Some applications of neural networks in microwave modeling

JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL. 13(1):39 -46, 2003 ©

Some Applications of Neural Networks in Microwave Modeling

Bratislav Milovanović, Vera Marković, Zlatica Marinković, Zoran Stanković

Abstract –This1 paper presents some apprilcations of neural

networks in the microwave modeling. The applications are related to modeling of either passive or active structures and devices. Modeling is performed 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, optimization and simulation of microwave circuits require reliable and efficient models of components and devices. The models based on the rigorous electromagnetic methods are very complex and time-consuming; and often, there is a requirement for powerful hardware platform. Sometimes, there is a lack of explicit mathematical expressions that describe the electro-magnetic nature of device. Alternatively, simple models are usually used in standard microwave circuit simulators but they have often limitations from the point of validity range and accuracy.

In many of 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 different data sets, [1]. Owing to this ability, they have been applied in a wide area of problems. Especially, they are interesting for problems not fully mathematically described. Once trained they can predict response with quite a good accuracy, even for the input values not presented in the training process, without changes in their structure and without additional knowledge of considered problem. Neural models are simpler than physically based ones but retain the similar accuracy. They require less time for response providing; therefore using of neural models can make simulation and optimization processes less time-consuming, shifting much computation from on-line optimization to off-line training.

Since the 1990s, neural network applications in the microwave engineering have been reported, [2]. Neural networks can be applied to the efficient modeling of 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 of microwave circuits, e.g., microstrip circuit design, automatic microwave impedance matching and Smith chart oriented

Faculty of Electronic Engineering, University of Niš, Beogradska 14, PO Box 73, 18000 Niš, Yugoslavia Phone: +381 18 529 303, Fax: +381 18 524 931 E-Mail: [bata,vera,zlatica,zoran]@elfak.ni.ac.yu

microwave circuit analysis and design [7]. Neural network modeling has also been applied to circuit optimization and statistical design, e.g., signal integrity analysis and optimization of very large-scale integrated circuit (VLSI) interconnects [8],[9], microwave circuit optimization 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 of antenna modeling (neural network based beam-forming,[11], neural network implementation of direction of arrival estimation problem,[12]) and radar technique (neural network for a radar signal characterization and automatic target recognition, [13]-[15]).

During last five years, the authors of 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 first 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 performed in MATLAB software 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 form of the cylindrical metallic cavities with various cross-sections (rectangular, circular, elliptical, etc). Spatially application in processes of dielectric material heating has the cavities loaded by plan-parallel homogeneous dielectric slabs. The knowledge of the mode tuning behavior as well as some other electrical parameters under loading condition has important significance.

Usually, a theoretical analysis of the cylindrical metallic cavities is based on the application of the transversal resonance method [16]. The resonant frequencies are determined from the characteristic equation, which is real transcendental for lossless dielectric slabs and complex transcendental for any lossy dielectric slab. An approximate numerical technique for resonant frequency determination and an efficient procedure for mode identification (especially in the case of a multilayer lossy load) are proposed, [17]. In order to avoid complex and time-consuming mathematical calculations authors have

40 MILOVANOVIĆ, B., ET AL. SOME APPLICATIONS OF NEURAL NETWORKS IN MICROWAVE MODELING

suggested an application of neural networks in resonant frequency determination, [18]-[24].

In Fig. 1 The cylindrical cavity with circular cross-section, loaded by dielectric slab (with small losses) of the thickness t, which is placed at the bottom of the cavity. Its resonant frequencies depend on relative dielectric permittivity and t/h ratio.

Fig.1. The cylindrical cavity with circular cross-section, loaded by dielectric slab (with small losses) of the thickness t, which is placed at the bottom of the cavity

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.2001.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

l

l

l

=2

=1

=0

k=2k=1

TM016

rr

rr

aa

aaTM

015TM014

TM013

TM012

TM011

TM010

f [G

Hz]

t/h Fig. 2. The resonant frequencies for TM01p mod family

In Fig. 2 the resonant frequencies for TM01p mod family are shown. Solid lines are related to the resonant frequencies of 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 of the cavity) and doted lines represent anti-resonant curves (monotonous increasing in air part and monotonous decreasing in dielectric part of the cavity). It can be seen that resonant frequency curve passes through the characteristic points rr and aa. rr (aa) points are cross points of the resonant (anti-resonant) curves in the air and resonant (anti-resonant) dielectric part of the cavity.

The first step in microwave cavity resonant frequencies modeling was based on use of 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 frequency, Fig. 3. Training data was obtained using transversal resonance method. Resonant frequencies determined using the trained models agreed to the ones obtained by transversal resonance method very well. The disadvantage of this method is a large number of training data required for the training process. Using appropriate non-uniform samples distribution

controlled by aa and rr points partly solved this problem [19]. Therefore, improved neural models are proposed.

