ANALYSIS OF NON-UNIFORM TRANSMISSION LINE USING THE DISCRETIZATION THEORY

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1 JAE, VOL. 5, NO., 3 ANALYSIS OF NON-UNIFORM TRANSMISSION LINE USING THE DISCRETIZATION THEORY L.Kumar*, H. Parathasarathy**, D.Upadhyay** Netaji Subhas Institute of Technology, Dwarka, University Of Delhi, India L.Kumar*, Persuing M.Tech(Signal Processing), Division of Electronics and Communication Engg, Dwarka, New-Delhi, India. lalitkumar_khanna@yahoo.com H. Parathasarathy**, Professor, Division of Electronics and Communication Engg, Dwarka, New-Delhi, India. D.Upadhyay**, Assitant Professor, Division of Electronics and Communication Engg, Dwarka, New-Delhi, India. Abstract In this paper, the analysis of voltage waves that propagate along a nonuniform transmission line is presented. For this analysis, the primary constants of non-uniform transmission lines are assumed to be slowly varying function of the distance (z) along the line from the source end. Now the first order linear partial differential equation for the voltage waves along the line is obtained by using KVL and KCL equations for the infinitesimal length Δz of the line. Further it is also seen that this pair of equations is a linear time invariant system but not shift invariant system w.r.t the distance (z) along the non-uniform transmission line. In order to solve this pair of system equations, we eliminate the current variable thereby obtaining a second order differential equation in z with variables coefficients. The partial derivatives are simply replaced by multiplication with j for going over to the Fourier domain w.r.t time. The final problem thus reduces to solving a linear, second order ordinary difference equation with non-constant coefficients and also boundary conditions determined at the source end and load end. Now the equation is approximated by a second order differential equation by the discretization approach w.r.t distance (z). Finally the effectiveness of the proposed analysis is simulated for different cases in MATLAB.

2 ANALYSIS OF NON-UNIFORM L.KUMAR, H. PARATHASARATHY, D.UPADHYAY. INTRODUCTION In [, ] a genetic algorithm for estimating the non-uniform transmission line parameters based on comparison of the theoretical scattering parameters and the actually measured scattering parameters is proposed. However they do not take into account slow variations of the line parameters along its length. Our method of analysis based on discretization theory takes into account possible variations and can be used to develop an algorithm for estimating the slow variation of the distributed parameters. An idea of how to go about this process is to assume a parameter model of the distributed parameters function like Equations () - (4). R z R A A n (,, ) () R A ( z) R ( z) () n k k Where R, {A k } are the unknown parameters and {R (z)} are known test functions and likewise for L( z, L, B B n ) (3) n k k L B ( z) L ( z) (4) k k G(z) and C(z). The theoretical parameters for this two port network are obtained by application of discretization theory. These parameters will be functions of A. A k, R, B. B k, L etc and can be matched with the experimentally measured parameters for estimation. In [3-5], non-uniformity of the line parameters has been considered. They assume a cubic polynomial variation of the characteristic impedance along the line. The expression z ( x ) = Lx ( ) Cx ( ) for the characteristic impedance at x meters from the source end gives the correct differential equation to be solved. The characteristic impedance z( x ) of the line appears in the equations given below. The authors have taken z ( x ) to be a third order polynomial in x. The corresponding second order differential equations are given below. V z x V L C V x z x x x.....

