Abstract
Orthogonal design method (ODM) is widely used in real world application while it is not used for antenna design yet. It is employed to optimize roughly designed antenna in this paper. The geometrical factors of the antenna are relaxed within specific region and each factor is divided into some levels, and the performance of the antenna is constructed as objective. Then the ODM samples small number of antennas over the relaxed space and finds a prospective antenna. In an experiment of designing ST5 satellite miniantenna, we first get a roughly evolved antenna. The reason why we evolve roughly is because the evolving is time consuming even if numerical electromagnetics code 2 (NEC2) is employed (NEC2 source code is openly available and is fast in wire antenna simulation but not much feasible). Then the ODM method is employed to locally optimize the antenna with HFSS (HFSS is a commercial and feasible electromagnetics simulation software). The result shows the ODM optimizes successfully the roughly evolved antenna.
1. Introduction
Orthogonal design method (ODM) has been widely researched. Literature [1, 2] surveys this method in both theoretical and applied way. The ODM samples a small number of evenly distributed points over a large search space. Then it statistically summarizes a prospective good solution. The application of this method is much wide, such as in, for example, chemical and biological fields [3, 4], image process [5], laser polishing [6], software testing technique [7], algorithm [8], semiconductor manufacturing [9], optics [10], and robust design [11]. Recent theory research on the ODM method can be found still, for example, the mixedlevel orthogonal array research in [12].
The current practice of designing antennas by hand is limited in its ability to develop new and better antenna designs because it requires significant domain expertise and experience and is both time and labor intensive. With this approach, an antenna engineer will select a particular class of antennas and then spend weeks or months testing and adjusting a design, mostly in simulation using electromagnetic modeling software. As an alternative, researchers have been investigating evolutionary antenna optimization since the early 1990s. For example, genetic algorithm/evolutionary algorithm ([13]) is adopted to optimize antenna [14–16], particle swarm optimization ([17–19]) to optimize antenna [20, 21], and differential evolution ([22, 23]) to optimize antenna [24].
A run of evolutionary algorithm in designing an antenna usually takes over electromagnetic simulations while a simulation usually takes minutes or even hours. Then fast computing and incomplete simulation are adopted to reduce running time. The output of such evolution can only be called a roughly evolved antenna.
This paper focuses on optimizing locally this kind of rough antennas. We first relax the geometrical factors of the rough antenna within specific regions, divide each factor into some levels, and construct objective by using the performance of the antenna. Then we sample small number of antennas over the relaxed space and find a prospective antenna. It is the method that we call orthogonal design method (ODM).
In an experiment of designing ST5 satellite antenna, we first get a roughly evolved antenna with NEC2. Then the ODM is employed to locally optimize the antenna with HFSS. The result shows the ODM method optimizes successfully the roughly evolved antenna.
The remainder of this paper is organized as follows. Section 2 introduces the principal of the ODM method by using an example. Section 3 presents the fact that the ODM method optimizes antenna locally. Designing NASA ST5 antenna is used to test the ODM method in Section 4. The paper is concluded in Section 5.
2. Principal of Orthogonal Design Method
2.1. An Example to Introduce Orthogonal Design Method
We use a concrete example in this subsection to introduce the basic concept of an “orthogonal design method”. For further details, see [25]. The example is concerned with the yield of vegetable growth. The yield of a vegetable depends on at least three factors:(1)the temperature,(2)the amount of fertilizer used,(3)the pH value of the soil. In this example, each factor has three possible values, as shown in Table 1. We say that each factor has three “levels.”
