estimation of permanent magnet motor parameters

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IbEE Industry Application Society

Annual Meeting

New Orleans, Louisiana, Octcber 5-9, 1997

Estimation of Permanent Magnet Motor Parameters

S . Weisgerber A. Proca A . KeyhaniThe Ohio State University

Electrical Engineering Department

Columbus, OH 43210

Phone (614) 292-4430 E-Mail: [email protected]

Abstract - To properly design and optimize a control system

of a permanent-magncf (PM) machine, the machine model

and its parameters mu,Ft be known. This study presents a

method for developing a model and estimating the

parameters of a PM nraclzine from standstill time-domain

step response data. A dq-axis equivalent circuit model with

a general number of da'mper windings is defined to describetlze behavior of the E'M machine. Standstill simulation

studies were performed on a known PM machine to

generate syntlzetic test data to be used for estimation

procedure verification. The stand-still dq-axis model is

subjected to a step-inp,rct signal and the resulting output is

used fo r parameter estimation. Th e output-error estimation

algorithm is used to evtimate the unknown parameters ofthe dq-axis model, Stand-still tests were then performed on

a 4 pole, 1 kW PM machine to obtain data. The PM

machine parameters were estimated and the model verijied

against experimental ifiput and output test data.

I. INTRODUCTION

Several different papers have been published that study

techniques used to estimate the parameters of synchronous

machines [13. This study focuses on model identification and

parameter estimation of synchronous permanent-magnet

(PM) machines using time-domain estimation techniques.

PM machine modeling is seen in such works as [3],[4], and

[ 5 ] . The estimation method used in this study [1][5], consists

of fixing the rotor of the PM machine in a specific position,

applying a simple DC voltage source, and collecting input

and output data to be used for estimation.

The present study focuses on a dq-axis circuit model of thePM machine with a k;eneral number of damper windings,

experimental technique: used to collect data, and m ethod of

applying the measured data to identify the correct m odcl andto estimate the m achine parameters.

11. PROBLEM DESCRIPTION

The objective of this study is to develop a method that can be

used to model the stand-still dq-axis model of a PM machine,

0-7803-4067-1 97/$10.00 0 1997 IEEE.

obtain input and output data from a PM machine, and

estimate the standstill dq-axis machine parameters.

To study this problem a PM m achine with known parameters

is simulated to create synthetic input and output test data to

be used during the identification process. Next, to evaluate

the effect of noise on the estimation method the test data iscorrupted with noise of a known distribution. Once the

noise-corrupted test data is obtained it can be treated as if it

came from a PM machine with an unknown system model

and parameters. A system model is then assumed and th e

output-error estimation method is used to estimate the

parameters of the unknown system.

Once the simulation studies verify that th e technique used to

estimate the parameters is valid, tests are performed on a

working 4 pole, 1 kW PM machine. Experimental input and

output data is collected from the PM machine to be used for

system model identification and parameter estimation. A

system model is assumed and the output-error estimationmethod is used to estimate the parameters. The assumed

model and estimated parameters are then validated against

the actual experim ental data. Next, if required, a different

model with more damper windings (section 111.) is chosen

and the same procedure is executed until the identified model

and estimated parameters are validated.

111. PERMANENT MAGNET MACHINE MODEL

The first step in identifying the permanent m agnet m achine is

to develop a generic dq-axis model that can be applied to any

PM machine over any operational frequency range. This

proposed generic model is derived from the PM machineshown in fig. 1 and defined by the electrical circuits shown in

Fig. 2. The circuits in Fig. 2 were derived by first

considering a PM machine such as the one shown in Fig. 1,

writing the voltage equations that describe the machine in theabc reference frame, and then transforming the equations tothe dq-axis reference frame [2] . Fig. 1 shows a generic two-

pole three phase PM machine with an unknown number of

damper windings in the rotor. The damper windings in the

rotor represent induced currents circulating in the rotor (eddy

29

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mailto:[email protected]



sensor

cs axis1 \ axis

Figure 2b. PM machine q-axis equivalent circuit

The state-space equations that describe the circuits in Fig. 2

where j represents either d o r q and p = &/dt :

y l = C,ixjv ( k ) ( 2 )

where,

x = [id i,, . . . i M d ] xq = [iq iZq . . . i N q ] (3)

Figure 1. Brushless permanent magnet machine

M = number of d-axis damper windings

N = number of q-axis dam per windings

currents), which are induced by harmonics in the applied

voltages and/o r oscillations in the rotor speed. Damper

windings are no t required to be built physically into the rotor

for purposes of creating starting torque since the motors in

question are controlled to be in synchronism at all speed

values. The circuits in Fig. 2 show the dq-axis of a PM

machine with M unknown d-axis damper windings, and N

unknown q-axis dam per windings. The number of damperwindings in the d-axis and the q-axis are determined through

the identification schem e proposed in this paper.

A mathematical representation is derived from the circuits

shown in Fig. 2. The number of damper windings to be used

is determined by using testing and identification procedures

over the PM mac hine's operational frequency range. The

equations shown below are given in state-space form based

on a general number of d-axis and q-axis damper w indings.

