G225159

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org Volume 2 Issue 2 ǁ February. 2013 ǁ PP.51-59
www.ijesi.org 51 | P a g e
Mathematical Modeling for Reliability Assessment of 250-5000
Watts Pulse Width Modulation Power Inverter in Nigeria
M.B. Adamu1
, I.G, Saidu2
, M.B. Abubakar3
, N.S. Jega4
, M.I. Ilyasu5
, A.Tukur
(MIEEE, MIET, MITP)6
, M.A. Yusuf7
1,6
(Physics department, faculty of science, Usman Danfodiyo University, Sokoto – Nigeria)
2,3,5
(physics unit, Sokoto state polytechnic, sokoto- Nigeria)
4
(School of nursing and midwifery, Birnin Kebbi, kebbi state-Nigeria)
7
(Electrical engineering department, Sokoto state polytechnic, Sokoto-Nigeria)
ABSTRACT: A power inverter, or inverter, is an electrical power converter that changes direct current (DC)
to alternating current (AC) the converted AC can be at any required voltage and frequency with the use of
appropriate transformers, switching, and control circuits. The paper presents mathematical modeling for
assessing reliability of PWM power inverter in Nigeria. The part stress method was used to predict the
reliability of the system. Data on the failure of PWM Inverter in Sokoto were used as a case study, with special
consideration given to factors like environmental impact, quality of power supply, service personnel, human
factors (over and under usage). A comparative assessment was made on the reliability and reliability indices of
the power inverter when operated within Sokoto environment and when operated within the environment for
which it was designed (china). The result shows that lower reliability level is associated with the use of PWM
power inverter in Sokoto state of Nigeria, as compared with the country for which it was designed.
Keywords: Failure rate. Inverter, Modeling, PWM, Reliability
I. INTRODUCTION
250-5000 watts PWM DC/AC 220V power inverter is as electrical device which is designed to coverts
direct current DC to alternating current AC with the use of a transformer, switching and control circuits
[13][14]. The reliability of this system, which has an original environment for which it was designed for,
becomes necessary to determine the degree to which it could be relied upon in the applied environment –Sokoto
state Nigeria [1][2][6][7].
1.1 Reliability
Reliability in engineering can generally be defined as the characteristic of an item expressed as the
probability that it will perform a required function understated conditions for a stated period of time [4].
Reliability predictions are an important tool for making design trade-off decisions and estimating future system
reliability [8][11]. This requires the understanding of probability and statistical concepts and has, therefore, been
found to be a very important tool in forecasting the patter of failure for systems and hence the reliability
assessment of the system at hand [10].
1.2 Part stress method
This is one of two methods used in assessing the reliability of electronic equipment. In the part stress
method, the effects of the various stresses on the actual hardware are put into consideration, with the
environmental factor and the quality of the utility. The parts count method (the other method) of assessing the
reliability of system is based on the number of different parts, quality level and application environment. The
aim in both cases is to determine the failure for a given system operating in a specific environment [9].
However, the part stress method is better method for mathematical assessing the reliability of the existing
system, based on the possibility of considering various stress peculiar to the equipment in a specific area of
application [4] and as such cold be relied upon to asses the reliability of the PWM power inverter.
1.3 The PWM power inverter system
Pulse-width modulation (PWM) is a commonly used technique for controlling power to inertial
electrical devices made practical by modern power switches. Power inverter is an electrical device that converts
direct current DC to attempting current AC with the use of a transformer, switching and control circuits. These
are two general types of power inverter: true sine wave or modified-sine wave (square wave). True sine wave

