An Introduced Neural Network-Differential Evolution Model for Small Signal Modeling of PHEMTs

Mazhar B. Tayel Amr H. Yassin Faculty of Engineering Faculty of Engineering Department of Electrical Engineering Alexandria University Alexandria University, Egypt . yassin_amr_eg@yahoo.de

Abstract –Since neural network algorithms are able to model nonlinear relations between different data sets, an introduced neural network model (INN) based on a generalized differential evolution training algorithm (INN-DE) is presented for pseudomorphic high electron mobility transistor (PHEMT). This global optimization algorithm is applied to avoid the local minima problem in the gradient descent-training algorithm and to achieve acceptable solution. The main advantage of this technique is its validation in wide range of frequencies and high accuracy for the small signal characteristics. The proposed (INN-DE) model is used to predict the scattering parameter values for various bias values different from the ones in the data set used for training. This model has been verified by comparing predicted and measured values of a PHEMT for a certain data set of S-parameters at different frequencies and bias points. Keywords: HEMT; S- parameters; small signal model; Neural networks.

INTRODUCTION It is important to develop an almost accurate small-signal model under variable conditions. Thus, the small-signal model results should be as close to the measured values as possible. One of the most important factors that are needed to optimize the equivalent circuit elements of PHEMT is the small-signal S-parameters. The S- parameters of microwave transistors are frequency, temperature and bias-dependent [2], and are considered as figure of merit for high frequency transistor performance. In general the scattering parameters data reflect the element values of an equivalent circuit model for PHEMT.

I. SMALL SIGNAL MODELING An introduced neural network model with differential evolution training algorithm (INN-DE) using Agilent ATF35143, a pseudomorphic AlGaAs/InGaAs/GaAs PHEMT is selected [3,4] because of its excellent room temperature noise properties. For this device the gate length is 0.5 μm and the gate width is 400 μm at 2GHz [4]. The small signal model has 3 input variables (frequency, Vds and Ids), and 8 output variables (magnitudes and phases of the scattering parameters). A large number of scattering parameters data, measured at various frequencies and

biases, was used to train this (INN-DE), using the strategy of differential evolution algorithm. Differential evolution (DE) can be classified as global stochastic population-based evolutionary optimization algorithm that has the ability of finding the global optimum of a given objective function [7]. The optimization goal here is to minimize an objective function E by applying the values of neural network weights w = ( w1, w2 , ……., wN ). The main idea of differential evolution (DE) is to generate trial parameters vectors. For each iteration the algorithm mutates vectors by adding weighted random vector differential to them .If the cost of the trial vector is better than that of the target, then the target vector is replaced by trial vector in the next generation [8]. This process is repeated until a stopping criterion is reached, which stops the algorithm as shown in figure (1).

Mutation

Crossover

obj Function min ?

Stop Algorithm

Initialize population

TrueFalse

Figure 1. Basic Structure of differential evolution algorithm. To apply DE in neural network training, the following steps are introduced: 1) Initialization: Start with a certain population of NP , NP > 3 of N dimensional randomly chosen weight vectors. 2) Mutation operation: For each weight vector g

iw with current generation or iteration number g , (g > 0) and a population member i , i =1,2,….NP ,a mutant vector

1giv + is formed according to the relation (1) [8,9]:

2009 International Conference on Electronic Computer Technology

978-0-7695-3559-3/09 $25.00 © 2009 IEEE

DOI 10.1109/ICECT.2009.149

499

11 2( )g g g g g g

i i best i r rv w w w w wμ+ = + − + − (1) where wr1 and wr2 are two randomly chosen population vectors, and the mutated vector vi is

formed in terms of their vector difference and gbestw

which is the best population element of the previous iteration , multiplied by μ the mutation constant , where μ ∈ [0,2]. 3) Crossover operation : is performed and a trial vector 1g

iu + is generated . For each integer Component j , ( j = 1, 2,….., N ) of the mutant weight vector 1g

iv + , a random real number “rand “ is generated from the interval [0,1]. This number is compared with the crossover constant ρ ( ρ ∈ [0, 1] ) as follows :

1, ,1

,, ,

(2)gi j i j randg

i j gi j i j rand

v if rand then j Iu

w if rand then j I

ρ

ρ

++

⎧ ≤ =⎪=⎨> ≠⎪⎩

If rand ρ≤ then the jth component of the trial

vector 1giu + gets the jth component of the mutant

vector 1giv + otherwise the jth component of the trial

vector 1giu + gets the value of the target vector

giw .In the previous equation r a n dI is a randomly

selected index that is used to ensure the trial vector has at least one component from the mutant vector . The trial vector is accepted for the next iteration if the value of the Error function E is reduced otherwise the previous value of the target vector g

iw is retained. 4) Selection process: in this step the fitness function starts to decrease at some iteration according to the following relation (equation 3):

( ) ( )( ) ( )

1 1

1

1 1(3)

g g gi i ig

i g g gi i i

u if E u E ww

w if E u E w

+ +

+

+ +

⎧ <⎪= ⎨≥⎪⎩

i.e. if the objective value of iu is less than iw then

iu replaces iw in the next generation otherwise iw is retained .The process is iterated until the termination criterion is satisfied.

