219 Computation of Different Parameters of Triangular Patch Microstrip Antennas using a Common Neural Model *Taimoor Khan and Asok De Department of Electronics and Communication Engineering Delhi Technological University (Formerly Delhi College of Engineering) Shahabad Daulatpur, Bawana Road, Delhi-110 042, India Tel: 011-2204846; Fax: 011-22048044; E- mail: ktaimoor@gmail.com Abstract- Artificial neural networks is being used with microstrip antennas since last one decade. Different neural models have been proposed till date for calculating the different parameters like resonant frequencies, physical dimensions etc. of different types of microstrip patches. Most of these have been used for calculating a single parameter (resonant frequency or physical dimensions) of a single microstrip patch whereas few models have also been used for calculating the single parameter of more than one type of patch simultaneously. But no single model has been proposed till date for calculating more than one parameter of the same microstrip patch like resonant frequency and physical dimensions. In this paper authors have proposed a single feed forward neural model with two hidden layers for calculating the resonant frequency and side-length of triangular patch. For calculating these two parameters, the proposed model is trained by four algorithms and three algorithms respectively. The results obtained in the calculation of these two parameters from a common neural model are in conformity with the theoretical results measured by some conventional approaches. Index Term- Feed forward neural networks, microstrip antennas, resonant frequency, side-length, training algorithms, triangular patch. I. INTRODUCTION The key advantages of microstrip antennas include low profile, conformable to planar and nonplanar surfaces, simple and inexpensive, mechanically robust if mounted on a rigid surface, light weight, and easy mountability but one of the main drawbacks of the microstrip antennas is low bandwidth usually ranges from one percent to several percent. Since the microstrip antennas operate only in the vicinity of the resonant frequency, this resonant frequency must be calculated accurately. Similarly, for designing the microstrip antennas, the calculations of side-length of the patch are of prime importance. There are many patches used with the microstrip antennas such as rectangular, square, triangular etc. Here an equilateral triangular patch microstrip antenna (ETMSA) is selected. Several conventional approaches are available in the literature [1-6] for calculating the resonant frequencies of the ETMSA. The resonant frequencies of the ETMSA using artificial neural networks (ANN) were computed [7-9]. S. Sagiroglu and K. Guney [7] proposed a multilayered perceptron ANN model of two hidden layers for calculating the resonant frequencies of the ETMSA and the number of neurons in the first and second hidden layers were taken as five and three, respectively. For training and testing data sets they referred to earlier studies [1-2]. They selected gradient-descent with momentum backpropagation algorithm for training their model. Using this algorithm they calculated the resonant frequencies of the ETMSA with a total absolute error (difference between measured values and the calculated values by neural model) of 23 MHz. In their proposed model, the epochs required were 25000. K. Guney and N. Sarikaya used two different neural approaches; one is based on ANFIS (artificial neural network and fuzzy interference system) method [8] and another based on CNFS (concurrent neuro-fuzzy system) method [9] for calculating the resonant frequencies of rectangular, triangular and circular microstrip antennas, simultaneously. In ANFIS method, they calculated the resonant frequencies of ETMSA with an absolute error of 27 MHz whereas in CNFS method, the minimum mentioned error was 2 MHz for the same training and testing data sets.
