JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL. 20:39-44, 2010©
Two Neural Approaches for
Small-Signal Modelling of GaAs HEMTs
Zlatica Marinković, Giovanni Crupi, Alina Caddemi, Vera Marković

Abstract — Focus of this paper is on the neural approach in
small-signal modelling of GaAs HEMTs. Two modelling
approaches based on artificial neural networks are discussed
and compared. The first approach is completely based on
artificial neural networks, while the second is a hybrid
approach putting together artificial neural networks and an
equivalent circuit representation of a microwave transistor.
Both models consider the device gate width and therefore both
are scalable. Results of modelling of three different
AlGaAs/GaAs HEMTs in a wide range of operating bias
conditions using the considered approaches are given.
Different modelling aspects are discussed. A special attention is
paid to the model development procedure and accuracy of the
models.
Index Terms—Artificial neural networks, HEMT, gate
width, small-signal modelling
R
I. INTRODUCTION
ELIABLE and accurate models of microwave devices
are necessary for the successful microwave circuit
design. Therefore, developing of such models is an
important and challenging task for researchers in the
microwave community. Here the attention is paid to the
modelling of the scattering (S-) parameters of microwave
HEMTs (High Electron Mobility Transistors). The Sparameters of a microwave transistor depend on the
operating bias condition as well as of the frequency. The
most common modelling approach is to represent a device
with a suitable equivalent circuit, whose ECPs are extracted
from the measured values of the S-parameters, and to
simulate the S-parameters in a microwave simulator.
Although the ECP extraction is usually an ill-conditioned
problem, there are efficient procedures for the ECP
extraction [1]-[5], as the one that is used in this work [5].
The common feature for all equivalent circuit based models
is that for each operating point it is necessary to repeat
extraction of the ECPs, which can be quite time consuming
This work was supported by “IMT-ARSEL” project prot. RBIP06R9X5
with financial support by Italian MIUR and by the project No TR-11033 of
the Serbian Ministry of Science and Technological Development.
Z. Marinković and V. Marković are with the Faculty of Electronic
Engineering, University of Niš, Aleksandra Medevedeva 14, 18000 Niš,
Serbia, (phone: +381-18-529303; fax: +381-18588399; e-mail:
zlatica.marinkovic@elfak.ni.ac.rs; vera.markovic@ elfak.ni.ac.rs).
G. Crupi and A. Caddemi are with the Dipartimento di Fisica della
Materia e Ingegneria Elettronica, University of Messina, Salita Sperone 31,
Messina
98166,
Italy,
(e-mail:
giocrupi@ingegneria.unime.it;
caddemi@ingegneria.unime.it).
DOI: 10.2298/JAC1001039M
if the ECPs should be extracted in a wide range of operating
conditions. Alternative approach in modelling of the Sparameters is based on application of artificial neural
networks (ANNs). In the last two decades artificial neural
networks have found their place as an efficient tool for
modelling of microwave devices [6], [7]. ANN models are
usually extracted from the measured data directly, without
need for detailed knowledge about device physics, therefore
they can encounter all effects contributing to the device
behaviour. As far as microwave transistors are concerned,
ANNs have been exploited to model their small-signal,
large-signal, and noise performance [8]-[12]. This paper is
focused on multi-bias small-signal modelling of GaAs
HEMTs differing in the gate width.
Recently, two different approaches based on the
application of ANNs have been proposed to model scaled
microwave FETs [11]-[12]. One approach is completely
based on the neural networks, i.e. it is a black box model,
while the other approach is a hybrid model, in which ANNs
are used to predict parameters of an equivalent circuit
representation. The aim of this paper is to compare both
approaches in order to highlight advantages as well as
disadvantages of both models in terms of modelling
efficiency and accuracy. This paper is an extended version
of the work presented in [13] giving more modelling details.
The paper is organized as follows. First, in Section II
basic background on artificial neural networks is given. In
Section III and IV the considered modelling approaches are
presented. Section IV gives also the description of the
analytical procedure used in the hybrid approach for the
extraction of the equivalent circuit parameters (ECPs) from
the measured scattering parameters of the modelled device.
The modelling results obtained by using the two mentioned
approaches on three AlGaAs/GaAs HEMT devices with
different gate width are presented in Section V. The main
conclusions are stated in Section VI.
II. ARTIFICIAL NEURAL NETWORKS
A widely used type of artificial neural networks is a
standard multilayer perceptron (MLP) ANN shown in Fig.
1, [6]. This network consists of an input layer (layer 0), an
output layer (layer NL), as well as several hidden layers.
