. The inductance estimation method is estimated in standstill.
The offline parameter identification method of an IPMSM using the Goertzel algorithm verifies feasibility through the simulation and experiment.
2. Resistance Estimation
 2.1 Modeling of IPMSM
Fig. 1
shows the structure of an IPMSM. The IPMSM has different inductance of d and q axis depending on the length of the effective airgap. The part having the pole in rotor is defined by daxis, and it is flux generating direction by the field winding. The qaxis is defined to be in the direction of the electrical angle of 90° from the daxis.
Structure of the IPMSM.
Fig. 2
shows the equivalent circuit of the IPMSM structure, and the voltage equations for each d and qaxes are shown in Eq. (1) and (2)
[15

17]
.
Equivalent circuit of the IPMSM.
where,
u
_{ds}
and
u
_{qs}
are the injected d and qaxis voltages,
i
_{ds}
and
i
_{qs}
are d and qaxis currents,
R
_{s}
is the stator resistance,
ω
is the angler velocity and
λ
_{ds}
and
λ
_{qs}
are the d and qaxis flux.
The backEMF is defined in (3) because the proposed algorithm estimates the parameters in standstill.
 2.2 Resistance estimation using daxis current injection
Fig. 3
shows the block diagram of the stator resistance estimation. The stator resistance estimation is composed of the coordinate transformation and current controller, and the current controller controls the current using a rotor position information obtained from an encoder.
Block diagram of the stator resistance estimation.
Generally, the stator resistance can be estimated using a relationship between the voltage, current and resistance
[13
,
14]
. However, the estimated values include the error of the nonlinear voltage curve of insulated gate bipolar transistors (IGBTs). In the case of the stator resistance estimation, the input current which is close to the rated current has to be applied since the nonlinear voltage curve should be considered
[8
,
9]
. The stator resistance estimation is defined as
where the estimated
R
_{s}
is the estimated stator resistance,
v
_{de}
* is the daxis synchronous reference frame voltage and
i
_{de}
* is the daxis synchronous reference frame current.
In this paper, the proposed method is standstill of motor. Since the daxis current is a reactive component which is in the flux generating direction, the daxis current close to the rated value will not generate the speed
[3
,
4]
. Therefore, daxis reference current used the value close to rated current, and qaxis reference current used the value close to zero.
3. Inductance Estimation
 3.1 Relationship between flux and inductance
The flux of the IPMSM is defined by the stator flux and flux by the permanentmagnet and which it is expressed as
where
λ
_{abcs(s)}
is the linkage flux in the stator winding and
λ
_{abcs(f)}
is the linkage flux by the permanentmagnet.
Each flux of the 3phase can be expressed to the d and qaxes fluxes in (6).
where
L_{ls}
is leakage inductance,
L_{m}
is magnetizing inductance of the d and qaxis and
L_{sf}
is inductance of permanentmagnet according to the position of rotor.
Fig. 4
is daxis equivalent circuit of inductance.
i_{di}
is the ironloss current,
R_{i}
is the ironloss resistance and
i_{dm}
is the magnetizing current daxis. Also, daxis current is stationary coordinate and it is made up by the sum of the ironlosses current and magnetizing current.
daxis equivalent circuit of the inductance.
Therefore,
Fig. 5
can be separated by
I
and
II
regions.
I
region is the magnetizing part and its slope shows linear inductance. Also,
II
region is the leakage part and
P
is the saturation point. The inductance in
II
region has a little value by saturation state.
Characteristic of the flux and current.
Eq. (7) and (8) show the saturation and linear state of the flux by the magnetizingcurrent, respectively.
Eq. (8) can be redefined to (9).
Hence, the daxis inductance by the magnetizing current is defined by (10).
However, the daxis inductance of (10) does not consider ironlosses components. Therefore, ironlosses components need to be added in this equation to obtain the correct value of the daxis inductance.
