Some Applications of Neural Networks in Microwave Modeling

Transcription

1 JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL. 13(1:39-46, 2003 Some Applications o Neural Networks in Microwave Modeling Bratislav Milovanović, Vera Marković, Zlatica Marinković, Zoran Stanković Abstract This 1 paper presents some apprilcations o neural networks in the microwave modeling. The applications are related to modeling o either passive or active structures and devices. Modeling is perormed using not only simple multilayer perceptron network (MLP but also advanced knowledge based neural network (KBNN structures. Keywords Neural network, modeling, microwave, microstrip gap, microwave cavity, microwave transistor, noise parameters, scattering parameters. I. INTRODUCTION Design, imization and simulation o microwave circuits require reliable and eicient models o components and devices. The models based on the rigorous electromagnetic methods are very complex and time-consuming; and oten, there is a requirement or powerul hardware platorm. Sometimes, there is a lack o explicit mathematical expressions that describe the electro-magnetic nature o device. Alternatively, simple models are usually used in standard microwave circuit simulators but they have oten limitations rom the point o validity range and accuracy. In many o these cases, neural modeling could be a good alternative to the classical modeling. As highly nonlinear structures, neural networks are able to model nonlinear relations between dierent data sets, [1]. Owing to this ability, they have been applied in a wide area o problems. Especially, they are interesting or problems not ully mathematically described. Once trained they can predict response with quite a good accuracy, even or the input values not presented in the training process, without changes in their structure and without additional knowledge o considered problem. Neural models are simpler than physically based ones but retain the similar accuracy. They require less time or response providing; thereore using o neural models can make simulation and imization processes less time-consuming, shiting much computation rom on-line imization to o-line training. Since the 1990s, neural network applications in the microwave engineering have been reported, [2]. Neural networks can be applied to the eicient modeling o microwave components, e.g., microstrip vias and interconnects [3], spiral inductors [4] and FET devices [2],[5],[6] as well as to the analysis and design o microwave circuits, e.g., microstrip circuit design, automatic microwave impedance matching and Smith chart oriented Faculty o Electronic Engineering, University o Niš, Beogradska 14, PO Box 73, Niš, Yugoslavia Phone: , Fax: [bata,vera,zlatica,zoran]@elak.ni.ac.yu microwave circuit analysis and design [7]. Neural network modeling has also been applied to circuit imization and statistical design, e.g., signal integrity analysis and imization o very large-scale integrated circuit (VLSI interconnects [8],[9], microwave circuit imization and statistical design with neural network models at either device or circuit levels [10]. Besides, encouraging results have been obtained using neural network modeling in the areas o antenna modeling (neural network based beamorming,[11], neural network implementation o direction o arrival estimation problem,[12] and radar technique (neural network or a radar signal characterization and automatic target recognition, [13]-[15]. During last ive years, the authors o the paper have been worked on neural network applications in the microwave circuits modeling. Neural networks are applied to the resonant cavity applicator modeling, [16]-[24] as well as to microwave MESFET and HEMT transistor modeling, [25]- [33]. Also, the irst steps in microwave discontinuities modeling using neural networks have been made [34]-[36]. This paper addresses to the achieved results. Neural network training, testing and simulation are perormed in MATLAB sotware package environment. Mostly, a multi layer perceptron network (MLP is implemented. Advanced neural structures based on existing knowledge are proposed and applied as well. II. MICROWAVE CAVITY MODELING Microwave applicators are used thermal dielectric material processing in industry, science and medicine. In the most cases these applicators have orm o the cylindrical metallic cavities with various cross-sections (rectangular, circular, elliptical, etc. Spatially application in processes o dielectric material heating has the cavities loaded by planparallel homogeneous dielectric slabs. The knowledge o the mode tuning behavior as well as some other electrical parameters under loading condition has important signiicance. Usually, a theoretical analysis o the cylindrical metallic cavities is based on the application o the transversal resonance method [16]. The resonant requencies are determined rom the characteristic equation, which is real transcendental or lossless dielectric slabs and complex transcendental or any lossy dielectric slab. An approximate numerical technique or resonant requency determination and an eicient procedure or mode identiication (especially in the case o a multilayer lossy load are proposed, [17]. In order to avoid complex and timeconsuming mathematical calculations authors have

