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1. Parameter Estimation for Single-Phase Induction
Motors using Test Measurement Data
Phuc Huynh
Dept. of ECE,
Univ. of Illinois, Urbana, IL, USA
Email: pthuynh2@illinois.edu
Hao Zhu
Dept. of ECE,
Univ. of Illinois, Urbana, IL, USA
Email: haozhu@illinois.edu
Dionysios Aliprantis
School of ECE,
Purdue Univ., West Lafayette, IN, USA
Email: dionysis@purdue.edu
Abstract—This paper presents a methodology to estimate the
parameters of a single-phase induction motor (SPIM) equivalent
circuit using a dc test, a locked-rotor test, and a no-load test.
By neglecting the core-loss resistance, the SPIM parameters can
be first directly calculated using a simplified equivalent circuit.
However, an equivalent circuit using the resultant parameter
values fail to match the power input measured at the motor
terminals, especially for the active power under the no-load test.
We include the core-loss resistance into the parameter estimation
framework and use the Newton-Raphson (N-R) algorithm to
improve the estimates obtained by the direct calculation method.
Experimental results on a laboratory SPIM demonstrate the
effectiveness of our proposed N-R based parameter estimation
scheme, in terms of excellent match with the active and reactive
power measurement data from the aforementioned tests, and
mismatch reduction at other operating conditions.
I. INTRODUCTION
Single-phase induction motors (SPIMs) are widely used,
especially for residential applications. These motors are usu-
ally at fractional horsepower level and thus are suitable for
deployment in residential appliances such as air-conditioners,
washing machines, dryers, and refrigerators. According to
the Residential Energy Consumption Survey by the U.S En-
ergy Information Administration [1], about 6% of the total
electricity consumption in the United States is directly at-
tributed to SPIMs of air-conditioners. This percentage could
be significantly higher in the southern states; e.g., 18% in
Texas, and 27% in Florida. Due to the widespread and large
energy consumption of SPIMs, it is important to improve
the parameter estimation techniques for SPIMs, either for
monitoring their health condition or to support the operations
of power systems during fault events; see e.g., [2], [3].
The estimation of SPIM model parameters is typically
conducted based on measurements collected from the steady-
state tests (i.e., dc test, locked-rotor test, and no-load test). The
IEEE Standard 114-2010 [4] has documented a standard for
testing SPIMs to evaluate the performance in terms of power,
torque, and speed. However, it does not describe the procedure
to obtain the parameters of the SPIM equivalent circuit.
Directly calculation of SPIM equivalent circuit parameters
using test data has been studied in [5] for typical SPIM, and
more recently in [6] for permanent-split capacitor-run SPIM.
These methods have neglected the core-loss resistance, which
is significantly larger than the impedance of other parameters,
to simplify the calculations. Nonetheless, it has been recog-
nized in [7] and [8] the importance of including the core-
loss resistance in SPIM modeling. The parameter estimation
methods therein require additional hardware deployment to
measure the rotor speed or motor torque. It is more practical
to develop advance numerical method that incorporates core-
loss resistance to fit the steady-state tests data at no additional
hardware complexity.
More recently, we have developed a non-intrusive approach
to estimate SPIM parameters using transient stator current
measurements collected during normal start-ups [2]. Compared
to existing test-based methods, this online approach is more
suitable for monitoring the motor capacitor, whose capaci-
tance can decrease gradually throughout the motor lifespan.
Nonetheless, the success of this method in tracking the capac-
itance value requires a good knowledge of other static motor
parameters such as winding resistance, leakage inductance,
and magenetizing inductance. Because of the parameter iden-
tifiability issue, the time-varying capacitance and the static pa-
rameters, especially the auxiliary winding leakage/magnetizing
inductance, cannot be determined simultaneously using only
transient measurements. This again speaks for the necessity
of improving the performance of test-based SPIM parameter
estimation techniques.
Towards this goal, this paper will develop a parameter
estimation methodology to better fit the active and reactive
power input measurements under various steady-state test
conditions by incorporating the core-loss resistance. With this
modification, we will improve the estimates obtained by the
direct calculation approach in [5] and [6] using the Newton-
Raphson (N-R) iterative updates. The objective of the proposed
two-step method using direct calculations and N-R iterations,
is to further tune up the parameters of the equivalent circuit
to match the active and reactive power input measurements.