Mathematical Modeling For Reliability Assessment Of 250-5000 Watts Pulse
inverter produce power that is either identical or sometimes slightly better to power from the public utility
power grid system [13].
Modified sine wave and square wave inverters are the most common types of power inverters on the
market. Modified sine wave produces a power wave that is sufficient for most devices. The power wave is not
exactly the same as electricity from the power grid. It has a wave form that appears as a choppy square off wave
when viewed through an oscilloscope [14].
The schematic circuit design is for a 250 watt output to increase the power of the circuit you have to
add more of the Q7 and Q8 transistor in parallel, each pair you add will increase your power by 250 watts.
If you increase the power transistors you have to enlarge the T2 transformer to match the new needs. The circuit
transformer is rated 25 amps to handle 220V.
220v/12v
0.5 amp
1 16
2 15
3 14
4 13
5 12
6 11
7 10
8 9
IN 7808 OUT
GND
Driver stage
2
1
3
4
T1 –


10uF
+
3.9k (1/2 watt)
BC327
1N4007
100K
10uF
+
+
104
104
104
10K
4.7K
4.7K
100K
R2
100K
C1
2.2K
R1 1K
+
+
+
100K
47uF
1uF
+Vcc
4700uF
20uF
3
D
1K
+
SG3524
Q2
BC337
100
1K
1K 100
Q1
BC337
Power stage
2N6277
2N3055
TIP122Q3
Q4
Q5
Q7
2N6277
2N3055
TIP122
Q6
Q8
+Vcc
D1
D1
T2 220v output
0.1/600V
2.2uF/400V
1 2
Filter
Dc voltage and transformer “t2” winding recommendation:
Power Supply Winding
750W 12VDC p: 24V “12 – 0 – 12” / S: 220V
1500W 24VDC p: 48V “24 – 0 – 24” / S: 220V
2250W 36VDC p: 72V “36 – 0 – 36” / S: 220V
3000W 48VDC p: 96V “48 – 0 – 48” / S: 220V
3750W 60VDC p: 120V “60 – 0 – 60” / S: 220V
4500W 72VDC p: 144V “72 – 0 – 72” / S: 220V
5250W 84VDC p: 168V “84 – 0 – 84” / S: 220V
The transformer should be “centre tapped” at the primary side.
- R1 is to set the PWM to 220V
- R2 is to set the frequency to 50 or 60Hz
- Wiring should be thick enough to handle the huge amps drain from the batteries.

- A cooling fan will be needed to reduce heat off the heat sinks and transformer, when you power up the
circuit the fan will start this will give you a simple way to know that 220V is present and everything is ok.
- Two circuit breakers are recommended instead of fusses, one on the DC side and one on the AC side,
depending on your design example; for a 24VDC (1500 watts design) put a 6 Amps breaker on the DC side and
a 6 Amp on the AC side. For every 1 Amp of 220AC you will be raining like 8 to 10 Amps from the 12V
battery, make your calculations.
- The two heat sinks should be big enough to cool the transistors, they are separate and should not touch
each other.
- Be caution when building this circuit it involves high voltage which is lethal.
1.4 Nigeria
There are three infrastructural indices of development. These are adequate, reliable and always
available energy supply, communication and transportation system. Any society that has got these thee indices is
generally characterized as developed society. Such societies includes USA, UK, Russia, Canada, France, Italy,
Germany and Japan – popularly called G8. Countries which have not attained the three perfect indices above are
generally termed developing countries. Example of such are Nigeria, Malaysia and so on.
Nigeria is mainly considered or at best, assemblers or product designed by developing nations. They inhabit
lands with tropical climates in Africa, Asia and Latin America [5]. The electrical power and hence, have
fluctuating voltages. Some of the region are periodically subjected to every dry dust (harmattan) which is
capable of introducing a high electric field strength of above 4000V/m and sometimes metallic objects moving
in this environment accumulate an average of 1000v/m charge [5].
II. MATERIALS AND METHODS
It is convenient to specify the reliability of electronic equipment by some probability parameters, which
give indication of the failure rate of such system or equipment and does not depend on the operating time [12].
By using such parameters, it is also possible to compare the performance between different systems with
different periods. Two of such parameters that are commonly used are the mean time between failure (MTBF)
and mean time to failure (MTTF) [12].
2.1 Mean Time between Failures (MTBF)
Reliability is quantified as MTBF (Mean Time between Failures) for repairable system. Avoiding
failure in a critical data centre is always a top priority; as such a correct understanding of MTBF is important.
Therefore systems users are usually concerned with the length of time that a system will run without failure.
This is a measure of the reliability of such a system [4][12]. The MTBF can be obtained by running a system for
a predetermined length of time under specified conditions. Hence for the failure rate  (the number of failure per
unit time) MTBF is given as [4]:
 
