Research Paper: 0 Minutes (IR 0 Minutes (IR T 0, After Applying A DC Step
Research Paper: 0 Minutes (IR 0 Minutes (IR T 0, After Applying A DC Step
Research Paper: 0 Minutes (IR 0 Minutes (IR T 0, After Applying A DC Step
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 > 0 minutes (IRt2 )
from t1 > 0 minutes (IRt2 ) 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.
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. Petráš, M.D. Ortigueira
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
IR10
P.I. = , (1.1)
IR1
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
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. Petráš, 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:
Y (s) K
G1 (s) = = , (1.2)
E (s) Ts + 1
for K, T > 0 and
Y (s) Kωn2
G2 (s) = = 2 ,
E (s) s + 2ζωn s + ωn2
K
= (1.3)
(T1 s + 1) (T2 s + 1)
for K, T1 , T2 , ωn > 0 and ζ > 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
T1 T2
y (t) = K 1 − e−t/T2 − e−t/T1 , t ≥ 0, (1.5)
T2 − T1 T2 − T1
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)
K
GF (s) = α
, (1.6)
Ts + 1
for K, T > 0 and where 0 < 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
dα
T α yF (t) + yF (t) = Ku (t) , (1.7)
dt
NOVEL POLARIZATION INDEX EVALUATION . . . 617
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 < 1 is [23]
∞ 1 α
k α
K α −T t K t
yF (t) = t = tα Eα,α+1 − , (1.8)
T Γ (αk + α + 1) T T
k=0
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 , RL , RG , R1 , R2 , · · · , Rk , C, C1 , C2 , · · · , Ck for 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. Petráš, 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
m
1 jωCk
Y (jω) = + jωC + , (2.2)
RL ||RG 1 + jωRk Ck
k=1
where ⎡ ⎤
1 R1 C1 ⎦
ω = [ωmin , ωmax ] ∈ ⎣ , m (2.3)
R1 C1 0.24
1+ϕ
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 . The symbol ∈ indicates that the lower and
upper limits of the frequency range are just approximate values, i.e. ωmin ≈
NOVEL POLARIZATION INDEX EVALUATION . . . 619
Time [mins.] L1-L2 Res. [MΩ] L2-L3 Res. [MΩ] L3-L1 Res. [MΩ]
0 61 67 67
1 193 213 216
2 232 249 252
3 248 264 267
4 258 271 273
5 266 276 277
6 270 278 278
7 275 280 279
8 286 281 281
9 283 283 281
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.
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
∞ 1 0.81
k
0.57 0.81 − 15 t
yF (t) = 500 t
15 Γ (0.81k + 0.81 + 1)
k=0
−0.0667t0.81 k
∞
0.81
= 19t
Γ (0.81k + 1.81)
k=0
= 19t0.81 E0.81,1.81 (−0.0667t0.81 ), (3.2)
NOVEL POLARIZATION INDEX EVALUATION . . . 621
250
200
Step response from the FOTF
150
Data from Fluke 1503
100
50
0
0 60 120 180 240 300 360 420 480 540 600
Time [s]
!$&%"!$$%! $
'#
'#
$
0.57
, GF,Hyp (s) = (4.1)
140s0.95 + 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
279.4
P.I. [GF,L3−L1 ] ≈ ≈ 1.2, (4.2)
225.0
while the P.I. value measured for GF,Hyp (s) is
267.2
P.I. [GF,Hyp ] ≈ ≈ 3.2. (4.3)
84.6
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
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