Efficient algorithms for the optimization of shielding

International Journal of Applied Electromagnetics and Mechanics 34 (2010) 141–154 DOI 10.3233/JAE-2010-1307 IOS Press

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Efficient algorithms for the optimization of shielding devices for eddy currents Alireza Yahaghiaa,c , Christian Hafnera , Arya Fallahia , Jasmin Smajicb,∗ , Bogdan Cranganu-Cretub and Ruediger Vahldiecka a ETH

Zürich, Lab. For Electromagnetic Field and Microwave Electronics, CH-8092, Switzerland Switzerland Ltd. Corporate Research, Dättwil, CH-5405, Switzerland c Department of Electrical Engineering, Shiraz University, Shiraz, Iran

b ABB

Abstract. Different optimization algorithms are applied to determine the best configuration of a shielding device for eddy currents. The algorithms include seven stochastic binary optimizers, which are based on the concept of genetic algorithms and evolutionary strategies, as well as one quasi-deterministic optimizer based on an algorithm inspired by hill-climbing strategies. The comparison of results obtained from each algorithm shows a very high probability of finding global optima when a special quasi-deterministic algorithm is applied. Keywords: Stochastic optimization, eddy currents, electromagnetic shielding

1. Introduction In the design of power devices such as power and industrial transformers, the magnetic shielding of the ferromagnetic conductive bodies is of paramount importance [1,4,5]. Namely, due to a modern tendency to produce very powerful compact devices, significant stray magnetic flux inevitably appears, inducing eddy currents in the conductive bodies. The corresponding level of eddy-current losses in ferromagnetic components of the power device can be rather high, causing overheating and radically reducing the expected life-time of the device. Besides reduction of the stray magnetic field sources this problem is tackled today usually either by designing a system of magnetic shunts or respectively of conductive shields [1,3,5]. The focus of this paper is on the design optimization of the magnetic shunts system, although the suggested optimization techniques are rather general and therefore can be successfully used for the design of conductive shields. The functionality of the magnetic shunts system is briefly explained hereafter. Highly permeable (µr in the order of at least several thousand) and low conducting (low eddy currents losses) material is used to produce relatively long plates (several meters) with a small cross section (several tens or hundreds of square centimeters). These highly permeable solid plates are called magnetic shunts. They are combined in a system and positioned inside the power device in such a way that they absorb the stray magnetic field and shunt it away from the protected ferromagnetic conductive body, which normally has significantly smaller magnetic permeability than the shunts. ∗

Corresponding author. E-mail: [email protected].

1383-5416/10/$27.50  2010 – IOS Press and the authors. All rights reserved

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A. Yahaghia et al. / Efficient algorithms for the optimization of shielding devices for eddy currents

As it has been already emphasized, a typical magnetic shunt is a plate-like elongated structure with much smaller cross section than its length. A group or cluster of such shunts are usually used to prevent the stray magnetic field from penetrating large and flat ferromagnetic conductive bodies (walls of the enclosures for example). Having such elongated structures it is rather logical to limit our analysis to 2D structures. This will significantly reduce CPU time needed for fitness evaluations (field computations) and allow us to define larger search spaces and afterwards to draw more reliable conclusions. However, all of the optimization algorithms are general and can be equally well used for design optimization of 3D structures. In order to keep the search space even more feasible, we have reduced the number of shunts (simpler topology). However, since the evaluation of the fitness function related to the total eddy-current losses is based on the electromagnetic field computation, such a fitness evaluation is very costly in terms of CPU time especially in the case when a high level of accuracy is required. On the other hand, without maintaining such an accuracy level the results will contain computational errors that will radically disturb the optimizer. Therefore it is important to find a compromise between the field solver accuracy (fitness evaluation) and chosen optimization algorithm (robustness) [2]. Recently the efficiency of different optimizers has been investigated for some photonic crystal structures [6], and frequency selective surfaces (FSSs) [7]. In this paper we follow the same approach to find efficient procedures for the optimization of a shielding device for low-frequency eddy currents. We present the algorithm for finding the optimal topology of the shunt system for a given configuration of the field source and ferromagnetic conductive body that should be shielded. The corresponding binary optimization problem is defined with two fundamental objective functions: the total eddy-current induced loss of the shielded body and total volume (material or cost) of the shunts’ system. The paper is organized as follows: In Section 2, the optimization model is defined. In addition, the analysis methods and the techniques used for decreasing the computation costs are explained briefly. The general process for evaluating an optimizer is also explained in this section. Next, the numerical optimizers are introduced in Section 3. In Section 4, the results are presented. The results in this study include the structures with best performance and the algorithms with best efficiency. 2. Definition of the problem Structure of the shielding device is shown in Fig. 1. Obviously a relatively simple configuration has been chosen. Namely, the two volume currents with rectangular cross section are used to simulate the whole current of primary and secondary coils as a field source (red rectangular conductors on the right-hand side of Fig. 1(a)). The ferromagnetic conductive plate (blue rectangular solid body on the left-hand side of Fig. 1(a)) is affected by the coils’ field and therefore significant eddy current losses are produced. The goal is to reduce the losses by introducing the system of magnetic shunts (shown in the middle of Fig. 1(a)) in the space between the bus-bars and ferromagnetic conductive plate. We have arranged the shunts on a rectangular grid with 6 rows and 5 columns. Therefore the topology of this system can be described by a 30-bit long binary string (1 or 0 means that a certain shunt is on or off respectively). To perform the eddy-currents analysis of the geometry shown in Fig. 1a, the following 2D boundary value problem must be solved: 1 ∂Az ∂ 1 ∂Az ∂ + − jωσAz = −JzS , onΩ ⊂ R2 (1) ∂x µ ∂x ∂y µ ∂y


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Fig. 1. (a) The binary optimization problem of the shunts’ topology is depicted; (b) The initial solution’s (without shunts) field distribution (scalar mag. potential) is shown; (c) The solution with all shunts placed is also presented. Shielding factor (Sh) is defined as: 1/(time average of resistive heating in ferromagnetic conductive bodies). The material properties of the ferromagnetic conductive plate are µr = 200, σ = 6.66 × 106 S/m and the material of the magnetic shunts has µr = 20000, σ = 0 S/m properties of the ferromagnetic conductive plate The frequency is 50Hz, and phase current density is 106 A/m2 RMS.

Az = 0, along Γ1 = ∂1 Ω

(2)

∂Az = 0, along Γ2 = ∂2 Ω ∂n

(3)

where Az is the z-component of the vector magnetic potential, µ is the magnetic permeability, σ is the specific electric conductivity, JSz is the z-component of the source current density, Ω is the 2D computational domain, and ∂ Ω is the boundary of the domain Ω. The coils are usually placed around a magnetic core with high magnetic permeability. If one considers the magnetic core linear we can avoid modeling it and use the normal-field boundary condition Eq. (3). Thus the computational domain and corresponding CPU time is reduced. The tangential – field boundary condition Eq. (1) is used to represent the boundary sufficiently far from the simulation objects. By using these boundary conditions in the simulation model shown in Fig. 1(a), our intention was to roughly resemble the situation in a power transformer and to keep the model small enough for numerous computations required by the optimizers. The induced eddy-current losses in the ferromagnetic plate are computed per unit length by integrating the loss density over the plate’s surface: Z Z Z Z 1 1 2 ′ σ · |Ez | dS = σ · ω 2 |Az |2 dS (4) PLOSS = 2 2 (SF e )

(SF e )

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Fig. 2. A typical finite element mesh used in simulation is presented (left). The detail of the mesh covering the ferromagnetic plate, magnetic shunts, and coils is shown (right). Triangular second order elements were used. The element sizes of 1mm, 10mm, and 10mm for the ferromagnetic plate, magnetic shunts, and coils were set respectively. The total number of elements was 87’579 and the number of degrees of freedom (DOFs) was 175’324.

where Ez is the z-component of the induced electric field and ω is the angular frequency. The presented analysis has a focus on the total power loss computed by the integral Eq. (4) rather than on local hot-spot losses. The reason for this is that local hot-spots are difficult to control in a consistent way, as they can significantly change their location and intensity with small geometrical variations. As opposed to the local values, the total power loss reveals more objectively the shielding efficiency and directions for improvement. However, the local hot-spot losses should be considered in future studies as they can radically reduce the expected life-time of a power transformer. 2.1. Electromagnetic calculations For electromagnetic field computation we have used COMSOL 3.4 (COMSOL 2008) a finite element based commercial software. Each fitness evaluation takes about 4 seconds, so the brute-force fitness evaluation of the entire search space which is necessary to do a complete comparison between optimizers would take more than 136 years. For this reason, we decided to first symmetrize the structure (by setting the coils in the symmetric position) and simulate half of it. In this way each fitness evaluation takes about 3.6 seconds (a finer mesh presented in Fig. 2 was used), also number of all possibilities reduces to 215 , therefore the whole computation time is about 32 hours. 2.2. The optimization procedure


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Fig. 3. Encoding the shielding device to a bit string is presented. The structure is divided into a 3x5 array of shunts.

As mentioned before, the shielding structure is made of magnetic shunts. For optimization purpose we have assigned a binary number to each combination of these magnetic shunts. As an example, the unit cell shown in Fig. 3 can be represented by the bit string 100011010001001. Next, a suitable fitness function for each bit string is defined. According to our experience, this is the most critical point in the optimization procedure. Since the shielding characteristic of the structure is much more important than its cost, we decided to do a two step optimization. In the first step, we search for a structure with the best shielding regardless of its cost. In the second step, we search among structures which have α percent of the best shielding – as determined is in the previous step – for minimum cost. The fitness functions that we have used in these two steps are: 1.

2.

F itness1 = Shielding = 1 time average of resistive heating in f erromagnetic conductive bodies F itness2 = floor (Nsh - α) + 1.1 · Nch + 0.1

Nsh =

shielding , normalized shielding max (shielding)

Nch =

length(a)-sum(a) , normalized cost length(a)

Where in this relation: “a” is the bit string corresponding to structure, length(a) and sum(a) are its length and bit sum respectively, “1.1” is a factor chosen to avoid zero values of the fitness function, and “floor” is a MATLAB function that rounds a number toward minus infinity. Since the “floor” function has a stepwise behavior, the fitness landscape is not smooth (Fig. 4(b)). To make it smoother we modified it as below: F itness2 = floor (Nsh - α) + 1 + Nsh · Nch + 0.1 · Nsh We are now prepared to connect the electromagnetic field solver, which evaluates the fitness function, with the optimizers. However, it is worthwhile to first mention some points about the optimization domain. To be able to compare and validate each optimizer, we first need an overall image of the optimization domain, which is provided by a brute-force simulation of all the possible cases. Only then can we decide which of the optimizers performs best. Based on the brute-force simulation, we are able to build a table containing all the possible cases and their corresponding fitness values. This eliminates the need to calculate the fitness functions each time an optimizer is tested. Figure 4 illustrates the fitness values of all the 32, 768 individuals, according to the definition of the fitness functions above. As can be seen from Fig. 4, the number of acceptable structures decreases when α increases. The existence of a large number

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Fig. 4. Fitness values of all the 32’768 individuals (a) according to the definition of Fitness1 ; (b) according to the first definition of Fitness2 with α = 0.95; (c) according to the second definition of Fitness2 with: α = 0.95, (d) α = 0.96, and (e) α = 0.99; are presented. The bit string that characterizes an individual is obtained by binary representation of the corresponding integer number.

of individuals with fitness values close to the maximum causes our algorithm to find high fitness values quickly and easily. Hence, this has a positive effect on the efficiency of the optimizers. However, it can also be deduced that in this case it is highly probable that an optimizer converges to a local maximum of fitness function and fails to find the global maximum. This preview of the optimization domain shows how important it is to apply a suitable algorithm for the optimization of a unit cell. 2.3. Comparison of the optimizers To allow a meaningful comparison between optimizers, the discriminating criteria must first be defined. As mentioned before, almost every optimization algorithm is based on random initialization. Therefore, it is possible for the optimizer to find the best case either in the first generations or after a large number of them. To obtain reliable information on a certain algorithm, each problem should be optimized by the same algorithm many times in order to obtain reliable statistical data. Furthermore, in each optimization algorithm the population size plays an important role and greatly affects the performance.


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Table 1 Population size for the seven stochastic optimizers and the quasy-deterministic one (RHC) are given. Npop denotes a certain population size which is used in the following tables Npop 1 2 3 4 5 6 7 8

Stochastic 4 5 7 10 14 19 25 32

RHC 1 2 4 7 11 16 22 29

There is no general rule to find the optimum population size for each problem. Hence, every algorithm is tested with a different population size, Npop . As mentioned before, seven stochastic and one quasi-deterministic algorithm are utilized in this investigation. Each algorithm is run 1000 times for different population sizes which are set according to Table 1. The reasons to choose the values for Npop are described in [6]. It should be mentioned that in the stochastic algorithms it is not useful to assume very low numbers for Npop such as 1 or 2. Hence, higher population sizes are assumed. In the optimizers applied here, calculated fitness values are saved in an incomplete fitness table. Hence, to find the fitness value for a bit string, the algorithm first searches for the solution in the incomplete fitness table. If the bit string is not found, the fitness is evaluated by the field solver. This avoids repeated simulation of the eddy currents problem and saves computation time. Therefore, this will be done for all the optimization algorithms. The effect of this table on the number of fitness evaluations in different optimizers is also investigated. Another parameter that also strongly affects the efficiency of the algorithm is the maximum number of fitness evaluations. Any algorithm that is not trapped in a local optimum will find the global optimum after a certain number of fitness evaluations, Neval . However, it is important to find the optimum with a low Neval . For this reason, the optimizers are stopped after 100, 200, 500, and 1000 fitness evaluations and the comparison criteria are calculated by taking the average over the above cases. The criteria to evaluate each optimizer are as follows. 1. From the brute-force simulations, we already know which structure is the optimal and its fitness value is normalized to 1. After running each optimizer 100 times, a probability of finding the global optimum can be defined as the number of times the global optimum was found, divided by 100. 2. The best fitness values found by the algorithm are averaged and defined as the average relative fitness. 3. Each algorithm is stopped as soon as it finds the global optimum or reaches Neval . Therefore, the average number of fitness calls is a suitable parameter for evaluating an algorithm. 4. The same value introduced above when no incomplete fitness table is used can also be used to gain an impression of the optimizer, and is referred to as average number of fitness calls without a table.

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3. Numerical optimizers In this study, the same optimizers are considered as those in [6,7]. The stochastic optimizers include statistical random search, three optimizers based on the micro-genetic algorithm and three optimizers based on mutation-based (evolutionary) strategies. The last optimizer is quasi-deterministic and is based on the hill-climbing algorithm with random reinitialization. There are extensive publications on the genetic algorithm (GA) and evolutionary strategies (ES) concepts. Moreover, details of the algorithms considered here are described in [6]. Therefore, only a short description of the algorithms is given here. The stochastic strategies considered all differ from the standard GA and binary ES. They are changed in such a way that their performances are better than the standard ones. Note that the goal is to achieve few fitness evaluations because of their long computation times. To this end, improving the efficiency of the optimizer through the use of an incomplete fitness table and the use of bit-fitness proportional mutation (BFP) (Hafner 2007) are added to the standard algorithms. The algorithms are as follows: 1. Statistical random search (STAT): (1) Perform random initialization and fitness evaluation of the generation. (2) Perform bit-fitness value evaluation (to learn more about the concept of the bitfitness evaluation and bit-fitness-based strategies the reader is referred to [6]). (3) Check whether all individuals are identical. If so, save the obtained result and restart step 1; otherwise, perform the following steps. (4) Generate the next generation using bit-fitness proportional (BFP) selections. (5) Evaluate bit fitness values and repeat step 2 until the maximum number of fitness calls is reached. 2. First micro-genetic algorithm (MGA0): (1) Perform random initialization and fitness evaluation of the generation. (2) Perform bit-fitness value evaluation. (3) Check whether all individuals are identical. If so, return to step 1; otherwise, perform the following steps. (4) Copy the best individual into the next generation (elitism). (5) Select pairs of parents and generate a pair of children per pair of parents using single-bit crossover. (6) Evaluate bit-fitness values and repeat step 2 until the maximum number of fitness calls is reached. 3. Second micro-genetic algorithm (MGA1): Similar to MGA0 but without elitism, only one child per pair of parents, and single-bit random mutation when both parents are identical. 4. Third micro-genetic algorithm (MGA2): (1) Perform random initialization and fitness evaluation of the first generation. (2) Perform bit-fitness value evaluation. (3) Check to see whether (a) all individuals are identical or (b) no new individuals were added to the incomplete fitness table during the last ngen generations. If so, generate a new generation using a bit-fitness-based algorithm and repeat step 2; otherwise, perform the following steps: (4) randomly select number of parents: two with probability pcross or one with probability 1-pcross . (5) If the number of parents is two and both parents are different, generate a child using single-point crossover; otherwise, mutate the parent. For the mutation, random mutation is selected with probability prand , and bit-fitness-based mutation is selected with probability 1- prand . (6) Evaluate fitness values and repeat step 2 until the maximum number of fitness calls is reached. 5. First mutation-based algorithm (MUT0): (1) Perform random initialization and fitness evaluation of the first generation. (2) Perform bit-fitness value evaluation. (3) Mutate all individuals using BFP and compute fitness values. (4) Repeat step 2 until the maximum number of fitness evaluations is reached. 6. Second mutation-based algorithm (MUT1): (1) Perform random initialization and fitness evaluation of the first generation. (2) Perform bit-fitness value evaluation. (3) Select the best individual as the parent for the next generation (strict elitism). (4) Generate a new generation using 1 bit mutations. (5) Repeat step 2 until the maximum number of fitness evaluations is reached.