N
tt
MTTF
n
t
i

 1
0
i.e.
     
N
tttttt
MTTF n 00201 


where t0 = starting (reference time)
(t1 – t0) = period to 1st
failure
(t2 – t0) = period to 2nd
failure
(tn– t0) = period to nth
failure
N = total number of failure components.
Consider the case in which a fixed number N0 of identical components are tested.
Let Ns = number surving up to time t.
Nf = number failed up to time t.
N0 = Ns + Nf = total number in operation at t=0
 Reliability at any time t becomes
 
0N
N
tR s
 (1)
The failure rate (t) is normally defined by the mathematical relation

 
t
N
Nt
t
f
s 




1
0
lim

dt
dN
N
f
s

1
 (2)
where Ns = number of serving items after a life test
Nf = number of failure item during the time interval, t.
Consider the case in which a fixed number N0 of identical components are tested,
Let Ns = number surving up to time t
Nf = number failed up to time t
N0 = Ns + Nf = total number in operation at t=0
 Reliability at anytime t becomes
 
0N
N
tR s

and failure rate (for constant failure rate)
dt
dN
N
f
f

1
 from Equation (2)
dt
dN
NN
f
s



0
1

or  
 f
s
dN
NN
dt
0
1

introducing the limits
 

fN
f
s
t
dN
NN
dt
0
0
0
1

    0loglog  oefoe NNNt





 

0
0
log
N
NN
t f
e
Thus
0
1
N
N
e ft

(3)
But from Equation (1)
  








00
0
1
N
N
N
NN
N
N
tR ss
o
s
(4)
Comparing equations (3) and (4), we have
  t
etR 
 (5)
The general expression for MTBF, m is (Epsma 2005, Oroge 1991, Susan, 2010)
 


0
dttRm (6)
For the case when  is constant from equation (5)
t
eR 
 so equation (6) becomes

 
 0
0
0
1
1
ee
e
dttRm
t
















1
m (7)
If failure are due to chance and if the failure rate  is constant, then
MTBF
1
 for repairable items
MTTF
1
 for non-repairable items
2.2 Equipment Availability
Mathematically, the utilization factor  can be express as [4]:
opidm
op
ttt
t

 (8)
As seen in equation (8), if the idle time is equal to zero, (i.e. tid = 0) and the maintenance time become as small
as possible, then utilization factor will approach its maximum value and can now be called availability of a unit
or system. Mathematically, this can be expressed as
opm
op
tt
t
(min)
(9)
Where top = the mean time before failure and tm(min) = the mean time to failure
MTTFMTBF
MTBF
A

 (10)
2.3 Mean time to failure (MTTF)
The mean time to failure is a term which applies to non-repairable items (such as resistors, capacitors,
electric bulbs and so on, which are disposed off as soon as they fail [12]. MTTF is the average time an item may
be expected to function before failure. This MTTF can be obtained by stressing a large number of components
under known conditions for a period of time and noting the number of failures [4][5]
2.4 Equipment Failure Profile
Over the years, complex equipment and components have been found to follow a familiar pattern of
failure, which has been well documented. Failure rates have been calculated for equal time intervals from
installation to replacement. When the failures rates is plotted against a time scale spanning the equipment life
time, the resulting graph, popularly known as “bathtub is obtained as shown in figure 2.1 [2][4].
It exhibits three district periods or zones, the infant mortality period, the constant failure rates period and the
wear out period.
Infant mortality period
This is the running-in period. During this period, the failure rate has been found to be high, due to other
design or manufacturing errors, misuse. It however, falls off rapidly with operation. Failures in this period can
be avoided during product development through the use of stimulated tests, or by vigorous stressing during
commissioning tests.
Constant Failure Rate Period
This period follows the running-in period. During this period, the failure rate is lowest and is a function
of the basic design. Failure results either through accidents or poor operation or maintenance. In this phase, the
mean time to failure (MTTF) is the reciprocal of the (constant) failure rate.
Wear Out Period