The objective is to find the best combination of levels for a maximum yield. We can perform an experiment for each combination and then select the combination with the highest yield. In the above example, there are combinations, and hence there are 27 experiments. In general, when there are factors, each with levels, there are possible combinations. When and are large, it may not be possible to perform all experiments. Therefore, it is desirable to sample a small, but representative, set of combinations, for the experimentation. The “orthogonal design method” was developed for this purpose [25], where an orthogonal array is constructed to represent the sampled set of combinations which evenly distribute over the experimentation space. An orthogonal array is a array with levels for each column, denoted by . We select combinations to be tested, where may be much smaller than . Equation (1) is an example of an orthogonal array where , , and . Figure 1 shows the representative combinations (marked with “Δ”) evenly distributed over all the combinations. Consider
The has three factors, three levels per factor, and nine combinations of levels. The three factors have respective levels 1, 1, and 1 in the first combination, 1, 2, and 2 in the second combination, and so forth. We apply the orthogonal array to select nine combinations to be tested. The nine combinations and their yields in the above example are shown in Table 2.
From the yields of the selected combinations, a promising solution can be obtained by the following statistical method.(1)Calculate the mean value of the yields for each factor at each level, where each factor has a level with the best mean value (cf. Algorithm 2). The mean yields of the temperature are at level 1(20°C), at level 2(25°C), at level 3(30°C). The mean yields of the fertilizer are at level , at level , at level . The mean yields of the PH value are at level , at level , at level . These mean yields are shown in Table 3.(2)Choose the combination of the best levels as a promising solution (cf. Algorithm 3). The temperature has the best mean yield, 4.76, at level 2 (i.e., 25°C). The fertilizer has the best yield, 4.73, at level 2 (i.e., ). The pH value has the best yield, 5.18, at level 3 (i.e., 8). We therefore consider (25°C, , ) to be a promising and robust solution. The solution may not be optimal when used with an orthogonal design. But for additive and quadratic models, it is provably optimal.
2.2. A Definition of Orthogonal Array
Definition 1 (an orthogonal array). is a array with levels for each column, denoted by . Denote the orthogonal array by as follows.(1)In any column , each of the symbols occurs the same number of times; that is, .(2)In any two different columns , each of the possible pairs , ,, occurs the same number of times .Every row of represents a different combination of levels, where means that the th factor in the th combination has a level value , and takes a value from the set .
3. Orthogonal Design Method Designing Antenna
3.1. Antenna Design Using Orthogonal Design Method
There are many antenna classes, such as reflector antennas (e.g., dish antennas), phased array antennas (consisting of multiple regularly spaced elements), wire antennas, horn antennas, and microstrip and patch antennas. Each of these classes uses different structures and exploits different properties of electromagnetic waves.
Using orthogonal design method to design antenna, three components are required to be determined: factors, levels, and optimization objective.
The geometrical structure of an antenna is usually regarded as factors. For example, a element Yagi antenna has element lengths (), spacing distances between elements (), and one wire radius (), giving factors total; a conventional rectangular microstrip patch antenna usually has factors: patch length () and width (), substrate height () and its dielectric constant (), and probe point (), the distance from the leftbottom corner of the rectangular microstrip.
Determination of levels for each factor depends on the design specification of the antenna and empirical design.
The optimization objective is to find an antenna best matching the specification (gain, VSWR, etc.). It is a function of the factors. The objective value of an antenna may be achieved by measuring the prototype which is expensive or by simulating the antenna by using electromagnetic simulation softwares The latter one is usually adopted to avoid expensive cost.
Then the antenna design using orthogonal design method is an optimization problem; the formulation of optimizing antenna is defined in the following.
Definition 2 (formulation of optimizing antenna using orthogonal design method). Suppose there are antenna design factors and each factor has levels , giving combinations total. Each combination determines an antenna; there are antennas in all, which is called search space denoted as . The performance of a combination (an antenna) is evaluated by the value of objective . The bigger the value of the better the performance of the antenna in maximization formulation, . In this way, the optimization is Max where .
Note. must be prime in this paper; see Section 3.2.
3.2. Creating Orthogonal Array
As we will explain shortly, the technique proposed in this paper usually requires different orthogonal arrays for different problems. The construction of orthogonal array is not a trivial task since we do not know whether an orthogonal array of given size exists. Many orthogonal arrays have been presented in the literatures. It is impossible, however, to tabulate them all. For the necessity of the technique in this paper, we introduce a simple permutation method that is derived from the mathematical theory of Galois fields (see [1]), to construct a class of orthogonal arrays . The , , fulfill the following: where is prime and is a positive integer.