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Table 2 Probabilities of finding global optimum in percent, averaged over Fitness1 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 1.28 1.38 1.02 0.975 1.10 1.05 0.950 1.07 1.10

MGA0 1.97 1.90 2.72 4.58 6.97 8.78 9.20 8.67 5.60

MGA1 1.62 2.00 2.00 2.03 2.90 2.50 2.78 2.72 2.32

MGA2 47.9 66.0 60.9 46.9 49.7 53.8 51.6 47.5 53.0

MUT0 1.50 1.42 1.30 1.42 1.22 1.25 1.43 1.43 1.36

MUT1 72.3 79.9 85.1 87.4 88.8 88.1 85.3 82.7 83.7

MUT2 71.5 79.7 85.7 87.6 88.5 86.6 84.8 82.9 83.4

RHC 98.7 100 88.1 41.9 17.2 10.0 8.05 5.85 46.2

Table 3 Average relative fitness in percent (fitness found by the algorithm vs. fitness of the global optimum), averaged over Fitness1 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 0.976 0.975 0.975 0.976 0.976 0.976 0.976 0.976 0.976

MGA0 0.977 0.978 0.979 0.980 0.981 0.980 0.980 0.978 0.979

MGA1 0.972 0.972 0.971 0.971 0.972 0.973 0.973 0.973 0.972

MGA2 0.992 0.995 0.994 0.992 0.992 0.992 0.991 0.990 0.992

MUT0 0.968 0.972 0.972 0.972 0.973 0.973 0.973 0.973 0.972

MUT1 0.997 0.998 0.998 0.998 0.998 0.998 0.997 0.997 0.997

MUT2 0.997 0.998 0.998 0.998 0.998 0.998 0.997 0.996 0.997

RHC 1.00 1.00 0.998 0.989 0.985 0.984 0.983 0.982 0.990

7. Third mutation-based algorithm (MUT2): Same as MUT1 but replace the mutation by a BFP-based mutation with probability pBF P . 8. Randomly initialized hill-climbing algorithm (RHC): (l) Perform random initialization and fitness evaluation of the first generation with Npop individuals. (2) Perform bit-fitness value evaluation. (3) Select the best individual (strict elitism) as the parent for the next generation with N (length of the bit string) individuals. (4) Generate child number n by flipping bit number n of the parent. (5) Repeat step 2 until the parent is better than all of its N children. When this happens, (6) reinitialize the first population using BFP and continue with step 2 until the stopping criterion is met. 4. Results Two types of results were obtained in this investigation. From the brute-force evaluation of all possible models, the best solutions can be found; with the comparison of the optimizers, the performance of the algorithms can be investigated. Both types of results are outlined in this section. 4.1. Performance of the optimizers The procedure to evaluate and compare optimizers was explained in Section 2. Following the same procedure for every fitness definition, the results of Tables 2–10 are obtained. In these tables, the defined criteria for evaluating each algorithm are tabulated for each optimizer and Npop. The average of the results is also outlined in the last row of each table. This gives quick, general information on

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A. Yahaghia et al. / Efficient algorithms for the optimization of shielding devices for eddy currents Table 4 Average number of fitness calls when an incomplete fitness table is used and the algorithm is stopped as soon as it finds the global optimum, averaged over Fitness1 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 448 448 450 454 455 456 467 463 455

MGA0 445 447 442 435 425 423 420 422 432

MGA1 446 443 443 445 442 443 442 443 443

MGA2 261 178 205 263 258 239 242 260 238

MUT0 434 446 446 446 447 446 446 446 445

MUT1 155 125 105 102 100 105 105 107 113

MUT2 155 127 106 104 102 106 106 106 114

RHC 33.4 49.0 108 270 396 422 426 440 268

Table 5 Average number of fitness calls when an incomplete fitness table is not used and the algorithm is stopped as soon as it finds the global optimum, averaged over Fitness1 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 454 453 455 460 460 461 472 468 460

MGA0 928 1040 1250 1470 1630 1720 1690 1680 1430

MGA1 1330 1610 2160 2950 3790 4620 5250 5360 3380

MGA2 584 435 609 889 927 886 859 872 758

MUT0 2510 490 487 486 486 483 481 481 774

MUT1 207 174 160 190 317 1270 2720 3580 1080

MUT2 208 177 161 197 313 1310 2510 3590 1060

RHC 64.5 66.3 193 420 428 434 435 448 311

the performance of the eight algorithms. From these tables it is deduced that RHC outperforms all the other algorithms, in the sense that it has the highest probability of finding the global optimum and lowest average number of fitness calls. MUT0 and STAT are the worst ones, which was also found in [6,7]. From Table 2 we can draw very useful conclusions about the optimization algorithms tested. Namely, the optimizer STAT has very low probability of finding global optimum for any population size tested. The optimizers MGA2 and MUT2 are much more efficient and it is possible to see that the optimal population sizes for them are 5 and 14 individuals respectively. Such a performance of MGA2 and MUT2 is not surprising considering the level of their complexity and advanced operators implemented (the bit-fitness-proportional operators related to the corresponding incomplete fitness table). The behavior of the method RHC is very impressive, as it by far outperforms the other algorithms by using a small population size. Surprisingly, its performance radically deteriorates as we increase the population size. This is explained by the fact that a larger population size increases the probability of finding a sharp local optimum. The local optimum found causes the algorithm to restart, preventing it from finding the global optimum. It is highly recommended, therefore, to work with a very small population (2 members seems to be the optimal value). According to Table 3, all of the algorithms are capable of finding a solution with relatively high fitness function, although, as Table 2 illustrates, with very different capabilites of finding the global optimum. Only the RHC algorithm has the average relative fitness of 1 for a small population size of 1 and 2 individuals. Tables 4 and 5 clearly illustrate the importance of maintaining an incomplete fitness table. This table radically reduces the required number of fitness calls for all of the algorithms. This is of paramount


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Table 6 Probabilities of finding global optimum in percent, averaged over first definition of Fitness2 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 6.17 6.22 6.48 6.70 5.93 6.70 6.83 6.80 6.48

MGA0 11.3 12.7 18.5 22.2 29.5 31.0 31.7 28.4 23.2

MGA1 8.40 11.8 20.7 31.9 43.8 53.1 56.1 56.4 35.3

MGA2 47.4 56.6 58.1 55.4 64.6 71.7 78.4 73.5 63.2

MUT0 5.70 4.05 4.40 5.32 5.08 4.77 4.92 4.92 4.89

MUT1 43.9 51.2 55.9 57.4 66.6 70.6 70.8 75.1 61.4

MUT2 47.3 52.7 55.0 56.7 67.6 73.7 72.7 76.7 62.8

RHC 96.8 66.1 55.3 58.6 53.0 40.5 36.6 31.2 54.8

Table 7 Probabilities of finding global optimum in percent, averaged over second definition of Fitness2 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 2.60 2.37 2.92 2.20 2.88 2.35 2.83 2.88 2.63

MGA0 4.60 4.95 7.45 11.5 15.8 19.4 18.8 17.8 12.5

MGA1 7.68 10.7 17.8 29.6 39.2 44.3 44.7 44.9 29.9

MGA2 53.2 67.6 66.2 58.6 61.9 66.1 67.4 62.5 62.9

MUT0 2.63 2.30 2.75 2.45 1.72 2.20 2.13 2.13 2.31

MUT1 60.3 68.0 73.1 75.5 80.2 79.8 72.7 68.7 72.3

MUT2 63.7 68.9 73.1 73.4 79.8 80.1 72.4 66.3 72.2

RHC 100 98.6 62.4 63.5 41.4 24.5 20.8 17.2 53.6

Table 8 Average relative fitness in percent (fitness found by the algorithm or fitness of the global optimum), averaged over second definition of Fitness2 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 86.4 86.7 86.5 86.5 86.6 86.7 87.0 86.9 86.7

MGA0 88.1 88.5 89.6 90.2 90.7 90.5 89.7 89.0 89.5

MGA1 89.0 90.0 91.4 92.7 93.7 93.9 93.6 93.2 92.2

MGA2 96.0 97.1 97.2 96.6 96.8 97.2 97.4 96.3 96.8

MUT0 84.5 84.5 84.8 85.1 85.3 85.4 85.8 85.8 85.1

MUT1 96.6 97.0 97.4 97.5 98.3 98.4 98.0 97.6 97.6

MUT2 96.7 97.1 97.3 97.4 98.3 98.4 98.0 97.6 97.6

RHC 100 99.9 95.6 95.4 93.8 92.3 91.6 91.1 95.0

importance for our electromagnetic applications, as we are always dealing with fitness functions which are very expensive in terms of CPU time. By comparing Tables 6 and 7, it is concluded that making the fitness function space smoother improves the performance of most of the optimizers. The interesting point about the RHC is that for first optimization step when Npop = 2, and for second optimization step when Npop = 1; one can be certain that the optimizer reaches the global optimum. However, it is possible that the efficiency varies for larger search spaces. The large number of local minima makes the use of an incomplete fitness table very important. A comparison of Tables 4 and 9 –against Tables 5 and 10 shows that all the algorithms tend to find fitness

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A. Yahaghia et al. / Efficient algorithms for the optimization of shielding devices for eddy currents Table 9 Average number of fitness calls when an incomplete fitness table is used and the algorithm is stopped as soon as it finds the global optimum, averaged over second definition of Fitness2 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 444 445 443 449 450 451 459 458 450

MGA0 434 435 428 412 395 384 381 387 407

MGA1 425 416 385 342 297 272 263 257 332

MGA2 240 178 179 217 199 183 180 188 196

MUT0 430 443 441 443 445 444 443 443 441

MUT1 204 171 150 145 130 121 122 108 144

MUT2 195 168 153 155 135 124 110 104 143

RHC 32.0 67.9 148 150 274 370 384 400 228

Table 10 Average number of fitness calls when an incomplete fitness table is not used and the algorithm is stopped as soon as it finds the global optimum, averaged over second definition of Fitness2 with 100, 200, 500, 1000 fitness evaluations, for all eight optimizers M 1 2 3 4 5 6 7 8 av

STAT 450 451 448 455 455 457 465 464 456

MGA0 904 1010 1210 1400 1500 1530 1480 1420 1310

MGA1 1290 1580 2050 2590 2880 2970 2860 2340 2320

MGA2 548 447 533 710 679 627 596 568 588

MUT0 2720 487 482 483 484 482 479 479 802

MUT1 282 254 267 391 784 2490 9680 12100 3280

MUT2 278 260 297 499 1150 4140 9700 13800 3760

RHC 32.0 114 252 226 297 388 399 412 265

values for similar test cases repeatedly; they would be much less efficient without an incomplete fitness table. This is very important for large “nBits”, where the number of fitness evaluations and accordingly their repetitions are higher. In addition, in such cases one should consider enough storage space for a much larger incomplete fitness table. The result obtained here regarding the performance of the optimizers is almost the same as the one obtained in [6,7] where high frequency applications were studied. According to the results presented in Tables 2 to 10, advanced optimizers MGA2 and MUT2 also perform very well for the eddy currents optimization problems at low industrial frequencies (50–60Hz). However, the highest level of optimization efficiency was again achieved by applying the RHC algorithm with a small population size, which makes this algorithm very suitable for 3D applications with very costly fitness evaluations. 4.2. Optimal solution of the test problem From the brute-force simulation, the global optimum of the symmetrized 15Bit structure can be found. In Fig. 5, the best structures corresponding to different values of are shown. Next, we applied the RHC optimizer to an unsymmetric problem. Some of the consequent geometries for along with their Shielding factors (Sh) are shown in Fig. 6. All resultant structures have acceptable characteristics with respect to their shielding performance and cost, but for example the configuration in Fig. 6(c) can be fabricated more easily. In all of this cases


153

Fig. 5. Optimal solutions of the symmetric problem corresponding to different values of are shown.

Fig. 6. Optimal solutions of the unsymmetric problem with (a) Shielding factor Sh = 0.04936, (b) Sh = 0.0502, and (c) Sh = 0.04905 are presented.

number of fitness evaluation was less than 2000, much less than the population of whole search space 230 , which clearly shows the efficiency of the RHC optimizer. Evidently, majority of the obtained solutions are counter-intuitive, i.e. there is very low probability of finding them without applying an optimization algorithm. Based on this analysis, it is possible to choose a suitable optimization algorithm and find very good solutions within an acceptable CPU time. This is very important from the industrial point o view, as the development time and cost are of paramount importance. However, to successfully apply such an optimization procedure directly connected with a field simulation solver requires from a designer relatively high level of simulation and optimization knowledge and experience, which could be considered an obstacle for wider acceptance of this approach.

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5. Conclusions Eight different two step optimization algorithms were applied to optimize a shielding device for an eddy currents driven problem. The comparison of the optimizers using brute-force simulation shows that a quasi-deterministic algorithm based on a randomly initialized hill climbing algorithm (RHC) outperforms the other algorithms. The effect of using an incomplete fitness table within the optimization algorithm was also investigated. It was shown that it drastically reduces the number of fitness calculations, which in turn reduces the optimization time. The RHC algorithm was applied to the unsymmetric problem where a brute-force search was not possible. The capability of RHC to find a sub-optimal solution by exploring a small fraction of the search space, i.e. within a small number of fitness evaluations, was confirmed and several interesting sub-optimal solutions were found. With such optimization efficiency RHC can be a very suitable method for large-scale optimizations of computationally very demanding 3D electromagnetic structures. References [1] [2] [3] [4] [5] [6] [7] [8]

B. Cranganu-Cretu, J. Smajic and Testin, (2007) Usage of Passive Industrial Frequency Magnetic-Field Shielding for Losses Mitigation: A Simulation Approach. In: Proceedings of the Advanced Research Workshop on Transformers (ARWtr 2007): 325–330, Baiona, Spain, 2007. P. Di Barba, (2010) Multiobjective Shape Design in Electricity and Magnetism. Springer, New York. J.J. Winders, (2002) Power Transformers: Principles and Applications, Marcel Dekker Inc, New York, pp. 69–116. Kennedy B W (1998) Energy Efficient Transformers. McGraw-Hill, New York. D. Vecchio, B. Poulin, P.T. Feghali, D.M. Shah and R. Ahuja, (2002) Transformer Design Principles CRC Press, Boca Raton. C.H. Hafner, C. Xudong, J. Smajic and R. Vahldieck, Efficient procedures for the optimization of defects in photonic crystal structures, J Opt Soc Am A 24 (2007), 1177–1188. A. Fallahi, M. Mishrikey, C.H. Hafner and R. Vahldieck, Efficient Procedures for the Optimization of Frequency Selective Surfaces, IEEE Trans Antennas Propag 56 (5) (2008), 1340–1349. Comsol Inc. FEMLAB Multiphysics Modeling (2008) Available via URL: http://www.comsol.com/products/femlab.


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Design of multi-DOF electromagnetic actuators using distributed multipole models and image method Hungsun Sona,∗ , Kun Baib , Jungyoul Limb and Kok-Meng Leeb a Department b Department

of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA

Abstract. Design and control of many electromagnetic actuators involve solving three dimensional (3D) magnetic fields of permanent magnets and/or electromagnets in the presence of magnetic conducting surfaces. This paper extends the distributed multipole (DMP) method, which offers compact but precise analysis for three dimensional magnetic fields in closed form, to account for the effects of magnetic conducting boundaries using the image method on the torque generated by electromagnetic actuators. We validate the proposed method referred to here as DMP-Image method by comparing the calculated torques against results computed by a finite element method (FEM). While two methods agree to within 5% in maximum torque, the DMP-image method takes less than 1% of the FEM computation time. Finally, we demonstrate the DMP-Image method to design a spherical motor in a class of multi-DOF actuators. While developed in the context of the multi-DOF actuators, the modeling methods presented in this paper are applicable to design of other PM-based actuators. Keywords: Magnetic field, dipole, magnetic conducting boundary, image method, spherical motor

1. Introduction Increasing demands for enhancing accuracy, high speed and flexibility of electromagnetic actuators can be found in numerous applications such as manufacturing, precision machining and robotics [1–3]. Most of the applications require orientation control of a tool and a workpiece. Recently, the growing interests in fuel-cell technology and low-cost electromechanical systems have motivated a number of researchers to develop compact and high efficient multi-DOF electromagnetic actuators. For the design and analysis of such novel electromechanical actuators, both accurate and fast computations of magnetic field distributions and force/torque analyses are often required. Existing techniques for analyzing magnetic fields and designing multi-DOF PM-based actuators primarily rely on three approaches; namely, analytic solutions to Laplace’s equation, numerical methods [4] and lumped-parameter analyses with some form of magnetic equivalent circuits (MEC) [5]. However, these existing approaches have difficulties in achieving both accuracy and low computation time simultaneously. In addition, many engineering problems with PMs or EMs are often required to solve the three dimensional (3D) magnetic fields with/without a magnetic conducting interface. These difficulties have led us to develop a new modeling method to derive closed-form field solutions for efficient design ∗

Corresponding author. Tel.: +65 6790 5508; Fax: +65 6792 4062; E-mail: [email protected].


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H. Son et al. / Design of multi-DOF electromagnetic actuators using DMP models and image method

and accurate motion control of the actuators. Recently, distributed multi-pole (DMP) model has been developed using magnetic pole models in [6]. In this research, the DMP modelling method was used to characterize the magnetic field distribution in open space. Since the DMP method is here based on the concept of magnetic dipole and a limited set of known field information to construct a distributed dipole model, the method offers a relatively complete formulation for deriving the closed-form and an effective means to characterize the magnetic fields and torque computations for design and control of electromagnetic actuators. However, the method in [7] mainly focused on characterizing the magnetic field of a PM (or an EM) in free space. When the magnetic field is involving a magnetic conducting material, the field of the PM or EM interacts with the material boundary. The change of the field distribution results in the consequent change of the magnetic force and torque. To account for the effects of magnetic conducting materials on the DMP method, an image method can be applied along with the DMP modeling method. As a basis problem-solving tool in electrostatics, the image method replaces the effects of the boundary on an applied field by adding and/or subtracting elementary fields behind the boundary line called image. For its simplicity but accuracy, the image method has been commonly used for analyzing boundary problems involving an eddy current in electromagnetic fields since the method provides certain solution forms for some important problems involving straight-line, circular and spherical boundaries in a simple manner, which decrease the need for formal solutions of Laplace’s and Poisson’s equations. The image method in [8] is used to analyze the unbounded magnetic field containing ferromagnetic materials by a numerical method (FEM). The magnetic field in two dimensional space to design a electromagnetic actuator are obtained in analytical forms using image and MEC methods [9]. Unlike the solutions in [9] with the first order accuracy, nonlinear approach is used to account for effects of eddy currents with magnetic conducting boundary [10]. However, the methods in [10] are mainly applicable for a simple structure of a conductive rod in a simplified geometry. In this paper, we explore the image method with the DMP method to handle magnetic conducting boundaries. The methods (referred to here as DMP-Image) offer a relatively complete solution of the magnetic field involving magnetic conducting boundaries. Emphases are placed on spherical iron boundaries for three-DOF spherical motor design. The remainder of this paper offers the following: 1. We formulate a class of spherical actuator problems with magnetic boundary conditions that appear to be difficult to satisfy if the governing Poisson’s or Laplace’s equation is to be solved directly. The conditions on the bounding surfaces in these problems are set up by appropriate image (equivalent) charges and solve using the image method. 2. To illustrate the DMP-Image, we model the magnetic field of a dipole (defined as a pair of source and sink with a well-defined separation) for three different boundaries; the dipole is (i) outside a grounded spherical rotor, (ii) inside a grounded hollow spherical stator, and (iii) in-between a pair of grounded spherical surfaces. With the field solutions from the DMP-Image, the torques can be calculated using one of the three methods; namely, the Lorentz force equation, Maxwell stress tensor, and force between magnetic charges. 3. We validate the DMP-Image method by comparing the calculated torques against results computed using FEM, which agree to within 5% in maximum torque. As will be shown, the DMP-Image method takes less than 1% of the FEM computation time. Finally, the effects of iron boundaries on the spherical motor were analyzed with the DMP-Image as an illustration of practical applications.


