Novel Polarization Index Evaluation Formula and Fractional-Order Dynamics in Electric Motor Insulation Resistance

Abstract

One of the common test metrics prescribed by IEEE Std 43 for testing motor insulation is the Polarization Index (P.I.) which evaluates the “goodness” of the machine’s insulation resistance by getting the ratio of the insulation resistance measured upon reaching t2 > 0 minutes ( IRt 2 ) from t1 > 0 minutes ( IRt 2 ) for t2 > t1 > 0, after applying a DC step voltage. However, such definition varies from different manufacturers and operators despite of decades of research in this area because the values of t1 and t2 remain to be uncertain. It is hypothesized in this paper that the main cause of having various P.I. definitions in literature is due to the lack of understanding of the electric motor’s dynamics at a systems level which is usually assumed to follow the dynamics of the exponential function. As a result, we introduce in this paper the fractional dynamics of an electric motor insulation resistance that could be represented by fractional-order model and where the resistance follows the property of a Mittag-Leffler function rather than an exponential function as observed on the tests done on a 415-V permanent magnet synchronous motor (PMSM). As a result, a new PMSM health measure called the Three-Point Polarization Index (3PPI) is proposed.

Full text

RESEARCH PAPER
NOVEL POLARIZATION INDEX EVALUATION
FORMULA AND FRACTIONAL-ORDER DYNAMICS
IN ELECTRIC MOTOR INSULATION RESISTANCE
Emmanuel A. Gonzalez 1,IvoPetr´aˇs2, Manuel D. Ortigueira 3
Abstract
One of the common test metrics prescribed by IEEE Std 43 for test-
ing motor insulation is the Polarization Index (P.I.) which evaluates the
“goodness” of the machine’s insulation resistance by getting the ratio of
the insulation resistance measured upon reaching t2>0minutes(IRt2)
from t1>0minutes(IRt2)fort2>t
1>0, after applying a DC step
voltage. However, such definition varies from different manufacturers and
operators despite of decades of research in this area because the values of
t1and t2remain to be uncertain. It is hypothesized in this paper that the
main cause of having various P.I. definitions in literature is due to the lack
of understanding of the electric motor’s dynamics at a systems level which
is usually assumed to follow the dynamics of the exponential function. As
a result, we introduce in this paper the fractional dynamics of an electric
motor insulation resistance that could be represented by fractional-order
model and where the resistance follows the property of a Mittag-Leffler
function rather than an exponential function as observed on the tests done
on a 415-V permanent magnet synchronous motor (PMSM). As a result,
a new PMSM health measure called the Three-Point Polarization Index
(3PPI) is proposed.
MSC 2010: Primary 26A33; Secondary 94C12, 47E05
c
2018 Diogenes Co., Sofia
pp. 613–627 , DOI: 10.1515/fca-2018-0033

614 E.A. Gonzalez, I. Petr´aˇs, M.D. Ortigueira
Key Words and Phrases: electric motors, fractional calculus, insula-
tion testing, permanent magnet motors, transfer functions, Mittag-Leffler
function
1. Introduction
The aging of the mechanical, electrical and chemical properties of elec-
tric motors is a very important consideration in determining its overall
health condition to avoid unwanted breakdowns and potential unsafe and
hazardous operations [14, 20, 21, 28, 8]. In fact, a low resistance value of
the electric motor could cause catastrophic failures in the electronic driver
circuit that supplies power to the machine. Damage in its thyristors, for
example, could cause heavy financial losses due to thyristor replacement
and circuit board repair—a well-known consequence of high repair cost in
industries using electric machines in general [4], for example, in an elevator
industry employing condition-based approaches in equipment maintenance
[9]. In practice, the most common measure being used in determining the
motor’s health is through the measurement of insulation resistance which,
based on fundamental physics, is proportional to the type and geometry of
insulation used, and is inversly-proportional to the conductor surface area.
There are many ways in determining the health of an electric motor
through its insulation resistance which are all discussed in the IEEE Rec-
ommended Practice for Testing Insulation Resistance of Electric Machinery,
IEEE Std 43-2013 [16]. One simple test that is employed is the Polarization
Index (P.I.) test which is normally defined in this standard as the ratio of
the insulation resistance measured upon reaching 10 minutes, i.e. IR10 ,
with respect to the insulation resistance measured upon reaching 1 minute,
i.e. IR1, which can be described mathematically as
P.I. =IR10
IR1
,(1.1)
from an input step voltage that is appropriate for the winding voltage rat-
ings. The IEEE Std 43-2013 recommends the following insulation resistance
test DC step voltages which is presented in Table 1 for convenience. In some
cases the time values of the insulation resistance measurement varies de-
pending on the specifications of the motor which is then prescribed by the
original equipment manufacturer (OEM), or based on the evaluation of the
experienced electric motor operators from the applied condition-based and
preventive maintenance programs. This is true especially for form-wound
stators employing modern insulation systems where tests could be done
between 1 to 5 minutes only. Such observation is also valid for generator
stator insulations [19, 25].