Fig. 3. MLP approach

As an illustration, in Fig. 4 a three-dimensional presentation of TM112 mode behavior obtained using M4-12-12 model (12 neurons in each of two hidden layers) is shown.

Fig. 4. Three-dimensional presentation of TM112 mode behavior obtained using M4-12-12 model

A. Difference-based approach

Size of the training set could be significantly reduced using difference-based method, Fig 4, [19]-[21]. Here, neural networks were trained to model the difference between values computed using approximate model [17] and actual values (computed by transversal resonance method). Therefore, neural networks have the same inputs as in MLP approach, and one output neuron corresponding to the mentioned difference.

Fig. 5. Difference-based approach

After the training, resonant frequency determination consists of approximate value computing and adding to the neural model output for the same input values. For achieving the same model accuracy this approach require significantly smaller training set. Fig. 6 presents a comparison of the referent resonant frequencies obtained by transversal resonance method and results obtained using difference based method with DB4-12-12 neural model for TM113

JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE 41

mode. DB4-12-12 model is trained on the training set of only 174 samples.

0.00 0.02 0.04 0.06 0.08 0.102.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 TM 113

Referent curve DB4-12-12

εr=80

εr=45

εr=8.1

εr=3.5

t/h

[GHz]

fr

Fig. 6. Comparison of the referent results and results for DB4-12-12 neural model for TM113 mode

B. Hybrid empirical neural approach

Hybrid empirical neural (HEN) approach is alternatively way to reducing of size of the training set could be significantly reduced, Fig 7, [22]. MLP neural networks are trained to model cavity resonant frequencies.

This MLP network has one input more than one in classical MLP approach. Namely, resonant frequency was computed using existing empirical model and this approximate values were lead to the network inputs. Training values for resonant frequency were computed by transversal resonance method. Figure R8 shows three-dimensional presentation of TM112 mode behavior obtained using H4-12-10 model (12 neurons in the firs and 10 neurons in the second hidden layer , observed from the input to the output).

Fig. 6. Hybrid empirical neural approach

Fig. 7. Three-dimensional presentation of TM112 mode behavior obtained using H4-12-10 model

C. Knowledge-based neural approach

The newest results in microwave resonant cavity modeling are addressed to the application of the knowledge based neural network (KBNN) structure, [23]-[24]. Since, there are explicate expressions for the resonant and anti resonant curves dependence on rε and t/h (shown in Fig. 2) [17], the basic idea for KBBN approach is implementation of this expressions as activation functions of some neurons in the neural network. In Figure 9, proposed KBNN architecture is presented.

KBNN structure is a modified MLP structure. Namely, a network is consists of neurons grouped into the layers. Beside layers of sigmoid neurons, there are so-called knowledge neurons. Activation functions are modified existing expressions of the resonant and anti-resonant curves.

Application of such structure leads to a increasing of network generalization capabilities yielding to a further reducing of required number of training samples. Furthermore, this approach eliminates need for the use of empirical model used in HEN approach making resonant frequency determination faster.

Fig. 9. Knowledge based neural network

In Figure 10 a three-dimensional presentation of TM112 mode obtained using KBNN3-4-16-16-16 model. This model has three sigmoid hidden layers (16 neurons in each layer), and four knowledge neurons. Training set consists of 80 samples.

42 MILOVANOVIĆ, B., ET AL. SOME APPLICATIONS OF NEURAL NETWORKS IN MICROWAVE MODELING

Fig. 10. Three-dimensional presentation of TM112 mode obtained using KBNN3-4-16-16-16 model

III. MICROWAVE TRANSISTOR MODELING

Microwave transistors are used in many microwave active circuits. Circuit design process requires reliable and efficient models of the microwave transistors. Most of the existing models refer to one bias point, and for any other bias point it is necessary to recalculate elements and/or parameters of the model. Neural network approach to the transistor modeling provides models valid in the whole operating range of biases, [25]-[33]. Microwave characteristics of transistors can be described by transistor scattering and noise parameters [33].

A. Modeling of transistor scattering parameters

In [25] neural modeling of HEMT/MESFET transistors’ scattering parameters are proposed. A standard MLP neural network is used for modeling dependence of S-parameters on biases and frequency, Fig 11. Therefore, three neural network input neurons corresponding to:

• dc drain-to-source voltage, • dc drain-to-source current and • frequency.

The network has eight outputs corresponding to the magnitudes and angles of four S-parameters.