197

Fig. 1. Multi-DOF actuator (spheriacal wheel motor) [12].

2. Magnetic field of spherical grounded boundary Figure 1 shows a multi-degree of freedom (DOF) electromagnetic actuator, called here as spherical wheel motor, which generates multi-degree of freedom motions in a single joint. The actuator mainly consists of three parts: a number of permanent magnets (PMs) in a rotor, electromagnets (EMs) in a stator with a certain pattern and a universal joint bearing at rotational centre of the rotor. The torque on the rotor is controlled by electric-current inputs through the EMs so as to control their orientation quickly, continuously, and isotropically in all directions. Although these kinds of multi-DOF electromagnetic actuators have the similar designs, their structure materials are often different; magnetic materials (iron) and/or non-magnetic materials (aluminium) [7, 8]. In the paper, we investigate an effect of materials on the magnetic field generated by both PMs and EMs as well as the torque and performance of the electromagnetic actuators. We consider here a class of electromagnetic problems where magnetic charges are in the presence of a magnetically grounded spherical boundary. Except at the point charges, the magnetic field is continuous and irrotational, for which a scalar potential Φ can then defined such that H = −∇Φ and B = µ0 H

(1a,b)

where µ0 is the permeability of free space. The formal approach for solving the magnetic field at every point outside the conducting boundary would be to solve the Laplace equation ∇2 Φ = 0 with the following boundary conditions: at points very close to the magnetic charge (source or sink), the potential Φ approaches that of the point charge alone; and at the grounded surface and points very far from ±m, Φ → 0. An alternative approach in lieu of a formal solution is the image method, which replaces bounding surfaces by appropriate image charges. As an illustration, Fig. 2(a) shows the magnetic charge ±m in the free space enclosed by the spherical boundary (of radius R) of very high permeability (µ → ∞, such as iron), where ± signs designate that the pole is a source or a sink respectively. The interest here is to determine the Φ distribution inside the grounded spherical surface due to the charge m. In Fig. 2(a) where XYZ is the reference coordinate system, m ¯ is the image of the charge m and lies along the radial line connecting m. The image charge must be outside the region in which the field is to

198


Fig. 2. Spherical magnetic boundary.

be determined, the parameters (m, a) and (m, ¯ a ¯) are related by (2): a ¯/R = −m/m ¯ = Λ where Λ = R/a

(2)

To facilitate the discussion, we define a local xyz coordinate system (shown in Fig. 2) such that m and m ¯ are on the y-axis at the vector positions, a and ¯ a , respectively. Any point x(x, y, z) can be expressed in spherical coordinates (r, θ, φ) with respect to the reference XYZ coordinate system: T x/ |x| = cos θ cos φ sin θ cos φ sin φ (3)

where θ = tan−1 (y/x); and φ = cos−1 (z/ |x|). Due to the symmetry of a sphere, the problem can be reduced to two dimensional (2D) in the yz plane. The potential at point p(x,y,z) for 0 6 p 6 R where p = |p| is given by   m 1 1  p Φ(p) = (4) −q 2 2 4π 2 p + a − 2p • a 2 (p/Λ) + R − 2p • a

where a = |a|; and a ¯ = |¯ a|. It can be seen from (4) that when |p| = R (on spherical surface), Φ vanishes. The solution is exactly the same as that between the charge m with the grounded boundary at R. 2.1. Images of a magnetic dipole Since magnetic poles exist in pairs, we define a dipole as a pair of source m and sink –m separated by a distance d. For the dipole (located at a1 and a2 ), the images of its source and sink are denoted as m ¯ 1 and − m ¯ 2 in Fig. 2(b). Using (2) and (4), the potential at p in the free space containing the dipole can be expressed as i h (5) Φ(p) = (m/4π) |r1 |−1 − |¯r1 |−1 Λ1 − |r2 |−1 − |¯r2 |−1 Λ2 where ri = p − ai ;

¯ ri = p − ¯ ai

(6a,b)


199

Fig. 3. Parameters of PM and EM and equivalent models.

ai / |ai | = ¯ ai / |¯ ai | = 0 sin φi cos φi

T

(7)

and i = 1, 2 denote the source and sink respectively. In general, if a1 6= a2 , m ¯ 1 6= m ¯ 2 ; the images of a dipole do not form another dipole, and do not satisfy the condition for continuous flow, ∇ • B = 0. The solution of the image method is invalid in the grounded sphere since the image dipole does not actually exist but the images are rather standing in for the magnetic densities induced on the magnetic boundary. 2.2. Image method for spherical EM actuator Without loss of generality, we consider in the following discussions PM-based electromagnetic systems which consist of axially magnetized PMs and air-cored EMs. For the purpose of deriving closed-form field solutions to facilitate design and control of PM-based spherical motors, we seek the field solutions outside the physical regions among the PMs, EMs and magnetic boundary. The parameters characterizing the geometries of the PM and EM are defined in Fig. 3(a) and (b) respectively. Using the DMP method [6], we use k circular loops of n equally spaced dipoles (of strength mk and parallel to the magnetization vector) on the circular loop of radius ρk to model the PM or EM as illustrated in Fig. 3. For practical applications, we consider the following cases in spherical coordinates (r, θ, φ): 2.2.1. Case 1 The dipole m (with its source and sink located at xr1 and xr2 respectively) is located outside the grounded spherical rotor of radius rr at |x ¯ri | /rr = −m ¯ ri /m = Λri where Λri = rr / |xri |

(8)

and

xri / |xri | = x ¯ri / |x ¯ri | = cos θi cos φi

sin θi cos φi

sin φi

T

; i = 1, 2

2.2.2. Case 2 The dipole m (with its source and sink located at xs1 and xs2 respectively) is located inside the hollow grounded spherical stator of radius rs at |x ¯sj | /rs = −m ¯ sj /m = Λsj where Λsj = rs / |xsj |

(9)

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Fig. 4. Electromagnetic system with 2 PM’s and 1 EM.

and

xsj / |xsj | = x ¯sj / |x ¯sj | = cos θj cos φj

sin θj cos φj

sin φj

T

; j = 1, 2

2.2.3. Case 3 The dipole m is in-between the grounded spherical rotor and stator, which are concentric. Each of the charges (source or sink) has two images located such that θi = θj = θ and φi = φj = φ: T where i, j = 1, 2 (10) x ¯ri / |x ¯ri | = x ¯sj / |x ¯sj | = cos θ cos φ sin θ cos φ sin φ

Case 3 is essentially a linear combination of Cases 1 and 2 since the Laplace equation is linear. With the specified magnetic dipole and boundary, the images of the source and sink can be calculated from (8), (9) or (10), Φ and hence H in the free space can be found from (5) and (1) respectively. 3. Illustrative examples The example considered here is an electromagnetic system consisting of two axially magnetized PM on the spherical rotor, and an air-cored EM on the inside surface of the hollow spherical stator. As shown in Fig. 4 where characteristic dimensions are defined, the two rotor-PM poles are identical but their magnetization vectors are in opposite directions. In Fig. 4, δr and δs define the magnetization vectors of the rotor-PM and stator-EM in their respective body coordinate frames. The interest here is to investigate the effects of the iron boundaries on the magnetic field distribution (in the region between the rotor and stator surfaces) and torques acting on the rotor (as a function of the separation angle γ ). For this, we compare four different design configurations (DC): DC1: Rotor and stator are non-magnetic boundaries. DC2: Only the rotor is a magnetically conducting sphere.


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Table 1A Comparisons of design parameters* Spheres Design A rr , rs , g 25.4, 64.3, 0.76 PM: l1 , D1 , Mo 12.7, 12.7, 1.34T EM: l2 , D2 , D3 25.4, 19.05, 9.525 N, I 1040, 4A *Geometrical dimensions are in mm; N = # of turns

Design B 38.1, 64.3, 0.76 12.7, 19.05, 1.34T 19.05, 20.32, 7.62 1040, 4A

Table 1B DMP parameters of PM Parameters d/l1 2ρo /D1 , mo (µAm) 2ρ1 /D1 , m1 (µAm) % error

Design A (n = 6, k = 1) 0.514 0, −22.9 0.5, 61.8 3.3

Design B (n = 6, k = 1) 0.3028 0, −47.2 0.5, 180.8 3.6

Table 1C Equivalent DMP parameters of EMs Parameters d/l2 , De 2ρo /De , mo (nAm) 2ρ1 /De , m1 (nAm) % error De : switching diameter

Design A (n = 6, k = 1) 0.9165, 18.59 0, 14.8 0.5, 108.3 5.7

Design B (n = 6, k = 1) 0.9501, 18.35 0, 5.6 0.5, 54.8 6.73

DC3: Only the stator is a magnetic conducting boundary. DC4: Both the rotor and the stator are magnetic boundaries. Once the magnetic field is known, its effects on the torque acting on the rotor can be investigated by comparing two design geometries; Designs A and B. The (PM and EM) geometries, which are based on the spherical motor in [14], and their corresponding DMP models are given in Tables 1A-C. Table 1A compares the two design geometries, where the differences are highlighted in bold; and g is the radial air-gap between the air-cored EM and rotor PMs. The values characterized the DMP models of the PM and EM are given in Tables 1B and 1C respectively, where the % error is DMP modeling error defined along the PM magnetization axis: R |Φ(z) − ΦA (z)| dz z R %error = 100 × |ΦA (z)| dz z

The corresponding images (location and strength) reflecting the source and sink of each dipole on the spherical boundaries can be derived from (5) with (8), (9) or (10). 3.1. No current flowing through the air-core EM The simulated magnetic fields are given in Figs 5 and 6, where the bold solid circles indicate the spherical boundaries (black for the rotor and red for the stator). Figure 5 shows the magnetic field in Design A due to the PM pair between two concentric grounded spheres. To visually illustrate the image method, we graph the effects of images of the dipoles in the grounded spheres. It must be emphasized

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H. Son et al. / Design of multi-DOF electromagnetic actuators using DMP models and image method Table 2 % Increase in maximum torque relative to DC1 Designs Design A Design B

DC2 (iron rotor) 3.2 % 8.68 %

DC3 (iron stator) 6.0 % 15.15 %

DC4 (both) 9.2 % 23.73 %

Fig. 5. PM Pair between two magnetic surfaces (Unit of colormap: Tesla (T)).

that the field distributions calculated using the image method are valid only in the free space between the spheres, and are invalid in the magnetically ground spheres where Φ and H are zero in iron (µ →∞) and are veiled in Fig. 6. Figure 6(a) shows the magnetic field of the DC1 (with no grounded boundaries) which provides a basis for comparison. The effects of the iron rotor and stator boundaries on the magnetic field are compared in Figs 6(b) and 6(c) respectively. As expected, the magnetic field is perpendicular to grounded spherical surface (Φ = 0). Similarly, the combined effect of both the iron rotor and stator boundaries on the magnetically field is graphically displayed in Fig. 6(d). 3.2. Effects of pole design with iron boundary on torque Using the DMP method with multi-dipoles for the PM and their images, the magnetic field in the air space between the two conducting surfaces and hence, the torque can be computed from the Lorentz force Eq. (A.1) or the forces between magnetic charges (A.3). Alternatively, if the total field (including both PM and EM) is known, the force on a body can also be computed from the surface integration in term of Maxwell stress tensor (A.2). For completeness, the equations for computing the magnetic torque are given in Appendix A. We examine the effects of iron boundaries on the magnetic torque by comparing the four design configurations for a given stator radius rs , each with two different design geometries. In each design, the computed torques for the four DC’s are plotted as a function of the separation angle γ (between the magnetization axes of the PM and EM) in Fig. 7. The % increases in the maximum torques relative to DC1 are compared in Table 2.


203

Fig. 6. Effect of boundaries on magnetic field (Unit of colormap: Tesla (T)).

Fig. 7. Effect of iron boundaries on torque.

In Design A, the combined rotor/stator irons (DC4) contribute to 9.2% increase in the maximum torque; two-thirds are from the iron stator shell and the remainder is from the iron rotor. The results are consistent with the predicted magnetic field distributions and can be explained with the aid of Fig. 6 as follows. DC4 has shortest magnetic flux paths as they enter perpendicularly into grounded boundary. The

204


shorter flux path in DC4 (relative to DC1) results in higher magnetic field intensity and thus larger force on the stator EM. In DC1, the magnetic flux paths between two EMs in the rs region are much longer than that between two PMs in the rr region; this suggests that the iron stator plays a more significant role in shortening the path lines than the iron rotor. 4. Simulation results and discussion In this section, we present and discuss the following torque results computed using the DMP-Image method: – Numerical validation and torque computation time – The DMP-Image method for analyzing designs of a spherical wheel motor (SWM). In the following design examples, both the rotor PM and stator EM are on iron conductors. All computation was performed on a Windows-based PC (dual core processor 2.21 Ghz CPU and 1GB memory). 4.1. Numerical validation To validate the torque computation using the magnetic fields from the DMP-Image, the computed torque for the electromagnetic system in Fig. 7(a) [13] where the geometries of the diagonally symmetric PM and EM pole-pairs are based on Design A in Table 1. In Fig. 8(a), the rotor with a pair of PMs rotates with respect to the stator with two EMs on the same plane. In this study, the torque about the axis perpendicular to this plane is computed as a function of the separation angle γ and compared against solutions numerically obtained using ANSYS (a commercial FE analysis package) in Fig. 8(b). 4.1.1. ANSYS model The ANSYS FE model uses cylindrical iron boundaries for simplicity due to the symmetry. The procedure for computing the electromagnetic torque using ANSYS can be found in [15]. In ANSYS, the iron boundary was modeled using the eight-node SOLID96 elements (µr = 1000 where µr is the relative permeability); the free space air volume was modeled using four-node INFIN47 elements; and the air-cored stator coil as SOURCE36 elements. For the PM, µ = Br /Hc where Br and Hc are the residual magnetization and the magnitude of coercive force vector respectively. In this computation, Hc = 795,770 A/m and Br = 1.34T. With the total magnetic flux density from ANSYS, the torque acting on the rotor is computed using (A.2) where Γ is a circular boundary enclosing the rotor including the PM. 4.1.2. Image method with DMP model (DMP-image) For the image method, the PM and EM pole-pairs are replaced by their respective DMP models, and their images are found following the procedure as discussed in Section 3. The resulting magnetic torque on the rotor can be computed using one of the three methods outlined in the appendix. Using the DMP-Image, the Maxwell stress tensor (A.2) and the magnetic charge equation (A.3) yield identical torque solutions, which agree (within 5% difference) with the ANSYS results shown in Fig. 8(d). Due to symmetry, there should be no torque generated when the separation angle γ is zero, which the PM and EM pole-pairs are aligned. The discrepancy (offset at γ = 0) in ANSYS could be because of the automatically generated FE mesh; this suggests that the quality of mesh could significantly affect the accuracy of the FE analysis. In addition, the DMP-Image with Lorentz force equation (A.1) took only 17 seconds to compute the torque curve while ANSYS requires 24.67 minutes to compute the 13 data. Note that (A.3) is in closed-form requiring relatively negligible computation time.


205

Fig. 8. Comparisons of computed torques (Computational time: ANSYS = 1,480 seconds. DMP = 17 seconds).

4.2. Effects of iron boundaries on the SWM design We illustrate the use of the DMP-Image for analyzing the magnetic torque of a spherical wheel motor (SWM) [15]. Unlike a variable-reluctance spherical motor (VRSM) [7] where the rotor PM and stator EM are placed on locations following the vertices of a regular polygon, the PM and EM of a SWM are equally spaced on layers of circular planes with their radial magnetization axes passing through the motor center. The PM and EM are grouped in pairs; and every two PM or EM pole-pairs form a plane providing symmetric forces electro-mechanically. The magnetization axes of the mr PM pole-pairs are given in rotor coordinates (x, y, z) by Eq. (11): T ri = (−1)i−1 cos δr cos θri cos δr sin θri sin δr (11)

where θri = (i − 1)θr ; and i = 1, 2,. . . , mr . Similarly, the ms EM pole-pairs in the stator frame (XYZ) are given by Eq. (12): T sj = cos δs cos θsj cos δs sin θsj sin δs (12)

where θsj = (j − 1)θs ; and j = 1, 2,. . . , ms . Unlike ms which may be odd or even, mr is always an even number. 4.2.1. Torque – Current relationship of a SWM The resultant torque of the spherical motor with linear magnetic properties has the following form: T T = TX TY TZ = [ K] u (13)

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H. Son et al. / Design of multi-DOF electromagnetic actuators using DMP models and image method Table 3A Geometries of the two SWM designs Parameters* SWM 1 ro , rs , rr , gap 76.2, 57, 38.5, 0.5 26◦ , 20◦ δs′ , δr (◦ ) D1 , ℓ1 ; M◦ 25.4, 12.7; 1.34T D2 , D3 , ℓ2 ; N 19.05, 6,12.7; 1040 ms , mr 10, 8 * Dimensions are in mm.

