S Parameter Computation and Their Use for Electromagnetic Energy Wireless Transmission

Marilena Stănculescu*, Mihai Iordache*, Dragoş Niculae*, Lavinia Iordache*, Victor Bucată* University Politehnica of Bucharest , Splaiul Independentei nr. 313, sector 6, Bucuresti, ROMANIA, marilena.stanculescu@upb.ro University Politehnica of Bucharest , Splaiul Independentei nr. 313, sector 6, Bucuresti, ROMANIA, mihai.iordache@upb.ro University Politehnica of Bucharest , Splaiul Independentei nr. 313, sector 6, Bucuresti, ROMANIA, dragos.niculae@upb.ro University Politehnica of Bucharest , Splaiul Independentei nr. 313, sector 6, Bucuresti, ROMANIA, laviniabobaru@yahoo.com ABSTRACT


INTRODUCTION
In order to characterize linear devices, in harmonic state, there are used, at low and medium frequencies, the following parameters: Z, Y, H, T etc. When these devices function at high frequencies, these parameters can no longer be used, because they require particular branches to be short-circuited or open so that the voltages and the current from the circuits to be measured or computed. For example, to compute the input impedance of a two-port circuit, one has to short-circuit the output port, which practically makes impossible the measurement at high frequencies. In this case, the equipment is not able to measure the total voltage and the total current from the ports of the circuit. Also, many active circuits, such as the transistors, tunnel diodes, often have a stable functioning when the circuit is short-circuited or open. The logic variables that should be used at these frequencies are the transversal waves.
The scattering parametersdenoted by S, are complex quantities, function of frequency, associated to a multi-port linear system functioning in harmonic state. Initially, S parameters have been used in long transmission line theory, for their definition being used the transmitted direct and inverse voltage wave. In general, S parameters can be defined in information transmission systems, such as microwave (waveguides) systems, where these parameters can be studies using the circuit theory. There are many ways to introduce these parameters [1 -3] which makes their interpretation and understanding often to be difficult.
S parameters do not have a direct correspondent in electric circuit theory, because this does not contain circuit elements in which there are as propagation means the waveguides. But, in electric circuit theory, there are reasonings similar to the ones for microwave circuits, by introducing the concept of power wave, terminology which comes from their significance that is related to the dependence between the active power absorbed by a load connected to a port and the working frequency [4]. There are conversion formulas between S parameters and the classical parameters corresponding to the circuit theory (impedance Z, admittance Y, H parameter, fundamental parameters), but the literature should be careful studied because, due to a wrong understanding of the significance, some formulas are useless (e.g. the formulas from [5], article criticized in [6]). The understanding of S parameters is especially important for high-frequency applications that include the active and passive components from the passive integrated circuits [7], inclusive the microelectromechanical systems (MEMS), as well as wireless power transfer systems [8]. This paper present the correct way of defining S parameters using electrical circuits theory, and the practical use of these parameters in obtaining efficient processes for transmitting the information and of the electromagnetic energy wireless transfer from emitter-receiver signal transmission point of view. There are also presented some procedures for computing these parameters. The computation procedures use the most advanced computation programs, such as Cadence [9,11,12,22], ADS [15,19,20,24], Ansoft Extractor Q3D [23,25], Feko [10] etc. In chapter 2 there is presented the correct formulation, based on electrical circuit theory, of S parameters for a linear passive two-port in harmonic state. Chapter 3 is dedicated to automated computation procedures for S parameters, one method based on modified nodal analyses and the other on state equations method. In this chapter there is also presented the practical use of this parameters for obtaining efficient processes for transmitting the information and for electromagnetic energy wireless transfer from emitter-receiver signal transmission point of view. Finally, there are presented a few illustrative examples that certify the validity of the used computation procedures.