_ _

Figure 2a. PM machine d-axis equivalent circuit

30

U , =[Vd 0 . . . olT Uq = [ v q . . . 0IT (4)

The system output is,

Yi = [ij] (7)

The inductance matrix Lj and the resistance matrix Rj are

generalized as,



O l

identiped=f(gl

rrrimuted

Rd =

f

R, =

-

R. , 0 . . 0

0 RI,

. .

0 . . . R,,-

. .R n , I

The off-diagonal terms in the Lj matrices are all Lmj and the

off-diagonal terms in the Rj matrices are all zero.

In the above equations the variables w(k) an d v(k) represent

the process noise and i.he measurement noise respectively.

The process noise is created by a naturally occurring

disturbance to the input sequence. Whereas the measurement

noise is present because all measurements are inherently

subject to error, the intensity of the measurement noise is

based on the quality of the sensor being used

The unknown parameter sets Bj are the parameter values to

be estimated from the stand-still step response input and

output data using the output-error estimation algorithm.

During standstill cond tions th e term s o A, an d o , h , a r e

equal to zero hence the unknown parameter sets for the d-

axis and the q-axis are,

IV . OUTPUT ERROR PARAMETER ESTIMATION

ALGORITHM

As stated this paper will focus its study on the evaluation of

the output error method (OE) for estimating PM machine

parameters. Th e oulput error estimation algorithm is

described in the flow cl-,art show n in Fig. 3 . The output errorestimation algorithm is based on minimizing a cost function

defined as,

1 N

N =O

V ( 0 )=- ( e ' ( k ) e ( k ) }

where e(k) is defined as the difference between the measured

output Y(k) and th e estiinated output ? ( k ) , and N is the total

number of measured data points.

I I

Figure 3 . The output error estimation algorithm

V. STUDY PROCESS

The study process consisted of first verifying the estimation

procedure through simulation and synthetic data and then

applying the technique to an actual PM machine. A test was

set up to subject a PM machine to a DC step input voltage

and collect the input and output data. To collect the

experimental data the following procedure was used. Using a

four pole, 1 kW PM machine, a DC step voltage signal was

applied to the machine for a period of time long enough so

that the system reached steady-state. During this time, the

step response input and output is collected as experimental

data to be used during the estimation procedure. To be able

to execute the described test, the PM machine must be setup

up specifically either for the d-axis test or for the q -axis test.

Fig. 4 and Fig. 5 show how the rotor should be positioned for

each test. As shown in Fig. 4, for the d-axis step response

test the q-axis is aligned with th e as-axis which represents a

rotor angle 0, of zero degrees. For the q-axis step response

test, the rotor angle 0, is 90" as shown in Fig. 5. Here the d-

axis is aligned with the as-axis. Based on the test setup

shown, the measured variables from the expe rimental test are

I and V. Both I an d V are variables in the 3-phase abc

reference frame. To apply the experimental data to the dq-

axis circuit model shown in Fig 2 the measured data must

undergo a transformation [2]. Based on the rotor position for

each test, d-axis or q-axis, the measured variablesI

and V canbe transformed using the following equations.

For the d-axis step response test,

JsI ( k ) = - - i d ( k)

2

31



Figurc 4. Rotor position for the d-axis step response test

and for the q-axis step respon se test,

cs a x i s &

Figure 5 . Rotor position for the q-axis step response test

J3I ( k ) = - i d ( k )

2

VI. ESTIMATION PROCEDURE

(17)As described previously and verified through simulation, the

estimation procedure can be generalized in the following

steps:

(i) Collect the standstill step response data using the testsetup described for both the d-axis and the q-axis.

(ii) Use the transformation equations (15) through (18) toconvert the test measurements to the dq-axis reference

frame.

(18)

32



Assume model with minimum number of damper

windings.

Using the standst 111 dq-axis equivalent circuit mode l and

the OE estimation algorithm, identify the standstill

model and estimate model parameters.

Simulate identified model with estimated parameters

using measured input voltage.

Validate the identified model and estimated parameters

against the experimental test data.

Increase number of damper windings and estimate the

parameters to tr,y to obtain a better fit between the

simulated and measured response.

d-axis

v a l u earam e t e r

RS 1 3 . 9 m R

R l d 6 8 4 m RR2d 3.6 m R

LIS 38.0 nH

Lmd 89.86 pH

Lld 1 .20 m H

L2d 251.6 m H

(viii) Revalidate the simulated data against the measured

test data.