Compared to the most related work in [7], in which the core-
loss resistance varies with the slip, our parameter estimates
can potentially be used for modeling the motor at different
operating conditions.
The rest of this paper is organized as follows. Section II
will introduce the steady-state equivalent circuit model of
SPIMs. In Section III, we will develop the aforementioned
methodology for parameter estimation. The validation of the
proposed method using real measurements is presented in
978-1-5090-3270-9/16/$31.00 ©2016 IEEE

2. TABLE I
LIST OF SPIM PARAMETERS
rs (rS) Resistance of main (auxiliary) winding
Lls (LlS) Leakage inductance of main (auxiliary) winding
Lms (LmS) Self inductance of main (auxiliary) winding
rr (r R) Resistance of rotor winding referred to
main (auxiliary) winding
L lr (L lR) Leakage inductance of rotor winding
referred to main (auxiliary) winding
C1 Capacitance of start capacitor
r1 Resistance of start capacitor
Fig. 1. The equivalent circuit of an SPIM with only the main winding.
Section IV, and the paper is wrapped up in Section V.
II. SPIM EQUIVALENT CIRCUITS
This section introduces the steady-state equivalent circuits
for SPIMs, which will be used in the proposed parameter
estimation method.
We consider squirrel-cage SPIMs. The stator has two wind-
ings, a main winding and an auxiliary winding, that are placed
90 electrical degrees apart in space. The two windings usually
have different number of turns. The external circuit typi-
cally consists of capacitors with/without a centrifugal switch,
connected in series with the auxiliary winding to enhance
the SPIM start-up torque characteristics. Depending on the
external circuit, the SPIMs can be classified into three types
of topology, namely, the capacitor-start (CS), the capacitor-
start–capacitor-run, and the permanent-split ones; see e.g., [9].
In this paper, we will focus on the CS topology because it is
the most popular type for residential applications such as air-
conditioner compressors. The list of parameters in the SPIM
steady-state equivalent circuit is summarized in Table I.
Development of the SPIM steady-state equivalent circuit
is very similar to the one for three-phase induction motor.
However, due to the winding arrangement and the single-
phase sinusoidal power supply, one has to consider two mmf
waves traveling in opposite directions for each of the stator
windings. If only the main winding is connected, the air-gap
mmf can be decomposed into two traveling waves: one travels
in the positive rotor angle direction, and the other one travels
in the negative direction. With sinusoidal current input, the
Fig. 2. The equivalent circuit of an SPIM with only the auxiliary winding.
fundamental component of magnetomotive force (mmf) in the
air-gap at time t and angular position θ can be written as:
Fag(t, θ) =
1
2
F0 cos(ωet − θ + φ)
+
1
2
F0 cos(ωet + θ + φ), (1)
where F0 is the peak of mmf at the fundamental harmonic in
the air-gap, φ is the phase angle of the main winding current,
and ωe is the electrical frequency (e.g., 120π Rad/s). The first
summand with ωet−θ is the mmf wave traveling in the positive
θ direction with the angular speed ωe, while the other one is
the mmf wave in the negative direction at the same speed. Let
s be the slip between the SPIM rotor’s mechanical speed and
the mmf wave in the positive direction. It can be shown that the
slip between the rotor and mmf wave in the negative direction
would be 2 − s. For each of the mmf waves, the reflected
effect of the SPIM rotor on each stator winding is similar to
that of the three-phase motor. Hence the equivalent circuit of
one SPIM winding consists of two equivalent circuits of three-
phase induction motor connected in series with the equivalent
parameters scaled by the 1
2 factor as described in Fig. 1 and 2.
The top parts in each of the equivalent circuits correspond to
the mmf wave in the positive rotor angle direction, and thus
are termed as the forward branch. Accordingly, the bottom
parts for the mmf in the negative direction are referred as
the backward branch. Last, note that in Fig. 2 the auxiliary
winding is connected in series to the external start-capacitor
circuit, consisting of C1 and r1 and a centrifugal switch.