2 40 MILOVANOVIĆ, B., ET AL. SOME APPLICATIONS OF NEURAL NETWORKS IN MICROWAVE MODELING suggested an application o neural networks in resonant requency determination, [18]-[24]. In Fig. 1 The cylindrical cavity with circular cross-section, loaded by dielectric slab (with small losses o the thickness t, which is placed at the bottom o the cavity. Its resonant requencies depend on relative dielectric permittivity and t/h ratio. controlled by aa and rr points partly solved this problem [19]. Thereore, improved neural models are proposed. Fig. 3. MLP approach As an illustration, in Fig. 4 a three-dimensional presentation o TM 112 mode behavior obtained using M model (12 neurons in each o two hidden layers is shown. Fig.1. The cylindrical cavity with circular cross-section, loaded by dielectric slab (with small losses o the thickness t, which is placed at the bottom o the cavity k=1 k= aa TM014 rr TM 015 TM TM 013 TM 012 aa rr l=2 2.5 TM l=1 TM 010 l= t/h g. 2. The resonant requencies or TM 01p mod amily In Fig. 2 the resonant requencies or TM 01p mod amily are shown. Solid lines are related to the resonant requencies o the lossy dielectric slab obtained using transverse resonance method [16]. Dashed lines represent resonant curves (monotonous increasing in air part and monotonous decreasing in dielectric part o the cavity and doted lines represent anti-resonant curves (monotonous increasing in air part and monotonous decreasing in dielectric part o the cavity. It can be seen that resonant requency curve passes through the characteristic points rr and aa. rr (aa points are cross points o the resonant (anti-resonant curves in the air and resonant (anti-resonant dielectric part o the cavity. The irst step in microwave cavity resonant requencies modeling was based on use o MLP neural networks, [18]. Used networks have two input neurons corresponding to a relative dielectric permittivity and t/h ratio, and one output neuron corresponding to a cavity resonant requency, Fig. 3. Training data was obtained using transversal resonance method. Resonant requencies determined using the trained models agreed to the ones obtained by transversal resonance method very well. The disadvantage o this method is a large number o training data required or the training process. Using appropriate non-uniorm samples distribution Fi Fig. 4. Three-dimensional presentation o TM 112 mode behavior obtained using M model A. Dierence-based approach Size o the training set could be signiicantly reduced using dierence-based method, Fig 4, [19]-[21]. Here, neural networks were trained to model the dierence between values computed using approximate model [17] and actual values (computed by transversal resonance method. Thereore, neural networks have the same inputs as in MLP approach, and one output neuron corresponding to the mentioned dierence. Fig. 5. Dierence-based approach Ater the training, resonant requency determination consists o approximate value computing and adding to the neural model output or the same input values. For achieving the same model accuracy this approach require signiicantly smaller training set. Fig. 6 presents a comparison o the reerent resonant requencies obtained by transversal resonance method and results obtained using dierence based method with DB neural model or TM 113

3 JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE 41 mode. DB model is trained on the training set o only 174 samples. [GHz] r 4.2 TM Reerent curve DB =3.5 =8.1 =45 = t/h Fig. 6. Comparison o the reerent results and results or DB neural model or TM 113 mode B. Hybrid empirical neural approach Hybrid empirical neural (HEN approach is alternatively way to reducing o size o the training set could be signiicantly reduced, Fig 7, [22]. MLP neural networks are trained to model cavity resonant requencies. This MLP network has one input more than one in classical MLP approach. Namely, resonant requency was computed using existing empirical model and this approximate values were lead to the network inputs. Training values or resonant requency were computed by transversal resonance method. Figure R8 shows three-dimensional presentation o TM 112 mode behavior obtained using H model (12 neurons in the irs and 10 neurons in the second hidden layer, observed rom the input to the output. C. Knowledge-based neural approach The newest results in microwave resonant cavity modeling are addressed to the application o the knowledge based neural network (KBNN structure, [23]-[24]. Since, there are explicate expressions or the resonant and anti resonant curves dependence on and t/h (shown in Fig. 2 [17], the basic idea or KBBN approach is implementation o this expressions as activation unctions o some neurons in the neural network. In Figure 9, proposed KBNN architecture is presented. KBNN structure is a modiied MLP structure. Namely, a network is consists o neurons grouped into the layers. Beside layers o sigmoid neurons, there are so-called knowledge neurons. Activation unctions are modiied existing expressions o the resonant and anti-resonant curves. Application o such structure leads to a increasing o network generalization capabilities yielding to a urther reducing o required number o training samples. Furthermore, this approach eliminates need or the use o empirical model used in HEN approach making resonant requency determination aster. Fig. 6. Hybrid empirical neural approach Fig. 9. Knowledge based neural network In Figure 10 a three-dimensional presentation o TM 112 mode obtained using KBNN model. This model has three sigmoid hidden layers (16 neurons in each layer, and our knowledge neurons. Training set consists o 80 samples. Fig. 7. Three-dimensional presentation o TM 112 mode behavior obtained using H model

