Reliability Simulation of an Antenna Array using MATLAB
By Juan Marin, Reliability Engineering Department • Lockheed Martin Missiles and Fire Control

Using MATLAB, a Monte Carlo simulation was performed on basic electrical circuitry found in a common array antenna. Three critical RF chains employed in an antenna array were simulated and the effective MTTF was determined for this combined circuitry. Only the complex RF chains used in a typical receive antenna array are considered in this analysis. Other items making up the antenna system such as power supplies and other support electronics were not included in the simulation analysis.

There are software packages available that utilize simulation to model systems and determine MTTF and other reliability parameters. However, it was found that some of these programs could not handle the complexity of the array or that editing a large number of the same component, as found in large antenna arrays, was very difficult or not possible. Managing and controlling all aspects of the simulation through the MATLAB programming environment was significant in developing the simulation used in this analysis.

Simulated Model
The receive array model simulated in this analysis is a general version of a phased array antenna. Phased array antennas can contain large number of components and be part of an overall complex system. A simplified version of an array is analyzed and component failure rates in the RF strings will be combined and analyzed as blocks in the reliability simulation (RF strings combined are single string items).

The general array system analyzed is one receive array face consisting of 48 subarrays and each of these subarrays contains 40 identical RF chains (the 1st RF string). The 2nd RF string that corresponds to one beam contains four 40:1 combiners that receive input from each of the 40 RF chains in a subarray. The output of the 2nd RF string (after some amplification) is input into the 3rd RF string containing 48:1 combiners that also filters and amplifies the signal before being processed by other system circuitry. Figure 1 shows a block diagram of this array system.

Each 1st, 2nd and 3rd RF string contains a group of components (recieve elements, amps and filters) in a series relationship. Failure rates for each chain are given in Table 1. Failure rates for each chain were based on MIL-HDBK-217F and other applicable methodologies. Each 1st RF chain has a total failure rate of 1.49 per million hours. There are 40 1st RF chains in each subarray and 1,920 1st RF chains (40 by 48) in the total array. Each 2nd RF chain has a failure rate of 0.6 per million hours and there are a total of 192 (4 by 48) 2nd RF chains in the total array. There are four 3rd RF chains with each having a failure rate of 0.69 per million hours. Note, the large number of RF chains and the complexity of the interconnections between RF chains. Traditional series and parallel relationships cannot model this system accurately. Additionally, the system is considered non-repairable and thus we solve for the effective MTTF of the simulated receive array (a repairable system would have Mean Time Between Failure (MTBF) as a reliability measure).

Simulated Model Assumptions
Each 40:1 combiner in a subarray corresponds to a received beam. For simplicity, this analysis assumes all four beams are required so failure of any 2nd RF chain will cause a corresponding subarray failure. In a detailed model, failure of any component (40:1 combiner, amp, etc.) comprising the 2nd RF chain will cause a corresponding subarray failure. Total array failure will be caused by failure of any 3rd RF chain. Note, if a 2nd RF chain fails then a total of one subarray or 40 receive modules or 1st RF chain failures occur. A 3rd RF chain failure would produce 1,920 1st RF chain failures.

This model also assumes a uniform planar array with equal gain and phase shifting across all elements. G/T (in dB) is used as the performance parameter used to determine array failure based on 1st RF chain failures. The G/T calculation is based on the number of failed receive modules or 1st RF chains in the total array. The degraded G/T or dB loss can be represented by equation (1) where NTOTAL is the number of total receive modules in the array and NFAILED is the number of total failed receive modules in the array.

Degraded G/T = 10*log [(NTOTAL – NFAILED)/NTOTAL] (1)

This degraded G/T value is the figure of merit threshold for the simulation analysis and will determine when the array antenna has failed to meet its operating requirements. This model assumes that a loss of -1dB for the array face causes total array failure. Given, there are 1,920 receive elements or 1st RF chains and using equation (1), it is calculated that 1,525 receive modules are needed for the array to be considered operational.

Simulation Methodology
The reliability simulation discussed implements in MATLAB the following steps:

1. Assign random failure times to RF chains in array.
2. Find the first RF chain failure (lowest Time to Failure (TTF)) in the array.
3. Calculate array performance parameter (G/T) given the failure.
4. Compare calculated array performance parameter with required specification.
5. If specification is met, find the next failure (or if antenna doesn’t meet specification then the TTF of the array is the TTF of the first failed RF chain).
6. Continue finding failures and comparing performance with G/T specification. When the specification is not met, the TTF of the last RF chain to cause the non-compliance is your system TTF (the process of taking fail ures in a timed order and comparing calcu lated G/T with specified performance con tinues until the array doesn’t meet the required specification).

Assigning Random TTF
Random failures are assigned based on the exponential distribution. Each RF chain (1st, 2nd and 3rd) has a failure rate and this failure rate is used to generate a random TTF for each chain. The following equation (derived by using the inversion method of determining random numbers) is used to generate the TTF for each RF chain:

TTF(1st, 2nd or 3rd) = -log(U)/l(1st, 2nd or 3rd) (2)

where U is a random number between 0 and 1 generated by the internal random number generator in MATLAB and l is the failure rate of a specific RF chain.