220 The lastly mentioned two methods (ANFIS method and CNFS method) were used for calculating only a single parameter i.e. resonant frequency of different types of patches, simultaneously. For calculating the side-lengths of the same ETMSA, R. Gopalkrishan and N. Gunasekaran [10] also used a multilayered perceptron ANN model of the similar structure as proposed by S. Sagiroglu and K. Guney [7]. They mentioned an absolute error of 0.0213 cm in calculating the side-lengths for three randomly selected data sets. They selected these three data sets during the testing of the neural model whereas in their proposed model the epochs required were 1, 00,000. S. Sagiroglu and K. Guney [7] calculated the resonant frequency for the given values of sidelength, L, thickness of the dielectric material, h, relative permittivity, ε r and mode of propagation, m and n whereas R. Gopalkrishan and N. Gunasekaran [10] calculated the side-length for the given values of resonant frequency f, thickness of the dielectric material, h, relative permittivity, ε r and mode of propagation, m and n. In both the approaches, total given parameters were five and the sixth parameter was calculated for the given values of five other parameters. No single model has been proposed till date for calculating more than one parameters of the same ETMSA. This paper suggests a common feed forward neural model based on multilayered perceptron ANN for calculating the two parameters (resonant frequency and side-length) of the same ETMSA. For calculating the resonant frequencies, the proposed model is trained by four algorithms whereas for the calculation of side-lengths, the same model is trained by three algorithms. Finally, the calculated resonant frequencies and the side-lengths from the common model are also compared with their corresponding theoretical counterparts. The training and testing data sets in the proposed model are similar to [7] and [10], respectively. II. GEOMETRY OF EQUILATERAL TRIANGULAR PATCH MICROSTRIP ANTENNA A microstrip antenna, in its simplest configuration, consists of a radiating conductive patch on one side of a dielectric substrate having a ground plane on the other side. The triangular patch microstrip antenna is one for which the patch conductor has a triangle shape. An equilateral triangular patch of sidelength, L, is mounted on a substrate of relative permittivity, ε r, and at the thickness, h, from the ground plane as shown in Fig. 1. Fig. 1: Geometry of equilateral triangular patch microstrip antenna The resonant frequency of the ETMSA is given [5] as: f = 3L 2c r, 1 2 2 [ m + mn n ] 2 mn + ε (1) where c is the velocity of electromagnetic waves in free space, L is the ective side-length of the patch, ε r, is the ective relative permittivity, and the integers (m and n) represent the mode of propagation. The ective side-length suggested by Helszain and James is given [5] as: h L = L + (2) ε r and the ective relative permitivity suggested by Bahl and Bhartia is given [3] as: r, 1 1 12h = ( εr + 1) + ( εr 1) 1 + 2 4 L 1 2 ε (3) It is clear from these three equations that the resonant frequency, f of the ETMSA can be calculated for the given values of length of the
221 patch, L, height of the dielectric material, h, relative permitivity, ε r and integers, m and n. The resonant frequencies for different combination of given parameters are illustrated in Table 1. Similarly, the side-length, L of the ETMSA can also be calculated for the given values of resonant frequency, f, height of the dielectric material, h, relative permitivity, ε r and integers, m and n. The data sets mentioned in Table 1 are used during training and testing of the proposed model for calculating the resonant frequencies whereas for side-lengths calculation the data sets are generated by inter changing the first and last column of Table 1. III. PROPOSED NEURAL MODEL AND ALGORITHMS A common neural model used for calculating the resonant frequencies and the side-lengths of the same ETMSA is shown in Fig. 2. I/Ps represent input parameter and O/P represents output parameter. Fig. 2: Neural model for resonant frequency/sidelength calculation After many trials it is found that the model configuration, 5-5-6-1, is suitable for calculating these two parameters of the same triangle patch. It means five neurons in first hidden layer and six neurons in the second hidden layer are required for calculating these two parameters for the proposed model. Logsigmoidal and Tansigmoidal are used as activation functions for these two hidden layers. The activation function in the output layer is pure linear whereas for the input layer no activation function is used. For calculating the resonant frequency, f, the parameters for the input layer are; L, h, ε r, m and n whereas for side-length, L calculations these parameters are; f, h, ε r, m and n. Total four algorithms based on scaled conjugate gradient (SCG) backpropagation, Bayesian regulation (BR) backpropagation, resilient (RP) backpropagation, and one step secant (OSS) backpropagation, are adopted for calculating the resonant frequencies whereas for the calculation of side-lengths, the proposed model is trained by conjugate gradient backpropagation with Powell- Beale restarts (CGB), Bayesian regulation (BR) backpropagation