Estimation of Induction Motor Equivalent Circuit Parameters from Nameplate Data

Keun Lee, Stephen Frank,

and Pankaj K. (PK) Sen

Division of Engineering

Colorado School of Mines

Golden, Colorado 80401

Email: kelee@mines. edu

Luigi Gentile Palese

Electricity, Resources,&

B uilding Systems Integration

National Renewable Energy Laboratory

Golden, Colorado 80401

Mahmoud Alahmad and Clarence Waters

Charles W. Durham School of

Architectural Engineering and Construction

University of Nebraska-Lincoln

Omaha, Nebraska 68182

Abstract-The induction motor equivalent circuit parameters are required for many performance and planning studies involving induction motors. These parameters are typically calculated from standardized motor performance tests, such as the no load, full load, and locked rotor tests. However, standardized test data is not typically available to the end user. Alternatively, the equivalent circuit parameters may be estimated based on published performance data for the motor. This paper presents an iterative method for estimating the induction motor equivalent circuit parameters using only the motor nameplate data.

I. INTRODUCTION

Induction motors are extensively used to drive mechanical

loads in commercial and industrial power systems due to

their low cost and reliability. Many engineering studies

including efficiency studies, fault studies, calculation of volt

age drop during motor starting, planning studies for power

factor correction, and the development of the motor torque

speed characteristic-require the induction motor equivalent

circuit model in order to evaluate motor behavior [1]-[3].

The induction motor equivalent circuit parameters are usu

ally computed from full load, no load, and locked rotor test

data as per IEEE Standard 112 [4]. For most commercially

available or previously installed motors, however, neither the

original test data nor the equivalent circuit parameters are

available from the motor manufacturer. In many cases, only

the motor nameplate data are available. These data include

the rated voltage, rated output power, speed, efficiency, and

power factor of the motor, as well as (in the United States) its

NEMA (National Electrical Manufacturers Association) design

characteristics. In this paper, we present a method to estimate

the induction motor equivalent circuit parameters from the

motor nameplate data.

Several previous papers have described methods to estimate

the induction motor equivalent circuit parameters given a set

of performance data [2], [5], [6]; these methods are reviewed

in Section II. Our method differs in that it requires only the

motor nameplate data. B ecause the nameplate is physically

affixed to the motor, the nameplate data are reliably available

even for already installed motors.

The estimation method proposed in this paper extends the

algorithm proposed by Haque [2] and consists of four steps:

978-1-4673-2308-6/12/$31.00 2012 IEEE

1) Computation of the motor full load and starting power

requirements;

2) Estimation of the motor losses from the nameplate data,

NEMA design characteristics, and published typical

values [4], [7];

3) Development of a set of simultaneous, nonlinear equa

tions that relate motor power and losses to the circuit

parameters; and

4) Solution of this system of nonlinear equations by an

iterative Gauss-Seidel method.

Section III provides a review of the induction motor equivalent

circuit, while Section IV discusses the proposed method in

detail. The proposed method converges reliably in very few

iterations and computes estimates of parameters very close to

the true values, as illustrated by the case studies in Section V.

II. PRIOR WORK

IEEE Standard 112 [4] outlines methods for determining

the rated losses and the various equivalent circuit parameters

of an induction motor. Some of the tests required include

A DC test for stator resistance,

One or more three-phase locked rotor tests (performed at

rated or reduced frequency),

A no load test, and/or

One or more load tests (performed at full or reduced

load).

These tests require controlled conditions and calibrated test

equipment [3]. Except for very large motors, manufacturers

do not typically provide the data from these tests. Moreover,

performing these or similar performance tests in the field is

both difficult and time consuming [2]. Therefore, the end user

does not have easy access to the data required to compute the

equivalent circuit parameters using standard methods.

Responding to this need, both Natarajan [5] and Haque [2],

[8] developed methods to estimate induction motor equivalent

circuit parameters from nameplate and published performance

data. The method of Natarajan requires both the motor name

plate data and specific performance data from the manufac

turer's catalog, including the motor full-load torque, starting

torque, and power factor and efficiency at 50%, 75%, and

R Load+Stray+Mech = RJC1-s)/s

Fig. 1. Equivalent Circuit Model for Induction Motor [3].

100% loading. The method uses a spreadsheet to solve a

system of linear equations that relate the circuit parameters

to these data.