System Representation in MATLAB
For the simulated array model, MATLAB is used to generate a 41-row by 48-column matrix of TTF’s for the 1st and 2nd RF chains. Each column in the matrix represents a subarray with rows 1 thru 40 representing the 1st RF chains in the subarray. Row 41 represents the 2nd RF chain in each corresponding subarray. A separate four-row by one-column matrix was created for the 3rd RF chain. Figure 2 shows a representation of the MATLAB generated matrices for the simulated array.

Using equation (2), the TTF for each RF chain value can be calculated. Equation (3) shows the calculation for generating the TTF’s in rows 1 through 40, and equation (4) computes the TTF for the 2nd RF chains in row 41. The 3rd RF chain TTF’s are calculated in the same manner but using the 3rd RF chain failure rate in the equation.

TTF(1st) = -log(U)/(1.43/1,000,000) (3)

TTF(2nd) = -log(U)/(0.6/1,000,000) (4)

Note, equation (3) uses the 1st RF chain failure rate and equation (4) uses the 2nd RF chain in the calculations. U is a different random number between 0 and 1 for each TTF calculated.

Once the matrices have been generated, the algorithm finds the lowest TTF in both matrices. For this analysis, it has been determined that 395 receive modules can fail and the array still be operational. Then, for each simulation run, the MATLAB program searches for the lowest TTF in the matrices and continues until 395 receive modules have failed. A minimum of 395 failed receive modules constitutes an array failure so the TTF for the 1st, 2nd or 3rd RF chain causing the 395th failure is the TTF for the array.

Figure 3 illustrates this principle using a smaller array of only nine RF chains. Assume the three-row by three-column matrix in Figure 3 represents TTF’s for the nine RF chains, and five of the receive modules is the minimum needed for an operational array. As can be seen, the TTF for this system is 193 hours, since this is the TTF for the receive module that would cause the sample array to fail.

Array Performance Parameter
The array performance parameter that determines array failure is G/T that is defined as the total decibel loss of the array. Knowing the maximum decibel loss allowed for an array, then the number of failed elements that will cause array failure can be determined using equation (1). Again, the decibel loss is not the only factor considered when determining array operability, but in order to keep the analysis as straightforward as possible, the other factors involved were not considered in the simulated model.

Simulation Analysis & Results
A simulation run consists of assigning random failure rates to each RF chain and then determining the TTF for that specific set of randomly assigned failure rates. In order to produce a MTTF for the system, more than one simulation run is needed. In general, the MATLAB program determines each TTF for a defined number of simulation runs and then calculates the MTTF by simply taking the average of the TTF for each simulation run.

The number of simulation runs needed for an accurate MTTF is easily determined. The MTTF can be plotted against the number of simulation runs and as the number of simulation runs increases the variance of the TTF’s are reduced. This method of determining a valid estimate of the MTTF is basic and no additional statistical calculations are used for this analysis. Figures 4 and 5 show plots of MTTF versus simulation runs for 100 and 5,000 simulation runs and clearly shows that 5,000 runs is more than adequate to determine a MTTF. Figure 5 shows convergence of the MTTF close to 96,000 hours.

The plots are intended to demonstrate the law of averages by showing that as the number of simulation runs increases a more accurate MTTF is produced. According to the general law of averages, a sample's mean will converge to its expected value as the sample size increases. In other words, a more accurate MTTF is calculated as the number of simulation runs increases.

Conclusions
The system analyzed in this article consisted of a large number of components and slight complexity. Other systems with larger or smaller number of components and more or less complexity can be modeled in the same manner using MATLAB. The number of simulation runs and MATLAB code needed to determine an accurate MTTF will vary depending on the number of components and system complexity. However, in all cases, convergence of the effective MTTF will occur as the number of simulations increase. Also, MATLAB contains many built-in functions that can be used in the code, and Graphical User Interfaces (GUI's) can be created that allow it to easily change system parameters or simulation runs.

Overall, a basic approach using simulation to determine MTTF was presented. It was shown that MATLAB can be used to model complex system relationships and perform simulations in order to determine MTTF. MATLAB is able to handle large arrays of data corresponding to large systems with many components, and is able to perform simple and complex mathematics on the data. Thus, this allows MATLAB to be considered as an excellent tool for performing reliability simulations of large complex systems including antennas.

References:
1. A. Brall, W. Hagen, H. Tran, “Reliability Block Diagram Modeling – Comparisons of Three Software Packages,” IEEE 2007
2. K. Murphy, A. Malerich, C. Carter, “What is Truth? A Practical Guide to Comparing Reliability Equation Answers to Simulation Results,” IEEE 2005
3. S. Asmussen, P. Glynn, “Stochastic Simulation,” Springer 2007

Juan Marin is part of the Reliability Engineering department at Lockheed Martin Missiles and Fire Control in Orlando, FL where he performs various reliability tasks. He has been involved in many aspects of systems and reliability engineering including: predictions, modeling/simulation, FMEA, parts stress derating analysis and reliability testing. Juan can be reached at juan.marin@lmco.com.

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