Methods for Analyzing the Reliability of Electrical Systems Used Inside Aircrafts

This chapter presents two solutions to perform reliability analysis of electrical systems installed on aircrafts. The first method for determining the reliability of electrical networks is based on an analogy between electrical impedance and reliability. The second method is based on application of Boolean algebra to the study of reliability in electrical circuits. By using these research methods we obtain information on operational safety of the electrical systems on board of an airplane, either for the entire system or for each of its components (Jula, 1986). The results allow further optimization of the construction of electrical system used on aircrafts (Aron et al., 1980), (Jula et al., 2008).


Introduction
This chapter presents two solutions to perform reliability analysis of electrical systems installed on aircrafts.The first method for determining the reliability of electrical networks is based on an analogy between electrical impedance and reliability.The second method is based on application of Boolean algebra to the study of reliability in electrical circuits.By using these research methods we obtain information on operational safety of the electrical systems on board of an airplane, either for the entire system or for each of its components (Jula, 1986).The results allow further optimization of the construction of electrical system used on aircrafts (Aron et al., 1980), (Jula et al., 2008).

Calculating electrical impedance and reliability -an analogy
Establishing the reliability of structures resulting from the analysis of electrical systems installed on board of aircrafts can be achieved by direct calculations, but involves a long working time as a result of taking into account all possible situations that can occur during system operation (Reus, 1971), (Hoang Pham ,2003), (Levitin, G. et al., 1997).
A more efficient calculation method for complex structures can be achieved by applying equivalent transformation methods in terms of reliability, similar to the transformation theorems for electrical circuits applied to determine the equivalent impedance between two nodes (Moisil, 1979), (Drujinin,1977), (Billinton, 1996).

Short presentation of the analogy method
To highlight the approximations introduced by this method of calculation consider a group of elements connected in series, with the likelihood of downtime q 1 , q 2 , ..., q n .Using transformation theorem for elements in series, these elements can be replaced with a resultant, a single item that has a probability of downtime q, (Drujinin,1977), given by: - The exact formula 1 1( 1) www.intechopen.com Recent Advances in Aircraft Technology

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-The approximation of order 1 The approximation of order 2 For the approximation of order 1, the error made is of the order of magnitude q i 2 , while for 2 nd order the approximation error is q i 3 , etc.
Therefore for order 1 approximation, the probabilities of downtimes q 1 , q 2 , ..., q n of elements connected in series are added together as if determining the equivalent impedance of a circuit with electrical components connected in series.
A group of elements connected in parallel with the probability of downtimes q 1 , q 2 , ..., q n can be replaced by one single element that has a probability of downtime: In this case, the equivalent probability of downtime is achieved as a product of individual probabilities; therefore the result in this case is different from the equivalent impedance of an electrical circuit made of components in parallel.
A group of elements with delta connection, with the likelihood of downtime q 12 , q 23 , q 31 may be replaced by another group of elements connected in star with the probability of downtime q 1 , q 2 , q 3 .The relations for transformation are: with an approximation error proportional with q 12 • q 23 • q 31 .Relation (5) was deducted under the assumption that the reliability of the circuit between two points, for example between point 1 and point 2 -Figure 1 -is the same for both connections in two borderline cases, namely:


The third point is offline,  The third point is connected to one of the first two.
Under these conditions the following relationships are obtained: qqq q qq q q q qqq qqq q q q qqq qqq q q q www.intechopen.comqq q q qq q qq q q qq q qq q q qq q       (7) It can be seen that the two systems described in ( 6) and ( 7) are incompatible.But if you take into account that the components used in electrical circuits on board of an aircraft are characterized by 1 q  , approximate solutions can be utilized (Aron & Paun, 1980).
Neglecting the smaller higher-order terms of the transformation delta-star, in this case the third order component, equations in (6) become: qq q q q q q q qq qq qqq q q q       (8) If the second equation is multiplied by (-1) and all the system equations are added, equation ( 9) is obtained: Applying the same methodology for the other two remaining equations in (8) results the below equivalence for delta-star transformation: From (7) and using the same methodology, relationships for star-delta transformation are obtained (Hohan, 1982): www.intechopen.com