and scaled conjugate gradient (SCG) backpropagation algorithms. The initial weight matrices selected randomly and rounded off between -0.02 and +0.01 whereas the initial biases are also selected randomly and rounded off between -0.01 and +0.03. IV. RESULTS AND CONCLUSIONS The resonant frequencies calculated by the proposed neural model are given in Table 2. This table shows that the resonant frequencies obtained from the neural model are closer to their corresponding theoretical resonant frequencies. A comparison of the resonant frequencies evaluated by the present method and the previous neural methods is given in Table 3. Table 3 shows that if the proposed model is trained with any of the above mentioned four algorithms then it shows lesser error for both training and testing data sets. The error is 0.5 MHz, 3.5 MHz, 4.6 MHz, and 7.6 MHz in case of scaled conjugate gradient (SCG) backpropagation, Bayesian regulation (BR) backpropagation, resilient (RP) backpropagation and one step secant (OSS) backpropagation algorithms, respectively. In previous works [7] and [8], the same error was calculated as 23 MHz and 27 MHz, respectively whereas in the model [9], the calculated error i.e. 2 MHz is also encouraging but the model [9] was used for calculating only one parameter i.e. resonant frequency. Similarly, the side-lengths calculated by the proposed neural model are given in Table 4 which shows that the calculated side-lengths from the same neural model are also closer to the corresponding theoretical side-lengths. A comparison of the sidelengths evaluated by the proposed model with their corresponding theoretical values is given in Table 5 for 15 data sets (Training + Testing). If the model proposed in the literature is trained with any of the above mentioned three algorithms, it also shows lesser error for both training and testing data sets.
222 The error is 0.0052 cm, 0.0124 cm, and 0.0161 cm when the model is trained by conjugate gradient backpropagation with Powell-Beale restarts (CGB), Bayesian regulation (BR) backpropagation, and scaled conjugate gradient (SCG) backpropagation, respectively. Table 6 shows a comparison of sidelengths for testing data sets only. It is clear from this table that the error in present method is only 0.0014 cm, 0.0040 cm, and 0.0096 cm for these three algorithms, respectively whereas in previous work [10] the same error was reported as 0.0213 cm. The proposed model is used for calculating two parameters of the same triangular patch whereas the previous neural models [7] and [10] were used for calculating a single parameter of the triangular patch as shown in Table 7. The total absolute deviations between the theoretical results and the results calculated by the proposed neural approach are more encouraging. Thus the proposed neural model is proved to be better than the previous models [7] and [10]. REFERENCES [1] J.S.Dahele, and K.F. Lee, On the resonant frequencies of the triangular patch antenna, IEEE Trans. on Anten. and Propag., Vol. AP-35, 100-101, 1987. [2] W. Chen, K. F. Lee, and J.S. Dahele, Theoretical and experimental studies of the resonant frequencies of the equilateral triangular microstrip antenna, IEEE Trans. on Anten. and Propag., Vol. 40, 1253-1256, 1992. [3] I. J., Bahl, and P. Bhartia, Microstrip Antennas, Artech House, Dedham, MA,1980. [4] R. Garg, and S. A. Long, An improved formula for the resonant frequency of the triangular microstrip patch antennas, IEEE Trans. on Anten. and Propag., Vol.36, 570, 1998. [5] J. Helszajn, and D. S. James, Planer triangular resonators with magnetic walls, IEEE Trans. on Microwave Theory and Tech., Vol. 26, 95-100, 1978. [6] D. Guha, and J.Y. Siddiqui, Resonant frequency of equilateral triangular microstrip antenna with and without air gap, IEEE Trans. on Anten. and Propag., Vol. 52, 2174-2177, 2004. [7] S. Sagiroglu, and K. Guney, Calculation of resonant frequency for an equilateral triangular microstrip antenna with the use of artificial neural networks, Microwave Opt. Technol. Lett., Vol.14, 89-93, 1997. [8] K. Guney, and N. Sarikaya, A hybrid method based on combining artificial neural network and fuzzy interference system for simultaneous computation of resonant frequencies of rectangular, circular, and triangular microstrip antennas, IEEE Trans. on Anten. and Propag. Vol. 55, 559-568, 2007. [9] K. Guney, and N. Sarikaya, Concurrent neuro-fuzzy systems for resonant frequency computation of rectangular, circular and triangular microstrip antennas, Progress In Electromagnetics Research, PIER 84, 253-277, 2008. [10] R. Gopalkrishanan, and N. Gunasekaran, Design of equilateral triangular microstrip antenna using artificial neural networks, IEEE International Workshop on Antenna Technology, IWAT, 246-249, 2005. [11] M. T. Hagan, and M. Menhaj, Training feed forward networks with the Marquardt algorithms, IEEE Trans. on Neural Networks, Vol. 5, 989-993, 1994. [12] H. Demuth, and M. Beale, Neural Network Tool Box for use with MATLAB, User s Guide,5 th ed., The Mathworks,Inc.,1998. [13] K. Levenberg, A method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math, Vol.2, 164 168, 1994. [14] D.W. Marquardt, An algorithm for least squares estimation of nonlinear parameters, SIAM J., Vol. 11, 431 441, 1963.