4 42 MILOVANOVIĆ, B., ET AL. SOME APPLICATIONS OF NEURAL NETWORKS IN MICROWAVE MODELING ones. Also, it should be noted that ater training o the networks, prediction o S-parameters does not requires network parameter changes. As an illustration, Fig. 12 shows prediction o S 11 parameter or a bias not used or the training. The black dots represent measured (reerence values and solid line with empty circles neural model output. A notation sp_10_10 represent neural network modeling s-parameters with ten neurons in each o two hidden layers. Fig. 10. Three-dimensional presentation o TM 112 mode obtained using KBNN model III. MICROWAVE TRANSISTOR MODELING Microwave transistors are used in many microwave active circuits. Circuit design process requires reliable and eicient models o the microwave transistors. Most o the existing models reer to one bias point, and or any other bias point it is necessary to recalculate elements and/or parameters o the model. Neural network approach to the transistor modeling provides models valid in the whole operating range o biases, [25]-[33]. Microwave characteristics o transistors can be described by transistor scattering and noise parameters [33]. A. Modeling o transistor scattering parameters In [25] neural modeling o HEMT/MESFET transistors scattering parameters are proposed. A standard MLP neural network is used or modeling dependence o S-parameters on biases and requency, Fig 11. Thereore, three neural network input neurons corresponding to: dc drain-to-source voltage, dc drain-to-source current and requency. The network has eight outputs corresponding to the magnitudes and angles o our S-parameters. V d s I d s Fig. 11. S-parameters modeling M a g ( S 1 1 A n g ( S 1 1 M a g ( S 1 2 A n g ( S 1 2 [ S ] M a g ( S 2 1 A n g ( S 2 1 M a g ( S 2 2 A n g ( S 2 2 Training data was taken rom the manuacturer Web site. The obtained neural models have an excellent prediction o the S-parameters not only or the biases used or the training but also or the biases completely dierent rom the training B. Modeling o transistooise Transistooise is described by the ouoise parameters (minimum noise igure, normalized equivalent noise resistance relection coeicient, and magnitude and angle o imum, [33]. The irst steps in neural network approach in transistooise modeling were so-called b -approach (biases & requency, [26]. Noise parameters dependence on biases and requency was modeled using MLP networks with two hidden layers, Fig ,0 S 11 0,5 1, reerence values sp_10_10 output 240 Fig. 12. S-parameters modeling V ds I ds Mag( Ang( Fig. 13. Neural model o noise parameters dependence on bias conditions and requency (b approach Networks have three neural network input neurons corresponding to: dc drain-to-source voltage, dc drain-to-source current and requency. Output layer consists o oueurons corresponding to our noise parameters

5 JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE 43 These b models have a good prediction o noise parameters. But, i any noise parameter (usually have irregular behavior i.e. requency characteristic is not smooth there is degradation in the prediction. Thereore, improved neural models are proposed. The irst solution is transistor scattering parameters adding as neural network inputs (Fig. 14 and such approach is denoted by "sb" (S-parameters, bias and requency, [27]. V ds I ds 8 [S] / Mag( Ang( Fig. 14. Neural model o noise parameters dependence on bias conditions, requency and S-parameters (sb approach parameters is enabled and on-line imization in circuit simulator is shited to o-line training o neural networks. [db] (a reerent data neural model b_10_10 neural model sb_10_10 Minimum noise igure The second way is model decomposition. Namely, a parameter with irregular behavior was modeled separately rom the other parameters, Fig. 15, [28], that yielded by better prediction o all parameters. I prediction o problematic parameter is still not satisying, S-parameters can be added in the input, Fig reerent data neural model b_10_10 neural model sb_10_10 V ds I ds b3 Mag( Ang( brn (b Equivalent noise resistance Fig. 15. Neural model ooise noise parameters: b3-brn approach V ds I ds b3 Mag( Ang( Fig. 16. Comparison o generalization capabilities o the b and sb models [S] / 8 / sbrn Fig. 15. Neural model ooise parameters: b3-sbrn approach As an illustration, comparison o generalization capabilities (noise parameters prediction or biases not used in the training o b, sb and b3-sbrn models is presented in Figures 16 and 17. During the testing process, a conclusion appeared, prediction o noise parameters is not signiicantly degraded when input values or S-parameters are obtained by neural model comparing to the case when the measured S- parameters are at the network input, [29]-[30]. Recently, neural networks have been applied to the wave approach to transistooise modeling, [31]-[32]. Wave approach is based on a noise wave representation o transistor intrinsic circuit. The noise wave temperatures are introduced as empirical parameters o the noise model and their requency dependence is modeled using neural networks. In that way requency extrapolation o noise [db] (a Minimum noise igure reerent values neural model b neural model b3