Input vectors are presented to the input layer and fed
through the network that then yields the output vector. The
l-th layer output is:
Yl  F ( Wl Yl 1  B l )
(1)
where Yl and Yl 1 are outputs of l-th and (l-1)-th layer,
respectively, Wl is a weight matrix between (l-1)-th and l-
40 MARINKOVIĆ Z, CRUPI G, CADDEMI A, MARKOVIĆ V, TWO NEURAL APPROACHES FOR SMALL-SIGNAL MODELLING...
th layer and B l is a bias matrix between (l-1)-th and l-th
layer.
each real and imaginary part of the S-parameters. The model
is illustrated in Fig. 2. The ANNs used in the model are
trained by using the S-parameters of the considered different
gate width devices measured at a certain number of biaspoints. Once the ANNs are trained properly, the Sparameters for any of the considered devices can be
determined directly from the ANNs, by finding the response
of the ANNs for the given input values.
Fig. 2. Black-box neural model
Fig. 1 MLP neural network
The function F is an activation function of each neuron
and, in our case, is linear for input and output layer and
sigmoid for hidden layers:
F (u )  1 /(1  e u ) .
(2)
The neural network learns relationship among sets of
input-output data (training sets). First, input vectors are
presented to the input neurons and output vectors are
computed. These output vectors are then compared with
desired values and errors are computed. Error derivatives
are then calculated and summed up for each weight and bias
until whole training set has been presented to the network.
These error derivatives are then used to update the weights
and biases for neurons in the model. The training process
proceeds until errors are lower than the prescribed values or
until the maximum number of epochs is reached (epoch is
the whole training set processing). Once trained, the
network provides fast response for various input vectors
(even for those not included in the training set) without
additional optimizations.
Usually, neural networks with different number of hidden
neurons are trained, tested and after their comparison, the
network with the best modeling results is chosen as the
neural model.
III. BLACK-BOX SMALL-SIGNAL NEURAL MODEL OF
MICROWAVE TRANSISTORS
The simplest type of ANN based models are the blackbox models consisting of one or more MLP ANNs trained to
predict dependence of the modelled parameters on the
desired input conditions. As far as small-signal model of
microwave FETs differing in the gate width is concerned,
the modelled parameters are the S-parameters, while the
inputs are: bias voltages (gate-to-source voltage, Vgs , and
drain-to-source voltage, Vds ), frequency as well as the gate
width, W [11]. Therefore, there are four neurons in the
input layer corresponding to the mentioned four input
parameters. There are eight outputs in total (real and
imaginary parts of four S-parameters). All eight outputs can
be modelled by a single ANN, but in order to achieve better
modelling accuracy it is recommended to train an ANN for
each of the S-parameters, even to use a separate ANN for
IV. HYBRID SMALL-SIGNAL NEURAL MODEL OF
MICROWAVE TRANSISTORS
As it was stated in Introduction, an equivalent circuit
representation requires repeated extraction of the ECPs for
each operating point. There are different extraction
techniques, either based on optimizations or on analytical
procedures. In order to avoid repeated extractions, one or
more ANNs can be trained to predict dependence of the
ECPs on the biases and the gate width [12]. For training of
the ANNs the ECP values should be extracted from the
measured S-parameters for certain number of different input
conditions by using an extraction procedure. After the
ANNs have been trained, the ECPs can be determined by
finding the response of the ANNs. Using the predicted
ECPs, the S-parameters are simulated from the equivalent
circuit, as represented in Fig. 3.
Fig. 3. Hybrid small-signal model
This approach is a hybrid technique since the proposed
model consists of ANNs, which are trained to predict the
ECPs, and an equivalent circuit. The developed model can
be used in the microwave circuit simulator as a library
element for the considered scaled devices.
gate Lg
Rg
Cgd
+
V_
Cpg/2
Cpg/2
Cgs
intrinsic circuit
Rgd
gme
Rgs
Ld drain
Rd
-j
V
Rds Cds
Cpd/2
Rs
Ls
source
Fig. 4. Small-signal equivalent circuit for FET
Cpd/2
JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL 20, 2010.
41
Equivalent circuit model considered in this work is shown
in Fig. 4. The eight ECPs of the intrinsic circuit ( C gs , Rgs ,
C gd , Rgd , g m ,  , Rds , Cds ) are dependent on the biases,
while the eight ECPs of the extrinsic circuit are assumed to
be bias independent.