3 JAE, VOL. 5, NO., 3 I z x x z x x x I.... L C. I Likewise they have expanded the voltage and current as a power series in x and obtained approximate expressions for the scattering parameters. However our method based on Discretization theory is more general and accurate since it can deal with situation when the parameters are not Taylor expandable or when a very large no of terms is needed to get a sufficiently good approximation in the Taylor series. Using appropriate test functions R k ( x ), L k ( x ) etc a very good approximation can be obtained using Discretization theory provided the step size is sufficiently small. Discretization theory converts the infinite dimensional differential equation model into a finite dimensional matrix problem. The only drawback with our discretization based approach is that it fails when the parameters variations are too rapid. The broad philosophy of our approach is to simulate the line dynamics for each choice of the unknown parameters using discretization and search for the optimal parameter values that give closest approximation for the measured scattering parameters. In [6-8], design of artificial transmission lines using monolithic microwave integrated circuits is proposed where in short subsections of different lines are connected to realize the required performance. The scattering matrix is then obtained by multiplication of the scattering matrices of each subsection. If the number of sections is too large, then it may be more comfortable to use the discretization procedure rather than multiplying an infinite sequence of matrices. Moreover, slow variations of the parameters in each subsection may be treated accurately by further discretization of each subsection. In [9, ], a study of the whole system of transmission lines is made. Each line in this interconnected network is assumed to have a constant R, L, G, and C per unit length. Our analysis of inhomogeneous lines based on discretization can be applied to more general networks in which each line in the network has inhomogeneous distributed parameters. 3

4 ANALYSIS OF NON-UNIFORM L.KUMAR, H. PARATHASARATHY, D.UPADHYAY In [], the equations of the non-uniform transmission line are transformed into those of a uniform transmission line. This method involves using distributed sources and complicated analytical tools are developed for this purpose. Our method based on the discretization approach is considerably simpler though less accurate since the discretization step size should be made sufficiently small in order to gain sufficient accuracy. The basic idea in [] is to replace the impedance and admittance per unit length by their average values and then to obtain expressions for the voltage and current along the line in the terms of the error occurred in replacing actual values by values averaged over the line. This algorithm can be iterated and it is expected to converge quickly provided that the variations in the impedance and admittance per unit line length are not too rapid. In [], non-uniform micro-strip lines are synthesized. The basic idea here is to develop the characteristic impedance along the line into a Fourier series with fundamental period being the total line length. The coefficients in this Fourier series are obtained by minimizing an error function at different frequencies. The error function is obtained from the input reflection coefficients at several frequencies. The authors have however not considered the analysis problem, namely the converse of the synthesis problem i.e., determining the voltage and current along the line in terms of the Fourier series coefficients. Our method can be adapted to solving the converse problem. In [3], a novel method has been proposed for solving for the voltages and currents along a non-uniform transmission line. The idea is to expand the impedance and admittance along the line as a power series in z d where z is the length along the line and d is the total length of the line. The voltages and currents along the line are also expanded as a Taylor series in z d. These are substituted into the differential equations of the line and coefficients of equal powers of z are compared resulting in iterative equations for the Taylor series coefficients of the voltages and current variables. This method however fails when there are sharp discontinuities in the distributed parameters that prevent a Taylor expansion or if there are singularities in the parameters. Our discretization based method will work even in such situation. 4

5 JAE, VOL. 5, NO., 3 In [4], the line parameters as well as the voltage and current along the line are expanded as a Fourier series in z with fundamental period d and substituted into the line differential equations. Comparing the differential coefficients of each harmonics in z gives a sequence of linear equations for the Fourier coefficients of the voltage and current with coefficients determined from those of the line parameters. This is an infinite sequence of linear relations and can only be approximately solved after truncation. In this paper we have done the analysis and simulation of voltage wave that propagate along a transmission line having non-uniform parameters. In this resistance, Inductance, capacitance and conductance per unit length (Respectively,R(z),L(z),C(z),G(z)) are assumed to be slowly varying functions of z, One end of the line is connected to a voltage source and the other end is terminated by a load. We start by writing down the basic KCL and KVL equations for an infinitesimal length Δz of the line. These equations are linear first order partial differential equations for the voltage v(t, z) and current i(t, z) along the line, owing to the variation in z of distributed parameters. The partial derivatives with respect to time are simply replaced by multiplication with j by going over to the Fourier domain w.r.t to time. This equation is approximated by a second order differential equation by the process of discretization w.r.t to z. This difference equation is rewritten as a matrix equation whose solution is obtained using MATLAB. We have in this work developed a technique for analyzing non-uniform transmission line, by fixing a temporal frequency, specially, we have developed an algorithm for computing the voltage phasor V (, z) along the line. The ratio H (, z) =V (, z)/v (, ) is the transfer function for the voltage between the source and the point z. The magnitude and phase of this transfer function has been plotted in simulations. Further, if the source voltage is a random process having power spectral density S(w,),then the voltage at z is a random process having power spectral density S(,z)=(H(,z)* H(,z)) S(,). In our simulations, we have estimated S(,z) for different z using time averaged correlations and have thus obtained estimates for (H(,z) * H(,z)).These values have then been compared with the theoretically predicted values. 5