223 Table 1: Measured resonant frequencies [1-2] L (cm) h (cm) ε r m n f 4.1000 0.0700 10.5000 1 0 1519.0000* 4.1000 0.0700 10.5000 1 1 2637.0000 4.1000 0.0700 10.5000 2 0 2995.0000 4.1000 0.0700 10.5000 2 1 3973.0000 4.1000 0.0700 10.5000 3 0 4439.0000 8.7000 0.0780 2.3200 1 0 1489.0000* 8.7000 0.0780 2.3200 1 1 2596.0000 8.7000 0.0780 2.3200 2 0 2969.0000 8.7000 0.0780 2.3200 2 1 3968.0000 8.7000 0.0780 2.3200 3 0 4443.0000 10.0000 0.1590 2.3200 1 0 1280.0000 10.0000 0.1590 2.3200 1 1 2242.0000 10.0000 0.1590 2.3200 2 0 2550.0000 10.0000 0.1590 2.3200 2 1 3400.0000 10.0000 0.1590 2.3200 3 0 3824.0000* First ten resonant frequencies measured by Chen et al [2] and remaining by Dahele and Lee [1] and * represents testing data sets. All frequencies are in MHz. Table 2: Measured and calculated resonant frequencies L (cm) h (cm) ε r m n f f scg f br f rp f oss 4.1000 0.0700 10.5000 1 0 1519.0000 1518.8000 1519.40000 1520.1000 1521.2000 4.1000 0.0700 10.5000 1 1 2637.0000 2637.0000 2637.30000 2636.8000 2637.0000 4.1000 0.0700 10.5000 2 0 2995.0000 2995.0000 2995.30000 2995.3000 2995.0000 4.1000 0.0700 10.5000 2 1 3973.0000 3973.0000 3973.20000 3972.6000 3973.0000 4.1000 0.0700 10.5000 3 0 4439.0000 4439.0000 4439.20000 4439.6000 4439.0000 8.7000 0.0780 2.3200 1 0 1489.0000 1489.0000 1489.30000 1488.8000 1488.5000 8.7000 0.0780 2.3200 1 1 2596.0000 2596.0000 2596.20000 2596.2000 2596.0000 8.7000 0.0780 2.3200 2 0 2969.0000 2969.0000 2969.20000 2968.7000 2969.0000 8.7000 0.0780 2.3200 2 1 3968.0000 3968.0000 3968.20000 3968.5000 3968.0000 8.7000 0.0780 2.3200 3 0 4443.0000 4443.0000 4443.20000 4442.4000 4443.0000 10.0000 0.1590 2.3200 1 0 1280.0000 1280.0000 1280.30000 1280.0000 1280.0000 10.0000 0.1590 2.3200 1 1 2242.0000 2242.0000 2242.20000 2242.1000 2242.0000 10.0000 0.1590 2.3200 2 0 2550.0000 2550.0000 2550.20000 2550.0000 2550.0000 10.0000 0.1590 2.3200 2 1 3400.0000 3400.0000 3400.10000 3400.0000 3400.0000 10.0000 0.1590 2.3200 3 0 3824.0000 3824.3000 3824.20000 3824.1000 3828.9000 f represent the measured resonant frequencies [1-2], f scg, f br, f rp, and f oss represent the calculated resonant frequencies when the model is trained by SCG, BR, RP and OSS backpropagation algorithms respectively. All frequencies are in MHz. Table 3: Comparison of resonant frequencies calculated by present method and previous neural method [7] f f scg f br f rp f oss f k f~f scg f~f br f~f rp f~f oss f~f k 1519.0000 1518.8000 1519.40000 1520.1000 1521.2000 1526.0000 0.2000 0.4000 1.1000 