In [5] an analytical technique for the direct extraction of
the ECPs of such circuit was proposed. The extrinsic ECPs
are extracted analytically from the S- parameters under
“cold pinch-off” condition. After removing their
contributions from the data, the intrinsic ECPs are
calculated straightforwardly at each bias point from the
intrinsic admittance (Y-) parameters.
Having in mind that the extrinsic ECPs are not bias
dependent, the ANN modelling them has only one input
neuron corresponding to the gate width, Fig. 5. The seven
outputs of the network correspond to the seven extrinsic
ECPs (it was assumed that C pg = C pd ). The ANNs
modelling the intrinsic ECPs have, beside the gate width,
the bias voltages V gs and Vds as inputs; therefore each of
them have three input neurons. There are eight output
neurons of this ANN model, corresponding to eight intrinsic
ECPs.
Fig. 6. Developed black-box model
four numbers N-H1-H2-M. N is a number of neurons in the
input layer, M is a number of neurons in the output layer,
while H1 and H2 are numbers of neurons in the first and in
the second hidden layer, respectively. More details about the
ANN training can be found in [11].
B. Hybrid model
The starting point for the hybrid model of the considered
devices was the equivalent circuit representation shown in
Fig. 4. The ECPs were extracted analytically in a way
described in [5].
The corresponding hybrid model with the information
about the trained ANNs chosen for the final model (using
the same notation as in the case of black-box model) is
depicted in Fig. 6.
Fig. 5. Hybrid model for the considered devices
V. MODELLING RESULTS
Both of the above mentioned approaches were applied to
on-wafer AlGaAs/GaAs HEMTs with different gate width
100, 200, and 300 µm [11]-[13]. The models were
developed from the S-parameters measured with an Agilent
E8364A PNA (precision network analyzer). The
measurements were done in 546 bias points.
A. Black-box model
As far as the black-box model is considered, the Sparameters were modelled by separate ANNs. For S11 and
S 22 , a single ANN is used to model both real and imaginary
parts of each parameter. But, for the two other parameters,
S 21 and S12 , due to their more complicated behaviour, the
real and imaginary parts of each parameter were modelled
by separate ANNs. Therefore, the model consists of two
two-output ANNs for S11 and S 22 and four one-output
ANNs for S 21 and S12 as it is shown in Fig. 6. Each ANN
was trained separately.
Details about the trained ANNs chosen for the final
model are given in Fig. 5, where each ANN is described by
Fig. 7. Developed hybrid model
It should be noted that in this case, g m and  were
modelled by separate one-output ANNs, while rest of the
parameters were modelled by a single six-output ANN.
More details on the model development are given in [12].
42 MARINKOVIĆ Z, CRUPI G, CADDEMI A, MARKOVIĆ V, TWO NEURAL APPROACHES FOR SMALL-SIGNAL MODELLING...
with the measured data, [13].
For the purpose of comparing the simulated and the
measured S-parameters at each particular bias point, the
percentage errors Eij were calculated as follows:
Eij 
1
Nf
1
 E  f   N 100
SijMEAS ( f )  SijSIM ( f )
ij
f
SijMEAS ( f )
(3)
where N f represents the number of frequency points per a
bias point, which is 101 in the present case.
The total percentage error of the model, ETOT , was
obtained by averaging the evaluated errors of the four Sij of
each device.
Since the modelling results of all three devices are very
similar, for sake of space, the comparative analysis is
focused on the device with 100 μm gate width. In Table I
there are the percentage errors averaged for all considered
bias points for the 100 μm device, for both neural models as
well as for the analytical approach. It can be seen that the
black-box model gives the smallest percentage errors, which
can be explained by the fact that the ANNs were trained
using the measured S-parameters without the constraint of
the specific equivalent circuit topology and therefore all
effects contributing to the device behaviour are included in
the model. On the other side, as can be expected, the hybrid
model gives slightly bigger percentage errors than the
analytical one. This is due to the fact that the ANNs in the
hybrid model are trained using the data obtained by the
analytical approach.
other referring to “pinch-off“ condition ( Vgs  1.5 V ,
Vds  2.3 V ). Note that difference between measured and
simulated S21 is quite small in both cases. Nevertheless,
Fig. 10 highlights that percentage error associated to S21
under pinch-off condition is much higher than under a
typical operating bias point. This is due to the small value of
S21 over the full frequency range, especially at low
frequency, under such bias condition.
