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.
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