NOVEL POLARIZATION INDEX EVALUATION . . . 615
Winding rated voltage Insulation resistance test DC voltage
<1000 V 500 V
1000 to 2500 V 500 to 1000 V
2501 to 5000 V 1000 to 2500 V
5001 to 12000 V 2500 to 5000 V
>12000 V 5000 to 10000 V
Table 1. Insulation resistance test DC voltages based on
electric motor winding rated voltage.
According to IEEE Std 43, to evaluate the health of an electric machine,
one must ensure the recommended minimum values of polarization index
depending on the thermal class ratings of the insulation used are avoided.
These values are presented in Table 1. For modern electric motors, if the
insulation resistance at 1 minute goes beyond 5000 MΩ, then the use of
the P.I. test may result in a wrong evaluation if used further since the
measured total current is already too small for proper analysis. In this
range, sensitivity issues from the test equipment itself on supply voltage,
humidity conditions and test setup could already compromise the readings
that is why the P.I. is not anymore recommended [16].
Thermal class rating Minimum P.I.
Class 105 (A) 1.5
Class 130 (B) and above 2.0
Table 2. Recommended minimum values of polarization
index for different types of insulation thermal class ratings.
One of the issues faced in using the P.I. as a test parameter is that
there is still no standard in determining the time intervals in the P.I. test.
Unfortunately, different organizations and OEMs would have their own
definitions. This then makes the minimum recommended P.I. values in
Table 1 somewhat unreliable in a sense. Furthermore, until now, there is
no pass-fail criteria which can be used from the P.I. value observed. As a
matter of practice, one could simply compute for the P.I. of the machine,
then makes an evaluation based on years of experience in determining if
a repair job, e.g. motor reconditioning or rewinding, needs to be done.
In effect, the standard in [16], when it comes to the P.I., could result in
an uncertainty where the operator has the opportunity to make his own
decisions beyond what is recommended by the standard.

616 E.A. Gonzalez, I. Petr´aˇs, M.D. Ortigueira
We assumed in this paper that one of the reasons on why the P.I. defini-
tion is not well-established is because there is no direct way of establishing
an appropriate relationship between the time duration for the P.I. test based
on the dynamics of electric motor. The dynamics of the electric machine
is not well-understood from the systems theory point-of-view, despite of
heavy research in the analysis of the total current measured in an electric
motor with respect to the motor’s absorption current, conduction current,
geometric capacitive current and surface leakage current—all well-explained
in [16, 2, 3]. However, one can simply assume that the relationship of the
output computed insulation resistance with respect to a step input dc volt-
age would follow a simple first-order or overdamped second-order system
of the forms:
G1(s)=Y(s)
E(s)=K
Ts+1,(1.2)
for K, T > 0and
G2(s)=Y(s)
E(s)=Kω2
n
s2+2ζωns+ω2
n
,
=K
(T1s+1)(T2s+1) (1.3)
for K, T1,T
2,ω
n>0andζ>1, respectively. If the dynamics of the electric
motor is assumed to follow the models in (1.2) and (1.3), then the insulation
resistance values with respect to time should follow the dynamics of the
usual exponential function in the form of
y(t)=K1−e−t/T ,t≥0,(1.4)
for the first order system in (1.2), and
y(t)=K1−T1
T2−T1
e−t/T2−T2
T2−T1
e−t/T1,t≥0,(1.5)
for the overdamped second-order system in (1.3). However, upon exper-
imentation on a low-power 415-V permanent magnet synchronous motor
(PMSM) used in a high-rise elevator, it was discovered that the insulation
resistance dynamics do not follow (1.4) nor (1.5). In fact, upon using frac-
tional calculus [24, 27], it was assumed that the model should appropriately
follow the fractional-order transfer function (FOTF)
GF(s)= K
Ts
α+1,(1.6)
for K, T > 0andwhere0<α<1 is the order of the model.
The above transfer function (1.6) corresponds to a simple fractional-
order differential equation (FODE) of the form
Tdα
dtαyF(t)+yF(t)=Ku(t),(1.7)