E21 [%]
step and -1.5 V < V gs < 0 V with 75 mV step) and compared
capability, while in the analytical approach the equivalent
circuit is extracted from the measurements at each specific
bias point without accounting for the device behavior under
the other bias points. In order to illustrate this more clearly,
Fig. 9 illustrates the real an imaginary parts of S21 obtained
by the black-box model for two bias points: one referring to
typical bias condition ( Vgs  0.6 V , Vds  2.5 V ) and the
140
120
100
80
60
40
20
0
0
-0.5
-1
Vgs [V]
E21 [%]
C. Model Comparison
Both models were implemented into the circuit simulator
by implementing the mathematical expression describing the
trained ANNs. After that, the S-parameters were simulated
over the frequency range extending from 0.5 up to 50 GHz,
at 546 different bias points (0 V < Vds < 2.5 V with 100 mV
2.5
1.5
1
10.2
6.2
11.0
E22 [%]
5.4
3.1
6.2
ETOT [%]
9.2
7.6
10.0
One can see that the S21 parameter in the case of the
black-box model exhibits a significantly larger error. In
order to explain this, Fig. 8 shows the bias dependence of
the percentage error associated to S21 for all three
approaches. It can be observed that in the case of the blackbox model the percentage error exhibits large values for the
biases where S21 is small (i.e., “cold” and “pinch-off”
conditions) compared to its high values under typical
operating conditions. This can be attributed to the fact that
the neural approach is strongly based on its generalization
0.5
Vgs [V]
hybrid
11.0
11.8
E12 [%]
(a)
2
-1
E21 [%]
E21[%]
2.5
2
1.5
1
Vds [V]
-0.5
PERCENTAGE ERROR AVERAGED OVER THE FULL BIAS RANGE
E11[%]
0.5
0
30
25
20
15
10
5
0
0
TABLE I
100 μm device
analytical black-box
11.0
2.5
10.2
18.7
-1.5
Vds [V]
-1.5 0
30
25
20
15
10
5
0
0
(b)
2.5
2
1.5
-0.5
1
-1
Vgs [V]
0.5
-1.5 0
Vds [V]
Fig. 8. Bias dependence of the percentage error associated to
(a) black-box, (b) hybrid, and (c) analytical approach
(c)
S21 :
JOURNAL OF AUTOMATIC CONTROL, UNIVERSITY OF BELGRADE, VOL 20, 2010.
Re (S21)
2.0
Im (S21)
2.5
VI. CONCLUSION
2.0
In this paper a comparison of two neural models of scaled
microwave GaAs FETs is given. A comprehensive analysis
is done, resulting in the following conclusions.
Both models provide a single bias-dependent model for
all three modelled devices that can be further used as the
circuit simulator library elements. This allows efficient
simulation of S-parameters with arbitrary steps of bias
voltages, which can be very important for building
subsequently a large signal model. Further, unlike the
standard modelling approaches that require the measured Sparameters for each considered bias point, both presented
models require the measured S-parameters only for certain
number of bias points to be used for the model
development.
Since the black-box model is trained using the measured
data, all effects contributing to the device behavior are
included in the model, therefore it exhibits high modelling
accuracy. On the other hand, the hybrid model retains the
advantages of the equivalent circuit based model (here the
analytical one), like the connection with the physical device
structure, which represents a useful feedback for a quick
and reliable optimization of the device fabrication
processing. The accuracy of the equivalent circuit is of
fundamental importance for developing a successful hybrid
model.
While the hybrid approach requires certain number of
extractions of the ECPs and then training of the
corresponding ANNs, the black box-model requires only
training of the ANNs. Hence, as far as the efficiency of the
model development procedure is concerned, the black-box
approach is simpler.
Both modelling approaches exhibit advantages and
disadvantages. The best solution is to choose the approach
depending on the specific case under study.
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
-0.5
-0.5
Re (S21)
-1.0
Im (S21)
2.5
-1.0
-1.5
-1.5
-2.0
-2.0
-2.5
-2.5
-3.0
-3.0
0
5
10
15
20
25
30
35
40
45
50
f [GHz]
(a)
0.3
0.3
Re (S21)
0.2
0.2
0.1
Im (S21)
0.0
0.0
Im (S21)
Re (S21)
0.1
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
0
5
10
15
20
25
30
35
40
45
50
f [GHz]
(b)
Fig. 9. Frequency dependence of S 21 : data simulated by black-box
approach (lines) versus measurements (symbols)
(a) typical bias condition ( V gs  0.6 V , V ds  2.5 V )
(b) “pinch-off” bias condition ( V gs  1.5 V , V ds  2.3 V )
E21 ( f ) [%]
15
10
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[2]
0
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10
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