 3.2 Inductance estimation using goertzel algorithm
FFT is popularly used in the signal analysis because it has a short calculation time. However, FFT requires the MCU of highperformance. On the other hand, the Goertzel algorithm is the DFT method. It has a long calculation time. However, the Goertzel algorithm has a short calculation time in the case of detecting the number of a small signal. Also, it is possible to implement through the MCU of lowperformance.
Table 1
shows the computation efforts of the Goertzel algorithm and FFT
[19]
.
Comparison of the computation effort (N is the number of sampling)
Comparison of the computation effort (N is the number of sampling)
The inputs of the Goertzel algorithm are the voltage and current of either the daxis or qaxis.
Fig. 6
shows the block diagram of the Goertzel algorithm
[1

5]
.
Block diagram of the Goertzel algorithm.
In this paper, highfrequency is injected because the parameters are estimated in the standstill. Hence, the voltage and current include highfrequency. Eq. (11) defines highfrequency voltage (
V_{hf}
).
where,
θ_{hf}
is angle of the highfrequency and
V_{Mag,hf}
is the magnitude of highfrequency signal.
The inputs of the Goertzel algorithm are the voltage and current including the highfrequency. The Goertzel algorithm can obtain the real and imaginary components from the signal including highfrequency, and which analyzes the continuous signal during the sampling number N. Hence, the outputs show the real and imaginary components. Also, the Goertzel algorithm can be indicate the magnitude and phase by the real and imaginary components, and the output is the same with an input highfrequency. However, the Goertzel algorithm includes errors in the magnitude and phase information because they include error values by N and the discrete time delay. Therefore, the Goertzel algorithm is required for selecting the appropriate number N, and it needs to consider error values of the discrete time delay.
However, in this paper, the voltage and current of the proposed Goertzel algorithm have the same time delay. Also, the better resolution can be obtained with the higher N, and N is selected from 50 to 100 considering the calculation time of a Texas Instruments TMS320F28335 digital signal controller (DSC) is used as the MCU.
To estimate the inductance, Inputs of the Goertzel algorithm are each of the current (
I_{ds}, I_{qs}
) and voltage (
V_{ds}, V_{qs}
) which contain highfrequency components. Inputs, in the case of the daxis inductance estimation, are the daxis stationary reference frame current and voltage, and are the qaxis stationary reference frame current and voltage in the case of the qaxis inductance estimation. The outputs of the Goertzel algorithm are the real and imaginary components as shown in
Fig. 6
.
Fig. 7
shows the simulation results of the Goertzel algorithm and the extracted voltage and current consist of only the frequency component of 1000 [Hz].
Estimated results of highfrequency components from the Goertzel algorithm: 1000 [Hz].
By using the real and imaginary components
[4
,
5]
, the magnitude (
I_{Mag,hj}
,
V_{Mag,hf}
) and phase (
θ_{I,hf}
,
θ_{V,hf}
) are calculated as
where, cos=cos(
θ_{hf}
/
dt
), cos2=2×cos, and sin=sin(
θ_{hf}
/
dt
).
Also, the magnitude of (12) is without the dead time compensation. Hence, the valid voltage (
V_{mag,hf,valid}
) is redefined as
V_{Dead}
is the compensation voltage of the dead time, and it is defined as
where
V_{DC}
is the DC link voltage, and
T_{sw}
is the switching period. Also,
T_{dead}
is the dead time and it was set 2 [μsec]
[12]
.
The total impedance (
Z
) can obtain from the magnitude and the phases of the voltage and current in the Goertzel algorithm. The total impedance (
Z
) can be divided into the resistance (
Z_{R}
) and inductance (
Z_{ωL}
) as shown in
Fig. 8
. Therefore, the magnitude (
Z_{Mag}
) and phase (
θ_{Z}
) of the total impedance is defined as
Total impedance: (a) Vectors of impedance; (b) Equivalent circuit of impedance.
The impedance of the real and imaginary components is defined as
The ironloss is defined by (18) from the magnitude and phase of the impedance, and it is used to define the inductance by (19)
[5

7]
.