Denote the th column of the orthogonal array by . Columns for are called the basic columns. The others are called the nonbasic columns. The algorithm first constructs the basic columns and then generates the nonbasic columns. The details are given in the Algorithm 1.



3.3. Determining the Size of the Needed Orthogonal Array
The constructed by Algorithm 1 has a size of combinations (antennas). It can be adopted for a problem with levels and factors where . The and in are determined by given and according to (2), while in an antenna design problem the number of level and the number of factor are given according to Definition 2. Then the is demanded to be determined to construct an orthogonal array for the problem.
combinations mean electromagnetic simulations each of which is time consuming. We choose the as small as possible. This can be done by choosing the as small as possible according to (2). Then the is determined by
The , constructed by Algorithm 1, has columns. For a problem with factors, we discard the last columns of and obtain an array which is still orthogonal according to Definition 1.
For the problem of vegetable growth, there are factors () and levels (). by (3) and the orthogonal array is with 4 columns (). We discard the last column of the and get the needed array :
3.4. Doing Orthogonal Experiments
The combinations (antennas) are evaluated by orthogonal experiments (electromagnetic simulations). We obtain objective values (performances of the antennas) denoted as where the objective has the value at the th combination. It is similar to fill out Table 2 for the above vegetable example.
The best combination (antenna) among the combinations, denoted as , is usually not a global optimal solution. A prospective better solution could be found by using statistical method in the following.
3.5. Calculating Mean Objective Value at Each Level of Each Factor
Denote as the mean objective value at the th level of the th factor; , . Consider where the orthogonal array has the value at the th row and th column; that is, the th factor has level in the th combination (experiment). The objective has value at the th combination, and implies the sum of where any satisfies for given . All those mean values compose a matrix .
For the vegetable example, this calculation will fill out Table 3. The details of the algorithm are shown in Algorithm 2.
3.6. Finding Prospective Good Solution
A best level can be found for each factor from the mean value matrix . The combination of the best levels is usually guessed better than the best combination among the simulated combinations. Actually, for additive or quadratic models, it is optimal. The details of calculating the combination of the best levels are given in Algorithm 3.
However, the goodness of the combination is only a guess. We must do experiment (electromagnet simulation) for it to verify its performance. Actually, it is possible that is worse than . In this way, the final output will be the better one between and .
4. Testing ODM by ST5 Antenna Design
NASA ST5 mission consists of three microsatellites successfully launched in 2006 [26]. The specification of their antennas is shown in Table 4. Antenna designed by evolutionary algorithm was very small and was successfully applied for this mission [27], which is the first application of evolutionary antenna in the region of space science.
In this paper, we first roughly evolved an antenna by using NEC2 to simulate since NEC2 computes fast in simulating wire antennas and its code is openly available. The roughly evolved antenna is shown in left plot of Figure 3. Its gain at frequency MHz is shown in the left plot of Figure 4, and the one at frequency MHz is in the left plot of Figure 5. Its VSWRs are shown in the left column of Table 7. The gains satisfy the specification, but the VSWRs do not.
NEC2 is not much feasible while HFSS is feasible and commercial. Then HFSS software is adopted for ODM to optimize the roughly evolved antenna locally.
4.1. Factors, Levels, and Objective
The antenna is generated by starting with an initial feeder and adding four identical arms. Antenna geometric structure is symmetrical about the axis and each arm rotated 90° from its neighbors. We only encode the arm in the first quadrant where , , and . After constructing the arm in the first quadrant, it is copied three times and these copies are placed in each of the other quadrants through rotations of 90°/180°/270°. Linking such four aims to antenna feeder, we get the complete antenna.
As shown in Figure 2, the arm including the feeder in the first quadrant is segments of conductors linked headtail. The geometric structure can be coded as follows: the initial feed wire is a thumbnail lead, starting by origin along the positive axis with end point . The other four ends of the wires are , , , and , respectively; see Figure 2. The radius of the wires is specified as mm. Then geometric structure of the antenna can be determined by 13 factors: , and .