S PARAMETER FORMULATION (DEFINITION)
The Scattering parameters S -are used to compute the efficiency of signal transmission for microwave networks and for Transfer Power Wireless Systems (TPWS). There are efficient techniques to measure the S parameters, such as Vector Network Analyzer (VNA)) [13 -15, 24, 25], reason that recommends the use of these parameters in obtaining an efficient information transmission and propagation and an efficient wireless electromagnetic energy transfer.
To define in a correct manner the scattering parameters S for a two-port structure, let's consider the circuit given in Fig. 1. where Z0 is a real positive variable, calledcharacteristic impedance.
By similarity with the wave equation, the solution 1 a ( 2 a ) represents the incident wave from port i' -i'' (e' -e''), and 1 b ( 2 b ) is the reflected wave at the same port. For linear circuits, the variables associated to each port can be considered as a superposition of incident (direct) waves and of reflected (indirect) waves [9,11,23].
The magnitudes of the new variables have dimension AV , which shows that the square of these modules have the dimensions of an electrical power. Usually, the reference impedance is equal to the module of the load impedance. From equation (1) results: and solving function of the new variables, we obtain: The scattering parameters S of a two-port structure ( Fig. 1) satisfy the following equations between the incident and the reflected signals: The four S parameters, associated to a linear two-port circuit ( Fig. 1), are defined below: The reflection parameter from port 1 -S11 is the voltage transfer factor (amplification), computed when at input port i -i'' is connected in series a t.e.m. Ei with input impedance Zi = Zc, and at output port 2 (e' -e'') is connected an impedance Ze = Zs = Zc; The transmission parameter from port 1 to port 2 -S12 where is the voltage transfer factor (amplification), when the characteristic (reference) impedance Zc is connected to the input port i -i'' (Ei being zero) and at output port e' -e'' e=is connected the impedance Zs = Zc in series with t.e.m. Eo; The transmission parameter from port 2 to port 1 -S21 is the voltage transfer factor (amplification) from output to input, under the circumstances mentioned in (8).

The reflection parameter from 2 -
is the voltage transfer factor (amplification) when at port i -i is connected the characteristic impedance Zc (Ei being zero) and at output port e' -e'' there is connected the impedance Zs = Zc in series with t.e.m. Eo. The reflexion factor S11 and the transmission factor S21 can be measured using Vector Network Analyzer (VNA)) [13-15, 24-27].
S parameters generation, for analogue linear circuits and for nonlinear analogue circuits, piecewise linear functioning point, in precise conditions given by polarization and temperature corresponding to electronic circuits, can take place by small signal simulations. [15,27].
S parameters are defined function of characteristic impedance which is in general equal to 50 Ω.
To generate S parameters for nonlinear circuits we use the simulator -Large-Signal S-Parameter Simulation (LSSP) [15], that uses the harmonic balance method. The simulation based o harmonic balance method is a large signal simulation for which the solutions include also the effects given by the nonlinearity of electronic components. S parameters for both small and large signals are defied as ratios between the incident and the reflected wave [15].
Usually, the most important S parameters are the reflexion parameter S11 and the transmission S21, because the reflected signals efficiency is

S PARAMETERS COMPUTATION PROCEDURE
To compute S parameters, for any two-port linear system or for different structures WPTS (Wireless Power Transfer Systems) one can use any analyses program for analogue circuits.
As follows, the parameters S will be automated generated, for any WPTS system, using either the software called Circuit Symbolic Analysis Program (CSAP), based on nodal modified equations, or the software SYmbolic STATE Equation Generation (SYSEG), which uses the state equations [9,11,12].
Because the results obtained using the two programs CSAP and SYSEG are identical, as follows, we present the computation algorithm for S parameters for series-series connection configuration resonator, when for nodal modified equation generation we used SCAP software [9,11,12]. For the other three connections of the resonator magnetically coupled: series-parallel, parallel-series, and parallel-parallel, when for nodal modified equation generation we used SCAP or SYSEG program, the algorithms for computing the S parameters have the same structure as the algorithm corresponding to the series-series connection. The validation of the results obtained with the programs SCAP and SYSEG took place by comparing the corresponding results with the ones obtained using ADS software [15].