NOVEL POLARIZATION INDEX EVALUATION . . . 617
Figure 1. Equivalent circuit of an electric motor as shown
in Fig. 1 of IEEE Std 43-2013, [16].
where yF(t) being an output function and u(t) being the unit-step function.
The solution of the FODE (1.7) for 0 <α<1is[23]
yF(t)=K
Ttα
∞

k=0 −1
Ttαk
Γ(αk +α+1) =K
TtαEα,α+1 −tα
T,(1.8)
where Eμ,β (z) is two-parameters Mittag-Leffler function [7].
The general theory of insulation resistance and reason why a fractional-
order system is suitable in modeling a PMSM is described in Section 2. The
development of a fractional-order model for a low-power PMSM is discussed
in Section 3, while discussion on its effects in polarization index is presented
in Section 4. A new insulation resistance measure called the Three-Point
Polarization Index (3PPI) is also presented in Section 4. Section 5 concludes
this paper with some additional remarks for further research.
This paper serves as an extended version of a previous conference paper
[10].
2. General theory of insulation resistance
The equivalent circuit of an electric motor is shown in Fig. 1, empha-
sizing on the current flowing into its branches. The total current, iT(t),
is a function of the following currents: the surface leakage current, iL(t),
the conduction current, iG(t), the geometric capacitive current, iC(t), and
the absorption current, iA(t). e(t) is the supplied voltage into the elec-
tric motor while RS,R
L,R
G,R
1,R
2,··· ,R
k,C,C
1,C
2,··· ,C
kfor k∈Z+
are resistors and capacitors representing the overall characteristics of the
electric motor.
From Fig. 1, it can be seen that the absorption current, iA(t), is de-
fined by an RC ladder where the values of its resistances and capacitances
depend on the properties, type, and most importantly the condition of the

618 E.A. Gonzalez, I. Petr´aˇs, M.D. Ortigueira
insulation system. In all of the currents flowing into the machine, the ab-
sorption current, iA(t), is considered to be the one that has the biggest
impact in defining how the insulation resistance would change with re-
spect to time. In IEEE Std 43-2013 [16], the absorption current, iA(t), is
normally expressed as an inverse power function of time. The absorption
current, iA(t), results from electron drift and molecular polarization of the
insulation [5, 26, 15, 33].
However, such similar phenomenon has been existing in many other
systems such as electrode/electrolyte-based systems including the Warburg
impedance and other similar interfaces [32, 30, 18], and those systems with
electrodes connected in porous channels [17], where the total impedance
follows a power law that is fractional degree in value. The realization of
such fractional-power-law impedances was generally described by Wang [31]
as a resistor-capacitor (RC) ladder network which can be mathematically
expressed using continued fraction expansions which would end up in a
circuit having a constant phase throughout the frequency spectrum, or
basically a fractional-order element with a fractional-order impedance of
the form
Z(jω)=(jω)α,(2.1)
where 0 <α<1. Practically, since the entire frequency spectrum cannot
be satisfied, approximation models by Valsa [29] were used where the RC
ladder network is truncated at a certain extent, making the circuit valid
for a certain limited range of frequencies [13]. Approaches in the imple-
mentation of such circuits where then attempted throughout the years for
industrial controllers [6, 12], design of signal processing filters [22], and
microelectronic circuits [1].
Combining all the resistances and capacitances in Fig. 1, except for
RS, would result in a similar case of an RC ladder with a fractional-order
impedance. In such case, the total admittance of the RC ladder with respect
to the resistors and capacitors in Fig. 1 becomes
Y(jω)= 1
RL||RG
+jωC +
m