6 ANALYSIS OF NON-UNIFORM L.KUMAR, H. PARATHASARATHY, D.UPADHYAY This paper is organized in such a way that section covers the discretization of nonuniform transmission line. Section 3 covers the simulated results with explanations. Section 4 covers the conclusion and Section 5 covers the references.. DISCRETIZATION OF NON-UNIFORM TRANSMISSION LINE Figure, shows the basic differential equations governing the voltage and current along a transmission line whose distributed parameters R, L, G and C which are varying along the line. The differential equations of non-uniform transmissions line are given in equations (5) and (6). Figure Discretization model of non-uniform transmission line From Equation (5) dv () z ( Rz jl( z)) I( z) dz (5) di() z ( Gz jc( z)) V ( z) dz (6) V () z I() z R( z) jl( z) (7) Substituting Equation (7) into Equation (6) d V () z ( G( z) jc( z)) V ( z) dz R( z) jl( z) V ( z) R ( z) jl ( z) V ( z) ( G( z) jl( z)) R( z) jl( z) R( z) jl( z) (8) (9) R ( z) jl ( z) V ( z) V ( z) (( G( z) jc( z))( R( z) jl( z)) V ( z) () R( z) jl( z) Where (( G( z) j C( z))( R( z) j L( z))) () 6

7 JAE, VOL. 5, NO., 3 Where Let Where Where = function of z R ( z) jl ( z) A (z) = R( z) jl( z) B (z) = R( z) R R sin( z) () L( z) L Lsin( z ) 6 (3) C( z) C Csin( z ) 6 (4) G( z) G G sin( z) (5) R L G C, R, L, G, C n Equation () can be represented given by Where ( ) ( ) ( ) ( ) ( ) (6) V z A z V z B z V z z n n,,... N N d After Discretization, this second order differential equation becomes a second order difference equation given by An V n V n Vn Vn Vn BnVn (7) V V V A V V B V (8) n n n n n n n n ( Rn jln) An ( Rn jln B G jc ( R jl ) Then Equation (8) will become n n n n n Vn ( Bn An ) Vn ( An ) Vn (9) Initial Condition V 7

8 ANALYSIS OF NON-UNIFORM L.KUMAR, H. PARATHASARATHY, D.UPADHYAY Final Condition (LOAD) Vn Z LOAD. I( d) But V ( d) Id ( ) () R( d) jl( d) Vn Vn ().( Rn jln) Vn Vn Vn Z LOAD. ().( R jl ) n n Z LOAD Vn Z LOAD. (3).( Rn jln ).( Rn jln ) Where V N V (4) N Z LOAD Z LOAD.( R jl ) N N 3.SIMULATED RESULTS We have simulated the propagation of voltage along a transmission line having non-uniform distributed parameters. We started by writing down the basic transmission line equations by applying KVL and KCL to infinitesimal section of the line. Then eliminating the current variable, we arrived at a single second order linear ordinary differential equation with non-constant coefficient for the voltage phase at a given frequency. We approximated that differential equation by a difference equation and solved it using MATLAB. Our simulation results show the voltage along the line as a function of the distance from the source and frequency variable. Figure shows the plot of voltage with distance(z) when the distributed parameters and frequency both are constant. This shows that the voltage is decreasing exponentially w.r.t to distance (z). Figure 3 shows the plot of voltage with distance(z) when the distributed parameters are non-constant but frequency is constant. This shows that the decrease in voltage variation w.r.t to distance (z) is exponential. Figure 4 shows the three dimensional plot of the voltage phasor as a function of both the variables frequency and distance from the source end. This shows that there is variation in voltage in normalized distance range of to 5 but in the normalized frequency range of to there is no more variation in voltage. This three dimensional plot can be used to 8