SWM 2 76.2, 48.3, 56.6, 0.5 18◦ , 25◦ 31.8, 12.7; 1.34T 25.4, 6, 12.7; 1040 8, 10

Table 3B DMP parameters of PM and EM Design SWM 1

PM EM k = 2, n = 6; d/l1 = 0.3100 k= 2, n = 6; d/l2 = 0.5833, De = 15.82 mk : 139.8/9.4/354.7 mk : 0.1471/0.0105/0.4536 SWM 2 k = 2, n = 6; d/l1 = 0.2984 k = 2, n = 10; d/l = 0.3163, De = 19.95 mk : 159.2/ 41.0/ 555.5 mk : 1.06/0.0775/3.031 De : switching diameter; mk : mo /m1 /m2 in µAm

where K ∈R

3×ms

= K1 · · · Kj · · · Kms

and

u = u1 · · · uj · · · ums

T

 m sj ×ri − Pr fˆ(γ) if sj × ri 6= 0 |sj ×ri | ; (14a,b) ; Kj = γ=γji  i=1 0 if sj × ri =0

(15)

Kj is the torque characteristic vector contributed by the j th EM. fˆ(γ) is a curve-fit function of the torque from Fig. 8(d) as a separation angle γ between a PM pole-pair and an EM pole-pair γji = cos−1 (sj • ri ) / (|sj | |ri |). u is current input vector. Since the SWM has more current inputs than the mechanical DOF, the actual current input vector u for a given torque is found by minimizing the input energy consumption subject to the desired torque constraint (13). Provided that the input currents are kept within limits, the optimal u can be solved using Lagrange multipliers. The optimal solution can be written in closed form [7]: −1 T [K] [K]T u = [K] T (16)

4.2.2. Application example As an illustration, we compare two different designs (denoted here as SWM1 and SWM2 in Fig. 9) for the same ro of 76.2 mm in Table 3 (at same size). Unlike the SWM1 [16], the SWM2 has 10 rotor PM pole-pairs mounted on the internal surface of the hollow hemi-sphere and 8 EM pole-pairs are on the external surface of the spherical stator. Specifically, the interest here is to determine the current inputs required to provide a specified torque of T = [0 1 0]T Nm. As shown in Fig. 9, due to symmetry, the current inputs are applied as follows: SWM1: u2 = u10 , u3 = u9 , u4 = u8 , u5 = u7 SWM2: u2 = u8 , u3 = u7 , u4 = u6


207

Fig. 9. Schematics illustrating SWM 1 and SWM2.

Figures 10 and 11 compare the current inputs and its norm defined as J = uT u. For the same coil resistances, the maximum current and the total energy input required by SWM 2 are significantly lower than that required by SWM 1. While (A.2) and (A.3) yield identical results, the average time for computing the applied torque at a specified orientation using the magnetic charge method requires only 0.25 seconds. The Maxwell stress tensor method, however, would require 220 seconds for the same calculation. 5. Conclusion We have shown how the DMP-Image method can be used to analyze a class of spherical motors where PMs and/or EMs are in the presence of magnetically grounded conducting boundary. The DMP-Image method, which solves the magnetic fields in closed form, has been validated by comparing the torques calculated using three different methods (Lorentz, Maxwell and magnetic charges) against the numerical results computed using FEM with Maxwell stress tensor. While the comparison agrees to within 5% in maximum torque, the DMP-Image method requires less than 1% of the FEM computation time. A

208


Fig. 10. Currents inputs.

Fig. 11. Comparison of total energy input.

relatively complete formulation has been presented for solving the magnetic field of a spherical motor with three different cases of magnetically grounded surfaces. As an illustration of practical applications, we demonstrate the effectiveness of using the DMP-Image solutions for analyzing the design of a magnetically linear spherical wheel motor. Appendix A. Torque calculation Three methods for computing the magnetic torque are given as follows: Lorentz force equation: Z ZZ ¯ T = − p × B × (Idℓ ) where I = − J • dS

(A.1)


209

where ℓ¯ is the normalized current direction vector. In (A.1), the current density vector J is directly used in the calculation and thus, it involves only modeling the B-fields of the permanent magnets. Maxwell stress tensor: Alternatively, if the total B field (including both PM and EM) is known, the force on a body can also be computed from the surface integration in term of Maxwell stress tensor Z 1 2 1 p × B(B • n) − B n dΓ (A.2) T= µ0 2 Γ

where Γ is an arbitrary boundary enclosing the body of interest; and n is the normal of the material interface. Force on magnetic charges [17] Figure A1 shows a dipole in the magnetic field, where HR+ and HR- are the magnetic field intensities acting on the magnetic source and sink of the dipole respectively; and R+ and R− are the corresponding distances from a field point. The force F on the magnetic dipole can be written (in analogy to that on the force on a stationary electric charge by the Lorentz law) as F = µo m [HR+ − HR− ]

(A.3)

Similarly, the torque on a dipole T = R+ × F+ + R− × F− = µo m [R+ × HR+ − R− × HR− ]

(A.4)

Fig. A1. Force on a dipole in the magnetic field.

References [1] [2] [3]

S. Sudo, K. Tsuyuki, T. Matsumoto, M. Yoshikawa, M. Watanabe and T. Honda, Biomimetic study of diving beetle robot propelled by alternating magnetic field, Int J Applied Electro and Mech 25 (2007), 601–606. E. Stump and V. Kumar, Workspaces of Cable-Actuated Parallel Manipulators, ASME, J of Mechanical Design 128 (2006), 159–167. Zhu, Yu-Wu Kim, Do-Sun; Moon, Ji-Woo and Cho, Yun-Hyun, Analysis of permanent magnet linear synchronous motor for advanced control, Int J Applied Electro and Mech 28 (2008), 283–289.

210 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

H. Son et al. / Design of multi-DOF electromagnetic actuators using DMP models and image method K.-M. Lee, Q. Li and H. Son, Effects of numerical formulation on magnetic field computation using meshless methods, IEEE Trans.on Magnetics 42 (2006), 2164–2171. H. Ghoizad, M. Mirsalim, M. Mirzayee and W. Cheng, Coupled magnetic equivalent circuits and the analytical solution in the air-gap of squirrel cage induction machines, Int J Applied Electro and Mech 25 (2007), 749–754. K.-M. Lee and H. Son, Distributed multipole model for design of permanent-magnet-based actuators, IEEE Trans on Magnetics 43 (2007), 3904–3913. H. Son and K.-M. Lee, Distributed Multi-Pole Models for Design and Control of PM Actuators and Sensors, IEEE Trans on Mechatronics 13 (2008), 228–238. L. Yan, I.-Ming Chen, G. Yang and K.-M. Lee, Analytical and Experimental Investigation on the Magnetic Field and Torque of a Permanent Magnet Spherical Actuator, IEEE Trans. on Mechatronics 11 (2006), 409–419. K. Lee and K. Park, Modeling eddy currents with boundary conditions by using Coulomb’s law and the method of images, IEEE Trans on Magnetics 38 (2002), 1333–1340. I. Dufour and D. Placko, Original approach to Eddy current problems through a complex electrical image concept, IEEE Trans on Magnetics 32 (1996), 348–365. Y. Saito, K. Takashi and S. Hayano, Finite element solution of unbounded magnetic field problem containing ferromagnetic materials, IEEE Trans on Magnetics 24 (1988), 2946–2948. G. Xiong and S.A. Nasar, Analysis of fields and forces in a permanent magnet linear synchronous machine based on the concept of magnetic charge, IEEE Trans on Magnetics 25 (1989), 2713–2719. L. Kapjin and P. Kyihwan, Modeling eddy currents with boundary conditions by using Coulomb’s law and the method of images, IEEE Trans on Magnetics 38 (2002), 1333–1340. K.-M. Lee, R.A. Sosseh and Z. Wei, Effects of the Torque Model on the Control of a VR Spherical Motor, IFAC J of Control Engineering Practice 12/11 (2004), 1437–1449. K.-M. Lee, H. Son and J. Joni, Concept Development and Design of a Spherical Wheel Motor (SWM), Proc of ICRA (2005), 3652–3657. K.-M. Lee, J. Joni and H. Son, Design Method for Prototyping a Cost-Effective Variable-Reluctance Spherical Motor (VRSM) Proc IEEE Conference on Robotics, Automation and Mechatronics Singapore 1 (2004), 542–547. D.B.H. Tellegen, Magnetic-dipole models, Am J Phys 30 (1962), 650–652.


181

Diffraction of electromagnetic plane wave from a PEMC strip Amjad Imrana,∗ , Q.A. Naqvia and K. Hongob a Department b 3-34-24,

of Electronics, Quaid-i-Azam University Islamabad, Pakistan Nakashizu, Sakura city, Chiba, Japan

Abstract. In this paper, diffracting behavior of a perfectly electromagnetic conductor (PEMC) strip has been studied. Both the E- and H-polarization are considered. The method of analysis is Kobayashi Potential (KP). Imposition of boundary conditions results into dual integral equations (DIEs). These DIEs are solved by using the discontinuous properties of Weber-Schafheitlin’s integral. The resulting expressions, finally, reduce to matrix equations with infinite number of unknowns whose elements are expressed in terms of infinite integrals. These integrals are hard to solve analytically so numerical simulation is done. Illustrative computations are presented for different parameters of interest especially the dependence of co-polarized and cross-polarized components on the admittance parameter.

1. Introduction Lindell and Sihvola [1] recently introduced the concept of perfectly electromagnetic conductor (PEMC) as a generalization of zero-impedance medium (PEC) and infinite-impedance medium (PMC). This medium is characterized by a single parameter M , PEMC admittance, which can vary from zero to infinity. A null admittance corresponds to a PMC medium and an admittance of infinity to a PEC medium [2]. This medium is isotropic. The most notable difference between this and an ordinary medium is its nonreciprocity [3]. Because after scattering from planar air-PEMC boundary, PEMC slab [4], PEMC sphere [5] and infinitely long circular PEMC cylinder [6], the electromagnetic wave also has cross-polarized component along with the co-polarized component. The possible realization of such type of materials has been discussed in [3]. As PEMC cannot let any electromagnetic energy to pass through it, therefore it can serve as a boundary material [7]. The boundary conditions to be satisfied on a PEMC surface can be written by using the PEC and PMC boundary conditions and the fact that PEMC is the generalization of PEC and PMC as follow n ˆ × (H + M E) = 0,

n ˆ .(D − M B) = 0

where M is defined as the PEMC admittance and n ˆ is the unit normal to the boundary. In the present investigation, we present the diffraction of electromagnetic plane wave from a PEMC strip of arbitrary admittance. The method of analysis adopted here is the Kobayashi Potential (KP) method. This method uses the discontinuous properties of Weber-Schafheitlin integral. The method has ∗



182

A. Imran et al. / Diffraction of electromagnetic plane wave from a PEMC strip

Fig. 1. Geometry of the problem.

been successfully applied to potential [8,9] as well as scattering problems for different geometries [10– 14]. Imposition of the boundary conditions result in dual integral equations (DIEs). These DIEs can be solved using the above properties of Weber-Schafheitlin integrals and projection method like the method of moment (MoM), in which Jacobi’s polynomials are used as the basis functions. Finally, the problem reduces to matrix equations whose matrix elements are the infinite integrals. These equations can be solved for the determination of unknown expansion coefficients. Numerical computations have been conducted for the parameters of interest. 2. Formulation and solution of the problem 2.1. E-polarization The geometry of the problem is shown in Fig. 1. It contains a PEMC strip of negligible thickness and width 2a. If φ0 is the angle of incidence, then the incident field Ezi , co-polarized component Ezd and cross-polarized component Hzd in the upper half space y > 0 can be written as h i Ezi = exp jk(x cos φ0 + y sin φ0 ) (1) Ezd

=

Hzd =

Z

0

Z

∞n

0

o h p i fe (ξ) cos (xa ξ) + ge (ξ) sin (xa ξ) exp − ξ 2 − κ2 ya dξ

∞n

o h p i fh (ξ) cos (xa ξ) + gh (ξ) sin (xa ξ) exp − ξ 2 − κ2 ya dξ

(2)

(3)

where κ = ka, xa = xa , ya = ay and k is the propagation constant of the free space. The fe,h (ξ) and ge,h (ξ) are the weighting functions to be determined from the boundary conditions. The required boundary conditions are given by (i) The fields are continuous at |xa | > 1 and y = 0. (ii) Hxt + M Ext = 0 and Hzt + M Ezt = 0 for |xa | 6 1. where superscript t stands for total. From (i), we have Z ∞ hp ih i ξ 2 − κ2 fe (ξ) cos(xa ξ) + ge (ξ) sin(xa ξ) dξ = 0, |xa | > 1 0

(4)


∞ hp

Z

0

ξ 2 − κ2

ih

183

i hh (ξ) cos(xa ξ) + gh (ξ) sin(xa ξ) dξ = 0, |xa | > 1

(5)

Boundary condition (ii) gives Z ∞ nh i h i o fh (ξ) + M fe (ξ) cos(xa ξ) + gh (ξ) + M ge (ξ) sin(xa ξ) dξ 0

Z

0

= −M exp[jκxa cos φ0 ]

(6)

∞p

h i ξ 2 − κ2 fe (ξ) − M Z 2 fh (ξ) cos(xa ξ)dξ Z ∞p h i 2 2 2 + ξ − κ ge (ξ) − M Z gh (ξ) sin(xa ξ)dξ 0

= jκ sin φ0 exp[jκxa cos φ0 ]

(7)

where Z is the impedance of free space. The above expressions are the dual integral equations. Equations (4) and (5) can be used to decide the nature of weighting functions fe,h (ξ) and ge,h (ξ) by making use of the discontinuous properties of Weber-Schafheitlin’s integrals, as follows fe (ξ) = p

1 ξ 2 − κ2

fh (ξ) = p

1 ξ 2 − κ2

∞ X

Am J2m (ξ),

m=0 ∞ X

Cm J2m (ξ),

m=0

ge (ξ) = p

1 ξ 2 − κ2

gh (ξ) = p

1 ξ 2 − κ2

∞ X

Bm J2m+1 (ξ)

(8)

∞ X

Dm J2m+1 (ξ)

(9)

m=0

m=0

where Jm (.) be the Bessel’s function of order m and Am , Bm , Cm and Dm are the expansion coefficients. Separating even and odd functions of the expressions Eqs (6) and (7) and then putting the values of fe,h (ξ) and ge,h (ξ), we get ∞ h i Z ∞ J (ξ) X p 2m Cm + M Am cos(xa ξ)dξ = −M cos(κxa cos φ0 ) ξ 2 − κ2 0 m=0 ∞ h iZ ∞ J X (ξ) p2m+1 sin(xa ξ)dξ = −M j sin(κxa cos φ0 ) Dm + M Bm 2 2 ξ − κ 0 m=0 ∞ h iZ ∞ X 2 J2m (ξ) cos(xa ξ)dξ = jκ sin φ0 cos(κxa cos φ0 ) Am − M Z Cm m=0 ∞ h X

m=0

0

Bm − M Z 2 Dm

iZ

∞

J2m+1 (ξ) sin(xa ξ)dξ = −κ sin φ0 sin(κxa cos φ0 )

0 ±1

Expanding the trigonometric functions in the above expressions in terms of Jacobi’s polynomials un 2 (x2a ) ±1

and vn 2 (x2a ) [15] and then using the orthogonal properties of these polynomials, we get ∞ h i Z ∞ J (ξ)J (ξ) X 2m 2n p dξ Cm + M Am 2 ξ − κ2 0 m=0

184


= −M J2n (κ cos φ0 )) ∞ h X

Dm + M Bm

2

Am − M Z Cm

J2m+1 (ξ)J2n+1 (ξ) p dξ ξ 2 − κ2 = −M jJ2n+1 (κ cos φ0 ) ∞

iZ

0

m=0

∞ h X

∞

0

m=0

∞ h X

iZ

2

Bm − M Z Dm

iZ

0

m=0

∞

(10)

(11)

J2m J2n+1 (ξ) dξ ξ = j tan φ0 J2n+1 (κ cos φ0 )

(12)

J2m+1 (ξ)J2n+2 (ξ) dξ ξ = − tan φ0 J2n+2 (κ cos φ0 )

(13)

n = 0, 1, 2, · · ·

which are the matrix equations for the expansion coefficients and can be written, for our convenience, as follow ∞ h X

(14)

m=0

i h i ih G(2m, 2n; κ) Cm + M Am = −M J2n (κ cos φ0 )

∞ h X

(15)

m=0

i h i ih G(2m + 1, 2n + 1; κ) Dm + M Bm = −M j J2n+1 (κ cos φ0 )

∞ h X

i ih H(2m, 2n + 1; κ) Am − M Z 2 Cm

m=0

∞ h X

m=0

h i = j tan φ0 J2n+1 (κ cos φ0 )

(16)

i ih H(2m + 1, 2n + 2; κ) Bm − M Z 2 Dm h

i = − tan φ0 J2n+2 (κ cos φ0 )

(17)

n = 0, 1, 2, · · ·

where G(α, β; κ) =

Z

∞ 0

Jα (ξ)Jβ (ξ) p dξ ξ 2 − κ2

(18)


H(α, β; κ) =

Z

Jα (ξ)Jβ (ξ) 2 sin[(α − β) π2 ] dξ = ξ π (α2 − β 2 )

∞ 0

185

(19)

In writing the Eqs (14)–(17), we have used the following relations r r πx πx J 1 (x), J 1 (x) sin x = cos x = 2 2 −2 2 ∞ √ 1 X √ 2(2n + m + 2 )Γ(n + m + 12 ) J2n+m+ 12 (ξ) m un (x) x−m/2 Jm (ξ x) = 1 Γ(n + 1)Γ(m + 1) ξ2 n=0 √ ∞ X 8(2n + m + 23 )Γ(n + m + 32 ) J2n+m+ 32 (ξ) m = vn (x) 3 Γ(n + 1)Γ(m + 1) 2 ξ n=0 The Eqs (14)–(17) may be solved to evaluate the expansion coefficients Am , Bm , Cm , Dm . The copolarized component Ezd and cross-polarized component Hzd for y > 0 may be computed from the Eqs (2) and (3). Using the saddle point method, far field expressions may be written as Ezd = C(kρ) Hzd = C(kρ)

∞ h X

(20)

m=0

i Am J2m (κ cos φ) + jBm J2m+1 (κ cos φ)

∞ h X

i Cm J2m (κ cos φ) + jDm J2m+1 (κ cos φ)

(21)

m=0

where C(kρ) =

q

π 2kρ

h i exp −j kρ + π4 and (ρ, φ) are the cylindrical coordinates of the observation

point. Far fields in the lower region can also be derived similarly. 2.2. H-Polarization

The field expressions corresponding to expressions Eqs (1)–(3) for H-polarization may be written as h i Hzi = exp jk(x cos φ0 + y sin φ0 ) (22) Ezd Hzd

=

Z

∞n

(23)

Z

∞n

(24)

0

=

0

o h p i fe (ξ) cos (xa ξ) + ge (ξ) sin (xa ξ) exp − ξ 2 − κ2 ya dξ

o h p i fh (ξ) cos (xa ξ) + gh (ξ) sin (xa ξ) exp − ξ 2 − κ2 ya dξ

All the notations used in the above expressions have the same meaning as described in last section. The boundary conditions remain the same. Imposition of boundary conditions gives Z ∞h i fe (ξ) cos(xa ξ) + ge (ξ) sin(xa ξ) dξ = 0, |xa | > 1 (25) 0