As mentioned in Section I, earlier work in [5] and [6]
ignores the core-loss resistance term rcs in these equivalent
circuits, assuming its value is significantly bigger compared
to those of other components connected in parallel. The rest
of the paper will develop the parameter estimation method
by accounting for the rcs term with improved performance in
fitting the test measurement data.
III. PARAMETER ESTIMATION METHOD
This section will present the parameter estimation method
using the tests with the main winding connected while the

3. Fig. 3. The equivalent circuit with only the main winding under the locked-
rotor test.
auxiliary winding disconnected; i.e., for the SPIM equivalent
circuit in Fig. 1. For the auxiliary winding circuit in Fig. 2,
one can first disconnect the external capacitor (with C1 and
r1 shorted). This way, the two equivalent circuits become the
same, and the method presented for only the main winding
can be directly adopted to estimate the motor parameters in
Fig. 2. The goal of the proposed estimation method is to
find the circuit parameter values in order to match the active
and reactive power measurements of the motor under various
test conditions. Certain parameters, such as the main winding
resistance rs, can be directly measured using the dc test.
For some of other parameters, including the inductance terms
Lls, Llr, and Lms, as well as the rotor resistance rr, one
can use additional test measurements to directly calculate a
rough estimate of their values. This stage is similar to the
method in [5] and [6] by ignoring the rcs term. To improve
the estimation accuracy, the Newton-Raphson (N-R) algorithm
will be adopted to fit all the test measurements by iteratively
updating the estimates for all these parameters together with
the core-loss resistance rcs.
A. SPIM tests
To collect the measurements, the following three tests will
be performed:
1) dc test: resistance rs is directly measured.
2) locked-rotor test: in this test, the rotor is locked from
rotation. An appropriate voltage source will be applied
to the stator winding such that the SPIM terminal current
is close to its rated value. The terminal rms voltage V m
LR,
rms current Im
LR, active power Pm
LR, and power factor
pfm
LR are measured.
3) no-load test: in this test, the rotor is rotated at the
synchronous speed with the assistance of an external
mechanical source. The applied voltage source to the
stator is at the rated level. Same measurements as those
in the locked-rotor test will be collected, denoted by
V m
NL, Im
NL, Pm
NL, and pfm
NL, respectively.
Fig. 4. The equivalent circuit with only the main winding under the no-load
test.
One can simplify the equivalent circuits in Figs. 1-2 under
the locked-rotor and no-load test conditions. The simplified
circuits make it possible to directly approximate some param-
eters such as Lls, Llr, Lms, and rr. The parameter estimates
obtained by this approach will be used as the initial guess for
the N-R updates later on.
The following assumptions can be made to simplify the
SPIM equivalent circuits:
• The core-loss resistance rcs is significantly larger than
the magnetizing reactance ωeLms. Both of them are much
larger than the rotor leakage reactance ωeLlr and the rotor
resistance rr.
• The stator leakage inductance and rotor leakage induc-
tance are the same; i.e., Lls = Llr.
• There is no coupling between the two stator windings.
Under these assumptions, the simplified equivalent circuit
for the locked-rotor test can be constructed using the fact that
the slip s = 1, as shown by Fig. 3. The core-loss resistance rcs
and the winding magnetizing reactance ωeLms are eliminated
from the simplified circuit. The resultant circuit consists of
three resistance and three inductance terms, all connected in
series.
Similarly for the no-load test, one can set the slip s = 0
and construct the simplified equivalent circuit as illustrated
in Fig. 4. As the rotor effective resistance in the forward
branch is inversely proportional to the slip s and goes to
infinity in this case, the main winding magnetizing reactance
ωeLms part needs to be kept with all other terms eliminated.
The simplification of the backward branch follows similarly
from the locked-rotor one. The resultant circuit consists of two
resistance and two inductance terms, all connected in series.
Having the simplified equivalent circuits for different tests,
the SPIM parameters can be calculate using the following

4. Fig. 5. SPIM equivalent circuits for (a) the locked-rotor and (b) the no-load
tests.
equations:
RLR =
Pm
LR
(Im
LR)2
= rs + rr (2)
ωeLls = ωeLlr =
1
2
(V m
LR)2
(Im
LR)2
− R2
LR, (3)
XNL =
V m
NL
Im
NL
= ωeLls +
ωeLlr
2
+
ωeLms
2
. (4)
Although the simplified equivalent circuits greatly facilitate
estimating several of the key parameters, the accuracy of
the estimate values is limited by the underlying assumptions.