Haque's method requires the nameplate data, the ratio of

starting torque to full load torque, and power factor and effi

ciency at 50% and 100% loading-fewer data than are required

Natarajan's method. Using these data, Haque develops a set

of nonlinear equations that relate the circuit parameters to the

motor input power and losses. These equations are solved

by an iterative Gauss-Seidel method. Haque later described

a similar method which also models deep bar or double cage

rotor construction [8].

The primary shortcoming of both these methods is the

requirement of catalog data for motor torque and performance

at other than full load. Manufacturers often do not provide

these data, particularly for smaller motors. Catalog data may

also be difficult to find for older motors.

III. BACK GROUND

A. Induction Motor Equivalent Circuit

Induction motors operate by inducing current and torque in

the rotor circuit via transformer action due to slip (difference

in frequency) between the rotor and the stator. Such motors

are typically modeled with the well-known per-phase induction

motor equivalent circuit, shown in Figure 1. Rl and Xl are the stator impedance, R2 and X2 are the rotor impedance as referred to the stator, Rc models the core loss, and XM represents the magnetizing reactance. The motor output power

is modeled by RLoad, which is a function of slip,

RLoad = R2 (1 - s )

(1) s

In this paper, RLoad models stray loss and mechanical loss (windage and bearing friction) in addition to the output power.

The induction motor equivalent circuit is further described in

many textbooks, such as [3].

The motor output power is

(2)

Typically, slip s varies approximately linearly from no load to

full load. At no load, s is nearly zero, such that RLoad is very large, 12 is very small, and the power in RLoad represents only mechanical and stray losses. At full load slip, RLoad decreases, lz increases, and the power in RLoad includes the rated output

Pout

Fig. 2. Power-How Diagram of an Induction Motor [3].

power in addition to losses. Typical full load slip values are in

the range of 0.03-0.05 p. u. [3], although newer motors may

have significantly lower full load slip.

B. Induction Motor Losses

Figure 2 shows the power flow diagram of an induction

motor. Each loss in the figure is modeled by a specific

resistance in the motor equivalent circuit; see Table I. The

stator and rotor resistive losses are modeled by Rl and R2, respectively. Rc models core loss, while stray and mechanical losses are included in RLoad.

TABLE I INDUCTION MOTOR Loss DEFINITION

Loss Type Circuit Element Stator Winding Resistive Rl Rotor Winding Resistive R2 Core Magnetic Re Stray Magnetic RLoad Friction & Windage Mechanical RLoad Stray and mechanical losses are accounted for in the load resistance.

In order to calculate the equivalent circuit parameters, it

is necessary to separate the resistive losses from the other

losses (core, stray, and mechanical). In the absence of test

data, these losses may be assumed as a ratio of the total

loss based on typical values. In modern motors, mechanical

losses account for approximately 14% of the total loss and core

losses account for approximately 12% [7]. Stray load loss is

higher for smaller machines. IEEE Standard 112 [4] provides

assumed values of the stray load loss as a function of the

machine power rating; see Table II. This paper uses assumed

values of the mechanical loss and stray load loss based on

these ratios to correct the power in RLoad and to calculate Re.

IV. PROPOSED MET HOD

The proposed method to estimate the circuit parameters

requires the following nameplate data:

1) Rated output power (Poud 2) Rated terminal voltage (VRated) 3) Full load efficiency (T))

TABLE II ASSUMED STRAY LOAD Loss AS A FRACTION OF RATED LOAD [4]

Motor Rating Stray Load Loss 0-90 kW 0.018 91-375 kW 0.015 376-1850 kW 0.012 >1850 kW 0.009

4) Full load power factor (PF) 5) Full load speed in RPM (N) and number of poles 6) NEMA design type

7) NEMA code letter

From these data, the method estimates all relevant circuit pa

rameters: RI, Xl, R2, X2, Ro, and XM. The equations in the

method description use the per-unit system. Any convenient

base may be used; the rated output power and terminal voltage

are one possibility.