This period manifests towards the tail end of the equipment component life. During this period, failure
is due to old age, various components are worn-out, metals become embrittled, and insulation dries out and so
on. Failure rates can only be reduced by preventive replacement of these components.
Generally in some systems, one or two of the phase (usually the ecouly failure and wear our failures) could be
more reduced or effectively absent. Therefore, estimates for parameters that affect equipment failure profile of
the constituent components, especially the length of the constant failure rate period and associated failure rates
are essential ingredients for predicting the reliability.
Fig.2. 1. Equipment failure profile – Bathtub-curve
2.5 RELIABILITY ASSESSMENT OF ELECTRONIC EQUIPMENT
The general expression for the parts stress method of mathematically assessing reliability is given as [8][10][15]
i = nB E A Q N (11)
Where
i = the failure rate ith
part
B = The basic failure rate obtained from derated data for each
generic part against normalized stress and temperature factors.
E = Account for the environmental factors other than
temperature
A = Account for secondary stresses (e.g. vibration, shock,
e.t.c.)
Q = Account for the degree of manufacturing control
N = Account for any additional factor that has not been taken care of above.
n = Number of particular component.
For the scope of this work, the above equation (11) for the part stress method was reduced to the expression of:
i = nBTK (12)
Where
i = The failure of the ith
part
n = number of particular component
B = Base failure rate obtained from the derated data for each generic part against normalized stress and
temperature factor.
T = Room temperature of the environment
K = Voltage stress ratio
Voltage stress (K) =
𝑀𝑒𝑎𝑠𝑢𝑟𝑑 𝑣𝑜𝑙𝑡𝑎𝑔𝑒
𝑅𝑎𝑡𝑒 𝑣𝑜𝑙𝑡𝑎𝑔𝑒
The failure rate of the regions under consideration, Sokoto-Nigeria and China are summed up to calculate the
inherent system reliability. These failures are constant and as best described by the exponential distribution law
for useful operation region.
Early failure
rate period
Infant
mortality
Constant failure rate period
Normal useful operation
period
Increasing failure
rate period
Wear out period
FAILURERATE
TIME IN SERVICE