The analogy method applied for electrical circuits used in aircrafts
Example 1.The diagram presented in Figure 2.a corresponds to a three-phase electrical generator, part of the airplane power system, powered by a three-phase electric motor, both having their stators with delta connection .The transformed version of the diagram according to the analogy method is shown in The transformation delta -star applied to q 1 , q 2 , q 3 and q 4 , q 7 , q 8 becomes a simple network configuration for which downtime can be established with the specific probability when applying the previously derived relations: () () 7813 5 47 23 6 48 12 q q q qqq qq qqq q Qq q      25 67 81 3 61 3 54 6 74 5 8 1 qq qq qq q qq q qq q qq q q Qq     If the components have the same probability q, then the probability of downtime Q is: 3 shows the diagram of a measurement instrument based on logometric principle, used to measure engine temperature or quantity of existing fuel in the plane tanks (Jula, 1986).The relations obtained for the probability of downtime Q after two transformations are: www.intechopen.com()() () () 123 7 6 14 127 5 24 3 6 14 5 24 67 123 147 356 q q qq qq q q q q q q q q qq qqq q Q qq qqq qqq qqq Q If the components have the same probability of downtime q, it results:   Whereby the result is identical to the one previously obtained, the calculation time is significantly reduced.

Conclusions regarding the analogy method
The method draws on the similarity between the calculus for the electrical impedance and the reliability one, allowing the use of simple relationships and reducing the number of equations to be solved.In case of complex networks other methods would lead to difficulties in obtaining results in short time, while the analogy method, with its rather low number of calculations ensures a time efficient way of finding the downtime probability of any electrical circuit.
If one or more circuit elements are less reliable than other parts of the circuit, and therefore its downtime probability is high, the transformation can get more accurate approximations of the real state of the system than other methods, mainly due to the multiplier effect contained.

The method based on Boolean logical structures
Large-scale systems reliability analysis is based on the quantification of the failure process at the structural level.Thus, any system downtime is a result of a quantified sequence of states in the failure process.The quantification level can be chosen in accordance with the desired goal and probability, down even to individual components of the system.The more detailed the quantification, the more accurate would be the resulting probability (Reus, 1971) (Muzi, 2008).
The conceptual representation of an emergent downtime is formed by a series of primary events, interconnected through different Boolean logical structures, which indicate the possible combinations of those elements having as result a system failure (Denis-Papin& Malgrange, 1970), (Chern & Jan, 1986).Thus determining the reliability of an aircraft electrical system using Boolean algebra actually means calculating the probability of a "failure" event.

Principles of the Boolean method
From the structural point of view, for the reliability analysis, we will use the terms: -Primary elements -components or blocks at the base level of the quantification, -Primary failures -primary elements failures, -Unwanted event -system failure state, -Failure mode -the set of primary elements that when simultaneously in failure mode, drives to a system failure -Minimal failure mode -the smallest set of primary components that when simultaneously in failure mode, drive to a system failure -Hierarchic level -all elements that are structurally equivalent and having equivalent positions in the system failure representation.
The method is based on binary logic.Thus, a system function is equivalent to a binary function, which variables are the events (the failures).
This binary function: is synthesized with logical elements AND/OR, using the following symbols and states: - (Reunion) for the function OR - (Intersection) for the function AND X i is 1 if the primary element is good and 0 otherwise, and Y is 1 if the system is good and 0 otherwise.
The method representation is depicted in Figure 6.For the reliability function indicators calculus, in the hypothesis of the failure intensity having an exponential distribution, we use the relations: where: 1 . 13) is used for the serial connection and relation ( 14) is used for the parallel connection of the elements.

Method application for determining the reliability of the aircrafts electric circuits
In order to exemplify the method for the reliability indicators determination, we will focus on the DC electrical power supply system of an aircraft.Figure 7 depicts the electric power supply system of an aircraft.
In principle, this electric power supply system is present (as the main electric power supply system) in a large number of military aircrafts ranging from the MiG family (21,23,27,29,31,35),Su (30,33,34,35,37) to Chengdu (J-10), Shenyang (J-11) and ORAO.The example refers only to a DC electric power supply system nevertheless the method can be used in alternative current and mixed systems set-ups.In Figure 7: The emerging failure state diagram using AND/OR elements is depicted in Figure 8.The failure event is the loss of voltage at the 28V bar.
For the failure intensity i  of the components we use the relation: where: k -maintenance and way-of-use coefficient (for aircraft components the coefficient varies between 120 and 160); 0  -failure intensity -manufacturer specific data.
The data relative to the electric power supply system are presented in Table 1.;; 1 1