NNet ) A n g ( 1 2 S ) M a g ( 1 2 S ) A n g ( 1 1 S ) M a g ( 1 1 S

) A n g ( 2 1 S ) M a g ( 2 1 S

) A n g ( 2 2 S ) M a g ( 2 2 S

d s V d s I f

] [ S

Fig. 11. S-parameters modeling

Training data was taken from the manufacturer Web site. The obtained neural models have an excellent prediction of the S-parameters not only for the biases used for the training but also for the biases completely different from the training

ones. Also, it should be noted that after training of the networks, prediction of S-parameters does not requires network parameter changes.

As an illustration, Fig. 12 shows prediction of parameter for a bias not used for the training. The black dots represent measured (reference 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 of two hidden layers.

11S

B. Modeling of transistor noise

Transistor noise is described by the four noise parameters (minimum noise figure , normalized equivalent noise resistance , and magnitude and angle of optimum reflection coefficient

minF

nr

optΓ ), [33].

The first steps in neural network approach in transistor noise modeling were so-called “bf”-approach (biases & frequency), [26]. Noise parameters dependence on biases and frequency was modeled using MLP networks with two hidden layers, Fig. 13.

0,5

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90

120

150

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210

240

270

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330

0,5

1,0

S11

reference valuessp_10_10 output

Fig. 12. S-parameters modeling

dsV

dsI

f

minF)(Mag optΓ

)(Ang optΓ

nr

NNet

Fig. 13. Neural model of noise parameters’ dependence on bias conditions and frequency (bf approach)

Networks have three neural network input neurons corresponding to:

• dc drain-to-source voltage, • dc drain-to-source current and • frequency.

Output layer consists of four neurons corresponding to four noise parameters.

JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE 43

These bf models have a good prediction of noise parameters. But, if any noise parameter (usually ) have irregular behavior i.e. frequency characteristic is not smooth there is degradation in the prediction. Therefore, improved neural models are proposed.

nr

The first solution is transistor scattering parameters adding as neural network inputs (Fig. 14) and such approach is denoted by "sbf" (S-parameters, bias and frequency), [27].

dsV

dsI

f

minF)(Mag optΓ

)(Ang optΓ

nr

NNet][S /8

Fig. 14. Neural model of noise parameters’ dependence on bias conditions, frequency and S-parameters (sbf approach)

The second way is model decomposition. Namely, a parameter with irregular behavior was modeled separately from the other parameters, Fig. 15, [28], that yielded by better prediction of all parameters. If prediction of problematic parameter is still not satisfying, S-parameters can be added in the input, Fig. 16.

nrNNetbfrn

dsV

dsI

f

minF)(Mag optΓ

)(Ang optΓ

NNetbf3

Fig. 15. Neural model for noise noise parameters: bf3-bfrn approach

nrNNetsbfrn

][S /8/

8

dsV

dsI

f

minF)(Mag optΓ

)(Ang optΓ

NNetbf3

/8/

8

Fig. 15. Neural model for noise parameters: bf3-sbfrn approach

As an illustration, comparison of generalization capabilities (noise parameters’ prediction for biases not used in the training) of bf, sbf and bf3-sbfrn models is presented in Figures 16 and 17. During the testing process, a conclusion appeared, prediction of noise parameters is not significantly degraded when input values for 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 transistor noise modeling, [31]-[32]. Wave approach is based on a noise wave representation of transistor intrinsic circuit. The noise wave temperatures are introduced as empirical parameters of the noise model and their frequency dependence is modeled using neural networks. In that way frequency extrapolation of noise

parameters is enabled and on-line optimization in circuit simulator is shifted to off-line training of neural networks.

0 1 2 3 4 5 6 7 8 9 10 11 120.0

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F min [d

B]

f [GHz]

referent data neural model bf_10_10 neural model sbf_10_10

(a) Minimum noise figure

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r n

f [GHz]

referent data neural model bf_10_10 neural model sbf_10_10

(b) Equivalent noise resistance

Fig. 16. Comparison of generalization capabilities of the bf and sbf models

0 1 2 3 4 5 6 7 8 9 10 10.0

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F min [d

B]

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referent values neural model bf neural model bf3

(a) Minimum noise figure

44 MILOVANOVIĆ, B., ET AL. SOME APPLICATIONS OF NEURAL NETWORKS IN MICROWAVE MODELING

0 1 2 3 4 5 6 7 8 9 10 10.00

0.05

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0.15

0.20

0.25

0.30

1

r n

f [GHz]

referent values neural model bf neural model bfrn neural model sbfrn

(b) Equivalent noise resistance

Fig. 17. Comparison of generalization capabilities of the bf and bf3-sbfrn models

IV. MICROSTRIP ELEMENT MODELING

In order to avoid a solving of a number of time-consuming complex electromagnetic equations needed for the analysis and optimization of microstrip circuits, a new approach using neural models of microstrip elements has been proposed. The neural modeling of 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 frequency, gap physical dimensions (gap width s and conductor width w) as well as on substrate characteristics (relative dielectric permitivity εr and substrate height h),[35]. Since the microstrip gap is a symmetric structure there is no need to model the all four scattering parameters; it is enough to model S11 and S12 parameters.

s

h

w

r

Fig. 18. Microstrip gap.

