Generalized Transmission Line Equations Research Articles

An enhancement of instruments for solution of general transmission line equations with method of lines, impedance-/admittance and field transformation in combination with finite differences

The Method of Lines in combination with impedance/admittance and field transformation is used to analyze electromagnetic waves. The used cases are waveguiding structures in microwave technology and optics. The core of the theory is the solution of generalized transmission line equations (GTL). In the case of complex structures, a combination with finite differences (FD) can be used. The quality of this solution essentially depends on the effectiveness of the used interpolation of the differences. The individual steps of the FD are permanently linked to the steps of the fully vectorial impedance/admittance and field transformation, so standard libraries cannot be used. Two approaches based on the linear and quadratic interpolation were built into the impedance/admittance and field transformation in the past. However, the degree of improvement due to one or another kind of interpolation depends on the concrete behavior of the solution sought. In the case of complex structures, choosing the appropriate type of interpolation should be an effective aid. In this paper, an extension of the family of built-in methods is proposed—with the possibility of being able to build any known numerical method from the class of one-step or multi-step methods into the GTL solution. These can be higher-order methods, including fast explicit methods, or particularly stable implicit methods. The transmission matrices for the impedance/admittance and field transformation serve as the building site. To illustrate the procedure, some different methods are integrated into the GTL solution. The efficiency of the solutions is tested on some test structures and compared with each other and with existing solutions. The relevant waveguide specifics were discussed. Initial systematics and recommendations for users were derived.

Optimized method of lines for non-linear waveguide problems

The Method of Lines is a semi-analytical versatile tool for the solution of partial differential equations. For the analysis of spatial complex linear waveguide structures, this method is combined with impedance/admittance and field transformation, as well as with finite differences. This paper extends this approach to the treatment of structures with non-linear dielectric materials. The non-linear generalized transmission line equations are derived. An iterative algorithm based on the impedance/admittance transformation with the field transformation obtains efficient and self-consistent solutions. For demonstration, a non-linear stripe waveguide is considered. The Kerr non-linearity was investigated, though the general case is treatable. As a criterion for the correctness of the algorithm, a second harmonic generation as well as a bidirectional, spatially, and temporally energy exchange between the harmonics was examined. The specific limiting factors for the algorithm were explored. The approach can be used for any spatial structure, including, for instance, photonic crystal waveguides and metamaterials.

The nonlinear simulation, design and experiments on 0.42 THz gyrotron with gradually tapered complex cavity

Gyrotron oscillators are one of promising sources to generate high power terahertz radiation. To achieve high power at high frequency, gyrotrons have to operate at high order modes to alleviate the problem of wall heating and the miniaturization of interaction structures at these frequencies. Due to high order modes, the mode spectrum is very dense and the electron beam is likely to interact with several modes. Mode competition is a major hurdle in achieving stable and efficient operation of a gyrotron at cyclotron harmonics. The gradually tapered complex cavity could suppress mode competition by locking a pair of operating modes in two single cavities. Starting from the general transmission line equations with an electron beam, self-consistent nonlinear simulations on the second harmonic complex cavity gyrotron operating at 0.42 THz are presented. The influences of the electron beam parameters on the interaction efficiency are analyzed. The experimental results show that the designed complex cavity gyrotron is operationally stable at TE17.4 with pulse output power of 22.25 kW when the beam current is 4.8 A, beam voltage is 45 kV, and applied magnetic field is 8.185 T. The corresponding frequency and interaction efficiency are 421.60 GHz and 10.27%, respectively.