Z

0

∞h

i hh (ξ) cos(xa ξ) + gh (ξ) sin(xa ξ) dξ = 0, |xa | > 1

(26)

186


Z

0

Z

0

∞ nh

i h i o fh (ξ) + M fe (ξ) cos(xa ξ) + gh (ξ) + M ge (ξ) sin(xa ξ) dξ i h = − exp jκxa cos φ0

(27)

∞p

h i ξ 2 − κ2 Y 2 fe (ξ) − M fh (ξ) cos(xa ξ)dξ Z ∞p h i + ξ 2 − κ2 Y 2 ge (ξ) − M gh (ξ) sin(xa ξ)dξ 0 i h = −jM κ sin φ0 exp jκxa cos φ0

(28)

where Y is the admittance of free space. Using the discontinuous properties of Weber-Schafheitlin’s integrals and incorporating the edge conditions for H-field, we get fe (ξ) =

∞ X

Am

∞ X

Cm

m=0

fh (ξ) =

m=0

J2m+1 (ξ) , ξ

ge (ξ) =

J2m+1 (ξ) , ξ

gh (ξ) =

∞ X

Bm

∞ X

Dm

m=0

m=0

J2m+2 (ξ) ξ

(29)

J2m+2 (ξ) ξ

(30)

Proceeding in a similar manner as in last section, we get the matrix equations ∞ h X

m=0

∞ h X

m=0

i ih K(2m + 2, 2n + 2; κ) Y 2 Bm − M Dm = M tan φ0 J2n+2 (κ cos φ0 )

(31)

i ih K(2m + 1, 2n + 1; κ) Y 2 Am − M Cm = −M j tan φ0 J2n+1 (κ cos φ0 )

(32)

∞ h X

(33)

m=0

i ih H(2m + 1, 2n; κ) Cm + M Am = −J2n (κ cos φ0 )

∞ h X

i ih H(2m + 2, 2n + 1; κ) Dm + M Bm = −jJ2n+1 (κ cos φ0 )

(34)

m=0

n = 0, 1, 2, · · ·

where K(α, β; κ) = H(α, β; κ) =

Z Z

∞ 0

0

∞

p

ξ 2 − κ2 Jα (ξ)Jβ (ξ)dξ ξ2

Jα (ξ)Jβ (ξ) dξ ξ

(35) (36)


187

How to compute the integral K(α, β; κ) is provided in the appendix and analytic expression for H(α, β; κ) can be seen in Eq. (19). The above expressions are the matrix equations and can be solved for the expansion coefficients Am , Bm , Cm , Dm by any standard method. Far diffracted fields for the co-polarized Hzd and cross-polarized Ezd components may be obtained from Eqs (23) and (24) by using the saddle point method. The final results are Ezd = C0 (kρ) Hzd = C0 (kρ)

∞ h X

(37)

m=0

i Am J2m+1 (κ cos φ) + jBm J2m+2 (κ cos φ) tan φ

∞ h X

i Cm J2m+1 (κ cos φ) + jDm J2m+2 (κ cos φ) tan φ

(38)

m=0

where C0 (kρ) = point.

q

π 2kρ

i h exp −j kρ − π4 and (ρ, φ) are the cylindrical coordinates of the observation

3. Results and discussions The far field patterns for co-polarized and cross-polarized components can be computed from Eqs (20) and (21) respectively for E-Polarized incident field and from Eqs (37), (38) for H-polarization case. The unknown expansion coefficients Am , Bm , Cm and Dm can be computed from the matrix equations Eqs (14)–(17) for E-polarization and from Eqs (31)–(34) for H-polarization. How to compute the integrals G(α, β; κ) and K(α, β; κ) are discussed in [10,16] in detail, however final expressions which were used to compute these integrals is given in the appendix. We have taken the matrix size (2κ + 1) × (2κ + 1) in our simulations. Since M , the admittance parameter is most important in our work, therefore we have tried to explore the dependance of field patterns on this parameter. Figures 2 and 3 show the variations in the field patterns as a function of M for φ0 = 70◦ , κ = 4 for E-polarization. It turns out that there exist no cross-polarized component Hzd for PMC and PEC case and it dominates for M = 1.0. The amplitude of co-polarized component Ezd slightly increases as we increase the value of M . Figures 4 and 5 show the field patterns for H-Polarization. Cross-polarized component Ezd follow the same trend as in case of E-polarization. We have also studied the dependance of co- and cross-polarized components on the strip width. Figures 6 and 7 present the same. From these figures, we can conclude that as we increase the strip width κ, side lobes start to appear for both the components. Similar trends can also be seen for H-polarization. Another important parameter is the angle of incidence φ0 . Figures 8 and 9 show how this parameter affects the field patterns. If we summarize the above results it can be said that the diffracted field has two components i.e coand cross-component. The effect of admittance parameter M on cross-component is much appreciable as compare to co-component. There exist no cross-component for PMC (M = 0) and PEC (M → ∞) and amplitude of the cross-component go on decreasing as we increase the value of M . The behavior of both the components, otherwise, is alike i.e both the components develop side lobes as the strip width increases, position and amplitude of the main lobe in both the components changes as the angle of incidence is varied.

188

A. Imran et al. / Diffraction of electromagnetic plane wave from a PEMC strip 3.5

DIFFRACTED FIELD

CO-COMPONENT

E-POLARIZATION

3.0

M= 0.0

2.5

M= 1.0 M= 3.0

2.0

M= 100.0 k= 4.0

1.5

f = 70 0

1.0

0.5

0.0

-0.5 -20

0

20

40

60

80

100

120

140

160

180

200

OBSERVATION ANGLE

Fig. 2. Variations of co-polarized component as a function of M. 1.8

E-POLARIZATION

CROSS-COMPONENT

1.6

DIFFRACTED FIELD

1.4

M= 0.0 1.2

M=1.0 M= 3.0

1.0

M=10.0 M=100.0

0.8

f = 70 0

0.6

k= 4.0

0.4

0.2

0.0

-0.2 -20

0

20

40

60

80

100

120

140

160

OBSERVATION ANGLE

Fig. 3. Variation of cross-component as a function of M.

180

200


189

3.5

DIFFRACTED FIELD

3.0

CO-COMPONENT

H-POLARIZATION

2.5

M= 0.0 M= 1.0

2.0

M= 3.0 M= 100.0

1.5

k= 4.0 f = 70 0

1.0

0.5

0.0

-0.5 -20

0

20

40

60

80

100

120

140

160

180

200

OBSERVATION ANGLE

Fig. 4. Dependence of co-polarized component on M. 3.5

DIFFRACTED FIELD

CROSS-COMPONENT

H-POLARIZATION

3.0

2.5

M =0.0 M =1.0

2.0

M =3.0 M =100.0

1.5

k= 4.0 f = 70.0 0

1.0

0.5

0.0

-20

0

20

40

60

80

100

120

140

160

OBSERVATION ANGLE

Fig. 5. Dependance of cross-polarized components on M.

180

200

190

A. Imran et al. / Diffraction of electromagnetic plane wave from a PEMC strip 2.8 2.6 k= 2.0

2.4

k= 4.0

H-POLARIZATION

2.2

DIFFRACTED FIELD

k= 8.0

2.0

M= 3.0

1.8

f = 90 0

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2

-20

0

20

40

60

80

100

120

140

160

180

200

OBSERVATION ANGLE

Fig. 6. variation of cross-polarized component as a function of strip width. 6.0 5.5

DIFFRACTED FIELD

5.0

H-POLARIZATION

k= 2.0

4.5

k= 4.0 k= 8.0

4.0

M= 3.0

3.5

f = p/2 0

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -20

0

20

40

60

80

100

120

140

160

180

OBSERVATION ANGLE

Fig. 7. Variation of co-polarized component as a function of strip width.

200


191

3.2 3.0 2.8

E-POLARIZATION

2.6

CROSS-COMPONENT

DIFFRACTED FIELD

2.4 2.2 2.0

f = 70 0

1.8

f = 80

1.6

0

f = 90

1.4

0

1.2

k= 4.0

1.0

M= 1.0

0.8 0.6 0.4 0.2 0.0 -0.2 -20

0

20

40

60

80

100

120

140

160

180

OBSERVATION ANGLE

Fig. 8. Cross-polarized component as a function of angle of incidence.

Fig. 9. Dependence of co-polarized component on the angle of incidence.

200

192


Appendix Detailed discussions on the above integrals can be seen in [10]and [16]. However we are giving here the final expressions which were used to compute these integrals through fortran

G(α, β; κ) =

Z

∞ 0

Jα (ξ)Jβ (ξ) p dξ ξ 2 − κ2

2p √ 21 (α+β)−1 (−1)p Γ 12 (α+β) − p Γ(2p+1) κ2 π X 1 1 1 = 1 2 Γ (α+β)+p+1 Γ (α − β)+p+1 Γ (α − β)+p+1 Γ(p+1)Γ − p 2 2 2 2 p=0 √ X ∞ κ α+β+2p (−1)p Γ(α+β +2p+1)Γ 21 (α+β)+p+ 12 π 1 −j 2 2 Γ(p+1)Γ(α+β +p+1)Γ(α+p+1)Γ(β +p+1)Γ (α+β)+p+1 2 p=0 ∞ κ α+β+2p (−1)p Γ(α+β +2p+1)Γ 12 (α+β)+p+ 21 1 X 1 − √ 2 π Γ(p+1)Γ(α+β +p+1)Γ(α+p+1)Γ(β +p+1)Γ 2 (α+β)+p+1 2 p=0 α+β +1 κ × 2 ln +2ψ(α+β +2p+1)+ψ +p − ψ(p+1) − ψ(α+β +p+1) 2 2 1 −ψ(α+p+1) − ψ(β +p+1) − ψ (α+β)+p+1 2

where ψ(.) and Γ(.) is the Di-Gamma and Gamma function respectively. Z ∞p 2 ξ − κ2 Jα (ξ)Jβ (ξ)dξ K(α, β; κ) = ξ2 0 β −1 p Γ α+β − p Γ(2p+1) √ α+ 2 κ 2p (−1) 2 π X = 2 α+β 1 1 3 4 p=0 Γ(p+1)Γ 2 +p+1 Γ 2 (α − β)+p+1 Γ 2 (β − α)+p+1 Γ 2 − p p Γ(α+β +2p+1)Γ α+β +p − 1 √ X ∞ κ α+β+2p (−1) 2 2 π =j α+β 4 2 +p+1 p=0 Γ(p+1)Γ(α+β +p+1)Γ(α+p+1)Γ(β +p+1)Γ 2 1 ∞ κ α+β+2p (−1)p Γ(α+β +2p+1)Γ α+β 2 +p − 2 1 X = √ 2 4 π p=0 Γ(p+1)Γ(α+β +p+1)Γ(α+p+1)Γ(β +p+1)Γ α+β +p+1 2 α+β 1 κ +p − × 2 ln +2ψ(α+β +2p+1)+ψ − ψ(p+1) − ψ(α+β +p+1) 2 2 2 α+β −ψ(α+p+1) − ψ(β +p+1) − ψ +p+1 (39) 2 References [1] [2]

I.V. Lindell and A.H. Sihvola, Perfect Electromagnetic conductor, J Electromag Waves Appl (2005), 861–869. I.V. Lindell, Differential Forms in Electromagnetics, NY: Wiley-Interscience, 2004.

A. Imran et al. / Diffraction of electromagnetic plane wave from a PEMC strip [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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I.V. Lindell and A.H. Sihvola, Realization of the PEMC Boundary, IEEE Trans on Antennas and Propagation 53(9) (Sep 2005), 3012–3018. Bernard Jancewics, Plane Electromagnetic Wave in PEMC, arXiv: physics/ 050823. R. Ruppin, Scattering of Electromagnetic Radiation by a Perfectly Electromagnetic Conductor Sphere, J Electromag Waves Appl 20(12) (2006), 1569–1576. R. Ruppin, Scattering of Electromagnetic Radiation by a Perfectly Electromagnetic Conductor Cylinder, J Electromag Waves Appl 20(13) (2006), 1853–1860. I.V. Lindell and A.H. Sihvola, Losses in the PEMC Boundary, IEEE Trans on Antennas and Propagation 54(9) (Sep 2006), 2553–2558. I. Kobayashi, Darstellung eines Potentials in Zylindrischen Koordinaten, das sich auf einer Ebene innerhalb und ausserhalb einer gewissen Kreisbegrenzung verschiedener Grenzbedingung unterwirft, Sci Rep Tohoku Univ., Ser.1, 20 (1931), 197– 212. I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966. K. Hongo, Diffraction of electromagnetic plane wave by a slit, Trans Inst electronics and Comm Engrg in Japan 55-B(6) (1972), 328–330. K. Hongo and Hirohide Serizawa, Diffraction of Electromagnetic plane Wave by a Rectangular Plate and a Rectangular Hole in the Conducting Plate, IEEE Trans on Antennas and Propagation 47(6) (June 1999), 1029–1041. A. Imran, Q. A. Naqvi and K. Hongo, Diffraction of Plane Wave by two parallel Slits in an infinitely long Impedance Plane using the Method of Kobayashi Potential, Progress in Electromagnetic research, PIER 63 (2006), 107-123. K. Hongo and Q.A. Naqvi, Diffraction of electromagnetic wave by disk and circular hole in a perfectly conducting plane, Progress In Electromagnetic Research, PIER 68 (2007), 113–150. A. Imran, Q. A. Naqvi and K. Hongo, Diffraction of Electromagnetic plane Wave by an Impedance Strip, Progress in Electromagnetic research, PIER 75 (2007), 303–318. W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, New York:Spinger-Verlag. Y. Nomura and S. Katsura, Diffraction of Electromagnetic Waves by Ribbon and Slit, J Phys Soc Japan 12 (1957), 190–200.


171

Fluid-induced frequency shift in a piezoelectric plate driven by lateral electric fields Bo Liua , Qing Jianga and Jiashi Yangb,∗ a Department b Department

of Mechanical Engineering, University of California, Riverside, CA 92521, USA of Engineering Mechanics, University of Nebraska, Lincoln, NE 68588-0526, USA

Abstract. We study thickness-shear vibration of a piezoelectric crystal plate whose one surface is in contact with a viscous fluid layer of finite thickness. The crystal plate is driven by a lateral electric field. The theory of piezoelectricity and the theory of Newtonian fluids are used. Both free and forced vibration solutions are obtained. An approximate expression for the frequency shift in the crystal plate due to the presence of the fluid is presented. The admittance of the structure is also calculated. The results are useful for acoustic wave fluid sensors.

1. Introduction A vibrating crystal (resonator) when put in contact with a viscous fluid changes its resonant frequencies due to the inertia and viscosity of the fluid. This effect has been used to make fluid sensors for measuring fluid viscosity or density [1–3]. These sensors belong to the general category of those acoustic wave sensors called quartz crystal microbalances (QCMs). More references can be found in relevant review articles [4,5]. For fluid sensor applications, vibration modes of a crystal body without a normal displacement at its surface are ideal and are of general interest. In these modes the surface of the body has no normal displacement so that no pressure waves are generated in the fluid. The fluid produces a tangential drag only on the body surface due to viscosity and the tangential motion of the surface, thereby causing a frequency shift in the body. Quartz is the main piezoelectric crystal for resonator and sensor applications. Thickness-shear vibration of a quartz plate is the most widely used structure and mode for QCMs. The sensitivity given in the classical reference [1] for a fluid sensor is based on an elastic analysis without piezoelectric coupling which is small in quartz and can usually be neglected in a free vibration frequency analysis but must be considered in an electrically forced vibration analysis to obtain the impedance or admittance of a device. In real device operations electrodes are necessary for generating electric fields or collecting charges (currents). Electrodes are usually deposited on the two surfaces of a crystal plate to produce a driving electric field in the plate thickness direction. This type of electrode arrangement is called thickness field excitation (TFE). Surface deposited electrodes are associated with a series of complicating effects including electrode inertia, stiffness, intrinsic stress, and thermal expansion which is usually incompatible ∗

Corresponding author: E-mail: [email protected].


172

Bo Liu et al. / Fluid-induced frequency shift in a piezoelectric plate driven by lateral electric fields x2

f =-0.5Vexp(i

t)

Fluid

f =0.5Vexp(i

H

b

Quartz

t)

x3

b

2c

Fig. 1. A crystal plate in contact with a fluid under lateral electric fields.

with that of the crystal plate, etc. [6–8]. These effects of the electrodes are usually undesirable, especially when the electrodes are on the surface of the crystal plate where sensing is performed. One way to avoid putting an electrode on the sensing surface of a crystal plate is to use side electrodes and the associated lateral or in-plane electric fields, called lateral field excitation (LFE) [9–12]. While at present there seem to be more devices with TFE than LFE, LFE offers a number of advantages over TFE [9,10]. For example, LFE can result in reduced aging, higher Q (quality factor) values and increased frequency stability because the regions of greatest vibrational motion are free of electrodes. This also makes LFE convenient for sensor applications in which the unelectroded active area of a resonator can be put in direct contact with measurands. Recently, due to the rapid development of resonator based mass, fluid, chemical and biological sensors, LFE has received more and growing attention [13–15]. However, there are very few theoretical results for devices driven by LFE in contrast to the vast literature on TFE. In this paper we study thickness-shear vibration of a quartz crystal plate with one surface in contact with a viscous fluid layer. The crystal plate is under LFE. The theory of linear piezoelectricity is used to model the crystal plate. The theory of Newtonian fluids is used for the fluid layer. For fluid sensor application we want to study fluid-induced frequency shift of the crystal plate from a free vibration analysis, and the capacitance of the plate from a forced vibration analysis. 2. Fields in different regions Consider the structure shown in Fig. 1 which is unbounded in the x1 direction. The crystal plate is of rotated Y-cut quartz which includes the widely used AT-cut quartz as a special case. The fluid is a linear Newtonian fluid. Whether the fluid is compressible or not does not matter because the motion to be considered is a pure shear without volume change. There are two edge electrodes at x3 = ±c. On these electrodes a driving voltage of φ = ±V exp (iω t) /2 is applied where φ is the electric potential. We assume thin plates with c >> b so that edge effects can be neglected and pure thickness-shear modes exist. We consider time-harmonic motions and use the usual complex notation. All fields have the same exp(iωt) factor which will be dropped in the following for simplicity. 2.1. Upper free space For thickness-shear modes independent of x1 and x3 in the plate, the corresponding governing equations of the electric field in the free space are

Bo Liu et al. / Fluid-induced frequency shift in a piezoelectric plate driven by lateral electric fields

D2,2 = 0, D2 = ε0 E2 , E2 = −φ,2 ,

173

(1)

where E is the electric field and D is the electric displacement. ε0 is the free-space permittivity. We consider the case when x2 = ±∞ are electrically open where D2 = 0. Since D2 is a constant in the free space as dictated by Eq. (1)1 , D2 ≡ 0 in the free space. The free space electric potential is simply φ = −Ex3 + C1 ,

(2)

where E = −V /2c is a constant, C1 is an arbitrary constant. Equation (2) implies that E3 = E and D2 = 0. The open circuit condition at x2 = +∞ is satisfied. 2.2. Fluid The fluid is assumed to be without electromechanical coupling. The electric fields are still governed by Eq. (1) but the free-space permittivity ε0 needs to be replaced by the fluid permittivity ε. The equation of motion for the fluid is [16] T21,2 = ρL v˙ 1 ,

(3)

where the shear stress is given by T21 = µ

∂v1 . ∂x2

(4)

µ and ρL are the viscosity and mass density of the fluid. v1 and T21 are the relevant velocity and shear stress components. The potential and velocity fields in the fluid are φ = −Ex3 + C1 , v1 = {C3 sinh [(1 + i) η (x2 − b)] + C4 cosh [(1 + i) η (x2 − b)]} ,

where C3 and C4 are integration constants, and r ρL ω η= . 2µ

(5)

(6)

The relevant stress and electric displacement components needed for boundary and continuity conditions are T21 = (1 + i) µη {C3 cosh [(1 + i) η (x2 − b)] + C4 sinh [(1 + i) η (x2 − b)]} , D2 = 0.