9 JAE, VOL. 5, NO., 3 determine spectral density of voltage waveform along the line when the input voltage is a stationary random-process. This can be used to estimate the distributed parameters from measurement of correlation of the voltage waveform along the line. We have considered six different cases for both fixed values and varying values of primary constants. The corresponding simulated results are given in Figures 4-9. The cases are given below Case : When /RC = / LC where variations are in the primary constants derived from Equations ()-(5) with normalized frequency and distance both. Result for the case is already discussed by using the Figure 4. Case : When /RC >> / LC where variations are in the primary constants derived from Equations ()-(5) with normalized frequency and distance both. Figure 5 illustrates three dimensional plot of the voltage phasor as a function of both the variables frequency and distance from the source end. This shows that there is voltage strength in normalized distance range from to 35 and negligible in remaining distance range but in the normalized frequency range of 55 to 75 voltage strength is decreasing, at starting and at the end, it is increasing. Case 3: When /RC << / LC where variations are in the primary constants derived from Equations ()-(5) with normalized frequency and distance both. From Figure 6 it is observed that the results are almost similar as in case or Figure 4. Case 4: When /RC = / LC for fixed primary constants with normalized frequency and distance both. Figure 7 illustrates a three dimensional plot of the voltage phasor as a function of the variables, frequency and distance from the source end. This shows that the voltage strength is compared to other cases mentioned above is high for all normalized distances. Case 5: When /RC >> / LC for fixed primary constants with normalized frequency and distances both. Figure 8 illustrates three dimensional plot of the voltage phasor as a function of both the variables frequency and distance from the source end. This shows that there is variation in voltage in normalized distance range of to 7 and above the distance range 7 voltage strength is negligible but in the normalized frequency range of to 9

10 ANALYSIS OF NON-UNIFORM L.KUMAR, H. PARATHASARATHY, D.UPADHYAY variations in voltage is negligible and after that there is a comparatively high strength. Case 6: When /RC << / LC for fixed primary constants with normalized frequency and distances both. From Figure 9 it is observed that the results are almost similar as in case 4 or Figure Voltage Voltage z Figure Voltage variations with normalized distance (z) for fixed frequency and primary constants > z(distance) Figure 3 Voltage variations with normalized distance (z) for fixed frequency but varying primary constants..5.8 Voltage Voltage z(distance) z(distance) w(frequency) Figure 4 Voltage variations with normalized frequency and distance for variable primary constants (/RC=/ LC). w(frequency) Figure 5. Voltage variations with normalized and distance for variable primary constants (/RC>>/ LC).