(a)
(b)
(a)
(b)
The code of the roughly evolved antenna is shown in Table 6. We relax each factor of the rough antenna upperoff or loweroff mm except the first feeder. Then each factor now takes the levels seen in Table 5. The search space is the set of all the possible combinations for all the factors at each level. The size of is , a big search space.
The objective takes a summary of the penalties of the gains and VSWRs according to the specification in Table 4.
The gains are sampled in 5° increments over region and . If a gain is less than dBic, a penalty value will be given as follows: where gain is the gain at direction , ; , .
Averaging the normalized penalties of the gains over the region, we get where .
Denote the average value at frequency MHz as and at frequency MHz as .
Regarding VSWR, a penalty value is given if the VSWR is larger than 1.5 at frequency MHz and larger than 1.2 at frequency MHz as follows:
Summarizing the penalties of both gain and VSWR at frequencies both MHz and MHz, we get objective where .
Note. The optimization is a minimization not maximization.
4.2. Orthogonal Experiments for ST5 Antenna Design
Given the level and number of factors , we have by (3). And by Algorithm 1 or (2), we have . Then by using ODM with experiments (that is 27 simulations of the antennas by using HFSS), a prospective combination (antenna) is found.
The code of the orthogonally designed antenna is shown in the last row in Table 6. The antenna is pictured in the right plot of Figure 3. The gains at frequency MHz are shown in the right subgraph of Figure 4, and the right subgraph of Figure 5 is the gains at frequency MHz. The VSWRs are shown in Table 7.
The objective (see (9)) is shown in Table 8. It shows that the orthogonally designed antenna is better than the roughly evolved one.
4.3. Discussion
Some comments on ODM in optimizing antenna design are given in the following.(1)Determining factors (variables) , levels of each factor , and objective: This is the preparation step. The size of the search space is exponent of the number of factors with the number of levels as base . The number of factors and the number of levels of each factor could not be too big. The search space of the ST5 application is where , .(2)Determining the size of orthogonal array : is given according to the above item. is chosen as smaller as possible to get the smallest orthogonal array since a simulation is time consuming. , in the ST5 application. Then simulations are done in the orthogonal experiments. It is far less than the search space . However, the objective value of the orthogonally designed antenna is better than that of the roughly evolved one.(3)Finding a prospective combination (antenna structure) by statistical summary from very small samples: In the case of linear or quadratic objective, the statistical summary is proven right. It is not proven right in other cases. But a derivable objective can be approached by a quadratic over a small neighbor region. That is, the statistical summary in this paper is reliable. Anyway, we must do experiment to verify the result finding from the summary. Fortunately, the orthogonally designed antenna is better than the roughly evolved one in the ST5 application problem. Figure 6 is the prototype of the orthogonally designed antenna.
5. Conclusion
The idea of the orthogonal design method in designing antenna is mainly as follows.(1)The geometrical structure of the antenna is parameterized into factors and each factor is quantized into discrete levels; the requirements specified for the antenna are functionalized as objective. The number of factors and number of levels should not be too big because of the exponent increase of the search space with the number of factors where the base is the number of levels.(2)Since an electromagnetic simulation lasts usually for minutes or even hours, smallest orthogonal array is taken. This paper offered an algorithm to create a class of orthogonal arrays and a way to determine the smallest orthogonal array for the ODM finishing in endurable time.(3)By using the ODM, a potential good antenna could be found with very small number of electromagnetics simulations over a very large design space.(4)The objective value of the orthogonally designed antenna is better than that of the rough one, and then the prototype was made by the orthogonally designed antenna.
Future work is explained as follows.(1)Antenna design is an expensive problem; see literature [28, 29]. ODM will be implemented in parallel for antenna design, and surrogateassisted model is another way to reduce running time.(2)Another future work is to compromise conflict between gains, VSWR, axial ratio, and so on in determining objective.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (nos. 61271140, 61203306, and 60871021).