6 44 MILOVANOVIĆ, B., ET AL. SOME APPLICATIONS OF NEURAL NETWORKS IN MICROWAVE MODELING reerent values neural model b neural model brn neural model sbrn s/w w/h MLP Mag( S 11 Ang( S 11 Mag( S 12 Ang( S Fig. 19. Classical MLP approach In order to improve the modeling, a hybrid empirical approach incorporating the existing empirical knowledge has been proposed, Fig (b Equivalent noise resistance Fig. 17. Comparison o generalization capabilities o the b and b3-sbrn models IV. MICROSTRIP ELEMENT MODELING s/w w/h Empirical Model [S] Parameters Mag( S 11 Ang( S 11 Mag( S 12 Ang( S 12 MLP Mag( S 11 Ang( S 11 Mag( S 12 Ang( S 12 In order to avoid a solving o a number o time-consuming complex electromagnetic equations needed or the analysis and imization o microstrip circuits, a new approach using neural models o microstrip elements has been proposed. The neural modeling o microstrip elements is presented through a microstrip gap scattering parameter modelling, [34]. A microstrip line gap is presented in Fig. 18. Its scattering parameters depend on requency, gap physical dimensions (gap width s and conductor width w as well as on substrate characteristics (relative dielectric permitivity and substrate height h,[35]. Since the microstrip gap is a symmetric structure there is no need to model the all our scattering parameters; it is enough to model S 11 and S 12 parameters. w s h r Fig. 18. Microstrip gap. First, a classical MLP approach has been applied. A MLP structure with our layers (two hidden layers has been used in the way shown in Fig 19. The input layer consists o our neurons corresponding to: s/w - gap width to conductor width ratio, w/h conductor width to substrate height ratio, substrate relative dielectric permittivity, and requency [GHz] and output layer o oueurons corresponding to the magnitude and angle o S 11 as well as S 12 parameter. The networks with dierent number o hidden neurons were trained using the training data obtained by the ADS circuit simulator, [36]. The obtained models were tested and the model M with 14 neurons in each o the two hidden layers gave the best testing results. Hybrid model Fig. 20. Hybrid empirical approach. Actually, a MLP structure is maintained, the output layer is the same but the input layer has got oueurons more than the previous structure corresponding to the magnitude and angle o S 11 and S 12 parameters computed using empirical equations that can be ound in [36]. Thereore, the new network has eight neurons in input layer. The training process was the same as in the previous case. The training set has reerred to the same values or gap physical dimensions, substrate relative dielectric permittivity and requency as the previous one but is extended with the corresponding values computed rom empirical equations. The including the additional inputs makes the training process less time consuming and the generalization accuracy better. The best testing results gave the model with 15 neurons in the irst and 11 neurons in the second hidden layer, observed rom the input to the output layer (H As an illustration, Figure 21 presents the comparison o gap scattering parameters dependence on requency or a speciied substrate and three values or s/w ratio. It should be noted that all these values are not used during the training process. Figure 21a reers to the magnitude o S 11 parameter, and Figure 21b its angle. The triangles denote reerence values, the doted lines - values generated by the M model and the solid lines - values generated by the H model. From these graphs it could be seen that the hybrid model gives better results then the classical one. Also, it should be noted that the hybrid approach requires smalleumber o epochs or achieving the same training goal making the training process aster. V. CONCLUSION Neural networks can be applied in a wide area o problems. This paper presents authors results in neural network applications in the ield o microwave. Neural networks are applied in the modeling o resonant cavity applicators and microwave transistors as well as o microstrip structures.

7 JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE 45 The existing methods or determination o microwave resonant cavity resonant requencies are based on solving o complex electromagnetic equations. There are some approximate methods but they are limited to some special cases. Neural networks provide ast and eicient resonant requency determination avoiding complex numerical calculation. Although the irst proposed approach based on multilayer perceptron networks gives very good determination results, its main disadvantage is a requirement or a large number o training samples. Thereore, there have been proposed some improved approaches such as dierence based, hybrid-empirical, and knowledge-based neural network methods that require smaller training sets. S 11 angle(s 11 [ degrees ] H M Reerent values s/w=0.15 s/w=0.95 s/w= H M Reerent values (a Magnitude o parameter S 11 s/w= s/w= s/w= (b Angle o parameter S 11 Fig. 21. Comparison o generalization capabilities o M and H ( =9,w/h=0.75 Microwave circuit design requires reliable and accurate models o microwave HEMT and MESFET transistors. Usually, the existing models reer to one bias point; thereore there is a need or recalculation o elements and parameters o the model or any other bias point. The proposed neural network approach has yielded models that can be used or the all operating bias conditions without additional changes. I some noise parameters have irregular behavior, improved approaches based on additional networks inputs adding and/or on model decomposition have been proposed. Also, a combination o neural network approach and wave approach to transistooise modeling has been proposed. Recently, the irst encouraged steps o microstrip elements modeling using neural networks have been made. REFERENCES [1] S. Haykin, Neural networks, New York, IEEE, [2] Q. J. Zhang, K. C. Gupta, Neural Networks or RF and Microwave Design, Artech House, 2000 [3] P. 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