II. EXPERIMENTAL PROCEDURES

The available ATF 35143 S-parameters data set is divided into three sets: training set, test set and data set. The training set is used to update the weights [5]. Patterns in this set are repeatedly presented in random order. The weight update equations are applied after a certain number of patterns. The test set is used to test the performance of the model. The validation set is used to decide when to stop training only by monitoring the mean square error (MSE). The MSE for both training data and test data is calculated by setting the neuron numbers to only 12 in the first hidden layer, 8 neurons in the second hidden layer, and 8 neuron in the output layer, as shown in figure (2).

F

Vds

Ids

input Layeroutput Layer

first hidden layersecond hidden layer1

12

1

8

S11 mag

S11 ang

S22 ang

Figure 2. The INN model. m = magnitude, a = angle

The learning rate is set to (0.06), and the momentum to 0.8, the squared error obtained is about 0.0016.The number of neurons in the hidden layers is selected close to the number of input and output variables so that to avoid overfitting (in case of too many neurons) and underfitting (in case of too few neurons) for the model. An initial pre-processing stage for the data sets is essential. Data must be scaled to the range used by the input neurons in the neural network. This is typically the range of –1 to 1 or zero to 1. Normalization of the input vectors is performed by calculating mean value and standard deviation for the input and output vectors. The total number of training epochs was 600.To apply DE optimization to this (INN-DE) the weight population size NP was chosen to be in the range of 2N where N is the dimension of the problem [9]. Mutation coefficient μ and crossover constant ρ are selected as 0.8 and 0.5 respectively. According to some experimental experience the value of NP will be acceptable in the range: 2N ≤ NP ≤ 4N so that to avoid overfitting or underfitting of the model [9]. In this section The modeling results were realized using MATLAB release 14 from Math works cooperation [6], and the time of executing the training cycle until convergence took about 4 minutes on a conventional 1.7 GHz PC for gradient based model but larger time for

500

the optimized (INN-DE) model. The outputs of the introduced INN-DE model appear to be very close with the original (measured) data.

III. EVALUATION METHOD The ability of the proposed INN-DE for prediction is measured by calculating average error (equation 4) of the training and testing samples (n samples):

Average error = 1

1 n

i ii

T Pn =

−∑ (4)

where Ti is the target ( measured ) value of the ith sample, and Pi is the output (predicted ) value of the ith sample .From table (1) , it is observed that the average error is less than 1% in some cases. Considering the rms errors:

2

2(5)

( ) ( )

( )

p mi j n i j n

ni j

mi j n

n

S S

S

ω ωε

ω

−=

∑

∑

where Sijp is the predicted S-parameters using the INN-DE model and Sijm is the measured S-parameters[10]. The correlation coefficient can be used as a measure of quality of the introduced INN-DE model and is given by:

( ) ( )

( ) ( )2 2(6)

i ii

i ii i

x x y yr

x x y y

⎛ ⎞− −⎜ ⎟

⎝ ⎠=⎡ ⎤− −⎢ ⎥⎣ ⎦

∑

∑ ∑

where i=1,2,…n , n is the number of test patterns, x and y are the measured and predicted values of S-parameters respectively. Only the overall correlation coefficient is calculated for each data set. The reported testing correlation coefficient is calculated by cross validating all the predictions for all the test data sets, therefore for each data set only one testing correlation coefficient is reported as indicated in table (2). It should be remarked here that the value close to one for the correlation coefficient indicates an excellent predictive ability for the proposed model [2,3].

Table 1. RMS error for the test pattern calculated from equation (5) (Pattern, which is not used in the training process)

RMS Error %

S-Parameters

Gradient descent training

GD

Differential evolution training

DE S11 1.1027 0.8943 S21 1.5569 1.6709 S12 1.9032 1.8944 S22 3.9206 2.7126

Table 2. Correlation coefficients for the test pattern (Pattern, which is not used in the training process)

Correlation coefficients

S-Parameters

Gradient descent training

GD

Differential evolution training

DE S11 0.9989 0.9996 S21 0.9998 0.9999 S12 0.9999 0.9999 S22 0.8026 0.8315

IV. RESULTS The introduced INN-DE model which is used to simulate the ATF-35143 PHEMT characteristics, is compared to the experimentally measured characteristics that were taken from the manufacture’s data sheet. The evaluation of the results can be seen in table1.and table2. The predicted and measured S-parameters for the PHEMT are shown in figures (3) and (4) for both gradient based and optimized models respectively, at a fixed bias. It can be seen that the predicted S-Parameters data from both models is very close with the measured values that obtained from the manufacture’s data sheet. Figures (3) and (4) indicate the comparison of predicted S-parameters from both models with the experimental values on the smith chart for Vds= 3v and, Ids=15 mA for gradient based model and INN-DE optimized model respectively. It can be seen that the proposed INN-DE model matches the measured test values with good accuracy as a result of calculating the average error (equation 4), rms error (equation 5) and correlation coefficients (equation 6). This verifies the proposed small signal model for the PHEMT.From equation (5), the rms errors are in the range 1.5%, and the resulting correlation coefficients from equation 4, are in the range of 0.9 which indicates a good prediction capability of the proposed INN-DE model.

