Neural 3-D Smith Chart

Smith chart is a graphical tool for solving transmission line problems which was developed by P. H. Smith [1]. Every book related on microwave theory or engineering field has had detailed information, solved problems and various applications after its ease of utilization improved on transmission lines [2]. Although analysis and design of microwave circuits are generally tedious with their complicated equations, Smith chart provides a very useful graphical tool or calculator which was improved by computer programmers for these types of the problems with its numerous applications. A microwave engineer having whole concept of the Smith chart in his mind has an ability to picture probable matching solutions for complex problems which includes extensive computation cost. In the literature, early studies on the computerized Smith chart have been taken place between 1992 and1995 by Prasad and her group in [3–5]. New generation RF/Microwave circuit designers use sophisticated computer-aided design (CAD) tools to decrease the computation time as much as possible. However, the developments in CAD tools do not eliminate the usage Smith chart in design problems. Especially, designers need to consider stability, gain and noise figure circles on a Smith chart for optimum solution options while designing matching networks. Famous design software packages serve design parameters to be figured in a Smith chart plane. Moreover, network analyzers have provided graphical outputs on a Smith chart (e.g. s-parameters applications). Presenting the negative real half of the impedance values is an important limitation of conventional twodimensional (2-D) Smith chart. In fact, some oscillator and microwave active filter circuit designs have involved negative resistances. However, it is not practical to manipulate both negative and positive real impedances on the same conventional Smith chart. There is a strong necessity to use separate charts one for the impedances having negative reel part and one for the impedances having positive reel parts. Zelly has proposed to use these two charts side-by-side like a mirror image to easily figure out whole domain in an attempt [6]. Lenzing et al. have suggested using “negative” Smith chart first for designing one-port amplifier [7]. When negative impedances are involved in design, visualizing the Smith chart as a three-dimensional (3-D) sphere rather than a 2-D circle the Smith chart can perform greater insight [6]. Thus, all possible impedances can be performed on the 3-D Smith chart. The power of the 3-D Smith chart has been strengthened with recent studies which are relevant to the theory of the spherical generalized omnipotent Smith charts [8–10]. Besides this, fractional Smith chart is a novel theory to represent fractional order circuit elements as well [11]. The conventional circular Smith chart on a flat 2-D plane has been modeled with Artificial Neural Networks (ANNs) and its impedance matching application in lownoise amplifier (LNA) has been presented [12-13]. In this paper, the 3-D Smith chart has been modeled with ANN to be utilized on the microwave circuitry with the purpose of enough accuracy having fast and practical in-use. Firstly, the mathematical basics of the 3-D Smith chart would have been explained in next section. Then the transformation of the conventional 2-D Smith in order to 3D spherical Smith chart using ANNs would have been expressed.


Introduction
Smith chart is a graphical tool for solving transmission line problems which was developed by P. H. Smith [1]. Every book related on microwave theory or engineering field has had detailed information, solved problems and various applications after its ease of utilization improved on transmission lines [2]. Although analysis and design of microwave circuits are generally tedious with their complicated equations, Smith chart provides a very useful graphical tool or calculator which was improved by computer programmers for these types of the problems with its numerous applications. A microwave engineer having whole concept of the Smith chart in his mind has an ability to picture probable matching solutions for complex problems which includes extensive computation cost.
In the literature, early studies on the computerized Smith chart have been taken place between 1992 and1995 by Prasad and her group in [3][4][5]. New generation RF/Microwave circuit designers use sophisticated computer-aided design (CAD) tools to decrease the computation time as much as possible. However, the developments in CAD tools do not eliminate the usage Smith chart in design problems. Especially, designers need to consider stability, gain and noise figure circles on a Smith chart for optimum solution options while designing matching networks. Famous design software packages serve design parameters to be figured in a Smith chart plane. Moreover, network analyzers have provided graphical outputs on a Smith chart (e.g. s-parameters applications).
Presenting the negative real half of the impedance values is an important limitation of conventional twodimensional (2-D) Smith chart. In fact, some oscillator and microwave active filter circuit designs have involved negative resistances. However, it is not practical to manipulate both negative and positive real impedances on the same conventional Smith chart. There is a strong necessity to use separate charts one for the impedances having negative reel part and one for the impedances having positive reel parts. Zelly has proposed to use these two charts side-by-side like a mirror image to easily figure out whole domain in an attempt [6]. Lenzing et al. have suggested using "negative" Smith chart first for designing one-port amplifier [7].
When negative impedances are involved in design, visualizing the Smith chart as a three-dimensional (3-D) sphere rather than a 2-D circle the Smith chart can perform greater insight [6]. Thus, all possible impedances can be performed on the 3-D Smith chart. The power of the 3-D Smith chart has been strengthened with recent studies which are relevant to the theory of the spherical generalized omnipotent Smith charts [8][9][10]. Besides this, fractional Smith chart is a novel theory to represent fractional order circuit elements as well [11].
The conventional circular Smith chart on a flat 2-D plane has been modeled with Artificial Neural Networks (ANNs) and its impedance matching application in lownoise amplifier (LNA) has been presented [12][13].
In this paper, the 3-D Smith chart has been modeled with ANN to be utilized on the microwave circuitry with the purpose of enough accuracy having fast and practical in-use. Firstly, the mathematical basics of the 3-D Smith chart would have been explained in next section. Then the transformation of the conventional 2-D Smith in order to 3-D spherical Smith chart using ANNs would have been expressed.