A. Derivation of Known Parameters

First, several intermediate data are derived from the name

plate data. The total input power and total loss at rated load

are

P _ POut In---TJ PLoss = PIn -POut

Similarly, the apparent and reactive input powers are

Is 1= POut In TJ' PF QIn = J'-IS-I-n-12---p-I-2n

The phasor input current is

(3)

(4)

(5)

(6)

I PIn -jQIn (7) Rated = V Rated The motor synchronous speed N s in RPM is derived from

the number of poles,

Ns = 1201 Number of Poles

(8)

where 1 is the system electrical frequency in Hz. Given the

synchronous speed and the full load speed, the full load slip

is

Ns-N s= Ns (9) The approximate locked rotor current hR can be deter

mined from the rated voltage, rated power, and NEMA code

letter. The NEMA code letter gives a range of starting kVA

values based on the motor horsepower rating, as shown in

Table III. As an approximation, the locked rotor kVA ISLRI may be set to the midpoint of the range corresponding to

the NEMA code letter. Then the corresponding locked rotor

current magnitude is

(10)

TABLE III TABLE OF NEMA CODE LE TTERS [3]

Code Letter Locked Rotor Code Letter Locked Rotor kVA/HP kVA/HP

A 0-3.15 L 9.00-10.00 B 3.15-3.55 M 10.00-11.20 C 3.55-4.00 N 11.20-12.50 D 4.00-4.50 0 12.50-14.00 E 4.50-5.00 P 14.00-16.00 F 5.00-5.60 R 16.00-18.00 G 5.60-6.30 S 18.00-20.00 H 6.30-7.10 T 20.00-22.40 J 7.10-8.00 U 22.40 and up

K 8.00-9.00

Next, the motor losses are segregated according to known

relationships and reasonable assumptions regarding the loss

distribution. The mechanical, stray, and core losses are as

sumed to be fixed fractions of the total loss (see Section III),

such that

PMech = PLoss' FMech PStray = POut' FStraY POore = POut FOore

(11)

(12)

(13)

The converted power, PConv, includes the output power, stray loss, and mechanical loss:

PConv = POut + PMech + PStray (14) The electromagnetically developed power, or air gap power,

PAG is

(15)

The stator and rotor resistive (copper) loss may be deter

mined from the other losses and the air gap power,

PSCL = PIn - PAG - PCore PRCL = PAG - PConv

(16)

(17)

The stator resistance can then be determined exactly from the

stator copper loss,

(18)

B. Development of Simultaneous Equations

After RI is determined from (18), the remammg

parameters-Xl, R2, X2, Rc, and XM-may be estimated by

solving a set of simultaneous nonlinear equations, developed

here.

Given the input current and an estimate of the stator

impedance, the air gap voltage E may be calculated,

(19)

The rotor current magnitude is

E h = -=----R2 .X - + J 2 s

Substituting Ihl from (20) into (15) yields

From (21), a quadratic expression in R2/ s is derived,

s

IE I2 J IE I4 - 4PAGX 2PAG

(20)

(21)

(22)

In practice, the larger root gives the correct value for the rotor

resistance. Therefore, for a given value of E, R2 is

(23)

(Here, R2 is written as a function of E rather than of h because the estimate of 12 depends strongly on R2 while the estimate of E does not. When used in an iterative method, (23) has superior convergence properties to an update using

only h) Assuming the value of R2 is similar at the locked rotor and

full load conditions, the locked rotor reactance XLR may be then estimated from

VRated = hR (R1 + R2 + jXLR)

I V;;;d 1 = V(R1 + R2 )2 + XZR XLR = IVRated

1 2 _ (R + R )2 (24)

IhR I2 1 2

where IVRatedl and IhR I are input data. It is known that XLR = Xl + X2, but the exact ratio of Xl to X2 cannot be determined from only the nameplate data. Instead, a ratio

is assumed from typical values based on the NEMA design

class [3], [4]; see Table IV. Defining Ratio = Xd XLR , the stator and rotor reactances are

Xl = XLR . Ratio X2 = XLR (1 - Ratio)

(25)

(26)

Once estimates for 11, h, and E are available and the rotor impedance is calculated, X M can be estimated from the

TABLE IV TYPICA L RATIO OF Xl AND X2 TO XLR [3), [4]

Rotor Design Xl X2 NEMA Design A 0.5 XLR 0.5 XLR NEMA Design B 0.4 XLR 0.6 XLR NEMA Design C 0.3 XLR 0.7 XLR NEMA Design D 0.5 XLR 0.5 XLR

require magnetizing reactive power Q M ,

(27)

(28)

Similarly, Rc is estimated based on the require core loss POore ,

IE I2 Rc = -POore (29)

A total of eight simultaneous equations are required to find

the unknown parameters, three of which are auxiliary equa

tions. The primary equations give estimates for the unknown

circuit parameters:

R2 from (23), Xl from (25), X2 from (26), XM from (28), and Ro from (29).