We have observed that the estimates obtained by the direct
calculation approach fail to satisfactorily match the active and
reactive power inputs measured by all the steady-state tests.
We will use these calculated values as the initial guess for the
ensuing Newton-Raphson (N-R) algorithm which can further
tune up the parameter estimates.
B. Newton-Raphson algorithm
Specifically, the N-R algorithm will be used to find the
parameter values to fit the active and reactive power measure-
ments collected by the locked-rotor and no-load tests. First, we
will represent the power inputs for locked-rotor and no-load
tests as a function of all the SPIM parameters. The equivalent
circuit in Fig. 1 can be represented by a single equivalent resis-
tance connected in series with a single equivalent reactance,
as shown in Fig. 5. The equivalent resistance and reactance
values under the locked-rotor test (denoted with the subscript
LR, eq) and those under the no-load test (denoted with the
subscript NL, eq) are given in Eqs. (5)–(8).
Upon defining these equivalent resistance and reactance
terms in (5)–(8), the functional form for active and reactive
power inputs for each of the tests become
PLR = (Im
LR)2
RLR,eq, (9)
QLR = (Im
LR)2
XLR,eq, (10)
PNL = (Im
NL)2
RNL,eq, (11)
QNL = (Im
NL)2
XNL,eq. (12)
Accordingly, the function values for the power inputs are
determined by
⎧
⎪⎪⎪⎨
⎪⎪⎪⎩
PLR = Pm
LR
QLR = (V m
LRIm
LR)2 − (Pm
LR)2
PNL = Pm
NL
QNL = (V m
NLIm
NL)2 − (Pm
NL)2.
(13)
Algorithm 1 Newton-Raphson Algorithm [10, Ch. 7]
Initialize x(0)
using the direct calculation estimates
Select stopping criterion
Set the iteration index v = 1
while ||f(x(v)
)||2 > do
Update J := ∂f(x)
∂x x=x(v)
Compute Δx = −J−1
f(x(v)
)
Update x(v+1)
= x(v)
+ Δx
Set the iteration v = v + 1
end while
x = x(v)
One can also compute the two reactive power input terms
in (13) using the measured power factor pfm
LR and pfm
NL,
respectively. Both approaches are shown to be consistent to
each other using our experimental results.
With the four equations in (13), the N-R algorithm can be
adopted to iteratively update the four unknown variables con-
tained by the vector x = rcs rr Lls Lms
T
. Different
from the direct calculation approach, the N-R algorithm can
include the core-loss resistance rcs as a parameter value to
improve the fitting performance. The algorithmic description
of the N-R updates is tabulated in Algorithm 1. It will solve the
system of non-linear equations f(x ) = 0 with the mismatch
function defined as
f(x) :=
⎡
⎢
⎢
⎣
PLR − Pm
LR
QLR − (V m
LRIm
LR)2 − (Pm
LR)2
PNL − Pm
NL
QNL − (V m
NLIm
NL)2 − (Pm
NL)2
⎤
⎥
⎥
⎦ .
Convergence of N-R algorithm depends on the the initial guess
x(0)
and the nonlinearity of function f(·). We have used
the estimated values by the direct calculation approach and
observed that the N-R algorithm would typically converge
within a dozen of iterations.
IV. EXPERIMENTAL RESULTS
The effectiveness of the proposed parameter estimation
method is verified using real measurement data collected from
an SPIM in our laboratory. The measurements from locked-
rotor and no-load tests are used to calculate the parameters
in the SPIM equivalent circuit. Improvements of the proposed
methodology over the direct calculation method are verified
from the measurements at different operating conditions. The
tested SPIM information on its nameplate is listed in Table II.
A variac is used to vary the voltage magnitude applied to the
tested SPIM while keeping the frequency at 60 Hz. The rotor
shaft is coupled with a dynamometer to lock the shaft under the
locked-rotor test, or to rotate it at the electrical frequency under
the no-load test. The power supply is connected to the motor
through a Yokogawa WT310 wattmeter so that the voltage,
current, active power, and power factor at the motor terminal
can be measured.