501

Sample results Gradient descent training

Figure 3a. Comparison between measured S11, S22 (solid line) and modeled S11, S22 (∆ ∆ line) on smith chart for vds= 3V,Ids=15 mA .

Figure 3b. Measured and predicted S11, S22 magnitude and angle – Frequency relations for vds=3V and Ids=15mA. ( __ ) measured data set, (--+--) predicted data set.

Freq S11m S11sim S22m S22sim 1 0.94 0.93558 0.57 0.54883 2 0.84 0.82689 0.5 0.48016 3 0.73 0.71725 0.44 0.41147 4 0.63 0.62951 0.37 0.35189 5 0.58 0.57266 0.31 0.29384 6 0.56 0.55232 0.24 0.23126 7 0.55 0.54745 0.19 0.17556 8 0.56 0.56322 0.14 0.12954 9 0.6 0.59694 0.11 0.092272 10 0.64 0.63473 0.09 0.093712

S11

S22

502

Sample results Gradient descent training

Figure 3c. Comparison between measured S21, S12 (solid line) and modeled S21, S12 (∆ ∆ line ) on smith chart for Vds=3V ,Ids=15 mA .

(The values on smith chart are : 0.1* S21 and 4* S12)

Figure 3d. Measured and predicted S21, S12 magnitude and angle – Frequency relations for vds=3V and Ids=15mA. ( __ ) measured data set, (--+--) predicted data set.

Freq S21m S21sim S12m S12sim 1 6.97 7.0663 0.042 0.03986 2 6.13 6.2047 0.074 0.071895 3 5.31 5.3047 0.102 0.094129 4 4.59 4.5667 0.107 0.10664 5 4 3.9734 0.115 0.11396 6 3.53 3.5176 0.121 0.12116 7 3.14 3.1044 0.125 0.125 8 2.81 2.8022 0.127 0.12644 9 2.5 2.5536 0.128 0.12849 10 2.26 2.3149 0.13 0.13048

0.1 x S21

4 x S12

503

Sample results Optimized (INN-DE) model

Figure 4a. Comparison between measured S11, S22 (solid line) and modeled S11, S22 (∆ ∆ line) on smith chart for vds =3V, Ids =15 mA.

Figure 4b. Measured and predicted S11, S22 magnitude and angle – Frequency relations for vds=3V and Ids=15mA. ( __ ) measured data set, (--+--) predicted data set.

Freq S11m S11sim S22m S22sim 1 0.94 0.94566 0.54883 0.55962 2 0.84 0.8412 0.48016 0.4872 3 0.73 0.72677 0.41147 0.41493 4 0.63 0.63385 0.35189 0.35612 5 0.58 0.57932 0.29384 0.29775 6 0.56 0.55541 0.23126 0.23927 7 0.55 0.54778 0.17556 0.18807 8 0.56 0.55935 0.12954 0.14433 9 0.6 0.59825 0.092272 0.10658 10 0.64 0.63925 0.093712 0.09829

S11

S22

504

Sample results Optimized (INN-DE) model

Figure 4c. Comparison between measured S21,S12 (solid line) and modeled S21,S12 (∆ ∆ line) on smith chart at for vds=3V,Ids=15 mA.

(The values on smith chart are : 0.1* S21 and 4* S12)

Figure 4d. Measured and predicted S21, S12 magnitude and angle – Frequency relations for vds=3V and Ids=15mA. ( __ ) measured data set, (--+--) predicted data set.

Freq S21m S12sim S12m S12sim 1 6.97 7.0721 0.042 0.041421 2 6.13 6.2491 0.074 0.073043 3 5.31 5.3815 0.102 0.093826 4 4.59 4.6595 0.107 0.10639 5 4 4.0315 0.115 0.11581 6 3.53 3.5602 0.121 0.1224 7 3.14 3.1084 0.125 0.12631 8 2.81 2.836 0.127 0.12621 9 2.5 2.5337 0.128 0.12773 10 2.26 2.2414 0.13 0.13033

0.1 x S21

4 x S12

505

V. CONCLUSION

In conclusion, the introduced INN-DE model has been used to model the small signal characteristics of HEMTs effectively. Generalization of model is proven by testing it with different sequence of data that have not participated during the training process. In this study, differential evolution has been used as a global optimization method for feed-forward neural networks. The simulation results in figures (3) and (4) show very slight differences (between differential evolution optimized INN-DE model compared with the gradient-based model). Optimized INN-DE model seems to give some better results according to rms error calculations. Differential evolution can rather be used in validation of reached the optimum value of objective function concerned in this work. The Pearson correlation coefficient was used to evaluate the ability of the INN-DE model to predict the test data. The modeled S-parameters are close to the original data of the manufacture, and the model show that fit of the angle of S22 is very good until about 8 GHz . It is not necessary to solve nonlinear regression equations that take a long running time .In this case using neural networks models is considered as an advantage.

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