Full-wave analysis of a compact bandpass meandering resonator filter using method of lines

Full-wave analysis based on the method of lines and the generalized transmission line equations is applied for the design of a small-size wide stopband bandpass filter. The analytical calculation in the direction of propagation and using nonequidistant discretization enables the analysis of the method of lines to be efficient for structures that can be divided into cascaded subsections along the propagation direction of the wave. The proposed filter consists of a pair of meandering uniform impedance resonators and feeding lines. Wide stopband and high selectivity as well as compact size of the filter are achieved due to the effect of multipath between the feeding points and the resonators that leads to suppression of the first and second harmonic of the filter. The modes distribution and the scattering parameters of the designed filter are computed and studied, leading to an achievement of a passband return loss greater than 18 dB and insertion loss about 1.7 dB. The designed filter is fabricated and tested, where good agreement is found between the numerical and experimental results.

High-Frequency Multiconductor Transmission-Line Theory

This work presents a thorough derivation of the full-wave transmission-line equations on the basis of Maxwell’s theory. The multiconductor system is assumed to be composed of nonuniform thin wires. It is shown that the mixed potential integral equations are equivalent to generalized telegrapher equations. Novel, exact, and compact expressions for the multiconductor transmission-line parameters are derived, and their connection to radiation effects is shown. Iteration and perturbation procedures are proposed for the solution of the generalized transmission-line equations.

Design of a periodically nonuniform coupled microstrip lines (PNCML) filter using the novel method of lines

AbstractA new eigenmode algorithm based on the method of lines and the generalized transmission line equations is applied on full‐wave analysis of (3‐D) MMIC components. The analytical calculation in the direction of propagation enables the analysis of structures of very short or long interconnections between single discontinuities. The analysis is developed by nonequidistant discretization to reduce the discretization window. In this article, a periodically nonuniform coupled microstrip lines filter is designed to offer a wide band performance with a small size structure and with harmonic suppression. The analysis of this filter is applied using the new technique of the method of lines. In this analysis, the scattering parameters of the designed filter are computed. The designed filter is fabricated and tested, where good agreement is found between the numerical and the experimental results. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2865–2870, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24763

A Stability Analysis for Time-Domain Method-of-Moments Analysis of 1-D Double-Negative Transmission Lines

The stability of time-domain simulations is examined for a generalized lossy double-negative transmission line (TL). This is accomplished using a "modified" version of the Courant-Friedrichs-Lewy stability criterion. The time-domain simulation is implemented using a method-of-moments formulation with the generalized TL equations. An analytic expression is derived by combining the Z-transform with von Neumann methods. The credibility of simulated time-domain results is verified through comparison with modal behavior determined using a frequency-domain treatment for a periodic structure.

Study on the Applicability of the Static Distributed Circuit Parameters of Non-uniform Microstrip Lines

The Generalized Transmission Line Equations are employed to analyze the circuit parameters of the microstrip lines. The static distributed capacitances and inductances of non-uniform microstrip lines are extracted. By comparing the differences between the port’s S-parameters respectively obtained by the extracted static circuit parameters using circuit method and by the Method of Moment directly, the applicable frequency range of the extracted static distributed circuit parameters can be found. The results show that the extracted static circuit parameters can be used in a frequency band from DC up to millimeter wave.

Method of Lines Numerical Analysis of Conformal Antennas

This paper presents a generalization of the method of lines (MoL) numerical algorithm for the analysis of integrated structures with a finite local curvature. Starting from the derivation of the generalized transmission line equations in a generic orthogonal reference system and in a bianisotropic material, it is shown the possibility of solving the electromagnetic problem associated to an integrated structure with arbitrary curvature and/or loaded by complex dielectric materials via a MoL approach. The introduction of new algorithms that extend and complete the original method is required in the most general case. A detailed description of this new extended version of the MoL is presented through the paper and its full coincidence with the original algorithm in the case of structures with simple loading materials is also shown. Finally, some numerical results obtained analyzing microwave conformal antennas and resonators are presented and compared with analyzes performed employing other numerical methods and available in the literature.