The 3-D Smith Chart
In this section, mathematical analysis of spherical 3-D Smith chart has been indicated briefly to get transformation 2-D Smith chart to 3-D Smith chart and numerical computations [8][9][10]. Initially, the transformation rule between rectangular normalized Zplane and polar Γ-plane, that is a one-to-one mapping of the resistance and reactance circles as seen in Fig. 1. The transformation rule has been presented as below As seen in Fig. 2, two 2-D Smith charts standing sideby-side provides the designer to easily monitor all possible solutions including positive and negative reel part of impedance applications [6,7].  The main idea of joining left-hand side and the right-hand side charts is necessity of overlapping the same circles (also points) like r=0 circles, as given in Fig. 2. Because they are identical and duplication has not been allowed. If the overlapping operation takes place on the 2-D plane, there will be ultimate complexity via mixing positive and negative reel parts.
Zelley has proposed a comprehensible transition way of transforming conventional 2-D Smith chart to 3-D spherical Smith chart without mathematical expressions [6]. Y. Wu et al. have improved mathematical analysis of the 3-D spherical Smith chart [8,9]. They have introduced two groups of transitions parameters to denote points that generate the resistance and reactance circles on the surface of the unit sphere. The representation of a data point on a resistance or a reactance circle can be developed briefly as follows. Γ x , Γ y and Γ z are the axes of the 3-D Cartesian coordinate system as illustrated in Fig. 3. The points O and A are the centers of the unit sphere and an arbitrary circle on a plane having angular position to Γ x -Γ y plane, respectively. The points B, C and D are on the surface of unit sphere and on the circle, too. So the |BD| line defines the diameter of the circle and it's perpendicular to the |OA| line. The two transition parameters are angles which can be defined as ABO   and BAC   . The length of the line |OB|=1 and the radius of the arbitrary circle is |AC|=|AB|=|OB|cosφ=cosφ. The Γ y axis value of the point C that is a random selected point on the circle can be obtained as below sin cos .sin The value of Γ z axis of point C can be obtained via the projection of |AC| to |AB| which is |AC| AB : All possible points on the surface of the unit sphere can be denoted by using transition parameters which are exactly providing angular intervals in radians as -π<φ≤π and -π/2≤θ≤π/2. Thus, whole surface of the unit sphere can be scanned by an infinite number of circles.  The poin circles of the renaming coo resistance circ reactance circl (resistance) p surface of the follows in (1) Some r a using (5) and which blue cir points.  ).tan r r  , 1 1 and inverse relationships of (9), (12) and (13) can be obtained as follows: arctan If a point on the surface of the 3-D Smith chart is necessary to be sketched, the parameters θ r and θ x would have been needed. It is known that a z=r+jx point has been sketched via r and x circle intersection on the conventional 2-D Smith chart. Consequently, Γ r , Γ i , Γ z of r and x circles have been equaled using (5) and (6) and by similar approach and omitting cosθ r in (16) and (17) it has been obtained as below 1 2arctan tan .cos As a result, all possible r and x circles from 2-D Smith chart can be transformed to related r and x circles on the 3-D Smith chart via equations (14), (15), (19) and (20). The inverse transformation has been possible via equations (9), (12) and (13).

ANN Model of 3-D Smith Chart
The advances in the computational sciences have made nonlinear learning machines possible, which enable to generalize discrete data into the continuous data domain. ANNs are fast and accurate nonlinear learning machines in their matured forms and capable of the parallel processing. Thus, they have found too wide applications in areas of science and engineering. Neural networks are also universal function approximators allowing reuse of the same modeling technology for both linear and nonlinear problems at both device and circuit levels. Neural network models are simple and model evaluation is very fast. The ANN architecture of the proposed "Neural 3-D Smith chart" has been demonstrated in Fig. 7. In this architecture, feed-forward Multilayer Perceptron Neural Network (MLPNN) has been used transforming (r, x) input values to (φ r , φ x , θ r , θ x ) output values. Because the MLPNNs, which have features such as the ability to learn and generalize, smaller training set requirements, fast operation, ease of implementation and therefore most commonly used neural network architectures. The next important concept is how to state the number of hidden layers for this transformation. With this purpose one hidden layer and two hidden layer MLPNNs, both having different number of neurons, have been tried to achieve small training and testing error. Two hidden layer MLPNN is able to learn selected data space more accurate which neurons of the two hidden layers have been activated by tangent-sigmoid function. The output neurons have linear functions for transferring output values of second hidden layer neurons. At last, suitable structure of the MLPNN has been 2-32-32-4.
The input-output data space has been mined using analytical equations (9), (10), (12) and (13) to train and test selected ANN structure. It has been guessed that the training data may be huge, but it must selected an interval for input parameters and it has been prefered an interval as (r, x)≤|10,000|. By the way, there has been no necessity any interval limitation for output parameters because they have trigonometrical periodicity. The data space has been divided into two parts of 50% training and 50% testing , , , d t a g a n t applications of microwave engineering designs via "Neural 3-D Smith Chart".