Modeling Method of a Low-Pass Filter Based on Microstrip T-Lines with Cut-Off Frequency 10 GHz by the Extraction of its Wave-Scattering Parameters from its Causal Bond Graph Model

Problem statement: This study presented a jointly application of bond graph technique and wave-scattering formalism for a new realization cal led scattering bond graph model which has the main advantage to show up explicitly the different wave propagation. Approach: For that, we proposed to find the scattering matrix from the cau sal bond graph model of a low-pass filter based on Microstrip lines and with cut-off frequency 10 GHz, while starting with determination of the integrodifferentials operators which is based, in their de termination, on the causal ways and causal algebrai c loops present in the associated bond graph model an d which gives rise to the wave matrix which gathers the incident and reflected waves propagatio n of the studied filter. Results: The scattering parameters, founded from the wave matrix, will be c h cked by comparison of the simulation results. Conclusion: Thereafter, we use a procedure to model directly th is scattering matrix under a special bond graph model form often called “Scattering Bond Graph Model”.


INTRODUCTION
The scattering or the wave-scattering formalism (Paynter and Busch-Vishniac, 1988) was used in vast physique fields such as the characterization of the electric circuits.
Several work, since the invention of the bond graph approach (Di-Filippo et al., 2002), showed that the scattering formalism (Newton, 2002) constitutes an alternative approach for the physical systems modeling (Kamel and Dauphin-Tanguy, 1993). They pointed out on the one hand some properties and in particular the orthogonality of wave matrix (Magnusson, 2001) respectively the scattering matrix (Ferrero and Pirola, 2006) which respects intrinsically the causal relations and includes explicitly the conservation laws (Pedersen, 2003). They showed in addition that the scattering representation exists for systems having neither impedance nor admittance such as the junctions of Kirchhoff, the gyrateurs and the transformers (Patrick and Adrien, 2008).
In addition, the bond graph language (Di-Filippo et al., 2002) is based on a graphic representation of the physical systems. These representations are based on the identification and the idealization of the intrinsic characteristics of the physical environments and on the structuring of a complex physical system in the networks form (Belevich, 1968). Moreover, in physics, the analogies theory allows bond graph technique (Di-Filippo et al., 2002) to generalize the representation networks with all the traditional physics fields of the systems with localized and/or distributed (based on Microstrip lines) parameters (Byrnes et al., 1999).
The propose of this study is to present and apply a new extraction method of the scattering parameters, which constitute the scattering matrix, of any physical systems while basing on its causal and reduced bond graph model (Taghouti and Mami, 2009).
At first and after having to present the method, we propose to use the causal bond graph model of a lowpass filter based on Microstrip lines (distributed elements) (Trabelsi et al., 2003) to find, on the one hand, the integro-differentials operators (Khachatryan and Khachatryan, 2008) which based on the causal ways and algebraic loops present in the causal bond graph model and, on the other hand, to extract the wave matrix (Magnusson, 2001) from these operators.
Then, we extract directly the scattering parameters (Newton, 2002) from the found wave matrix (Magnusson, 2001) and, at the aim to validate the found results; we make a comparison by the simulation of these scattering parameters under a simple program and under the classic techniques of conception and simulation of the microwave circuits by using the HP-ADS software (Vendelin et al., 2005).
Finally, we propose to build a special form of bond graph model often named "Scattering Bond GraphModel" (Kamel and Dauphin-Tanguy, 1996) which able to highlight these transmission and reflexion coefficients (Scattering parameters) (Duclos and Clement, 2003).