These are supplemented by auxiliary equations for E, 1 121, and XLR:

E from (19), 1 121 from (20), and XLR from (24).

C. Iterative Solution Method

The set of eight simultaneous equations may be solved

via an iterative, Gauss-Seidel type algorithm. The Gauss

Seidel solution method improves upon an initial estimate by

sequentially solving each equation using the current estimate

of all parameter values in order to obtain an updated estimate

of a given parameter value. For reliable convergence, the

method requires that

1) The initial estimate of the parameters is reasonably close

to their true values, and

2) The right-hand-side value of each equation is not a

strong function of the parameter being updated.

The iterations continue until all the process converges within

a specified tolerance.

The proposed iterative method is as follows:

1) Compute known motor powers and currents as described

in Section IV-A.

2) Compute R1 from (18).

3) Define initial estimates for air gap voltage E and rotor current magnitude I2,

E VRatedLO II21 Re{h}

(These are similar to the initial estimates proposed by

[2]. ) Initialize the values of the five unknown circuit

parameters to zero.

4) Store the present estimates of Xl> R2, X2, Re, and XM for later comparison.

5) Update R2 from (23) using the present estimates for E and X2. (An initial zero value for X2 has minimal impact on this step because the rotor impedance is

mostly resistive at full load. )

6) Compute XLR from (24) using the present estimate of R2.

7) Update Xl and X2 from (25)-(26) using the present estimate of XLR and the reactance ratio derived from the NEMA design type.

8) Update XM from (28) using the present estimates of E, Xl, and X2.

9) Update Re from (29) using the present estimate of E. 10) Check for convergence by comparing the updated values

of Xl, R2, X2, Re, and XM with their previous values. a) If all parameters have converged within a specified

tolerance, STOP.

b) Otherwise, update E and II21 from (19)-(20) using the updated parameter values. Then, return to Step

4.

11) Save final parameter values and display results.

At the end of the iterative procedure, the validity of the

computed parameters may be checked by solving the full load

powers in the resulting induction motor equivalent circuit and

comparing to known values, such as PIn, PAG, and PConv.

D. Limitations

As with previous methods of this type [2], [5], the proposed

method has a number of limitations.

The rotor resistance and reactance are assumed identical

under the locked rotor conditions as at full load. This

is not the case for deep bar and double cage rotors,

which are designed to experience significant skin effect

at high slip. Caution should therefore be used when using

the computed parameters for starting and pull-out torque

calculations. Methods are available to correct the model

for rotor skin effect [8], [9], but they require additional

data beyond the nameplate data.

The ratio between stator and rotor reactance is assumed,

rather than determined from calculation.

The parameters are fit to the full load condition only.

Stray loss, core loss, and mechanical losses are assumed

to be fixed ratios of the full load loss based on typical

values [4], [7].

The core loss is placed in the stator, when in reality it is

distributed between stator and rotor.

The proposed method makes no special provisions for

single-phase machines.

These limitations are a compromise required by the limited

set of data.

V. NUMERICAL RESULTS

The proposed method was implemented as a MATLAB

script and tested using nameplate data from various motors

available online. In all cases, the procedure converged within

a 0.001 per unit tolerance in five or fewer iterations. In order

to verify the accuracy of the computations, the method was

also tested for a motor with known parameters: a textbook

example from [3]

Chapman [3, Example 7-3] provides an example of a three

phase induction motor with known circuit parameters and loss

breakdown. Table V provides the data for this motor. In the

TABLE V EXAMPLE MOTO R DATA [3, EXAMPLE 7-3]

Rated voltage VRated 460 V Rated power output POut 25 HP (18.64 kW) Mechanical Loss PMech 1100 W Core Loss Peore OW Stray Loss PStmy OW Stator Resistance RI 0.641 n Stator Reactance Xl 1.106 n Rotor Resistance R2 0.332 n Rotor Reactance X2 0.464 n Magnetizing Reactance XM 26.3 n

example, the full load slip is not provided. It is therefore

calculated at s = 0.04189 by determining the slip at which

the output power equals the rated value. Similarly, appropriate

values for the rated power factor and efficiency were computed

by solving the circuit at rated slip.

The parameters for this motor were calculated using the

proposed method but using exact values for the locked rotor

current, reactance ratio, and loss distribution. Table VI com

pares the results of the parameter estimation method with the

actual circuit parameters.