S parameters computation algorithm description for series-series configurarion (RpoCMSS)
P1. Computation of parameters S11_ss and S21_ss Fig. 2: The circuit used to generate the parameters S11_ss and S21_ss.
In the input file RCMSS_S11_S21.smbfor SCAP (RCMSS_S11_S21.datfor SYSEG) were kept Ri and RL to be able to compute the efficiency of the active power transmission from port 1 to port 2: ç21_ss = 100*PRL_ss/P1_ss, where PRL _ss= RL*IRL_ss^2 and P1_ss = Re(Ei*conjugate(Ii_ss)). The signals transmission efficiency from port 1 to port: 2 , then by using the definition formula S11_ss = 1.0 -2.0*Zc*Yii it is obtained the expression of reflection coefficient from port 1, function of the parameters of the circuits from Fig. 2 and of the frequency f; P1.2. To compute the S21_ss parameters, one generates, for the circuit given in Fig. 2, the voltage transfer factor (amplification) , then using the definition formulae S21_ss= -2.0*Aoi results the expression of the transmission coefficient from port 1 to port 2, function of the parameters given in Fig. .2 Fig. 3 a and b, the variations with respect to the frequency of the modules of the parameters S11_f_ss and S21_f_ss, computed using the programs ASINOM and SYSEG -Fig 3. a and resulted after using ADS program - Fig. 3, b. In Fig. 4, a and  P2. Computation of parameters S12_ss and S22_ss Fig. 5: The circuit used to determine the coefficients S12_ss and S22_ss.
For scattering parameters computation S12_ss = -2.0*Aio and S22_ss= 1.0 -2.0*A2o is analysed in harmonic state, using either SCAP or SYSEG program, the circuit given in Fig.5, powered at port e'-e'' with condition that Ri = RL = Zc (Rc). In the file RCMSS_S12_S22.smb (RCMSS_S12_S22.dat) were kept Ri and RL in order to be able to compute the active power transmission efficiency from 2 to port 1 ç12_ss = 100*P1_ss/P2_ss, where P1 _ss= Ri*I1^2 and P2_ss = Re(Eo*conjugate(I2)). The signals transmission efficiency from port 2 to port 1 * ss _ S _ S S . 12 12 12 0 100 12     is computed for Ri = RL = Zc and it is obviously not identical to the active power transmission efficiency ç12_ss.
P2.1. To generate the parameter S12_ss we compute first the voltage amplification , then by using the definition formula S12_ss= -2.0*Aio results the transmission coefficient expression from port 2 to port 1, function of the circuit parameters from Fig. 5 and the frequency f. P2.2. To compute the parameter S22_ss is generated from the circuit Fig. 5, the voltage transfer , then using the definition S22_ss= 1.0 -2.0*A2o results the expression of the reflexion form port 2, function of the circuit parameters from Fig. 5 and the frequency f. In Fig.10, a-f and in Fig. 11

CONCLUSIONS
Transfer Power Wireless Systems (TPWS) is a new technology, used when electromagnetic energy transmission is not possible for certain reasons (difficult allowable places, recharging implant batteries, etc.), through the conductors. The energy transfer using this procedure can take place at any distance for which the electromagnetic field is strong enough, such that to allow a reasonable energy. This is possible if booth the emitter and the receiver operate at resonance, because the resonant systems exchange energy much efficient that the non-resonant ones.
Modern applications of telecommunications systems ( transfer of information) is based on the propagation of electromagnetic waves, but the radiating antenna technology is not suitable for power transfer efficiency because its efficiency is highly reduced ( a large part of the energy is lost through dispersion into the environment ).
The scattering parameters S are very useful for computing the signals transmission efficiency between two magnetic coupled resonators: In general, the active power transmission efficiency 21