2.2000 7.0000 2637.0000 2637.0000 2637.30000 2636.8000 2637.0000 2637.0000 0.0000 0.3000 0.2000 0.0000 0.0000 2995.0000 2995.0000 2995.30000 2995.3000 2995.0000 2995.0000 0.0000 0.3000 0.3000 0.0000 0.0000 3973.0000 3973.0000 3973.20000 3972.6000 3973.0000 3973.0000 0.0000 0.2000 0.4000 0.0000 0.0000 4439.0000 4439.0000 4439.20000 4439.6000 4439.0000 4439.0000 0.0000 0.2000 0.6000 0.0000 0.0000 1489.0000 1489.0000 1489.30000 1488.8000 1488.5000 1478.0000 0.0000 0.3000 0.2000 0.5000 11.0000 2596.0000 2596.0000 2596.20000 2596.2000 2596.0000 2596.0000 0.0000 0.2000 0.2000 0.0000 0.0000 2969.0000 2969.0000 2969.20000 2968.7000 2969.0000 2969.0000 0.0000 0.2000 0.3000 0.0000 0.0000 3968.0000 3968.0000 3968.20000 3968.5000 3968.0000 3968.0000 0.0000 0.2000 0.5000 0.0000 0.0000 4443.0000 4443.0000 4443.20000 4442.4000 4443.0000 4443.0000 0.0000 0.2000 0.6000 0.0000 0.0000 1280.0000 1280.0000 1280.30000 1280.0000 1280.0000 1280.0000 0.0000 0.3000 0.0000 0.0000 0.0000 2242.0000 2242.0000 2242.20000 2242.1000 2242.0000 2242.0000 0.0000 0.2000 0.1000 0.0000 0.0000 2550.0000 2550.0000 2550.20000 2550.0000 2550.0000 2550.0000 0.0000 0.2000 0.0000 0.0000 0.0000 3400.0000 3400.0000 3400.10000 3400.0000 3400.0000 3400.0000 0.0000 0.1000 0.0000 0.0000 0.0000 3824.0000 3824.3000 3824.20000 3824.1000 3828.9000 3829.0000 0.3000 0.2000 0.1000 4.9000 5.0000 Total Absolute Deviations (MHz) 0.5000 3.5000 4.6000 7.6000 23.0000 f represent the measured resonant frequencies, f scg, f br, f rp, and f oss represent the calculated resonant frequencies by present method and f k represents previous neural results [7].
224 INTERNATIONAL JOURNAL OF MICROWAVE AND OPTICAL TECHNOLOGY, Table 4: Measured and calculated side-lengths f (MHz) h (cm) ε r m n L L cgb L br L scg 1519.0000 0.0700 10.5000 1 0 4.1000# 4.1000 4.1001 4.1016 2637.0000 0.0700 10.5000 1 1 4.1000 4.1009 4.1000 4.0997 2995.0000 0.0700 10.5000 2 0 4.1000 4.1004 4.1000 4.1000 3973.0000 0.0700 10.5000 2 1 4.1000 4.1005 4.1000 4.1001 4439.0000 0.0700 10.5000 3 0 4.1000 4.0983 4.1000 4.1002 1489.0000 0.0780 2.3200 1 0 8.7000# 8.7007 8.7083 8.6955 2596.0000 0.0780 2.3200 1 1 8.7000 8.7001 8.7000 8.7000 2969.0000 0.0780 2.3200 2 0 8.7000 8.6999 8.7000 8.7000 3968.0000 0.0780 2.3200 2 1 8.7000 8.7000 8.7000 8.7000 4443.0000 0.0780 2.3200 3 0 8.7000 8.7000 8.7000 8.7000 1280.0000 0.1590 2.3200 1 0 10.0000 10.0000 10.0000 10.0000 2242.0000 0.1590 2.3200 1 1 10.0000 10.0002 10.0000 10.0001 2550.0000 0.1590 2.3200 2 0 10.0000 9.9999 10.0000 10.0000 