k=1
jωCk
1+jωRkCk
,(2.2)
where
ω=[ωmin,ω
max]
∈⎡
⎣1
R1C1
,R1C1
0.24
1+ϕm⎤
⎦(2.3)
is the range of frequencies in rad/s, m>0 is the number of RC branches
in the ladder network, and ϕ>0 is a certain small value of phase ripple
in degrees which is allowed to satisfy the approximation in the constant
phase region from ωmin to ωmax.Thesymbol
∈indicates that the lower and
upper limits of the frequency range are just approximate values, i.e. ωmin ≈

NOVEL POLARIZATION INDEX EVALUATION . . . 619
1/R1C1and ωmax ≈R1C1(0.24/1+ϕ)−m. The readers are referred to
[13] for the derivation of (2.2) and (2.3). Combining RSand the fractional-
order element from the combination of RL,RG,R1,R2,··· ,R
k,C,C1,
C2,··· ,C
k, which is described by the admittance in (2.2) would then result
in a resistor-fractional-order-element series circuit with a total impedance
of Z(s)=RS+Msα,
where M>0 is a certain resistive magnitude. Discussion on the properties
of such circuit are presented in [11]. This then justifies the mathematical
basis for assuming that a PMSM could be represented by a fractional-order
dynamical model as discussed in the next section.
3. Development of a fractional-order dynamical model
for a PMSM
Time [mins.] L1-L2 Res. [MΩ] L2-L3 Res. [MΩ] L3-L1 Res. [MΩ]
061 67 67
1193 213 216
2232 249 252
3248 264 267
4258 271 273
5266 276 277
6270 278 278
7275 280 279
8286 281 281
9283 283 281
10 284 289 281
Table 3. Insulation resistance values gathered on a 415-
V PMSM used in a high-risk elevator system between two
different lines.
Throughout this section, we focus on the resistance data obtained in L3-
L1. Without using any sophisticated system identification algorithm and
from a simple trial-and-error routine, the FOTF model identified results in
GF,L3−L1(s)= 0.57
15s0.81 +1,(3.1)
where the order of the system becomes α=0.81. The step response of
(3.1) from 500-Vdc step input is shown in Fig. 3 together with the data in
Table 3 for L3-L1. In Fig. 3 a plot of (3.1) is discretized every 60 seconds
for compatibility with the data obtained through the Fluke 1503 insulation
resistance tester.

620 E.A. Gonzalez, I. Petr´aˇs, M.D. Ortigueira
         











Figure 2. Insulation resistance measured on different time
values from 0 to 10 mins from an input DC step voltage of
500 Vdc. Blue line with square markers are for L1-L2. Red
line with circular markers are for L2-L3. Black line with
cross markers are for L3-L1.
PMSM Specifications Data
Insulation class F (Thermal class 130)
Moment of intertia [kg·m2]4.0
Rated voltae [V] 380 to 480 ±10%
Rated current [A] 0.65
Degree of protection IP 21
Table 4. Specifications of the PMSM tested for insulation resistance.
The data gathered from a 415-V PMSM through a Fluke 1503 insulation
resistance tester are shown in Table 3. Plots of the data are also presented
in Fig. 2. The specifications of the motor tested are shown in Table 4.
Using (1.8), the step response of (3.1) from a 500-Vdc step input yields
yF(t) = 500 0.57
15 t0.81
∞

k=0 −1
15 t0.81k
Γ(0.81k+0.81 + 1)
=19t0.81
∞

k=0 −0.0667t0.81k
Γ(0.81k+1.81)
=19t0.81E0.81,1.81 (−0.0667t0.81 ),(3.2)