First, a classical MLP approach has been applied. A MLP structure with four layers (two hidden layers) has been used in the way shown in Fig 19. The input layer consists of four neurons corresponding to: • s/w - gap width to conductor width ratio, • w/h – conductor width to substrate height ratio, • εr – substrate relative dielectric permittivity, and • f – frequency [GHz]

and output layer of four neurons corresponding to the magnitude and angle of S11 as well as S12 parameter. The networks with different number of hidden neurons were trained using the training data obtained by the ADS circuit simulator, [36]. The obtained models were tested and the model M4-14-14 with 14 neurons in each of the two hidden layers gave the best testing results.

MLP)Ang( 12S)Mag( 12S

)Ang( 11S)Mag( 11Ss/w

w/h

fεr

Fig. 19. Classical MLP approach

In order to improve the modeling, a hybrid empirical approach incorporating the existing empirical knowledge has been proposed, Fig. 20.

Empirical Model

MLP

][SParameters )Ang( 12S

)Mag( 12S)Ang( 11S)Mag( 11S

s/ww/h

f

)Ang( 12S)Mag( 12S

)Ang( 11S)Mag( 11S

Hybrid model

εr

Fig. 20. Hybrid empirical approach.

Actually, a MLP structure is maintained, the output layer is the same but the input layer has got four neurons more than the previous structure corresponding to the magnitude and angle of S11 and S12 parameters computed using empirical equations that can be found in [36]. Therefore, the new network has eight neurons in input layer. The training process was the same as in the previous case. The training set has referred to the same values for gap physical dimensions, substrate relative dielectric permittivity and frequency as the previous one but is extended with the corresponding values computed from 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 first and 11 neurons in the second hidden layer, observed from the input to the output layer (H4-15-11).

As an illustration, Figure 21 presents the comparison of gap scattering parameters dependence on frequency for a specified substrate and three values for s/w ratio. It should be noted that all these values are not used during the training process. Figure 21a refers to the magnitude of S11 parameter, and Figure 21b its angle. The triangles denote reference values, the doted lines - values generated by the M4-14-14 model and the solid lines - values generated by the H4015011 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 smaller number of epochs for achieving the same training goal making the training process faster.

V. CONCLUSION

Neural networks can be applied in a wide area of problems. This paper presents authors’ results in neural network applications in the field of microwave. Neural networks are applied in the modeling of resonant cavity applicators and microwave transistors as well as of microstrip structures.

JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE 45

The existing methods for determination of microwave resonant cavity resonant frequencies are based on solving of complex electromagnetic equations. There are some approximate methods but they are limited to some special cases. Neural networks provide fast and efficient resonant frequency determination avoiding complex numerical calculation. Although the first proposed approach based on multilayer perceptron networks gives very good determination results, its main disadvantage is a requirement for a large number of training samples. Therefore, there have been proposed some improved approaches such as difference based, hybrid-empirical, and knowledge-based neural network methods that require smaller training sets.

1 2 3 4 5 6 7 8 9 10.970

0.975

0.980

0.985

0.990

0.995

1.000

1.005

0

s/w=0.95

s/w=0.55

s/w=0.15

H4-15-11 M4-14-14 Referent values

|S11

|

f [GHz] (a) Magnitude of parameter S11

1 2 3 4 5 6 7 8 9

-15

-10

-5

10

s/w=0.95

s/w=0.15

s/w=0.55

angl

e(S 11

) [ d

egre

es ]

f [GHz]

H4-15-11 M4-14-14 Referent values

1 2 3 4 5 6 7 8 9

-15

-10

-5

10

1 2 3 4 5 6 7 8 9

-15

-10

-5

10

(b) Angle of parameter S11

Fig. 21. Comparison of generalization capabilities of M4-14-14 and H4-15-11 (εr=9,w/h=0.75)

Microwave circuit design requires reliable and accurate models of microwave HEMT and MESFET transistors. Usually, the existing models refer to one bias point; therefore there is a need for recalculation of elements and parameters of the model for any other bias point. The proposed neural network approach has yielded models that can be used for the all operating bias conditions without additional changes. If 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 of neural network

approach and wave approach to transistor noise modeling has been proposed.

Recently, the first encouraged steps of microstrip elements modeling using neural networks have been made.

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