All measurement data under each of the two tests with
only main winding or auxiliary winding connected is recorded

5. RLR,eq = rs +
rr
ω2
e L2
ls+r 2
r
+ 1
rcs
ωeLls
ω2
eL2
ls+r 2
r
+ 1
ωeLms
2
+
rr
ω2
eL2
ls+r 2
r
+ 1
rcs
2 , (5)
XLR,eq = ωeLls +
ωeLls
ω2
eL2
ls+r 2
r
+ 1
ωeLls
ωeLls
ω2
eL2
ls+r 2
r
+ 1
ωeLms
2
+
rr
ω2
eL2
ls+r 2
r
+ 1
rcs
2 , (6)
RNL,eq = rs +
2
rcs
4
ω2
eL2
ms
+ 4
r2
cs
+
2
rcs
+
8rr
4ω2
eL2
ls+rr
2
2
ωeLms
+ 8ωeLls
4ω2
e L2
ls+rr
2
2
+ 2
rcs
+
8rr
4ω2
eL2
ls+rr
2
2 , (7)
XNL,eq = ωeLls +
2
ωeLms
4
ω2
eL2
ms
+ 4
r2
cs
+
2
ωeLms
+ 4ωeLls
4ω2
e L2
ls+rr
2
2
rcs
+
4rr
4ω2
eL2
ls+rr
2
2
+ 2
ωeLms
+ 4ωeLls
4ω2
e L2
ls+rr
2
2 (8)
TABLE II
TESTING SPIM NAMEPLATE
MOD 5KC33FN33G
HP 1/3 HZ 60
V 115/230 PH 1
RPM 3450 CODE L
A 5.6/2.8 SF 1.00
SF A FR 48
AMB 40C INSUL CLASS B NEMA DESIGN
TIME RATING CONT
TABLE III
LOCKED-ROTOR AND NO-LOAD TEST MEASUREMENTS
Connected winding Test V m [V] Im [A]
Main
Locked-rotor 31.399 5.4865
No-load 118.68 4.4585
Auxiliary
Locked-rotor 53.486 4.9211
No-load 118.36 2.5865
Connected winding Test P m [W] pfm
Main
Locked-rotor 105.03 0.6105
No-load 85.79 0.1608
Auxiliary
Locked-rotor 227.11 0.8708
No-load 78.18 0.2567
in Table III. For each test, the applied voltage and winding
current are kept at the rated value.
Parameter values are first estimated using the direct calcula-
tion approach, and improved using the N-R algorithm later on.
The estimation results are listed in Table IV and V. Compared
with the direct calculation results, the N-R based estimates
are more significantly different in the rotor resistance rr and
the stator winding inductance Lms. The estimated core-loss
resistance values by the N-R algorithm are indeed quite large
compared to other parameters. The error performance of N-
R algorithm is illustrated in Fig. 6, which plots the residual
norm ||f(x)||2 versus the iteration. Residual norm of the initial
guess is plot at iteration 0. The algorithm achieves satisfactory
error norm within just 3-4 iterations of N-R updates.
To validate the accuracy of each set of parameter estimates,
TABLE IV
PARAMETERS VALUE FOR THE SPIM WITH ONLY MAIN
WINDING
Parameter Direct Calc. N-R Alg.
rs [Ω] 1.705 1.705
rr [Ω] 1.784 1.965
rcs [Ω] – 488.5592
Lms [mH] 123.159 122.434
Lls [mH] 6.016 6.087
Llr [mH] 6.016 6.087
TABLE V
PARAMETERS VALUE FOR THE SPIM WITH ONLY AUXILIARY
WINDING
Parameter Direct Calc. N-R Alg.
rS [Ω] 6.099 6.099
rR [Ω] 3.279 3.514
rcS [Ω] – 684.189
LmS [H] 220.907 216.119
LlS [H] 7.287 7.258
LlR [H] 7.287 7.258
we compute their corresponding active/reactive power inputs,
as listed in Table VI, VII, and VIII. The N-R estimates result
in perfect match in terms of active and reactive power inputs
with the actual measurements, significantly outperforming the
results obtained by the direct calculation approach. The latter
has experienced much larger mismatch with the active power
measurement under the no-load test. For example, for the no-
load test with only main winding connected, the measured
active power is 85.79W, but the one obtained by the direct
calculation approach is only 41.97W. However, the N-R es-
timates can perfectly fit the measurement value at 85.79W.