(7)

We note that the continuity of φ and D2 between the upper free space and the fluid are already satisfied. 2.3. Crystal plate In the crystal plate, due to the presence of E3 , we begin with the following trial fields. They will be shown to satisfy all governing equations and boundary/continuity conditions later. u1 = u1 (x2 ),

u2 = u3 = 0,

φ = φ(x2 ) − Ex3 + C2 ,

(8)

174


where C2 is an undetermined constant. The nontrivial components of the strain, electric field, stress, and electric displacement components are, correspondingly, 2S12 = u1,2 ,

E2 = −φ,2

E3 = −φ,3 = E,

T31 = c56 u1,2 + e25 φ,2 − e35 E, D2 = e26 u1,2 − ε22 φ,2 + ε23 E,

T21 = c66 u1,2 + e26 φ,2 − e36 E, D3 = e36 u1,2 − ε32 φ,2 + ε33 E.

(9) (10)

The equation of motion and the charge equation of electrostatics take the following form: T21,2 = c66 u1,22 + e26 φ,22 = −ρω 2 u,1 D2,2 = e26 u1,22 − ε22 φ,22 = 0.

(11)

The displacement and potential fields determined from Eq. (11) are u1 = C5 sin [ξ (x2 − b)] + C6 cos [ξ (x2 − b)] , φ=

e26 {C5 sin [ξ (x2 − b)] + C6 cos [ξ (x2 − b)]} + C7 (x2 − b) − Ex3 + C2 , ε22

(12) (13)

where C5 , C6 and C7 are undetermined constants, and ξ2 =

ρ 2 ω . c¯66

(14)

The stress and electric displacement components are T21 = c¯66 ξ {C5 cos [ξ (x2 − b)] − C6 sin [ξ (x2 − b)]} + e26 C7 − e36 E,

(15)

D2 = −ε22 C7 + ε23 E, ε32 D3 = e36 − e26 ξ {C5 cos [ξ (x2 − b)] − C6 sin [ξ (x2 − b)]} − ε32 C7 + ε33 E, ε22

(16) (17)

2 ), k 2 = e2 /(ε c ). The free charge Q on the edge electrode at x = c per where c¯66 = c66 (1 + k26 22 66 e 3 26 26 unit length along x1 , the current I that flows into this electrode, and the admittance Y of the structure are given by Rb Qe = −b −D3 dx2 , (18) I = Q˙ e = iωQe , Y = I/V.

2.4. Lower free space For the lower free space we have φ = −Ex3 + C8 ,

where C8 is an arbitrary constant. The open circuit condition D2 = 0 at x2 = −∞ is satisfied.

(19)


175

3. Boundary and continuity conditions The top of the fluid layer is traction-free, i.e., T21 (b + H) = 0.

(20)

At the interface between the fluid and the top of the crystal plate, we have the continuity of the velocity, electric potential, shear stress, and normal electric displacement: u˙ 1 (b− ) = v1 (b+ ) , φ (b− ) = φ (b+ ) , T21 (b− ) = T21 (b+ ) , D2 (b− ) = 0.

(21)

At the interface between the bottom of the crystal plate and the free space below it, we have: φ (−b− ) = φ (−b+ ) , D2 (−b− ) = 0, T21 (−b− ) = 0.

(22)

We note that although there are eight equations in Eqs (20)–(22), they are effectively seven because D2 in the crystal plate as given in Eq. (16) is a constant and D2 (±b− ) = 0 are effectively one condition. Substitution of the relevant fields into Eqs (20)–(22) gives the following seven equations for C1 through C8 : C3 cosh [(1 + i) ηH] + C4 sinh [(1 + i) ηH] = 0, 26 C6 = C2 , C1 − εe22 ε22 C7 = ε23 E, C4 − iωC6 = 0, − (1 + i) µηC3 + c¯66 ξC5 + e26 C7 = e36 E, c¯66 ξ [C5 cos (2ξb) + C6 sin (2ξb)] + e26 C7 = e36 E, e26 ε22 [C5 sin (2ξb) − C6 cos (2ξb)] + C7 (2b) + C8 = C2 .

(23)

Effectively C1 − C2 and C8 − C2 are two constants. Therefore the unknown constants in Eq. (23) are also seven. 4. Vibration analysis In Eq. (23) E is the only driving term. In the following we consider free and electrically forced vibrations separately. 4.1. Free vibration For free vibrations we set E = 0. Equation (23) becomes homogeneous. For nontrivial solutions the determinant of the coefficient matrix of Eq. (23) has to vanish, which gives the following frequency equation: r r ρ ωρL µ tan 2bω tanh [(1 + i) ηH] . (24) = − (1 − i) c¯66 2ρ¯ c66

176


Quartz has a small piezoelectric coupling. If we neglect the small piezoelectric coupling by setting e26 = 0 (in this case c¯66 = c66 ) and consider the limit when H → ∞, Eq. (24) reduces to r r ρ ωρL µ , (25) = − (1 − i) tan 2bω c66 2ρc66 which is the frequency equation in [1]. On the other hand, if we neglect the drag due to fluid viscosity in Eq. (24) by setting µ = 0, the right-hand side of Eq. (24) vanishes and the left-hand side can be factored into two equations. One is sin(ξb) = 0 which is not of interest because it determines modes that are symmetric about x2 = 0. What is used in devices is the other equation cos(ξb) = 0 which determines modes antisymmetric about x2 = 0. In this case, ξb = nπ/2, n = 1, 3, 5, . . ..

(26)

Corresponding to Eq. (26), from Eq. (14) we obtain the following frequencies for antisymmetric thickness-shear modes when the fluid is not present as our reference frequencies: r nπ c¯66 (n) ω0 = . (27) 2b ρ We now return to the general frequency Eq. (24). Consider the case of a low viscosity fluid, we look for approximate roots of Eq. (24) by letting nπ ξb ∼ − ∆(n) , (28) = 2 where ∆(n) is small. Substituting Eq. (28) into Eq. (24), for small viscosity, we obtain s (n) 1 − i ρL µω0 (n) tanh[(1 + i)η0 H], (29) ∆(n) ∼ = 2 2ρ c¯66 q (n) (n) where η0 = ρL ω0 /(2µ). With ∆(n) known from Eq. (29), from Eq. (14) we obtain the relative frequency shift as s (n) (n) (n) ω − ω0 2 (n) 1 − i ρL µω0 n) ∆Ω(n) = = − ∆ = − tanh[(1 + i)η0 H]. (30) (n) nπ nπ 2ρ c ¯ 66 ω 0

Equation (30) is more general than the classical result in [1] by including the effect of piezoelectric coupling in c¯66 and the effect of the finite fluid layer thickness H . When e26 = 0 and H = ∞, Eq. (30) reduces to s (n) ρL µω0 1 − i (n) (n) ω0 . (31) ∆ω (n) = ω (n) − ω0 = − nπ 2ρc66 When n = 1, the real part of Eq. (31) gives the classical result of [1] for the frequency shift of the fundamental thickness-shear mode in a crystal resonator due to contact with a viscous fluid. The real part of Eq. (31) is negative, indicating that the fluid drag lowers the frequency. From Eq. (31) higher-order modes with larger n seem to have smaller frequency shifts. The imaginary part of Eq. (31) represents the damping effect due to the fluid viscosity.


177

Fig. 2. ∆Ω(1) versus the fluid layer thickness H.

Fig. 3. ∆Ω(3) versus the fluid layer thickness H.

4.2. Forced vibration For forced vibrations Eq. (23) is inhomogeneous. Under a real driving frequency the coefficient matrix does not vanish. A solution always exists, is unique, and can be obtained directly on a computer. 5. Numerical results (1)

Consider a resonator of AT-cut quartz with b = 0.58 mm so that ω0 = 9 × 106 1/s. The fluid layer thickness is fixed to H = 2b except in Figs 1 and 2. For the fluids we use ethanol with density ρL = 0.78522 g/cm3 and viscosity µ = 1.04 mPa·s, or toluene with ρL = 0.8669 g/cm3 and µ = 0.5503 mPa·s, or chloroform with ρL = 1.483 g/cm3 and µ = 0.542 mPa·s. Figure 2 shows the effect of the fluid layer thickness H on frequency shifts due to different fluids for the fundamental mode with n = 1. The fluid lowers the frequency as expected. For small H the frequency shift is proportional to H . There is a maximal frequency shift of the order of 10−4 when H is somewhat less than b, half the thickness of the crystal plate. This is considered strong and clear signals

178


Fig. 4. Effects of fluid density on ∆Ω(1) .

Fig. 5. Effects of fluid viscosity on ∆Ω(1) .

because typical thermal noises in quartz resonators are of the order of 10−6 . When H > 2b the frequency shift becomes constant. In this case the fluid layer can in fact be treated as a half space. Figure 3 shows similar behaviors of the third overtone mode with n = 3, with smaller frequency shifts and quicker decay of fields in the fluids (smaller penetration depth). To examine the effect of the fluid density individually, we artificially vary the density of ethanol and plot the result in Fig. 4. A heavier fluid with a larger density causes more frequency shift as expected. The relationship between ρL and ∆Ω(n) is essentially parabolic as suggested by Eq. (30). In Fig. 5 we artificially vary the viscosity of ethanol and observe similar effects. Figure 6 is from the forced vibration analysis. It shows the admittance per unit length of the plate in the x1 direction. At resonance the admittance assumes maximum. (a) shows an isolated resonance which is ideal for resonant acoustic wave sensors. (b) is a magnified picture of (a) locally near resonance. 6. Conclusion An exact solution is obtained for thickness-shear vibrations of a rotated Y-cut quartz plate in contact with a fluid driven by a lateral electric field. An approximate expression for the frequency shifts due to the fluid is presented. It includes the classical result in [1] as a special case. The fluid density and


179

Fig. 6. Admittance versus driving frequency.

viscosity tend to lower the frequencies of the crystal plate. Higher-order modes are less sensitive to the fluid than lower-order modes. The relative frequency shift is of the order of 10−4 . The results obtained are fundamental and useful for the understanding and design of quartz crystal fluid sensors driven by lateral electric fields.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

K.K. Kanazawa and J.G. Gordon II, The oscillation frequency of a quartz resonator in contact with a liquid, Analytica Chimica Acta 175 (1985), 99–105. F. Josse, Z.A. Shana, D.E. Radtke and D.T. Haworth, Analysis of piezoelectric bulk-acoustic- wave resonators as detectors in viscous conductive liquids, IEEE Trans Ultrason, Ferroelec, Freq Contr 37 (1990), 359–368. C.E. Reed, K.K. Kanazawa and J.H. Haufman, Physical description of a viscoelastically loaded AT-cut quartz resonator, J Apl Phys 68 (1990), 1993–2001. E. Benes, Improved quartz crystal microbalance technique, J Appl Phys 56 (1984), 608–626. E. Benes, M. Gröschl, W. Burger and M. Schmid, Sensors based on piezoelectric resonators, Sensors and Actuators A48 (1995), 1–21. E.P. EerNisse, Simultaneous thin-film stress and mass-change measurement using quartz resonators, J Appl Phys 43 (1972), 1330–1337. H.F. Tiersten and B.K. Sinha, Temperature dependence of the resonant frequency of electroded doubly-rotated quartz thickness-mode resonators, J Appl Phys 50 (1979), 8038–8051. H.F. Tiersten and B.K. Sinha, Intrinsic stress in thin films deposited on anisotropic substrates and its influence on the natural frequencies of piezoelectric resonators, J Appl Phys 52 (1981), 5614–5624. A. Ballato, E.R. Hatch, M. Mizan and T.L. Lukaszek, Lateral field equivalent networks and piezocoupling factors of quartz plates driven in simple thickness modes, IEEE Trans Ultrason, Ferroelec, Freq Contr 49 (2002), 922–928. A. Khan and A. Ballato, Piezoelectric coupling factor calculations for plates of langatate driven in simple thickness modes by lateral-field-excitation, IEEE Trans Ultrason, Ferroelec, Freq Contr 35 (1988), 435–436. R.C. Smythe and H.F. Tiersten, An approximate expression for the motional capacitance of a lateral field resonator, IEEE Trans Ultrason, Ferroelec, Freq Contr 35(3) (1988), 435–436. P.C.Y. Lee, Electromagnetic-radiation from an at-cut quartz plate under lateral-field excitation, J Appl Phys 65(4) (1989), 1395–1399. Y.H. Hu, L.A. French, Jr., K. Radecsky, M.P. da Cunha, P. Millard and J.F. Vetelino, A lateral field excited liquid acoustic wave sensor, IEEE Trans Ultrason, Ferroelec, Freq Contr 51(11) (2004), 1373–1380.

180 [14] [15] [16]

Bo Liu et al. / Fluid-induced frequency shift in a piezoelectric plate driven by lateral electric fields D.F. McCann, J.M. McCann, J.M. Parks, D.J. Frankel, M.P. da Cunha and J.F. Vetelino, A lateral-field-excited LiTaO3 high-frequency bulk acoustic wave sensor, IEEE Trans Ultrason, Ferroelec, Freq Contr 56(4) (2009), 779–787. W.Y. Wang, C. Zhang, Z.T. Zhang, T.F. Ma and G.P. Feng, Energy-trapping mode in lateral-field-excited acoustic wave devices, Appl Phys Letter 94 (2009), 192901. R.L. Panton, Incompressible Flow, John Wiley and Sons, New York, 1984.


155

Design of current pulse for electromagnetic railguns Lizhong Xu∗ and Yanbo Geng Mechanical engineering institute, Yanshan University, China Abstract. In this paper, using the differential equation of the armature motion for the electromagnetic Railguns, influences of the driving current distribution on the velocities of the projectile are investigated. Several kinds of the typical energy losses are considered. The influences of the driving current distribution on the transfer efficiency of the railgun system are analyzed. For a given exit velocity of the projectile and rail length, a proper relative time ratio of the pulse currents is determined which corresponds to the optimum system transfer efficiency. The results can be used to design pulse current distribution of the railgun system. Keywords: Railgun, velocities of the projectile, transfer efficiency, pulse currents, relative time ratio

Abbreviations a b c f F h he if I I0 l L′r L0 m Ra S t t1 , t2 and t3 T ∆t ∆t1 ∆t2 ∆t3 ∗

is the acceleration of the projectile. is the distance between two rails. is the viscous friction coefficient between the rail and the armature. is the sliding friction coefficient. is the Lorentz force. is the height of rail. is equivalent height of the rail. is the rail current as shot exit. is the current intensity flowing in the armature. is the value of constant current. is the length of the rail. is the induction gradient of the rail. is the induction of source. is the mass of the armature and projectile. is the armature resistance. is the equivalent area of the partial rail through which the currents pass. is time. is the time of the different current stages, respectively. is the total time. is some time. is some time at range from 0 to t1 . is some time at range from t1 to t2 . is some time at range from t2 to t3 .



156

v v0 v1 , v2 and v3 vf w W0 Wa Wa1 , Wa2 and Wa3 Wf Wf 1 , Wf 2 and Wf 3 Wi Wk Wr Wr1 , Wr2 and Wr3 x x0 x1 , x2 and x3 xf α and α′ δ δ¯ η µ0 ρ τ1 , τ2 and τ3

L. Xu and Y. Geng / Design of current pulse for electromagnetic railguns

is the velocity. is the initial velocity. is the velocity of the different current stages, respectively. is the exit velocity of the projectile. is the width of rail. is the total energy. is armature energy loss. is armature resistance loss of the different current stages, respectively. is friction energy loss. is friction energy loss of the different current stages, respectively. is remaining electromagnetic energy loss. is the kinetic energy of the projectile. is rail resistance loss. is rail resistance loss of the different current stages, respectively. is the acceleration length. is the initial position. is armature displacement of the different current stages, respectively. is the exit displacement of the projectile. are coefficients. is the maximum skin depth. is the average skin depth. is the efficiency. is the magnetic conductivity of free space. is the resistivity. is the relative time of the different current stages, respectively.