NOVEL POLARIZATION INDEX EVALUATION . . . 621
060 120 180 240 300 360 420 480 540 600
0
50
100
150
200
250
300
Time [s]
Insulation Resistance [MOhms]
Step response from the FOTF
Data from Fluke 1503
Figure 3. 500-V step response of the FOTF in (3.1) with
respect to the data plot from Table 3 for L3-L1. Dashed-
blue line with circular markers represents Table 3 L3-L1 data
plot while the solid-black line represents the step response
in (3.1). Unit-step response is from a 500-Vdc step input.
by letting K=0.57, T= 15 secs., α=0.81, and by scaling the entire
function with 500 units to incorporate the effect of the 500-Vdc step input
voltage. It can be concluded from the point of view of L3-L1, the motor
insulation resistance is a 0.81-order system.
4. Characteristics of the fractional-order dynamics
in relation to polarization index
The P.I. method is known as a good indicator in determining if the
insulation of the PMSM is already at fault. However, the type of fault may
not necessarily be identified from the value itself. In fact, the severity of the
fault could cause various result in the P.I. value. Although it is understood
that the P.I. alone may not be able to provide a detailed picture of the
condition of the motor, it is still a parameter that has to be identified in
relation to the dynamics of the PMSM’s insulation resistance over time due
to practical reasons.
The P.I. formula in (1.1) itself could be misleading because it only mea-
sures the 1st and 10th minute insulation resistance values without regards
to the dynamics. To illustrate this situation, consider the approximate
FOTF in (3.1) and another hypothetical FOTF

622 E.A. Gonzalez, I. Petr´aˇs, M.D. Ortigueira
         







 $
!$&%"!$$%! $
'#





'#

Figure 4. 500-V step responses of (3.1) and (4.1). The
solid-blue line represents GF,L3−L1while the dashed-black
line represents GF,Hyp.
GF,Hyp (s)= 0.57
140s0.95 +1,(4.1)
both of its step responses with a 500-Vdc step input voltage shown in Fig. 4.
At a first glance, one can see that the two responses are different in shape:
GF,L3−L1(s) looks “healthier” than GF.Hyp (s) because the insulation resis-
tance of GF,L3−L1(s) has reached a stabilizing point upon reaching around
120 seconds. On the other hand, the step response of GF,Hyp (s) has sig-
nificant and highly-sloped monotonically growth throughout the 10-minute
period. The graphical evaluation is counterintuitive when compared to
their respective P.I. values. For GF,L3−L1(s), the measured P.I. value is
P.I. [GF,L3−L1]≈279.4
225.0≈1.2,(4.2)
while the P.I. value measured for GF,Hyp (s)is
P.I. [GF,Hyp]≈267.2
84.6≈3.2.(4.3)
Using the IEEE Std 43-2013 [16] and by looking at the P.I. values alone, one
could say that GF,L3−L1(s) failed the P.I. test while GF,Hyp (s) passed the
test, since the minimum P.I. prescribed in Table 1 for a Class B insulation
system is 2.0.