RELIABILITY ASSESSMENT OF 250-5000 WATTS PWM DC/AC POWR INVERTER
The design criteria presented above will be used to assess 250-5000 watts PWM DC/AC power inverter
in Sokoto-Nigeria and in the country for which it was designed for (China). Hence, we shall have designed
failure rate, as it will be applicable to the system operating in the environment for which it was designed and a
relative failure rate as it will be applicable to the system operating in the Sokoto state Nigerian environment [5].
The conclusion arrived at will be used assess the reliability of the PWM power inverter. The components of the
250 to 5000 watts PWM power inverter with the generic failure rate, which has been taken care of the
environmental factors and the application stress factor and the results of the failure rates are shown in Table 3.
Table 1: failure rates of the pwm dc/ac power inverter
Circuit
Ref
Component
description
Qty
(n)
TD
(0
C)
TN
(0
C)
Voltage stress
ratio K
BD
(x10-6
/hr)
BN
(x10-6
/hr)
eff
D
=
(nKTBD)
eff
(nTKBN)
T1 Transformer 1 25 27 0.1167 0.011 0.019 0.0320925 0.0598671
D1 General purpose
diode (germanium)
1 25 27 0.006 0.0032 0.102 0.00048 0.016524
D2 -do- 1 25 27 0.03875 0.0032 0.102 0.0031 0.1067175
D3 -do- 1 25 27 0.1755 0.0032 0.102 0.01404 0.483327
D4 -do- 1 25 27 0.0125 0.0032 0.102 0.001 0.034425
R1 -do- 1 25 27 0.8 0.0219 0.105 0.438 2.268
R2 -do- 1 25 27 2.0 0.0219 0.105 1.095 5.67
R3 -do- 1 25 27 0.05 0.0219 0.105 0.027375 0.14175
R4 -do- 1 25 27 1.90 0.0219 0.105 1.04025 5.3865
C1 Ceramic capacitor 1 25 27 0.0625 0.033 0.141 0.0515625 0.2379375
C2 -do- 1 25 27 0.045 0.033 0.141 0.037125 0.171315
C3 -do- 1 25 27 0.0512 0.033 0.141 0.04224 0.1949184
IC1 IN 7808 1 25 27 0.125 0.007 0.91 0.021875 3.07125
R5 Film resistor 1 25 27 6.10 0.105 0.0219 16.0125 3.60693
D5 LED 1 25 27 0.0025 0.102 0.0032 0.006375 0.000216
C4 Electrolytic capacitor 1 25 27 0.563 0.033 0.171 0.464475 2.599371
C5 -do- 1 25 27 0.0085 0.033 0.171 0.0070125 0.0392445
Q2 Transistor 1 25 27 0.0975 0.0144 0.860 0.0351 2.26395
R6 Film resistor 1 25 27 0.0340 0.0008 0.011 0.00068 0.010098
D6 Diode 1 25 27 0.0052 0.0066 0.51 0.000858 0.071604
D7 -do- 1 25 27 0.0066 0.0066 0.51 0.001089 0.090882
C8 Film capacitor 1 25 27 0.1145 0.0008 0.011 0.00229 0.0340065
R8 Film resistor 1 25 27 0.0110 0.0008 0.011 0.00022 0.003267
Circuit
Ref
Component
description
Qty
(n)
TD
(0
C)
TN
(0
C)
Voltage stress
ratio K
BD
(x10-6
/hr)
BN
(x10-6
/hr)
eff
D
=
(nKTBD)
eff
(nTKBN)
R9 Film resistor 1 25 27 0.0340 0.0008 0.011 0.00068 0.010098
R10 Film resistor 1 25 27 0.0348 0.0008 0.011 0.000696 0.0103356
IC2 SG3524 1 25 27 0.03875 0.011 0.15 0.1065625 1.569375
R11 Film resistor 1 25 27 0.528 0.0008 0.011 0.01056 0.156816
R12 Variable resistor 1 25 27 0.5097 0.086 1.3 1.095855 17.89047
Q2 Power transistor 1 25 27 3.3 0.0085 0.078 0.70125 6.9498
Q3 -do- 1 25 27 3.34 0.0085 0.078 0.70975 7.03404
Q4 -do- 1 25 27 2.77 0.0085 0.078 0.588625 5.83362
Q5 -do- 1 25 27 8.35 0.0085 0.078 1.774375 17.5851
Q6 -do- 1 25 27 11.80 0.0085 0.078 2.5075 24.8508
Q7 -do- 1 25 27 2.74 0.0085 0.078 0.58225 5.77044
Q8 -do- 1 25 27 1.76 0.0085 0.078 0.374 3.70656
Q9 -do- 1 25 27 2.13 0.0085 0.078 0.452625 4.48578
D7 Diode 1 25 27 0.0036 0.0066 0.51 0.000594 0.049572
D8 -do- 1 25 27 0.0018 0.0066 0.51 0.000297 0.024786
T2 High power x4mer 1 25 27 0.1043 0.003 0.047 0.0078225 0.1323567
C10 Polyester film
capacity
1 25 27 0.22 0.0008 0.011 0.0044 0.06534
C11 -do- 1 25 27 0.1167 0.0008 0.011 0.002334 0.0346599
D9 - D12 Bridge rectifier 1 25 27 0.0192 0.0066 0.51 0.003168 0.264384
TOTAL 28.46851425 124.099455