TABLE VI COMPUTED CI RCUIT PA RAMETE RS FO R EXAMPLE MOTO R [3, EXAMPLE

7-3]

Parameter Exact Proposed Method Proposed Method + Exact Motor Data

RI 0.641 0.6573 0.6408 Xl 1.106 1.0983 1.1061 R2 0.332 0.3332 0.3320 X2 0.464 0.4607 0.4640 Rc 00 1721 00 XM 26.3 26.03 26.30 Units are n

The results demonstrate first that the proposed method with

the proposed assumptions returns results very close to actual

values, and second that the proposed method recovers the exact

-- Exact

- - - Exact + Proposed

....... Proposed

200

150 E 6 " " e-o

I-100

50

0L---------000= 0 160 01800 200 400 600 800 1 000 12 14

Mechical Speed (r/min)

Fig. 3. Torque Vs. Speed Curve

circuit parameters when the loss distribution and reactance

ratio assumptions are replaced with the actual motor data.

Figure 3 shows the torque vs. speed curves of the example

with the given information and Table VI. The solid line,

dashed line, and dotted line are the curves using the parameters

from the exact, proposed and exact, and proposed methods,

respectably. Since their parameters are very close to each

other, the curves appear overlapped, but the curve using the

parameters from the proposed method is slightly different in

the range of the pullout torque.

VI. CONCLUSION

This paper proposes a new iterative method to estimate

the induction motor circuit parameters using only the name

plate data and typical assumptions regarding motor behavior.

The use of only the nameplate data is an advantage over

previous methods which require additional performance data.

The proposed method converges reliably and calculates circuit

parameters very close to the true values.

NOTATION

Rated voltage

Stator current of full load input current

Locked rotor current

Stator resistance due to stator copper loss

Stator reactance

R2 X2 Rc XM Ratio

RLoad

s

Rotor resistance due to rotor copper loss

Stator reactance

Core loss resistance

Magnetizing reactance

Ratio of Xl to XLR Equivalent load resistance

and stray loss

Full load motor slip

including mechanical

SIn Input apparent power POut Output power PLoss Total motor loss PMech Mechanical loss, including windage and bearing

FMech PStray FStray PCore

friction loss

Fraction of mechanical loss to total loss

Stray loss

Fraction of stray loss to output power

Core loss

F Core Fraction of core loss to output power PConv PAC PRCL PSCL E

Converted power

Air gap power

Rotor copper loss

Stator copper loss

Air gap voltage

ACKNOWLEDGMENT

The research presented in this paper resulted from work

performed under direction of the National Renewable Energy

Laboratory (NREL) in Golden, CO, with funding from the

B onneville Power Administration, TI Project No. 192, Con

tract No. 51353, and Interagency Agreement No. IAG-ll-

1801, which the authors gratefully acknowledge.

REFERENCES

[I] IEEE Recommended Practice for Industrial and Commercial Power Systems Analysis, IEEE Std. 399, 1997.

[2] M. H. Haque, "Estimation of three-phase induction motor parameters," Electric Power Systems Research, vol. 26, no. 3, pp. 187-193, 1993.

[3] S. Chapman, Electric Machinery Fundamentals, 4th ed. New York, NY: McGraw-Hili Education, 2005.

[4] IEEE Standard Test Procedure for Polyphase Induction Motors and Generators, IEEE Std. 112, 2004.

[5] R. Natarajan and V. Misra, "Parameter estimation of induction motors using a spreadsheet program on a personal computer," Electric Power Systems Research, vol. 16, no. 2, pp. 157-164, 1989.

[6] G. Rogers, "Demystifying induction motor behavior," IEEE Computer Applications in Power, vol. 7, no. I, pp. 29-33, Jan. 1994.

[7] R. Saidur, "A review on electrical motors energy use and energy savings," Renewable and Sustainable Energy Reviews, vol. 14, no. 3, pp. 877-898, 2010.

[8] M. Haque, "Determination of nema design induction motor parameters from manufacturer data," IEEE Transactions on Energy Conversion, vol. 23, no. 4, pp. 997-1004, Dec. 2008.

[9] J. Pedra and L. Sainz, "Parameter estimation of squirrel-cage induction motors without torque measurements," IEEE Proceedings - Electric Power Applications, vol. 153, no. 2, pp. 263-270, march 2006.