Analysis of a Double Folded Stub Microstrip Structure using Generalized Transmission Line Equations Method

This paper presents a generalized transmission-line theory which is useful to describe the local radiation effect along almost arbitrary three-dimensional wire structures. For the generalized transmission-line theory, the mathematical structure of the classical telegrapher equation is preserved, while the coefficients (the per-unit-length parameters) are generalized in order to represent the intrinsic behavior of the wire structure. Essentially, the generalized transmission line equations (GTLEs) is a novel kind of telegrapher equation. The equation coefficients of the generalized transmission line equations (GTLEs) are determined from full-wave numerical solutions in this paper. In comparison with the conventional telegrapher equation, the GTLEs has two additional terms — dependent series voltage and shunt current sources — which represent local radiation effect from discontinuities of the nonuniform transmission-line itself. For the double folded stub microstrip structure, the novel 1D-GTLEs whose coefficients are determined by dynamic numerical methods can have broadband frequency characteristics. To the investigated microstrip structure as a band stop filter, the S parameters for a frequency band from 3 GHz to 18 GHz can be calculated by using only one group of the 1D-GTLEs at a frequency of 12 GHz, while 1D-GTLEs is only established on the central transmission line rather than on the whole structure with the stubs. Furthermore, the 2D-GTLEs developed from the 1D-GTLEs is sought for the whole structure with stubs and the model of 2D-GTLEs is validated to be more accurate than that of 1D-GTLEs through numerical experiments in this paper

Analysis of a double step microstrip discontinuity using generalized transmission line equations

In this paper, a new analysis for a double step microstrip discontinuity is presented by using generalized transmission line equations . The new analysis is based on a new concept of a finite-length transmission line integrated with its double step discontinuity so that the double step discontinuity is regarded as a finite-length nonuniform transmission line and the generalized equations could be directly implemented. Since the generalized equation coefficients are determined by dynamic numerical methods and the coefficients are invariant with the line excitation and load, the equations are dynamic rather than TEM. The interested discoveries are that the generalized equations for the whole double step structure or a partial double step structure can give us the same results and the generalized equations have broadband frequency characteristics. For the double step structure used in this paper for the analysis, the S-parameters for a frequency band from 0.5 GHz to 10 GHz can be well calculated by using only two separately generalized equations at frequency of 3 GHz and 8 GHz.

Generalized transmission line equations for bianisotropic materials

The generalized transmission line equations for bianisotropic materials are derived in this paper. These equations are obtained in a general orthogonal curvilinear reference system, taking properly into account the metric factors and decomposing the field vectors into two components: a transverse component and a longitudinal one, both with respect to a reference coordinate direction. Such a general formulation is very useful in order to analyze metallic/dielectric uniform bended waveguides. On the other hand, it is very appropriate when studying conformal integrated circuits by using the method of lines (MoL). Finally, the formulation here presented leads to a straightforward derivation of the transmission line equations in the spectral domain when studying conformal integrated antennas. As particular cases of the theory here presented, transmission line equations in orthogonal curvilinear coordinates are also derived for anisotropic, bi-isotropic, and isotropic materials.

Modeling of waveguide circuits based on generalized transmission line equations and impedance/admittance transformation concept

Algorithms for the modeling of waveguide circuits are described. The material parameters can be anisotropic. Starting with generalized transmission line equations, suitable expressions for the analysis are derived. The introduction of the admittance/impedance transformation concept guarantees numerical stable expressions even in case of long devices or high numbers of concatenated sections. The presented algorithm is validated by comparing numerical results with those obtained by measurement and other numerical methods.

The Method of Lines: A Versatile Tool for the Analysis of Waveguide Structures

Algorithms for the determination of eigenmodes and for the analysis of longitudinally varying structures are presented. The media may be anisotropic and the devices may consist of an arbitrary number of layers or waveguide sections, respectively. Starting with generalized transmission line equations, suitable expressions are derived which are implemented in the method of lines, a semianalytical algorithm. To ensure the numerical stability, a transformation of the reflection coefficient is performed. Numerical results are presented for a Bragg-grating with a very high number of periods, for an electro-optical switch, and for an anisotropic groove waveguide. The results prove stability of the algorithm, and comparisons with other methods show the accuracy.