Fig. 9
shows the total block diagram of the proposed parameter estimation method.
The total block diagram of the proposed parameter estimation method.
4. Simulation and Experimental Results
 4.1 Simulation
The simulation is performed using the PSIM tool. The rated power, speed and current (RMS) of the motor are 11 [kW], 1750 [rpm] and 19.9 [A] respectively.
Table 2
shows parameters of the proposed parameter identification method, and
Table 3
shows the motor specifications. The daxis reference currents are values close to the rated value in the case of the resistance estimation. However, d and qaxes reference current are values close to zero in the case of the inductance estimation which applies the reference voltage of the highfrequency.
Parameters of the proposed algorithm
Parameters of the proposed algorithm
Motor specifications
Also, the Goertzel algorithm can obtain better resolution by an increasing number of sampling, and outputs can bring better exact values. The simulation results of
Fig. 10
select the frequency of 1000 [Hz] and the number of the sampling is 50.
The proposed parameter identification in standstill: (a) daxis; (b) qaxis.
In this paper, the proposed offline parameter identification method was performed the simulation for the right calculation amount of MCU. Hence, the simulation was performed in the same condition as the experiment, and the validity of the proposed algorithm was verified by the simulation.
Fig. 10
shows the simulation result of the proposed parameter identification method. It shows estimated results of both resistance and inductance and these estimation modes can be checked through a flag sequence. The error range of parameter estimation is set to 5 [%]. If the flag sequence is 1, the proposed method of parameter identification estimates the value of stator resistance. Also, if the flag sequence is 2, the proposed method of parameter identification estimates the values of d and qaxes inductance. The value of the estimated stator resistance can be retained during estimating mode of inductance. Further, estimating mode of inductance using an estimated value of stator resistance is conducted by using the highfrequency voltage and output of goertzel algorithm.
Fig. 11
shows the comparison result of the estimation performance according to the changing highfrequency.
Fig. 11(a)
is the case of daxis, and
(b)
is the case of qaxis for the range of the changing frequency from 100 [Hz] to 1000 [Hz]. The daxis inductance was the verified approximated value in the case of injected components of from 100 to 300 [Hz], and the qaxis inductance was verified approximated value in the case of components ranging from 500 [Hz] to 1000 [Hz]. The performance of the Goertzel algorithm is determined by the highfrequency and the number of sampling. Therefore, the highfrequency is set such that it does not cause any influence to the fundamental frequency.
Estimation performance according to the high frequency: (a) L_{d}; (b) L_{q}.
Also, the Goertzel algorithm can obtain better resolution by an increasing number of sampling, and outputs can bring better exact values.
In this paper, the proposed offline parameter identification method was performed the simulation for the right calculation amount of MCU. Hence, the simulation was performed in the same condition as the experiment, and the validity of the proposed algorithm was verified by the simulation.
 4.2 Experimental results
Fig. 12
shows the experimental set of a threelevel NPC inverter, control board and MG set.
The experiment equipment: (a) Threelevel inverter and control board; (b) MG set.
Fig. 13
shows the resistance estimation. The estimated resistance values are used for the inductance estimation. Therefore, it should be an accurate estimation. Also, the MCU must consider the calculation time because it exceeds the calculation amount by the size of the array from the Goertzel algorithm.
Experimental result of the resistance estimation
Fig. 14
shows the highfrequency voltage analyzed from the Goertzel algorithm. The inputs of the Goertzel algorithm set by each of the d or qaxis current and voltage.
Figs. 14(a)
and
(b)
each show the d and qaxes highfrequency voltage, and also show the outputs of the analyzed result of the highfrequency voltages from the Goertzel algorithm.
Highfrequency voltage analysis of the Goertzel algorithm: (a) daxis; (b) qaxis.
Fig. 15
shows the estimated d and qaxes inductance.