3400.0000 0.1590 2.3200 2 1 10.0000 9.9999 10.0000 10.0000 3824.0000 0.1590 2.3200 3 0 10.0000# 9.9996 9.9960 10.0093 L represent the measured side-lengths [1-2], L cgb, L br, and L scg represent the calculated side-lengths by present method when the model is trained by CGB, BR and SCG backpropagation algorithms respectively and # represent the testing results. All side-lengths are in cm. Table 5: Comparison of measured and calculated side-lengths for 15 data sets (Training + Testing) f (MHz) h (cm) ε r m n L L cgb L br L scg L~L cgb L~L br L~L scg 1519.0000 0.0700 10.5000 1 0 4.1000 4.1000 4.1001 4.1016 0.0000 0.0001 0.0016 2637.0000 0.0700 10.5000 1 1 4.1000 4.1009 4.1000 4.0997 0.0009 0.0000 0.0003 2995.0000 0.0700 10.5000 2 0 4.1000 4.1004 4.1000 4.1000 0.0004 0.0000 0.0000 3973.0000 0.0700 10.5000 2 1 4.1000 4.1005 4.1000 4.1001 0.0005 0.0000 0.0001 4439.0000 0.0700 10.5000 3 0 4.1000 4.0983 4.1000 4.1002 0.0017 0.0000 0.0002 1489.0000 0.0780 2.3200 1 0 8.7000 8.7007 8.7083 8.6955 0.0007 0.0083 0.0045 2596.0000 0.0780 2.3200 1 1 8.7000 8.7001 8.7000 8.7000 0.0001 0.0000 0.0000 2969.0000 0.0780 2.3200 2 0 8.7000 8.6999 8.7000 8.7000 0.0001 0.0000 0.0000 3968.0000 0.0780 2.3200 2 1 8.7000 8.7000 8.7000 8.7000 0.0000 0.0000 0.0000 4443.0000 0.0780 2.3200 3 0 8.7000 8.7000 8.7000 8.7000 0.0000 0.0000 0.0000 1280.0000 0.1590 2.3200 1 0 10.0000 10.0000 10.0000 10.0000 0.0000 0.0000 0.0000 2242.0000 0.1590 2.3200 1 1 10.0000 10.0002 10.0000 10.0001 0.0002 0.0000 0.0001 2550.0000 0.1590 2.3200 2 0 10.0000 9.9999 10.0000 10.0000 0.0001 0.0000 0.0000 3400.0000 0.1590 2.3200 2 1 10.0000 9.9999 10.0000 10.0000 0.0001 0.0000 0.0000 3824.0000 0.1590 2.3200 3 0 10.0000 9.9996 9.9960 10.0093 0.0004 0.0040 0.0093 All side-lengths are in cm. Total Absolute Deviations (cm) 0.0052 0.0124 0.0161 Table 6: Comparison of measured and calculated side-lengths for 3 data sets (testing data sets) f (MHz) h (cm) ε r m n L L g L~L cgb L~L br L~L scg L~L g 1519.0000 0.0700 10.5000 1 0 4.1000 4.1013 0.0009 0.0000 0.0003 0.0013 1489.0000 0.0780 2.3200 1 0 8.7000 8.7100 0.0001 0.0000 0.0000 0.0100 3824.0000 0.1590 2.3200 3 0 10.0000 9.9900 0.0004 0.0040 0.0093 0.0100 Total Absolute Deviations (cm) 0.0014 0.0040 0.0096 0.0213 Table 7: Comparison of model configuration and number of calculated parameters Method Model Configuration Calculated Parameters Present Method 5-5-6-1 2 (Resonant Frequency + Side-length) Previous Method [7] 5-5-3-1 1 (Resonant Frequency) Previous Method [10] 5-5-3-1 1 (Side-length )