(ix) Continue adding damper windings until error betweenthe estimated and measured data is acceptable or does

not reduce significantly with the addition of new

dampers.

q-axis

parameter v a l u e

RS 73.9 m R

R l q 0 .4 1 3 R

R2q 67.0 m i 2

R3q 7.603 RLIS 38.0 n H

Lmq 92.34 p HLlq 0 .788 p H

L2q 0.108 m H

L3q 7.409 pH

VII. SIMULATION STUDIES AND RESULTS

rigma V a u e

parameter ,ialue

0.:!91612

0.80330

5.3900

0.0002H

Lmd 22450 H

L ld 0.11262H

L2d 0.0044H

The defined estimation procedure was first validated using

simulation studies. A PM machine was simulated to create

synthetic data with noise. Fig. 6 shows the step response for

the defined d-ax is test with synthetic noise added at a signal-

to-noise ratio o f 200:1. Table 1 shows the estimation results

using two damper windings. It can be seen from the table

that the simulation results validate the PM model

identification an d parameter estimation procedure.

Estimated Value

estimated value % error

0.2915 < l 0.03

0.77970 2.94

5.22180 3.12

0.0003 H 7 . 5 3

2.2 I70 H 1.25

0.0254 H 2.91

0.0042 H 5 16

"0 0.05 0.1 0.15

0oise,

Amps

10.05 0.1 0.15

Figure 6 . d-axis riynthetic test data w ith noise a dded

-0.5-0

time

VIII. EX PERIMENT AL STUDIES AND RESULTS

As mentioned, tests were performed on a 4 pole 1 kW PM

machine. The identification procedure was initialized,

including the inductances and resistances, based on apriori

information such as motor design data. The initial guess for

the motor stator winding resistance r , can be determined

based on the steady-state value of the current.

The PM machine q-axis model was identified using three

damper windings, whereas the d-axis model was identified

with two damper windings. The parameters of the model are

presented in Table 2.

TABLE 2. ESTIMATED DQ-AXIS PARAMETERS

The parameter estimation validation was performed for

several sets of data. Figs. 7 an d 8 represent the d-axis and q-

axis model validation for one of the sets. The input to the

model (top picture on each graph) differs from a typical dc

step, the cause being the use of a car battery for dc supply.

The input was intentionally kept in this form due to its

richness in frequencies. An alm ost perfect fit is observed in

the d-axis model. For the q-axis model, there is a m ismatchbetween the measured a nd estimated data that appeare d in all

data sets. Further study will be necessary to determine its

nature.

IX. CONCLUSIONS

The present paper focuses on the modeling of a permanent

magnet machine. First, simulation studies were conducted to

establish the modeling procedure and the effect of noise on

parameter estimation. An expe rimental setup was built to test

the machine. The testing consisted of aligning the rotor to

the d-axis and q-axis of the stator reference and applying

voltage steps to the stator windings. The win ding current and

the input voltage were recorded. Next, the model structurewas identified through several iterations on the number of

damper windings. Each iteration consisted of parameter

identification and comparison to the measured responses. In

this case, the number of damper windings for the d-axis andq-axis was different. The last part of the study was modelvalidation, in which different sets of input-output data were

used. The model was subjected to the measured input and the

model output was compared to the machine output.

33



10 I I I

35

I

30-5 - - 1 1 .

- - measured1imulated

820 ; ;5 / I ’

z 1 5 - l l ‘ .

iime in seconds

lime in seconds

Figure 7. d-axis validation

14 .....,.. i.,.. ..... ...........j ....: . . . . ........ . : ............ . . . . . . . .

0 00 1 0.02 0.03 0.04 0.05 006 0.07 0.08

ilme ~n econds

40

35

30

$25

E$2 0

$ 1 5

10

5

0

0 001 OW 003 004 005 006 007 00 8

ilme in seconds

Figure 8. q-axis validation

Future work will be developed following two main paths.

First, a complete dynamic model will be built for working

conditions. The structure of the model will have to include

the permanent magnet as a source of magnetic flux and the

back emf on the d-axis and q-axis (for the stand still testbeing zero). The present model will serve as a starting point

for it. Second, Finite Element Analysis (F EA) will be used to

determine the machine parameters and compare them to the

parameters obtained through the present method.

X. ACKNOWLEDGMENTS

This work is supported in part by Delphi Saginaw Steering

Systems.

XI. REFERENCES

[4

15

34

[l ] Keyhani andS .

I. Moon, “Maximum likelihoodestimation of synchron ous mach ine parameters an d study

of noise effect from flux dec ay”, IEE Proc., vol. 139, no.

1, pp. 76-80, Jan. 1992.

[2] P. C. Krause and Oleg Wasynczuk, ElectromechanicalMotion Devices. N ew York: McG raw-Hill, 1989.

[3] T. Sebastian, M.A. Rahman, “Modeling of PermanentMagnet Synchronous Motors”, IEEE Transactions on

Magnetics, vol. MAG-22, no. 5 , pp. 1069-1071, Sept.

1986.

T. Sebastian, G.R. Slemon, “Transient Modeling and

Performance of Variable-Speed Permanent Magnet

Motors”, IEEE Transactions on Industry Applications,

vol. 25, no. 1, Jan/Feb 1989.I. Kamwa, P. Viarouge, M. Ferfra, “Modeling and

Identification of Permanent Magnet Synchronous

Machines from Standstill Time Response Tests Using a

Non-Linear Method”, IEEE 1993

estimation of permanent magnet motor parameters

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