Under various operating conditions, the active and reactive
power produced by the N-R method are more consistent with
respect to the real measurement data as compared to the direct
calculation ones. For example, when slip is 0.035, the direct
calculation method results in the mismatch errors of 74.71
W and 55.22 VAR, while the N-R method leads to 39.34

6. 0 1 2 3 4
0
5
10
15
20
25
Iteration index
Residualnorm||f(x)||2 Main winding
Auxiliary winding
Fig. 6. Iterative residual error norm obtained by the N-R algorithm for the
main and auxiliary winding tests, respectively.
TABLE VI
ACTIVE AND REACTIVE POWER COMPARISON FOR THE SPIM
WITH ONLY MAIN WINDING
Test
Active power [W]
Measured Direct Calc. N-R Alg.
Locked-rotor test 105.03 100.07 105.03
No-load test 85.79 41.97 85.79
Test
Reactive power [VAR]
Measured Direct Calc. N-R Alog.
Locked-rotor test 136.55 135.15 136.55
No-load test 522.13 528.20 522.13
TABLE VII
ACTIVE AND REACTIVE POWER COMPARISON FOR THE SPIM
WITH ONLY AUXILIARY WINDING
Test
Active power [W]
Measured Direct Calc. N-R Alg.
Locked-rotor test 227.11 222.01 227.11
No-load test 78.18 45.96 78.18
Test
Reactive power [VAR]
Measured Direct Calc. N-R Alog.
Locked-rotor test 133.04 133.76 133.04
No-load test 295.99 305.94 295.99
W and 19.86 VAR. Although it is reasonable to neglect the
core-loss resistance as corroborated by its large value, it is
truly necessary to include this term for better fitting the motor
terminal power measurements under various steady-state test
conditions.
V. CONCLUSIONS
In this paper, we have presented a methodology to es-
timate the parameters for the SPIM steady-state equivalent
circuit. Based on the motor terminal active and reactive power
measurements under different tests, it is possible to directly
calculate several parameter values by eliminating the core-
loss resistance in the equivalent circuit. Although the resultant
estimates provide a good initial guess, there exists a noticeable
error in matching the active power output under the no-load
test. To tackle this issue, we solved the parameter estimation
problem with the core-loss term as an additional unknown
TABLE VIII
ACTIVE AND REACTIVE POWER COMPARISON UNDER
DIFFERENT OPERATING CONDITIONS
Slip Measured Direct Calc. N-R Alg.
Active
power
[W]
0.0083 202.75 124.50 164.30
0.0150 280.28 188.85 227.56
0.0231 362.37 263.83 301.18
0.0306 364.30 328.11 363.11
0.0350 444.58 369.87 405.24
Reactive
power
[VAR]
0.0083 445.76 124.50 164.30
0.0150 433.06 188.85 227.56
0.0231 426.65 263.83 301.18
0.0306 422.47 328.11 363.87
0.0350 425.09 369.87 405.23
using the Newton-Raphson iterative updates. Experimental
results demonstrated that the N-R estimates improve the fit
of both real and reactive power measurements upon the direct
calculation ones at different operating conditions.
Future research directions include extension of the current
steady-state testing framework to the dynamic setting because
the dynamic transient behavior of SPIMs have increasingly af-
fected the operations of power systems under fault conditions;
see e.g., [3]. We are interested in investigating the performance
analysis and equivalent circuit parameter estimation of SPIMs
during start-up time. Those investigations are anticipated to use
the transient measurement data under normal motor operating
conditions in a non-intrusive fashion.
VI. ACKNOWLEDGMENT
This research is partially supported by an unrestricted gift
from Texas Instruments (TI). P. Huynh and H. Zhu also would
like to thank Dr. R. Narasimha at the TI ESP group for
valuable inputs and discussions.
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