Extraction method of the scattering parameters:
The new extraction method of the scattering parameters (Newton, 2002) is an analytical method which makes it possible to establish, for a linear complex system, the scattering relations between a fixed entry and exit. However, this method implies the succession of the following stages: • Decomposition of the system (causal bond graph model) in subsystems (causal bond graph submodel) put in cascades which are characterized by their respective wave matrix (Magnusson, 2001) • Calculating the total wave matrix (Magnusson, 2001) of the whole system by carrying out the product of the elementary wave matrices • Finally, the application of a linear transformation to this total matrix for extracting the scattering parameters (scattering matrix) (Newton, 2002) characterizing the complex system The step that we propose was thought in this objective and consists in establishing a systematic method which binds the bond graph technique (Di-Filippo et al., 2002) to the wave-scattering formalism (Paynter and Busch-Vishniac, 1988). This method is based on an algebro-graphic procedure (Amara and Scavarda, 1991) which uses the causal ways notions and the Mason's rule (Bolton, 1999) applied to a causal bond graph transformed and reduced (Taghouti and Mami, 2009).
Wave-scattering decomposition and representation of complex system: Generally, we can represent any complex system functioning in low or high frequency by the following model of the Fig. 1 where the process is represented by the quadripole Q with different wave scattering. These three subsystems (source, quadripole (Q) and load) are inter-connected and communicate between them by the means of a power transfer which is done in a continuous way from the source to the load as Fig. 1 indicates it.
It is considered that the process, in its quadripole form, is in complex form and can be decomposed to subsystems which are connected by the intermediary of the wave-scattering variables (Paynter and Busch-Vishniac, 1988) as Fig. 2 indicates it.
The incident and reflected waves distribution, which is propagated from the source towards the load through the quadripole Q, can be translated by a matrix representation of the wave scattering variables like Fig. 3 indicates it, where W is the wave matrix.
Let us consider the two processes A and B with share where the signal entering B is directed in the same direction as the outgoing signal of process A, a similar way, the outgoing signal of B is in the same direction as the signal entering a as Fig. 3 indicates it. If these two processes are coupled, the assumption of the power continuity (Paynter and Busch-Vishniac, 1988) will imply: The a iA and a iB quantities entering the A and B processes are called incident waves in the same way, the quantities b iA and b iB associated with the signals leaving the A and B processes are called reflected waves (Duclos and Clement, 2003).
Classically, we express the power circulating in a bond and connecting two systems in the shape of a product of the two variables effort (noted: ε) and flow (noted: ϕ) in reduced form (Amara and Scavarda, 1991): So we can introduce the following linear transformation: The linear opposite transformation of the H transformation allows the passage of the intrinsic variables effort and flows (ε,ϕ) with the wave variables (a i , b i ) as the following relation indicates it: The processes A and B constitute two processes with 2-ports of entry and exit whose wave matrices are: The chain of n processes with 2-ports of entry and exit, like Fig. 3 indicates it, constitutes a process with 2-ports of entry and exit where the global wave matrix W is: So: The scattering parameters are given by the following scattering matrix: The relations between these matrixes are given by these following equations: Wave scattering parameters and causal bond graph model: It is considered that the process, in its quadruple form and which inserted between two particular ports P 1 and P 2 which represent respectively the entry (source) and the exit (load) of the complex system can be represented by the following bond graph model transformed and reduced such us: • ε 1 and ε 2 are respectively the reduced variable (effort) at the entry and the exit of the system • φ 1 and φ 2 are respectively the reduced variable (flow) at the entry and the exit of the system These are the reduced Effort (e) and Flow (f) with respect to R 0 (scaling resistance). and to establish the output-input analytical relations, the bond graph model of the studied system must be transformed, reduced and especially be causal since these relations rest on the concepts of causal way and causal algebraic loops which can comprise the reduced bond graph model. The causality assignment to the reduced bond graph model of Fig. 4 enables us to notice that they are four different cases of causality assignment in inputoutput of the process (Amara and Scavarda, 1991).
For each type of reduced and causal bond graph model given below, we will have one matrix which connects the reduced variables to the integrodifferentials operators H ij .
From each matrix, we can deduce the corresponding wave matrix by referring to the Eq. 5, 6 and 10.
These wave matrices can give us the corresponding scattering parameters by referring to the Eq. 13 and the following equations: Reduced bond graph model with flow-effort causality (Case 1): The causality assignment on the model of Fig. 4 makes it possible to find the first case (Case 1) given by Fig. 5       From this model, we deduce the H matrix of integro-differentials operators and his corresponding wave matrix W:

Reduced bond graph model with effort-effort causality (Case 4):
The causality assignment on the model of Fig. 4 makes it possible to find the fourth case (Case 4) given by Fig. 8 Now, if we like to find the scattering matrix we need to use the Eq. 13.
We note that H ij are the integro-differentials operators which are based, in their determination, on the causal ways and algebraic loops present in the associated bond graph model (Amara and Scavarda, 1991): Where: ∆ = The determinant of the causal bond graph H ij = Complete gain between P j and P i P i = Input port P j = Output port N = Total number of forward paths between P i and P j G k = Gain of the k th forward path between Pi and P j L i = Loop gain of each causal algebraic loop in the bond graph model L i L j = Product of the loop gains of any two nontouching loops (no common causal bond) L i L j L k = Product of the loop gains of any three pair wise nonteaching loops ∆ k = The cofactor value of ∆ for the k th forward path, with the loops touching the k th forward path removed; i.e., Remove those parts of the causal bond graph which form the loop, while retaining the parts needed for the forward path

Application to a low-pass filter based on microstrip lines:
In this paragraph, we will try to apply and check the procedure described previously to a low-pass filter with distributed elements (based on Microstrip lines) like Fig. 9 indicates it. This filter is a Tchebychev low-pass with order 4 and having the following characteristics: • Cut-off frequency (f c ) 10 GHz • Sensibility: k = 0.5 • Attenuation: Amax = 0.1dB, Amin = 20dB The bond graph model of this studied filter is given by Fig. 10.