11 JAE, VOL. 5, NO., 3.8 Voltage Voltage z(distance) z(distance) w(frequency) Figure 6 Voltage variations with normalized Frequency and distance for variable primary Constants (/RC<</ LC). w(frequency) Figure 7 Voltage variations with normalized and distance for fixed primary constants (/RC=/ LC). 4.CONCLUSION This work deals with the analysis of transmission lines having inhomogeneous distributed parameters. A simple method is proposed for the analysis of non-uniform transmission line by using the discretization method. In this paper several cases have been discussed to see the effectiveness of the proposed analysis. It is observed that the case 4 for fixed primary constants with the conditions of /RC / LC, the voltage strength is available on the whole frequency range as compared to the other cases. Therefore such type of transmission line is always preferable for longer distance communication. Further it is also observed that the case /RC >> / LC for fixed primary constant is preferable only up to normalized distance (z) = 35. But the other case /RC << / LC with the variation in the primary constants is preferable only for the normalized distance range of distance (z) = 6 to 7.This method is much simpler and can be used for any non-uniform transmission line. It is hoped that this method would lead to algorithms for estimating the distributed parameters of the system from voltage measurements. REFERENCES [] Jianmin.Zhang, Marina Y. Koledintseva, James L. Drewnik, Extracting R, L, G, C Dispersive Planar Transmission Lines from Measured S-Parameters using a genetic Algorithm IEEE International Symposium on Electromagnetic Compatibility Rolla, MO, vol.,pp Aug 4. [] J.Zhang, M.Y.Koledintseva, G. Antonini, K.N. Rozanov, J. L. Drewniak, and A.Orlandi, Reconstruction of the parameters of Debye and Lorentzian dispersive

12 ANALYSIS OF NON-UNIFORM L.KUMAR, H. PARATHASARATHY, D.UPADHYAY media Using Genetic algorithm, Proc of the 3 IEEE International Symposium on Electromagnetic Compatibility Boston, MA, pp Aug 3. [3] Le Roy M, A. Perennec, S. Toutain, L.C. CALVEZ, A New Design of Microwave Filters By Using Continuously Varying Transmission line IEEE MIT- S International Microwave Symposium Denver, CO, vol., pp jun 997. [4] C.W,Hsue and C.D. Hechtman, Transient analysis of non-uniform High-pass transmission line, IEEE Trans. Microwave Theory Tech., vol. 38, pp.3. [5] C.W.Hsue, Time-domain scattering parameters of an exponential transmission line, IEEE Trans. Microwave Theory Tech., vol.39, pp , Nov. 99. [6] Xu Yansheng and Rento G. Bossio, Analysis and Design of Novel Structures of Artificial transmission lines for MMIC/MHMIC Technology IEEE Trans. Microwave Theory Tech, vol.47,pp.99-,jan.999. [7] T.Hiraoka, T. Tokumitsu, and M. Aikawa, Very small wide-band MMIC magic-t using micro-strip lines on a thin dielectric film, IEEE Trans. Microwave Theory Tech., vol.3, pp , Oct [8] M Gillick, and I. D. Robertson, Ultra low impedance CPW transmission lines for multilayer MMIC s, IEEE MTT-S Int. Microwave Symp. Dig., 993, pp [9] El-Deeb,Nabil.A, Member, IEEE, Esmat A. F. Abdallah, and Mohamed B. Saleh, Design Parameters of Inhomogeneous Asymmetrical Coupled Transmission Lines IEEE Trans. on Microwave Theory and Tech, vol.mit-3,pp , July [] T.G Phillips and K. B. Jefferts, A low temperature bolometer heterodyne receiver for millimetre wave astronomy, Reu. Sci. Inst., vol. 44, pp 9-4, Aug [] Khalaj-Amirhosseini, M., Analysis of non-uniform transmission lines using the equivalent sources, Progress In Electromagnetics Research. PIER 7, 95 7, 7. [] Khalaj-Amirhosseini, M., Wideband or Multiband complex impedance matching using microstrip non-uniform transmission lines, Progress In Electromagnetics Research. PIER 66, 5 5, 6. [3] Khalaj-Amirhosseini, M., Analysis of coupled or single non-uniform transmission lines using the Taylor s series expansion, Progress In Electromagnetics Research. PIER 6, 7-7, 6.

13 JAE, VOL. 5, NO., 3 [4] Khalaj-Amirhosseini, M., Analysis of periodic and aperiodic coupled nonuniform transmission lines using the Fourier series expansion, Progress In Electromagnetics Research. PIER 65, 5-6, 6. 3