NOVEL POLARIZATION INDEX EVALUATION . . . 623
From this particular example, it is clear that there should be a certain
parameter that would describe the dynamics in between the 1- and 10-
minute insulation resistance measurements. It is therefore recommended to
have a third measure which results in the proposed three-point polarization
index (3PPI) definition below.
Definition. Let IR10 >IR
θ>IR
1be the insulation resistances
measured in 10 mins., θmins. and 1 min., respectively, where 10 >θ>1.
The Three-Point Polarization Index (3PPI) is defined as
3PPI 
=γIR10 −IRθ
IRθ−IR1
+(1−γ)IR10
IR1
,(4.4)
for 0 <γ<1.
The proposed 3PPI is a function of two parameters: the usual P.I. on
the second term of (4.4) which is defined in the IEEE Std 43-2013, and a
certain ratio between three insulation resistance measures in the first term
of (4.4). When γ= 0, then the 3PPI definition because the usual P.I.
definition. However, when γ= 1, the 3PPI definition changes into a ratio
of the between the difference of IR10 and IRθ, and the difference of IRθ
and IR1. The following research questions are then formed:
(1) What is the optimal value of θfor certain types of insulation sys-
tems?;
(2) What is the optimal value of γ?; and
(3) How low should (IR10 −IRθ)/(IRθ−IR1) be to say that the
PMSM is in “good” condition?
A conjecture is then formed based on the definition of 3PPI:
Conjecture. A PMSM is “healthier” as 3PPI →0.
5. Conclusion and further research
This paper has shown that a permanent magnet synchronous motor
(PMSM) could have insulation resistance dynamics that is fractional-order
in nature. As a consequence, the polarization index (P.I.) measure based
on the IEEE Std 43-2013 definition may provide a misleading result if
the dynamics of insulation resistance is not well-understood. The authors,
therefore, recommend to measure a third insulation resistance value in be-
tween IR1and IR10 whichisdefinedasIRθresulting in the proposed new
“goodness” measure called Three-Point Polarization Index (3PPI) which is
believe to be a better representation of how “good” or “healthy” a PMSM
is.
To further advance this study, the authors recommend looking into how
to find the optimal values of θand γwhichisassumedtobeafunctionofthe

624 E.A. Gonzalez, I. Petr´aˇs, M.D. Ortigueira
type of insulation resistance of the PMSM. To identify these optimal values,
a series of exhaustive tests should be done by the research community using
various test voltages and PMSMs.
Moreover, for further research would be better to use some exact method
for instance maximum likelihood estimation method for estimating the
model parameters θand γ. We will use proposed technique for a class
of the motors.
This paper has also shown the importance of the mathematics of frac-
tional calculus and the application of fractional-order systems in the study
of PMSM insulation resistance dynamics. A fractional-order system, in this
context, is a generalization of the dynamical model of the PMSM in the
assumption that we are constrained with the order at 0 <α<1.
Acknowledgements
The authors would like to thank Engr. Michael John B. Castro, Engr.
Rolly S. Presto and Engr. Michael Angelo D. Abalos of Jardine Schindler
Elevator Corporation, 20/F Insular Life Corporate Center, Filinvest Ala-
bang, Muntinlupa City, Philippines, and Marwan Radi of Jardine Schindler
Group, 29/F Devon House, Taikoo Place, 979 King’s Road, Quarry Bay,
Hong Kong, for the series of discussions regarding the data obtained and
presented in this paper. The authors also highly appreciate the guidance
and help provided by Greg Stone of the P43 Working Group of the IEEE
Std 43-2013 who has been instrumental in providing clarity about Polar-
ization Index.
This work was supported in part by Portugese National Funds through
the FCT–Foundation of Science and Technology under the Project PEst-
UID/EEA/-00066/2013, the Slovak Grant Agency for Science under Grant
VEGA 1/0908/15; the Slovak Research and Development Agency under
the Contract: No. APVV-14-0892; Project ARO W911NF-15-1-0228 from
USA; and COST Action CA15225 – a network supported by COST (Euro-
pean Cooperation in Science and Technology).
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NOVEL POLARIZATION INDEX EVALUATION . . . 625
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1Schindler Elevator Corporation
1530 Timberwolf Dr, Ste. B, Holland, OH
43528-9161, USA
e-mail: emmgon@gmail.com
2Technical University of Koˇsice
Faculty of BERG, B. Nˇemcovej 3
042 00 Koˇsice, SLOVAK Republic
e-mail: ivo.petras@tuke.sk Received: February 20, 2017
3UNINOVA and DEE/ Faculdade de Ciˆencias e Tecnologia da UNL
Campus da FCT, Quinta da Torre
2829-516 Caparica, PORTUGAL
e-mail: mdo@fct.unl.pt
Please cite to this paper as published in:
Fract. Calc. Appl. Anal.,Vol. 21, No 3 (2018), pp. 613–627,
DOI: 10.1515/fca-2018-0033