1. Introduction For many years, the utility of electromagnetic guns has been explored. The railgun is a kind of the electromagnetic gun (see Fig. 1). It consists of two parallel conductors rails across which an armature makes electrical contact. The armature is an integral part of the projectile assembly. When current flows in the circuit, a magnetic field is established in the space between the rails. This field interacts with the current to produce the Lorentz or J·B force, which both accelerates the projectile and produces a mutually repulsive force on the rails. Velocities up to 6000 m/s have been reported, making the railgun become an attractive approach. In 1978, S. C. Rashleigh and R. A. Marshall built an inductively driven rail-gun macroparticle accelerator in which velocities of 5.9 km/s were obtained using an arc as the driving armature [1]. The results lead people to believe that higher velocities can be obtained and a lot of studies on the railgun were done. One of the important problems in developing the railgun is design of the rail and the related elements. The current in the rails and associated magnetic field extend from the breech to the armature location. Temporal changes occur as the current provided by the pulsed power system varies during the inbore cycle. The rail is subjected to a hostile thermal environment. So, the rail should have high ability against these complicated loads [2,3]. The stress fields, temperature fields and electromagnetic fields of the rail and the related elements are investigated [4–6]. The other problem is increase of the energy transfer efficiency for the railgun system. Usually, the transfer efficiency is relatively low and the increase of the


157

Fig. 1. Schematic of railgun and its magnetic field. (a) schematic of railgun (b) geometric parameters and coordinate system (c) magnetic field.

efficiency is often contradictory to increasing velocities of the projectile. So, a lot of research is done to increase velocities of the projectile and the transfer efficiency simultaneously. Fitch O and Rose M F investigated effects of the current modes on the transfer efficiency, and found that the transfer efficiency can be increased effectively when the current becomes zero at shot exit [7]. Thus, three modes of the operating current were proposed. The first is the constant current mode, the second is the rapid rise-slow set current mode, and the third is the rapid rise-constant-slow set current mode (see Fig. 2). For the first mode, a large muzzle velocity of the projectile can be obtained easily for a given rail length, but about 50% of the breech electric energy remains in the railgun after shot exit, so the efficiency of the railgun system is relative low. For the second mode, the breech electric energy vanishes after shot exit, so the efficiency of the railgun system is relative high, but a large muzzle velocity of the projectile can not be obtained easily for a given rail length. For the third mode, the efficiency of the railgun system is relative high, and a relatively large muzzle velocity of the projectile can be obtained for a given rail length that is a proper operating current mode. For reducing energy loss, some electric sources were developed and the distributed-current-feed and distributed-energy-store railguns were presented [8–11]. Using these electric sources, the breech electric energy and the resistance energy loss are reduced, so

158


Fig. 2. The current operating mode.

the rail length can be increased. Besides the electric sources, the structure of the railgun was improved to increase velocities of the projectile and the transfer efficiency. For example, the Multistage Rail gun was proposed. Here, The rail is divided into several individual sections so that the rail heat is reduced and the efficiency is increased [12,13]. For the same purpose, several types of the augmented electric gun systems were proposed [14–16]. Of course, improving the electric sources is more convenient than changing railgun structure. For these novel electric sources, the rapid rise-constant-slow set current mode is accepted. However, it has not been studied how the relative time ratio of the pulse current is determined to obtain the optimum transfer efficiency for a given exit velocity of the projectile and rail length. In this paper, using the differential equation of the armature motion, influences of the driving current distribution on the velocities of the projectile are investigated. Several kinds of the typical energy losses are considered. The influences of the driving current distribution on the transfer efficiency of the railgun system are analyzed. A proper time ratio of the driving currents at different stages is proposed so that a large velocity of the projectile and relatively high transfer efficiency can be obtained simultaneously. The results are useful for design and application of the railgun system.

2. The differential equation of armature motion As shown in Fig. 1, two copper rails are arranged in parallel. The rails are copper strips h × w and of lengths L. The distance between the two copper strips is b. The armature together with the projectile can run along the rails. The current flowing in the rails produces a magnetic flux density B between the rails, and this magnetic field interacts with the current flowing in the armature. The resulting Lorentz force F accelerates the armature together with the projectile along the rails. The differential equation of armature motion is m

d2 x dx +c −F =0 2 dt dt

(1)

Where m is the mass of the armature and projectile, c is the viscous friction coefficient between the rail and the armature, t is time, the Lorentz force F = 12 L′r I 2 , I is the current intensity flowing in the armature, L′r is the induction gradient of the rail.


For the third mode of the operating current, the operating current can be written as  (0 6 t 6 t1 )  I = αt I = I0 (t1 6 t 6 t2 )  I = I0 − α′ t (t2 6 t 6 t3 )

Where I0 is a constant current, α and α′ are coefficients. Combining Eq. (2) with F = 12 L′r I 2 , yields  (0 6 t 6 t1 )  F = 12 L′r α2 t2 1 ′ 2 (t1 6 t 6 t2 ) F = 2 Lr I0  F = 12 L′r (I0 − α′ t)2 (t2 6 t 6 t3 )

159

(2)

(3)

Substituting Eq. (3) into Eq. (1), yields m

1 dx1 d2 x1 = L′r α2 t2 (0 6 t 6 t1 ) +c 2 dt dt 2

(4a)

m

d2 x2 dx2 1 +c = L′r I02 (t1 6 t 6 t2 ) dt2 dt 2

(4b)

m

2 1 dx3 d2 x3 = L′r I0 − α′ t (t2 6 t 6 t3 ) +c 2 dt dt 2

(4c)

1 The initial conditions are x1 |t=0 = 0 and dx dt |t=0 = 0, the continuous conditions are x2 |t=t1 = x1 |t=t1 , v2 |t=t1 = v1 |t=t1 , x3 |t=t2 = x2 |t=t2 and v3 |t=t2 = v2 |t=t2 . Using these initial conditions and the continuous conditions, the solutions of the Eq. (4) can be given as below

x1 = − x2 = (

L′r m3 α2 L′r m3 α2 − c t L′r α2 3 L′r mα2 2 L′r m2 α2 + e m + t − t + t c4 c4 6c 2c2 c3

(5a)

L′r m3 α2 L′r mI02 − c (t−t1 ) L′r m3 α2 − c t1 L′r mα2 2 L′r m2 α2 m e t + t − + )e m − + 1 1 c4 2c2 c3 c4 2c2

L′r I02 L′ mI 2 L′ α2 (t − t1 ) + r t31 − r 2 0 (5b) 2c 6c 2c ′ 3 2 c L′r m3 α2 L′r mα2 I02 Lr m α − c t1 L′r mα2 2 L′r m2 α2 ′ e m − t1 + t1 − + e− m t2 − x3 = x2 − 4 2 3 4 2 c 2c c c 2c ′ 3 2 Lr m α − c t1 L′r mα2 2 L′r m2 α2 L′r m3 α2 L′r m2 I0 a′ m3 L′r a′2 m − e t + t − − + − 1 1 c3 c4 c4 2c2 c3 c4 c L′r mα2 I02 L′r m2 a′ I0 L′r m3 a′2 − c (t−t2 ) L′r a′2 −m t2 (t − t2 )3 − + e e m + + 2c2 c3 c4 6c

L′r a′ I0 L′r ma′2 + 2c 2c2

2

(t − t2 ) +

L′r I02 L′r ma′ I0 L′r m2 a′2 + + 2c c2 c3

(t − t2 )

(5c)

160


v1 = −

L′r m2 α2 L′r m2 α2 − c t L′r α2 2 L′r mα2 m + t − e t + c3 2c c2 c3

(6a)

c L′r m2 α2 − c t1 L′r α2 2 L′r mα2 L′r m2 α2 L′r I02 L′r I02 −m (t−t1 ) m v2 = − e t + t − + (6b) e − + 1 c3 2c 1 c2 c3 2c 2c ′ 2 2 c Lr m α − c t1 L′r α2 2 L′r mα2 L′r ma′ I0 L′r m2 α2 L′r α2 I02 t2 −m m − − v3 = − e t + t − − − e 1 c3 2c 1 c2 c3 2c c2

′ ′ Lr a I0 L′r ma′2 L′r m2 a′2 − c (t−t2 ) L′r a′2 2 m + (t − t ) − + e (t − t2 ) + 2 c3 2c c c2 ′ 2 L′ mI0 a′ L′r m2 a′2 Lr I0 + r 2 + 2c c c3

(6c)

3. Energy loss and efficiency of railgun system From Newton’s second low, the motion parameters of the railgun can be expressed by the electric parameters, i.e. L′r I 2 − gf 2m Z t L′ gI (t) − gf t adt = v0 + r v = v0 + 2m 0 Z t L′ hI (t) gf t2 − vdt = x0 + v0 t + r x = x0 + 2m 2 0 a=

(7a) (7b) (7c)

Where f is the sliding friction coefficient between the rail and the armature,a is the acceleration of the projectile, R t v is its velocity, xRist its acceleration length, x0 is its initial position, v0 is its initial velocity, gI (t) = 0 I 2 dt and hI (t) = 0 gI (t)dt. In the railgun system, six kinds of the energy losses occur. They are energy loss on the rail resistance, energy loss on the armature, remaining electromagnetic energy loss after shot exit, the friction energy loss between the rails and the projectile, air resistance loss in front of the projectile, and the plasma resistance loss. The last two kinds of the energy losses are relatively small and are not considered here. The other four kinds of the energy losses can be calculated as below ! Z t 2 Z t Z t 2 I 2 ρgf t2 I ρx I 2 ρL′r h(t) I ρx p p − dt (8a) Wr = 2 dt = 2 dt = 2 S mhe πρt/µ0 mhe πρt/µ0 0 0 0 he δ Wa =

Z

0

t

I 2 ρb dt = S

Z

t 0

2I 2 ρb p dt h πρt/µ0

(8b)


161

1 (L0 + L′r L)If2 2 Z t Z l (mgf v + cv 2 )dt (mgf + cv)dx = Wf = Wi =

(8c)

(8d)

0

0

Where Wr , Wa , Wi and Wf are rail resistance loss, armature resistance loss, remaining electromagnetic energy loss and friction energy loss, respectively, Ra is the armature resistance, L0 is the induction of source, l is the length of the rail, f is the friction coefficient, if is the rail current as shot exit, S = he δ , it is the equivalent area of the partial rail through which the currents pass, he = h + 2 × 0.8w + 0.4h, it is q πρt δ 1 ¯ equivalent height of the rail, δ = = , it is the average skin depth, δ is the maximum skin depth, 2

2

µ0

ρ is the resistivity, here the eddy current effect is considered, it is because the current and magnetic field do not have time to diffuse completely into the rails and are concentrated in a skin layer near the rail surfaces due to the rapid motion of the armature along the rails. Thus, the efficiency η of the railgun system can be given as η=

Wk Wk = W0 Wr + Wa + Wi + Wf + Wk

(9)

Where Wk is the kinetic energy of the projectile. From the operating current mode as shown in Fig. 2, these energy losses and the efficiency can be calculated as below: Letting x0 = v0 = 0(projectile starts moving from rest), and substituting Eq. (7) into Eq. (8), yields Wr = Wr1 + Wr2 + Wr3

(10a)

Wa = Wa1 + Wa2 + Wa3

(10b)

Wf = Wf 1 + Wf 2 + Wf 3

(10c)

L′2 I 4 Wi = r 0 ∆t21 + 4∆t1 ∆t2 + 6∆t22 48m 4 L′2 r I0 (∆t1 + 3∆t2 ) 12m

2t t2 − +1 + ∆t23 ∆t3

(10d) 6 5 4 3 4 2t2 6t 15t 16t t3 t L′2 I − − + − +t + r 0 + 6t2 48m ∆t43 ∆t33 ∆t3 ∆t23 ∆t3 ∆t23 5

5

R ∆t I 2 ρL′r hI (t) R ∆t I 2 ρx1 ρL′r I04 ∆t12 4ρgf I02 ∆t12 p p p dt = − dt = 2 0 1 Where Wr1 = 2 0 1 S mhe πρt/µ0 39mhe πρ/µ0 9he πρ/µ0 Z ∆t1 4 gf ∆t21 ′ 2 c∆t31 L′2 gf L′r I02 r I0 2 2 2 (mgf v1 + cv1 )dt = Wf 1 = (Lr I0 − 12mgf ) + − +g f 24 3 84m2 5m 0 s Z ∆t1 2 ρb 2I µ0 ∆t1 Ra I 2 dt = 0 Wa1 = 5h πρ 0

162


Wr2 = 2

R ∆t2 0

2I 2 ρ p 0 he πρt/µ0 

L′ I 2 t2 gf t2 L′r I02 ∆t1 t − gf ∆t1 t + r 0 − dt+ 6m 4m 2 

  ρgf I02 ∆t21 ρL′r I04 ∆t21  s s −  dt  ∆t1 ∆t1 +t µ0 2he πρ +t µ0 24mhe πρ 2 2 r ′ 4 √ √ Lr I0 µ0 ρ 3/2 5/2 = (80∆t1 ∆t2 + 72∆t2 + 15∆t21 4∆t2 + 2∆t1 − 15∆t21 2∆t1 )− 180he m π  R ∆t2  2 0   

gf I02 15he Wf 2 = =

R ∆t2 0

r

√ √ µ0 ρ 3/2 5/2 (40∆t1 ∆t2 + 12∆t2 + 15∆t21 4∆t2 + 2∆t1 − 15∆t21 2∆t1 ) π

(mgf v2 + cv22 )dt

gf L′r I02 mg2 f 2 cL′2 I 4 (2∆t1 ∆t2 +3∆t22 )− (2∆t1 ∆t2 +∆t22 )+ r 20 (∆t21 ∆t2 +3∆t1 ∆t22 +3∆t32 )− 12 2 36m cgf L′r I02 (∆t21 + 4∆t1 ∆t2 + 3∆t22 ) + g2 f 2 (∆t21 + 2∆t1 ∆t2 + ∆t22 ) 3m

Wa2 =

Z

0

Wr3 =

∆t2

p 2I 2 ρb p Ra I 2 dt = p 0 ( ∆t1 + ∆t2 − ∆t1 ) h πρ/µ0

L′r I04 ρ p 6mhe πρ/µ0

8∆t1 t3/2 10t9/2 32t7/2 12t5/2 16∆t1 t5/2 + 8∆t2 t3/2 + + − − − 3 7∆t3 5 5∆t3 3∆t23

48∆t2 t5/2 12t11/2 8∆t1 t7/2 24∆t2 t7/2 2t13/2 − + + + 5∆t3 11∆t33 7∆t23 7∆t23 13∆t43

!

ρgf I 2 − p 0 he πρ/µ0

4 ∆t1 t3/2 + 3

4∆t2 t7/2 2t9/2 4 2 8∆t1 t5/2 8∆t2 t5/2 4t7/2 4∆t1 t7/2 + + ∆t2 t3/2 + t5/2 − − − + 3 5 5∆t3 5∆t3 7∆t3 7∆t23 7∆t23 9∆t23 +

!

ρI 2 L′ I 2 p 0 [ r 0 (∆t21 + 4∆t1 ∆t2 + 6∆t22 ) − gf (∆t21 + ∆t22 + 2∆t1 ∆t2 )]{(2∆t1 + he πρ/µ0 12m

2∆t2 +4t)1/2 −

1 1 [ (2∆t1 +2∆t2 +4t)3/2 − (∆t1 + ∆t2 )(2∆t1 +2∆t2 +4t)1/2 ]+ ∆t3 6

1 1 1 1 (∆t1 + ∆t2 )(2∆t1 +2∆t2 +4t)3/2 + (∆t1 + ∆t2 )2 [ (2∆t1 +2∆t2 +4t)5/2 − 2 12 4 ∆t3 80 √ √ √ 2 2 2 2 (∆t1 + ∆t2 )3/2 − 2(∆t1 + ∆t2 )1/2 } (2∆t1 +2∆t2 +4t)1/2 ]− (∆t1 + ∆t2 )5/2 − 2 3∆t3 15∆t3


Rt

Wf 3 =

0

163

(mgf v3 + cv32 )dt

t3 L′r gf I02 3t2 t4 mg2 f 2 − = + (2∆t1 t + 2∆t2 t + t2 )+ ∆t1 t + 3∆t2 t + − 6 2 2 4∆t23 ∆t3 4 4 3 3 4 cL′2 r I0 2 t+6∆t ∆t t+ ∆t1 t − 2∆t1 t +3∆t t2 +9∆t2 t+ 3∆t2 t − 6∆t2 t +9∆t t2 + ∆t 1 2 1 2 2 1 36m2 ∆t3 ∆t3 2∆t23 2∆t23 t7 L′r gf I02 t6 3t5 9t4 3 + 3t ∆t21 t + 3∆t22 t + t3 + 4∆t1 ∆t2 t + 2∆t1 t2 + − − + − 3m 7∆t43 ∆t33 ∆t23 2∆t3 3 (∆t1 + ∆t2 )t4 (∆t1 + ∆t2 )t3 3t4 t5 2 f 2 (∆t2 t + ∆t2 t + t + 2 − + g 3∆t2 t + − + 2 1 ∆t3 4∆t3 5∆t23 3 4∆t23 2∆t1 ∆t2 t + ∆t1 t2 + ∆t2 t2 ) Rt

Wa3 =

0

Ra I 2 dt

4 I02 ρb 2(∆t1 +∆t2 +t)1/2− [(∆t1 +∆t2 +t)3/2−3(∆t1 +∆t2 )(∆t1 +∆t2 +t)1/2 ]+ = p 3∆t3 h πρ/µ0 2 [3(∆t1+∆t2 +t)5/2−10(∆t1+∆t2 )(∆t1+∆t2 +t)3/2+15(∆t1+∆t2 )2 (∆t1+∆t2 +t)1/2 ]− 15∆t23 8 16 5/2 (∆t +∆t ) 2(∆t1 +∆t2 )1/2 − (∆t1 +∆t2 )3/2 − 1 2 3∆t3 15∆t23

Here, ∆t1 = t1 , ∆t2 = t2 − t1 , ∆t3 = t3 − t2 , t is some time at range from t2 to t3 . The kinetic energy of the projectile is 1 mv 2 2 3 ′ 2 2 Lr I0 t3 1 3t2 + 3t − gf (∆t1 + ∆t2 + t) ∆t1 + 3∆t2 + = m − 2 6m ∆t23 ∆t3 3 4 4 3t2 L′2 t L′2 r I0 r I0 2 2 − = (∆t1 + 6∆t1 ∆t2 + 9∆t2 ) + (∆t1 + 3∆t2 ) + 3t + 72m 36m ∆t23 ∆t3 (11) 6 ′ 2 5 4 3 ′2 4 t gf Lr I0 6t 15t 18t Lr I0 (∆t1 + 3∆t2 )(∆t1 + ∆t2 + t)− + 9t2 − − + − 72m ∆t43 ∆t33 ∆t3 6 ∆t23 gf L′r I02 (∆t1 + ∆t2 )t3 3(∆t1 + ∆t2 )t2 t4 3t3 2 + 3(∆t1 + ∆t2 )t + + 3t + − − 6 ∆t3 ∆t23 ∆t23 ∆t3

