200 1 IEEE - Russia Conference : MEMIASOO 1 99

DIRECT RECONSTRUCTING VOLTAGE-CURRENT DISTRIBUTION IN ARBITRARY NONUNIFORM

TRANSMISSION LINE B.Yu. Kapilevich

Department of Applied Electromagnetics, Siberia State University of Telecommunications and Informatics, Novosibirsk, Russia

Abstract - Reconstructing voltage-current distribution in arbitrary nonuniform transmission line is presented on the basis of direct solution of generalized telegraphic equation with proper boundary condition. The details of reconstructing procedure are also given. Wronskian determinant behavior is used to estimate an accuracy of the reconstruction process.

1. Introduction.

Nonuniform transmission lines (NTL) are widely used in different RF and Microwaves devices. Their study is basically carried out using nonlinear Riccati equation [ 11 that can be solved analytically in very limited impedance profiles such as exponential NTL’s and power-law NTL’s. As a result, the development of numerical technique oriented to NTL analysis and synthesis is playing an important role to provide designers with better understanding of their properties and behavior. In this letter a universal numerical approach is proposed for direct reconstructing voltage-current distribution along arbitrary NTL. It is based on direct solution of a proper boundary problem associated with NTL’s telegraphic equation. Both lossless and lossy NTL’s can be investigated without any limitations on profiles of the impedance and propagation coefficient. As an illustrative example, voltage distributions are presented for parabolic impedance profiles of microstrip lines.

2. Formulation of the Problem

Consider an arbitrary NTL characterized by position dependent characteristic impedance Zc(<) and propagation coefficient P(<) where <=dL is normalized longitudinal coordinate, L is a length of NTL section, Fig. 1.

region I P1, z1

region I11

;=0 I <=1

e - Fig. 1. A section of nonuniform transmission line, <= x/L

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The voltage of NTL within interval O<C;<l is a solution of generalized telegraph equation written as

Assuming unit amplitude of the voltage generated by a source, the solutions of (1) in region I, I1 and I11 of Fig.1 can be presented in the form [2]:

V(C;) = exp(-jp~LC;) + S I I exp(-jpLC;) in iegion I (2)

V(<) = A fl(Q + B f2(Q (3)

VC;) = S21 e~p(-jP2L [C;-11) in region I11 (4)

in region I1

where fl(C;) and f2(C;) are the two linear independent solutions of (1) satisfying to the initial conditions:

fl(0) = 1, fl’(0) = 0, f2(0) = 0, fi(0) = 1 (5)

Applying the boundary conditions to V(Q and V’(Q at C; = 0 and C; = 1, the linear system of 4-th order for unknown amplitudes S11, S21, A and B can be written. Solving the system, above coefficients can be determined. In order to reconstruct a voltage distribution in the region I1 coefficients A and B should be calculated:

The following procedure is recommended for reconstructing process:

1. Specify impedance and propagation constant laws within region 11; 2. Solve numerically differential equation (1) to determine fi, fi‘, fi, f2’ in the discrete number of points C;

taking into account the initial conditions (5) ; 3. Calculate V(C;) at the same points using (3), (6) and (7).

3. Examples and Discussion

Lets us consider a NTL using the microstrip line with the dielectric constant of a substrate E~ = 10, height h = lmm and length L = 35mm. The impedance profile is described by a parabolic law corresponding to the maximum impedance O f Zmax = 100 Ohms at the center of NTL:

Fig. 2 shows a behavior of w/h ratio and teff as a function of normalized coordinate C; for the above profile of NTL. The standard 4-5th order Runge-Kutta technique has been applied to find a solution of differential equation (1) at the specified discrete points 0 I C;i 5 1 ( i = 25 ). Then, the calculation of V(C;) has been done at these points using (3), (6) and (7) in order to reconstruct a voltage distribution along NTL. Fig.3 illustrates the normalized voltage distributions at the resonance frequency fo ( I S11 I = 0 ) and frequencies lying upper fup ( I S11 I = 0.717 ) and lower fi,, ( I S11 I = 0.828 ) of the resonance. When NTLs is mismatched the distributions of a voltage are far from “regular” behavior depending on the frequencies and level of reflections.

200 1 IEEE - Russia Conference : MEMIA200 1 101

wlh

0 0.2 0.4 0.6 0.8 6

Fig.2. Behavior of Z, E,ff, w/h as a hnction of < for NTL with “normal” parabolic law Z(Q = Zmin - (Zhn - Z-)[ 1 - 4(0.5 - <)*I, Zmin = 20 Ohm and Z,, =, 100 Ohm.

Fig.3. Distribution of normalized voltage VNO along microstrip NTL with “normal” parabolic profile at the frequencies:

fup = 3 GHz ( I SI1 I = 0.717), fo = 2.37 GHz ( I S11 I = 0 resonance) and fi, = 1.8 GHz ( 1 S11 1 = 0.828)

0 35mm I

Although the reconstructing process has been presented here for simple parabolic profile of the NTL, but it is directly applicable to various practical cases described by much more sophisticated laws inherent, for instance, inhomogeneous filled wavegiudes [3], Moreover, similar approach can be applied to reconstructing current distribution along a NTL if the eq.( 1) is changed by its “current” equivalent. It can be envisaged that the same reconstructing process is valid for lossy NTLs [4].

Finally, let us estimate an accuracy of the reconstructing process using Wronskian determinant W = fl(l). &’(1) - f2(1). fl’(1) [2] . For the initial conditions (5) it is equal 1. It has been estimated within the frequency range 0.5 - 10 GHz as a result of numerical solution of (1) by means of 4-5 order Runge-Kutta method. The result is shown in Fig.4 demonstrating that the deviation of the Wronskian determinant from 1 was not exceed 5. loe7 increasing toward higher frequencies.

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4. Conclusion

The proposed technique for reconshucting voltage distribution in NTLs being simple, universal and very accurate provides designers with the effective tool to study real physical processes in arbitrary nonuniform lines. It can be easy extended to lossy NTLs with arbitrary impedance profile and position dependent propagation coefficient.

References

[ 13. R.E.Collin, Foundation for Microwave Engineering, McGraw-Hill, 1966. [2] T.W.Kao, “Reflection and transmission of electromagnetic waves in inhomogeneous dielectric filled rectangular

waveguide”, IEEE Trans. Microwave Theory Tech., vol. MTT-17, pp.639-641, Aug.1969. [3] B.Yu.Kapilevich and N.S.Simin, “Reflection from a dielectric wedge in a rectangular waveguide”, Izvestia VUZ

Radiofizika( USSR), vo1.19, no. 1, pp. 135-140, 1976. English translation in Radiophys. & Quantum Electronics (USA). B.Yu.Kapilevich, “Direct S-matrix analysis of nonuniform transmission lines” Applied Microwave & Wireless (USA), no.10, October, 1999, pp. 68-76.

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