Where
BN = Base failure rate of the inverter in Sokoto-Nigeria
BD = Base failure rate of the inverter in designed environment (China)
Toc
(D) = Average room temperature of the designed environment (China)
Toc
(N) = Average room temperature of Sokoto-Nigeria
K = Voltage stress ratio
effD = Effective base failure rate for designed environment (China)
effN = Effective base failure rate for the applied environment (Sokoto-Nigeria)
Reliability can be assessed from the failure rates obtained from table 3 for China and the applied environment
(Sokoto-Nigeria).
Table 3: Reliability result
Year 1 Year 2 Year 3 Year 4 Year 5
China 0.779284 0.607283 0.473246 0.368793 0.2873943
Nigeria 0.337190 0.1136968 0.0383374 0.012927 0.0043588
Table 4: Percentage of reliability
Year 1 Year 2 Year 3 Year 4 Year 5
China 77.9284% 60.7283% 47.3246% 36.8793% 28.73943%
Nigeria 33.7190% 11.36968% 3.83374% 1.2927% 0.43588%
Example, for one year, = 365 days x 24 hours = 8760hrs
R = ē(28.4685E – 06 x 8760)
= 0.779284
The reliability result taken over five years for the two environments were plotted against time
0
10
20
30
40
50
60
70
80
90
0 1 2 3 4 5 6
China
Nigeria
Fig.2.2: Graph of reliability Vs year
III. DISCUSSION OF RESULT
From fig.2.2, we can appreciate and compare the reliability of the country for which the system was
designed for and that of Sokoto-Nigeria. And from it we see that the power inverter under consideration has a
relatively higher reliability in the designed country (China) than the applied environment (Sokoto-Nigeria).
The exponentially decaying reliability function graph above shows that the system has a higher failure
rate in Nigeria, due to factors, which are associated with the environment like voltage fluctuation, surge
Year
Reliability

frequency, high relative humidity, among others. The ratio of the failure rate of Sokoto-Nigeria to China is
approximately 4:1.
Comparatively, the rate of failure of the power inverter in the designed country is much less than the Nigerian
case. From the failure rate of the system obtained for the two environments, we obtain the mean time to failure
(MTTF) of the system as follows
MTTF = 1
effD
MTTFChina = 4.0099 yrs.
MTTFsokoto = 0.92 yrs.
The mean time the system is expected to function before failure (MTTF) in Sokoto-Nigeria is 0.9 years as
against 4.0 years for the designed country. The rate is about four times higher than the Sokoto-Nigerian case.
IV. CONCLUSION AND RECOMMENDAITONS
The comparative results for the system at hand, taken over five years between the country for which the
system was designed for and that of Nigeria, showed that lower reliability is associated with the use of the
system in Nigeria than the designed country (China).
The following steps are recommended for the reliability of the power inverter to be higher in Nigeria:
a. Provision of a parallel configuration in the design so as to reduce the rate of failure of the components in the
system.
b. The design of power inverter that will consider the environmental as well as stress factor in Nigeria.
REFERENCES
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M.Sc. Thesis Usmanu Danfidiyo University Sokoto, Physics department.
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Managers. Vol. 5, 2005. PP 1 – 11, Department of Library and Information Science, Ahmadu Bello University Zaria.
[3]. Arsenault, J.E. (1980) “Reliability and Maintainability of Electronics Systems” Pitman Ltd., United Kingdom. PP 162 – 187.
[4]. Oroge, C.A. (1991) Fundamental of Reliability and Testing Methods; Sooji Press Ltd., Kaduna, Nigeria.
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