Symbol Description
Thus, we have which is similar to Considering the failure intensity as exponential distribution, the system failure probability is given by the following relations:
Thus, mean time between failures in the non improved system may be approximated as follows 1070 hours.MTBF 

Reliability optimization of electric power supply in the aircraft industry
We can improve the electric power supply system reliability using a redundant (reserve) subsystem.The proposed improved electric power supply, including the back-up subsystem (dotted lines) is depicted in Figure 9.
Further on we will analyze the improved electric power supply system reliability, using the Boolean method presented in chapter 3.2.This analysis also allows a determination of a relation between the system reliability and the system weight.Such a relation is useful when emphasizing the variation of the system reliability with the total weight of system components.
Through a compared analysis of different reliability improving variants, imposing as minimum condition the component weight, we can obtain an optimal solution.The logic structure that drives to the system failure status (for the improved system schematics) is depicted in Figure 10.
Table 2 presents the values of the failure intensity for the supplementary components from the back-up system, in the exponential distribution hypothesis.

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The Boolean function in this case is: Transforming in algebraic form, we have:  From ( 24) we can determine the system failure probability

Influence of the maintenance and way-of-use coefficient k on MTBF
Taking into account the characteristics of the system failure probability -() Ft and reliability () Rt as in Figure 7 and 9, a simulation was made using a Matlab program (Jula et. Al., 2008), which presents the time evolutions of the variables.
Coefficient k from the equation ( 15) has the starting value k =160.For this value MTBF was calculated both for the initial and the improved systems.The Matlab program helps conduct a complex analysis of the influence of coefficient k on system failure's probability, its reliability and MTBF.Figures 11 to 13 present the results for the initial system.As it can be seen, the increase of k is directly proportional with function () Ft and inversely proportional with the reliability function () .

Rt
Mean time between failure (MTBF) is bigger for small values of the coefficient k.
The same analysis will be conducted for the improved system, in order to compare results.The graphic characteristics are the presented in Figures 14 to 16, while the obtained values both for initial system and improved system are presented in Table 3.A comparative presentation of the two systems' reliability for different values of k is depicted in Figure 17 (for initial system with blue lines and red for the improved system).
For the five analyzed values of coefficient k, the improved electric supply with a redundant (reserve) subsystem is characterized by superior values of MTBF compared to the initial system (fig.18).In Figure 18 the evolution of MTBF for the initial system is represented by a dashed line, while the evolution of MTBF for the improved system is represented by a continuous line.www.intechopen.com Methods for Analyzing the Reliability of Electric Systems Used Inside Aircrafts 379

Conclusions regarding the Boolean method
From the analyzed examples and then results obtained for MTBF, we can conclude that the method can be successfully used in the aircraft industry for determining the reliability of the electrical systems.The MTBF influencing parameters in the main system nodes (power supply bars and distribution panels) can be calculated and compared.
Through the failure related logic function analysis we can determine the circuits that can improve the system reliability.In the case presented, through the introduction of the components 60E, 61E and corresponding contacts, substantial increase of the reliability (approximately 6 times higher) was obtained for the 28V DC power supply bar.
We have conducted a complex analysis of the influence of the maintenance and way-of-use coefficient k on system failure probability, system's reliability and MTBF.
The diagram in Figure4corresponds to an aircraft specific electromagnetic system powered by multiple nodes.

Fig. 5 .
Fig. 5.A version of the final state after the transformation.A relation for this state is: Fig. 6. a) The general concept of the method based on Boolean algebra (1, 2,..., n are independent primary events); b) the schematics of the logic function AND; c) the schematics of the logic function OR.
Fig. 7.The electric power supply diagram for a DC main electric supply system aircraft (fragment).

Fig. 8 .
Fig.8.The logic structure that drives to the system failure status.In these conditions, the Boolean function associated to the logic structure depicted in Figure8has the following form:    71 2 123456 891 01 1 Y XX XXXXXX XXX X      (16)To transform the logic equation into algebraic form we use the following relations Fig. 9. Electric power supply system of an aircraft including the back-up subsystem (fragment).

Fig. 10 .
Fig. 10.The logic structure of the electric system presented in fig.9.

Fig. 11 .
Fig. 11.System failure probability F(t) for different values of k (initial system).

Fig. 14 .
Fig. 14.System failure probability for different values of k (improved system).

Fig. 15 .
Fig. 15.System's reliability for different values of k (improved system).

Fig. 17 .
Fig. 17.Comparative analysis of the two systems' reliability for different values of k.