Fig. 15(a)
is the estimated daxis inductance, and
Fig. 15(b)
is the estimated qaxis inductance. The estimation method of inductance uses the estimated resistance value. The estimated resistance value is approximately 0.352 [Ω], and the estimated inductance includes some errors by the estimated resistance. Also, the estimated inductances are influenced by the highfrequency and resolution of the Goertzel algorithm. Therefore, the estimated value of daxis inductance is approximately 0.01321 [H] and the estimated value of qaxis inductance is approximately 0.01554 [H].
Experiment result of the estimated inductance (f_{sw} : 10 kHz): (a) L_{d}; (b) L_{q}.
Additionally,
Fig. 16
shows the estimated d and qaxes inductance in the switching frequency 1 [kHz]. The control period was set to 500 [μsec]. The initial values of d and qaxes inductance are 50 [%] of the actual value. The experimental result of
Fig. 16
shows performance of the proposed parameter identification method by the initial values and switching frequency. The estimated result values of daxis inductance and qaxes inductance are approximately 0.01336 [H] and 0.01578 [H].
Experiment result of the estimated inductance (f_{sw} : 1 kHz): (a) L_{d}; (b) L_{q}.
In this paper, the proposed parameter identification method in standstill was verified through the experimental results.
5. Conclusion
The proposed algorithm is divided into two modes in this paper. First mode estimates the stator resistance values and second mode estimates the inductance values. The stator resistance was estimated exactly in the first mode using a relationship between the voltage, current and resistance, and the first mode injects the daxis current of the reactive component. Also, the qaxis current is the value close to zero. Hence, the daxis reference current was set to the value close to the rated current, and the qaxis reference current was set to the zero value because the motor and generator are at standstill.
On the other hand, the second mode injected the highfrequency voltage, and analyzed the voltage and current of the d and qaxes by the Goertzel algorithm. Hence, the second mode estimates the total impedance using a magnitude and phase of the highfrequency components. Also, this mode can obtain the ironloss using an estimated total impedance, and it estimates inductance of the d and qaxes using an outputs of the Goertzel algorithm.
In this paper, the performance of the proposed algorithm has been validated through the simulation and experimental results.
Acknowledgements
This work was supported by the Human Resources Development program (No.20134030200310) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy.This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2A10006090).
BIO
JaeSeung Yoon received the B.S. and M.S. degrees in Electrical and Computer Engineering from the Ajou University, Korea, in 1996 and 2013, respectively. He is currently working toward the Ph.D. degree at Ajou University, Korea. His research interests include electric machine drives and switched reluctance motor drives.
KyoungGu Lee received his B.S. degree in Mechatronic Engineering from Korea Polytechnic University, Siheung, Korea, in 2012. He received his M.S. degree at Ajou University, Suwon, Korea, in 2015. He is assistant research engineer of the WOOJIN Industrial Systems Co., Ltd. Korea. His research interests include electric machine drives, switched reluctance motor drives and energy storage system.
JuneSeok Lee received the B.S. and M.S. degrees in Electrical and Computer Engineering from the Ajou University, Korea, in 2011 and 2013, respectively. He is currently working toward the Ph.D. degree at Ajou University, Korea. His research interests include gridconnected systems, multilevel inverter and reliability.
KyoBeum Lee received the B.S. and M.S. degrees in electrical and electronic engineering from the Ajou University, Korea, in 1997 and 1999, respectively. He received the Ph.D. degree in electrical engineering from the Korea University, Korea in 2003. From 2003 to 2006, he was with the Institute of Energy Technology, Aalborg University, Aalborg, Denmark. From 2006 to 2007, he was with the Division of Electronics and Information Engineering, Chonbuk National University, Jeonju, Korea. In 2007 he joined the School of Electrical and Computer Engineering, Ajou University, Suwon, Korea. He is an associated editor of the IEEE Transactions on Power Electronics, the Journal of Power Electronics, and the Journal of Electrical Engineering & Technology. His research interests include electric machine drives, renewable power generations, and electric vehicle applications.
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