Wk =

mg2 f 2 mg2 f 2 2 (∆t1 + ∆t2 )2 + mg2 f 2 (∆t1 + ∆t2 )t + t 2 2

164


Letting t = ∆t3 and substituting it into Eq. (10), yields Wr = Wr1 + Wr2 + Wr3 5 5 r 4ρgf I02 ∆t12 L′r I04 µ0 ρ ρL′r I04 ∆t12 3/2 5/2 p p − + (80∆t1 ∆t2 + 72∆t2 + = π 39mhe πρ/µ0 9he πρ/µ0 180he m r p √ gf I02 µ0 ρ 3/2 5/2 2 2 30∆t1 ∆t2 + ∆t1 /2 − 15∆t1 2∆t1 ) − (40∆t1 ∆t2 + 12∆t2 + 15he π √ √ 64 L′r I04 ρ 3/2 p ( ∆t1 ∆t3 + 15∆t21 4∆t2 + 2∆t1 − 15∆t21 2∆t1 ) + 6mhe πρ/µ0 105

3376 ρgf I 2 32 64 32 3/2 5/2 3/2 3/2 ∆t2 ∆t3 + ∆t3 ) − p 0 ( ∆t1 ∆t3 + ∆t2 ∆t3 + 35 15015 105 105 he πρ/µ0 ′ 2 2 16 ρI Lr I0 5/2 ∆t3 ) + p 0 (∆t21 + 4∆t1 ∆t2 + 6∆t22 ) − gf (∆t21 + ∆t22 + 315 he πρ/µ0 12m 1 1 1 2 / 2∆t1 ∆t2 )] {(2∆t1 +2∆t2 +4∆t3 ) − (2∆t1 +2∆t2 +4∆t3 )3/2 − ∆t3 6 1 1 1/2 (2∆t1 +2∆t2 +4∆t3 )5/2 − (∆t1 + ∆t2 )(2∆t1 +2∆t2 +4∆t3 ) + ∆t23 80 1 1 3/2 2 1/2 (∆t1 +∆t2 )(2∆t1 +2∆t2 +4∆t3 ) + (∆t1 +∆t2 ) (2∆t1 +2∆t2 +4∆t3 ) − 12 4 √ √ √ 2 2 2 2 5/2 (∆t1 + ∆t2 )3/2 − 2(∆t1 + ∆t2 )1/2 } (∆t1 +∆t2 ) − 2 3∆t3 15∆t3

(12a)

Wf = Wf 1 + Wf 2 + Wf 3 =

gf ∆t21 ′ 2 c∆t31 L′2 I4 gf L′r I02 gf L′r I02 (Lr I0 − 12mgf ) + ( r 02 − + g2 f 2 ) + (2∆t1 ∆t2 + 3∆t22 )− 24 3 84m 5m 12 cgf L′r I02 cL′2 I 4 mg2 f 2 (2∆t1 ∆t2 + ∆t22 ) + r 20 (∆t21 ∆t2 + 3∆t1 ∆t22 + 3∆t32 ) − (∆t21 + 2 36m 3m L′ gf I02 4∆t1 ∆t2 + 3∆t22 ) + g2 f 2 (∆t21 + 2∆t1 ∆t2 + ∆t22 ) + r (∆t1 ∆t3 + 6 (12b) 4 3∆t23 cL′2 mg2 f 2 r I0 2 2 3∆t2 ∆t3 + )− (2∆t1 ∆t3 + 2∆t2 ∆t3 + ∆t3 ) + (∆t1 ∆t3 + 4 2 36m2 9∆t2 ∆t23 9∆t33 L′ gf I02 3∆t1 ∆t23 + 9∆t22 ∆t3 + + )− r (∆t21 ∆t3 + 6∆t1 ∆t2 ∆t3 + 2 2 14 3m 9∆t33 5∆t1 ∆t23 9∆t2 ∆t23 3∆t22 ∆t3 + + 4∆t1 ∆t2 ∆t3 + + ) + g2 f 2 (∆t21 ∆t3 + 20 4 4 ∆t33 + 2∆t1 ∆t2 ∆t3 + ∆t1 ∆t23 + ∆t2 ∆t23 ) ∆t22 ∆t3 + 3


165

Wa = Wa1 + Wa2 + Wa3 r √ 2I02 ρb µ0 √ I 2 ρb (5 ∆t1 + ∆t2 − 4 ∆t1 ) + p 0 = {2(∆t1 + ∆t2 + ∆t3 )1/2 − 5h πρ h πρ/µ0 (12c) 2 4 [(∆t1 + ∆t2 + ∆t3 )3/2 − 3(∆t1 + ∆t2 )(∆t1 + ∆t2 + ∆t3 )1/2 ] + [3(∆t1 + 2 3∆t3 15∆t3 5 2 ∆t + ∆t ) / − 10(∆t + ∆t )(∆t + ∆t + ∆t )3/2 + 15(∆t + ∆t )2 (∆t + ∆t + 2

3

1

2

1

2

3

1

2

1

2

16 8 (∆t1 + ∆t2 )3/2 − ∆t3 )1/2 ] − 2(∆t1 + ∆t2 )1/2 − (∆t1 + ∆t2 )5/2 } 3∆t3 15∆t23

It should be noted that remaining electromagnetic energy loss Wi after shot exit becomes zero. Here, remaining electromagnetic energy is zero, and the kinetic energy of the projectile is Wk =

4 L′2 r I0 (∆t21 + 9∆t22 + ∆t23 + 6∆t1 ∆t2 + 2∆t1 ∆t3 + 6∆t2 ∆t3 )− 72m

gf L′r I02 (∆t21 + 3∆t22 + ∆t23 + 4∆t1 ∆t2 + 2∆t1 ∆t3 + 4∆t2 ∆t3 )+ 6 mg2 f 2 (∆t21 + ∆t22 + ∆t23 + 2∆t1 ∆t2 + 2∆t1 ∆t3 + 2∆t2 ∆t3 ) 2 The exit velocity and displacement of the projectile is vf =

L′r I02 (∆t1 + 3∆t2 + ∆t3 ) − gf (∆t1 + ∆t2 + ∆t3 ) 6m

xf =

L′r I02 (∆t21 + 6∆t22 + 4∆t1 ∆t2 + 4∆t1 ∆t3 + 12∆t2 ∆t3 + 3∆t23 )− 24m

(13)

(14)

(15)

gf (∆t21 + ∆t22 + 2∆t1 ∆t2 + 2∆t1 ∆t3 + 2∆t2 ∆t3 + ∆t23 ) 2

Substituting Eqs (10), (11) or (12) into Eq. (9), the efficiency of the railgun system can be given. When the total time is given, the different ratios of ∆t1 , ∆t2 and∆t3 have obvious effects on the efficiency of the railgun system. Letting T denote the total time, and then ∆t1 = T − ∆t2 − ∆t3 . ∆t2 ∆t3 1 Letting τ1 = ∆t T , τ2 = T andτ3 = T , and then τ1 = 1 − τ2 − τ3 . Substituting it into above related equations, yields xf =

gf 2 L′r I02 T 2 (1 + 3τ22 + 2τ2 + 2τ3 + 6τ2 τ3 ) − T 24m 2

(16)

vf =

L′r I02 T (1 + 2τ2 ) − gf T 6m

(17)

166

L. Xu and Y. Geng / Design of current pulse for electromagnetic railguns 5

5

5

5

5

4ρgf I02 T 2 (1 − τ2 − τ3 ) 2 L′ I 4 T 2 ρL′r I04 T 2 (1 − τ2 − τ3 ) 2 p p − + r 0 Wr = 180he m 39mhe πρ/µ0 9he πρ/µ0 3/2 10τ3 )τ2

r

+ 15(1 − τ2 − τ3

√

)2 [

2 + 2τ2 − 2τ3 −

p

r

µ0 ρ {8(10 − τ2 − π 5

gf I02 T 2 2(1 − τ2 − τ3 )]} − 15he

p √ µ0 ρ 3/2 {4(10−7τ2 −10τ3 )τ2 +15(1−τ2 −τ3 )2 [ 2+2τ2 −2τ3 − 2(1−τ2 −τ3 )]}+ π 5

5

64 3376 5/2 ρgf I 2 T 2 64 L′r I04 ρT 2 3/2 3/2 p (1 − τ2 − τ3 )τ3 + τ2 τ3 + τ3 ) − p 0 ( 35 15015 6mhe πρ/µ0 105 he πρ/µ0

(18)

5 2

(

L′ I 2 32 16 5/2 ρI 2 T 32 3/2 3/2 [ r 0 (1 + 3τ22 + τ32 + (1 − τ2 − τ3 )τ3 + τ2 τ3 + τ3 ) + p0 105 105 315 he πρ/µ0 12m

1 1 2τ2 − 2τ3 − 2τ2 τ3 ) − gf (1 − 2τ3 + τ32 )]{(2 + 2τ3 )1/2 − [ (2 + 2τ3 )3/2 − (1 − τ3 ) τ3 6 1 1 1 1 [ (2 + 2τ3 )5/2 − (1 − τ3 )(2 + 2τ3 )3/2 + (1 − τ3 )2 (2 + 2τ3 )1/2 ]− 2 12 4 τ3 80 √ √ √ 2 2 2 2 (1 − τ3 )5/2 − (1 − τ3 )3/2 − 2(1 − τ3 )1/2 } 2 3τ3 15τ3

(2+2τ3 )1/2 ]+

Wf =

gf T 2 (1 − τ2 − τ3 )2 ′ 2 cT 3 (1 − τ2 − τ3 )3 L′2 I4 gf L′r I02 (Lr I0 − 12mgf ) + ( r 02 − + g2 f 2 )+ 24 3 84m 5m mg2 f 2 T 2 cL′2 I 4 T 3 gf L′r I02 T 2 (2τ2 + τ22 − 2τ2 τ3 ) − (2τ2 − τ22 − 2τ2 τ3 ) + r 0 2 (τ2 + τ22 − 12 2 36m 2τ2 τ3 + τ2 τ32 − τ22 τ3 − 2τ23 + 3τ33 ) − g2 f 2 T 2 (1 − 2τ3 + τ32 ) +

L′r gf I02 T 2 24

cgf L′r I02 T 2 (1 + τ32 + 2τ2 − 2τ3 − 2τ2 τ3 )+ 3m

(4τ3 + 8τ2 τ3 − τ32 ) −

mg2 f 2 T 2 2

(19)

(2τ3 − τ32 )+

4 3 cL′2 r I0 T (14τ3 + 56τ22 τ3 + 2τ33 + 56τ2 τ3 − 7τ32 + 42τ2 τ32 )− 504m2

L′r gf I02 T 3 1 (20τ3 + 4τ33 + 40τ2 τ3 − 15τ32 − 20τ2 τ32 ) + g2 f 2 T 3 (τ3 + τ33 − τ32 ) 60m 3 Wa = Wa1 + Wa2 + Wa3 2I 2 ρb = 0 5h

r

p µ0 T p I 2 ρb (5 (1 − τ3 )−4 (1 − τ2 − τ3 ))+ 0 πρ h

r

4 µ0 T {2 − [1−3(1 − τ3 )]+ πρ 3τ3

2 16 8 (1 − τ3 )3/2 − [3 − 10(1 − τ3 )+15(1 − τ3 )2 ]−2(1 − τ3 )1/2 − (1 − τ3 )5/2 } 2 3τ 15τ3 15τ32 3

(20)


167

Table 1 The Parameters used in simulation L′r (µH/m) 0.5

m(kg) 0.02

I0 (KA) 300

g(m/s2 ) 10

c 0.1

f 0.1

Fig. 3. Changes of the velocity and kinetic energy along with τ2 and τ3 .

Wk =

4 2 gf L′r I02 T 2 mg2 f 2 T 2 L′2 r I0 T (1 + 4τ22 + 4τ2 ) − (1 + 2τ2 ) + (1 + 4τ2 ) 72m 6 2

(21)

4. Results and discussions The Parameters used in simulation is given in Table 1. Changes of the exit velocity and kinetic energy along with relative time τ2 and τ3 are given in Fig. 3(a) and (b). Changes of the energy losses along with relative time τ2 and τ3 are given in Fig. 4(a), (b), (c) and (d). Changes of the system efficiency along with relative time τ2 andτ3 are given in Fig. 5. Figs.3–5 show: 1. The exit velocity and kinetic energy of the projectile grow with increasing relative time τ2 . The relative time τ3 do not have nearly effects on the exit velocity and kinetic energy of the projectile. However, due to τ1 + τ2 + τ3 = 1, for a larger τ3 value, the span of the relative time τ2 becomes smaller. So, for getting large exit velocity and kinetic energy of the projectile, relative time τ3 and τ1 should be taken as small values as possible. 2. The resistance energy loss of the railgun system grows obviously with increasing relative time τ2 . The relative time τ3 do not have nearly effects on resistance energy loss of the railgun system. For getting small resistance energy loss, relative time τ2 should be taken as small values as possible. 3. The friction energy loss of the railgun system grow with increasing relative time τ2 and τ3 . The relative time τ2 has more obvious effects on friction energy loss than relative time τ3 . For reducing friction energy loss, relative time τ2 should be taken as small value. The friction energy loss is much smaller than resistance energy loss.

168


Fig. 4. Changes of the energy losses along with τ2 and τ3 .


169

Table 2 Changes of the efficiency η along with τ2 and τ3 τ3 τ2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.63805 0.63236 0.61544 0.59558 0.57688 0.56111 0.54906 0.54101 0.53693 0.53646 0.53673

0.2 0.56636 0.56900 0.56483 0.55692 0.54759 0.53808 0.52904 0.52064 0.51160 – –

0.4 0.53034 0.53574 0.53725 0.53510 0.53068 0.52462 0.51650 – – – –

0.6 0.51714 0.52308 0.52609 0.52540 0.52138 – – – – – –

0.8 0.51173 0.51715 0.51932 – – – – – – – –

1 0.50795 – – – – – – – – – –

Fig. 5. Changes of the system efficiency along with τ2 and τ3 .

4. The armature energy loss is much smaller than resistance energy loss as well. Changes of the armature energy loss along with relative time τ2 and τ3 are similar to ones of the friction energy loss. 5. Changes of the total energy loss along with relative time τ2 and τ3 are similar to ones of the resistance energy loss. It shows the resistance energy loss is the most principal energy loss in the railgun system. 6. The maximum efficiency of the railgun system occurs at τ2 = τ3 = 0(it means τ1 = 1). Here, the exit velocity is the minimum. It is not proper for operation of the railgun system. 7. For given total time, the different τ2 causes the different changes of the efficiency with τ3 . For τ2 = 0, the efficiency of the railgun system drops obviously with increasing relative time τ3 . For τ2 larger than 0.8, the efficiency of the railgun system grows obviously with increasing relative time τ3 . For τ2 smaller than 0.8 but larger than 0, the efficiency first grows, gets to its maximum value, and then drops with increasing the relative time τ3 (details see Table 2). 8. For given total time and different τ3 , the efficiency of the railgun system drops obviously with increasing relative time τ2 .

170


Under condition that the exit velocity and rail length are given, the relative time τ1 , τ2 and τ3 can be determined from Eqs (16) and (17), Table 2, and Fig. 5. From Table 2 and Fig. 5, the ratio τ2 /τ3 for the maximum efficiency can be obtained, and then substituting the ratio into Eqs (16) and (17), the relative time τ1 , τ2 and τ3 can be determined. Thus, the optimum efficiency is obtained for a given exit velocity and rail length. 5. Conclusions In this paper, using the differential equation of the armature motion, influences of the driving current distribution on the velocities of the projectile and the transfer efficiency of the railgun system are investigated. For a given exit velocity of the projectile and rail length, a proper relative time ratio of the pulse currents is determined which corresponds to the optimum system transfer efficiency. For relative time τ2 larger than 0.8, when τ3 is as large as possible and τ1 is as small as possible, the transfer efficiency is near its maximum value. For relative time τ2 smaller than 0.8, when τ3 is near τ1 , the transfer efficiency is near its maximum value. The results can be used to design pulse current distribution of the railgun system. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

S.C. Rashleigh and R.A. Marshall, Electromagnetic acceleration of macroparticles to high velocities, J Appl Phys 49 (1978), 2540–2542. T. Benton, F. Stefani, S. Satapathy, K.T. Hsieh, Numerical modeling of melt-wave erosion in conductors, IEEE Transactions on Magnetics 39(1) (January 2003), 129–133. C. Persad, Railgun tribology – Chemical reactions between contacts, IEEE Transactions on Magnetics 43(1) (January 2007), 391–396. J.T. Tzeng, Structural mechanics for electromagnetic railguns, IEEE Transactions on Magnetics 41(1) (January 2005), 246–250. J.C. Nearing, Skin and heating effects of railgun current, IEEE Transactions on Magnetics 25(1) (Jan 1989), 381–386. M.P. Galanin, Y.A. Khalimullin, A.P. Lototsky and K.K. Milyayev, 3-D modeling of electromagnetic fields in application to electromagnetic launchers, IEEE Transactions on Magnetics 39(1) (January 2003), 134–138. O. Fitch and M.F. Rose, Limiting factors in the performance of rail guns. 4th IEEE pulsed power conference, Albuquerque, New Mexico, 1983, 75–79. R.A. Marshall and W.F. Weldon, Analysis of performance of railgun accelerators powered by distributed energy stores. 14th IEEE pulsed power modulator symposium, Orlando, Florida, 1980, 318–322. L.D. Huland, Distributed-current-feed and distributed-energy-store railguns, IEEE Transactions on magnetics 20(2) (1984), 272–275. R.A. Marshall, The distributed energy store railgun, its efficiency, and its energy store implications, IEEE Transactions on Magnetics 33(1) (1997), 582–588. J.V. Parker, Electromagnetic projection utilizing distributed energy sources, Journal of Applied Physics 53(10) (1982), 6710–6723. J.V. Parker, SRS railgun: A new approach to restrike control, IEEE Transactions on Magnetics 25(1) (1989), 412–417. Antonino Musolino, The Multistage Rail gun, IEEE Transactions on Magnetics 37(1) (2001), 445–447. D.A. Fikse, J.A. Ciesar, H.A. Wehrli, H. Riemersma, E.F. Docherty and C.W. Pipich, The HART I augmented electric gun facility, IEEE Transactions on Magnetics 27(1) (1991), 176–180. J.H. Beno and W.F. Weldon, Railgun current guard plates: active current management and augmentation, IEEE Transactions on Plasma Science 17(3) (1989), 422–428. J.H. Beno and W.F. Weldon, Investigation into the potential for multiple rail railguns, IEEE Transactions on Magnetics 25(1) (1989), 92–96.

Efficient algorithms for the optimization of shielding

Efficient algorithms for the optimization of shielding

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