2D generalized transmission‐line equations for a microstrip cross

AbstractIn this paper, a rigorous analysis for a microstrip cross is presented by using 2D generalized transmission‐line equations. The 2D generalized transmission‐line equation is developed from the 1D generalized transmission‐line equation in [1], which was derived from circuit theory, while the equation coefficients are determined from full‐wave numerical solutions. Although 1D generalized transmission‐line equation can well model many microstrip structures, such as discontinuities and low‐pass filters, it is confined to be used for those structures which can be regarded as 1D finite‐length nonuniform transmission‐line. For a microstrip cross, the novel 2D generalized transmission‐line equation whose coefficients are determined by dynamic numerical methods can have broadband frequency characteristics. For the investigated microstrip cross, the S parameters for a frequency band from 4 GHz to 24 GHz can be calculated by using only one group of the 2D generalized equation at frequency of 14 GHz. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 35: 378–380, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10613

Broadband property and time‐domain application of general transmission‐line equations on microstrip low‐pass filter

AbstractGeneral transmission‐line equations based on the theory of differential equations [1] were introduced in [2] and [3]. The equations' dispersion characteristic of a microstrip low‐pass filter is studied. The parameters L, C, α, and β of general transmission‐line equations are found to be almost invariant to frequency in the passband of the microstrip low‐pass filter. By restoring current and voltage distributions with the use of equations in both frequency and time domains, the broadband property of these equations on the microstrip low‐pass filter is evident. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 33: 123–126, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10250

Dispersion Characteristic and It's Application of General Transmission Line Equations On Microstrip Step Structures

The differential equations invariant of termination conditions of thin wire structures and transmission line have been derived. New transmission line equations we called general transmission line equations based on the theory of differential equation are described. The dispersion characteristics of the general transmission line equations on microstrip step structures are studied. We found the general transmission line equations on some of microstrip step structures such as cascade double step structures and low-pass filters have the property of broadband in frequency-domain. Using an equivalent circuit modal, we can restore current and voltage distribution in frequency-domain.

A new time‐domain method based on the general transmission‐line equations

AbstractGeneral transmission‐line equations that can describe both uniform and nonuniform transmission line are derived in brief. By using the algorithm analogous with a one‐dimensional FDTD in the time domain, we obtain a new time‐domain method we call the on‐line FDTD method. Frequency‐domain results are compared with those computed by the MoM to show the validity of our new method. © 2002 John Wiley & Sons, Inc. Microwave Opt Technol Lett 32: 46–51, 2002.

Determining reflections for beam propagation algorithms

Reflections occur whenever different waveguide sections are concatenated. In this paper, we will give approximate formulas for determining the reflection coefficient. Starting with generalized transmission line equations we will derive general formulas where suitable approximations are introduced. These approximations allow the determination of the reflections with low numerical effort. The expressions were implemented in an FD-BPM algorithm to determine the field distribution in waveguide devices. For an application in CAD systems the reflection coefficient of the fundamental mode was computed with the approximate formulas. Numerical results were obtained for various concatenations of different waveguide sections. A comparison with the method of lines, a semi-analytic algorithm, shows a good agreement.

A finite difference beam propagation algorithm based on generalized transmission line equations

A wide angle beam propagation algorithm is presented, which is based on generalized transmission line (GTL) equations. Besides the discretization, no further approximation is introduced. In principle, a full vectorial analysis is possible, and anisotropic material can be taken into account. The algorithm has been verified for planar 2D-waveguide devices. The wide angle characteristic has been examined by determining the propagation of tilted Gaussian beams in homogeneous media. Further studies were performed for tilted waveguides and for taper structures. Results for the latter devices were compared with other methods showing a very good agreement.