Extraction of the scattering parameters from the bond graph model:
So that, to extract the scattering parameters from the bond graph model and by using the new extraction method which is described previously; we must transform the bond graph model given by Fig.10 into a causal bond graph model often named reduced bond graph model like Fig.11 indicate it (Amara and Scavarda, 1991;Taghouti and Mami, 2009) only containing the reduced variables with respect to a scaling resistance R 0 (internal source resistance) such us: The bond graph model given by Fig. 11 can be broken up into tow cells (sub-model) put in cascade form while respecting the assumption of the power continuity (Paynter and Busch-Vishniac, 1988) between all submodels.
Each cell is made up with an impedance z in parallel with an admittance y often noted [z---y], if the studied filter is with T form, or [y---z] if the studied filter is with Π form (type).
So we have the first sub-model given by Fig. 12, this model is in conformity with case 1 described previously. So we have the integro-differentials operators by taking account to the previously equations:  From these operators, we can deduce directly the wave matrix W (1) of the first sub-model of Fig. 12 by taking account to equations of case 1: (1) 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 z y z y 2 z y z y W z y z y z y z y and now we have the second sub-model given by Fig. 13 and in the same manner we can extract the second wave matrix W (2) such us: and the second wave matrix is: (2) 2 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 z y z y 2 z y z y W z y z y z y z y The total wave matrix W (T) is given by the product of the first and the second wave matrix such us: (T) (1) (2) 11 12 So the corresponding scattering matrix S (T) is given below: From this matrix we can deduce these following scattering parameters: 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 11 z z y y z y (y z ) z (y y 1) z (y y 1) y y s d(p) The Fig. 11, represent the reduced bond graph model of the studied filter before decomposition.
The Fig. 12 and 13 represent respectively the first and the second sub-model of the studied filter; they are given by the decomposition of the reduced bond graph model of Fig. 11 at the appropriate bond (place).

RESULTS
Thus, the validation is carried out by simulate the scattering parameters of Eq. 37-39.
The Fig. 18 thus represents the system's model studied with adapted entry and exit and the numerical values of its elements necessary for simulation.
The simulation of the low-pass filter above gives the graphical representation of the reflexion and transmission coefficients S ii and S ij (i ≠ j and i, j = 1...2) according to the frequency like Fig. 19 indicate it. By observing Fig. 19 and Fig. 14-17, we can say that our method of extracting the wave matrix from a reduced bond graph model is validated because it give us the same simulation results.
Modeling of the scattering parameters with bond graph technique: We noted that we will use the method which is developed by Kamel and Dauphin-Tanguy (1993;1996). Procedure used to model the scattering matrix by the bond graph technique: The scattering matrix of our studied process is a 2-2 matrix having a particular form whatever the expressions complexity of the series impedance or parallel admittance; it is orthogonal since the process is considered without loss and it admits the following general form: We note that: Indeed, if we consider the scattering matrix form found above for the process alone, we can say that it is not a true transfer matrix (Belevich, 1968) moreover, it is not in the adequate form since its various S ii parameters and sometimes S ij have the numerator's degree equal to that of denominator and that poses a major problem to determinate the scattering bond graph model of any physical system studied in a general way.
The solution with this problem is to regard the scattering matrix of a process as a transfer matrix from an input-output point of view, connecting the incident and the reflected waves in a symbolic system form.
We start by carrying out an Euclidean division of each term of the numerator matrix (scattering parameters) by the common denominator d(p) what leads to the new shape of the scattering matrix (Breedveld, 1985) such as: S' = The new scattering matrix with degrees in the numerator at most one less than that of d(s) D = Direct transmission matrix: Thereafter we seek for the new matrix S' its development in continuous fraction in alpha-beta starting from the Routh method (Shamash, 1980) and build the corresponding bond graph model, since it is about a multivariable system (Molisch et al., 2002) while being based on the systematic procedure according to: • Calculate the α- Routh table from  It is interesting to notice that the structure of the scattering bond graph of the process remains the same whatever the degree of the common denominator of scattering matrix. The only thing that changes is the corresponding number of I and C linked to the α-Routh expansion (Kamel and Dauphin-Tanguy, 1993;1996).
The alpha-beta coefficients are then: From d(p) we have: , , and β β β β

DISCUSSTION
We sought to highlight the incident and reflected waves and their propagation on a bond graph called "scattering bond graph" which, in addition to the simplicity of its structure which remains unchanged about or the complexity of the studied system, that it has a physical interpretation and easier to handle than an abstracted mathematical model and then it proposes a temporal approach phenomena usually modeled with the frequency tools.
Moreover all the panoply of the techniques and properties of a bond graph can be used in a unified way with the service of a formalism which used from the low frequencies to highest. Moreover, this methodology which makes it possible to study simultaneously a given system with two complementary formalisms can supports us a broader comprehension of its behavior.

CONCLUSION
In this study, we tried to present a method which appears new for the determination of the scattering parameters of any physical system functioning in high frequency.
Then, we applied this technique to a low-pass filter based on Microstrip lines.
Lastly, we validated the results found by a simple comparison between two methods of simulation: simulation by the traditional methods used in microwave under the HP-ADS software and simulation by our own method of the reduced bond graph which is based on the causal and simplified bond graph model of the studied system like on the minimum of the causal ways and loops present in this model often decomposed to sub-models as we showed previously.
Generally, this new analysis method lead us, to use this new method which combines at the same time the bond graph technical and the scattering formalism for modeling and simulation of the scattering matrices of any electrical circuits often functioning in high frequency and based on localized or distributed elements giving rise to the famous model often named: Scattering bond graph.
This new type of modeling will enable us to capture the power transfers in a simple and direct manner at the same time and it proposes us a temporal approach of the phenomena usually modeled with the frequencies tools.