Accurate Analysis of Tapered Planar Transmission Lines for Microwave Integrated Circuits

A broad class of nonuniform transmission lines is analyzed through the method of coupled modes accompanied by the spectral domain solution of uniform lines. This combination offers efficient computation of the coupling coefficients. Small coupling allows the rigorous field theory of the spectral domain approach to give explicitly the scattering matrix of the taper. Several structures, including microstrip to coplanar waveguide taper and waveguide to fin-line taper, are successfully analyzed. The computed values of reflection coefficients are compared with the values derived by a simple impedance method and with the measured values. Results from the experimentally investigated tapers show that the theory is in good agreement.     

Analysis of coupled nonuniform transmission lines using Taylor's series expansion

A new method is introduced for the frequency domain analysis of arbitrarily loaded lossy and dispersive nonuniform coupled transmission lines. We assume that the per-unit-length matrices are known and can be expressed by a converged Taylor's series. All known per-unit-length matrices, as well as the voltages and currents, are expanded in Taylor's series. The solutions of voltages and currents are obtained after finding the unknown coefficients of the series. The validity of the method is verified using the analysis of some special kinds of coupled lines.     

Nonuniformly coupled microstrip transversal filters for analog signal processing

The Fourier transform relationship between frequency response and impedance profile for single nonuniform transmission lines is used to derive the time-domain step response of single and coupled nonuniform lines. The expression for the step response of a characteristically terminated nonuniformly coupled transmission line structure is shown to correspond to the characteristic impedance profile. By using this relationship, any arbitrary step response can be realizing by utilizing nonuniformly coupled strip or microstrip lines for possible applications as waveform-shaping networks and chirp filters. A numerical procedure to compute the step response of the nonuniform coupled line four-port is also formulated in terms of frequency-domain parameters of an equivalent cascaded uniform coupled line model with a large number of sections. Sinusoidal and chirp responses are presented as examples that are readily implemented using coupling microstrip structures. The step response of an experimental nonuniformly coupled microstrip structure is presented to validate the theoretical results.<>     

Coupling Between Dissimilar Waveguides

Investigators have used coupled-mode theory to analyze the coupling between identical waveguides; in such cases the coupling coefficients are found to be identical. If the waveguides differ, the coupling coefficients are asymmetrical and difficult to evaluate by strictly theoretical methods. An alternate approach to this case is considered in the present work. A pair of coupled-mode equations is first developed from a consideration of the permissible fields within the device. This clarifies the relationship between the coupled-mode theory and the more general classical electromagnetic theory by giving a careful definition of the coupled and the normal modes of a coupled structure. It is shown that the coupled-mode equations are an exact representation of the waveguide fields, although for engineering purposes it is often convenient to use approximate values of the coefficients of these equations. The mutual coupling coefficients are obtained from a two transmission-line model of the structure, with the actual coupling mechanism represented by a mutual impedance common to the two lines. For dissimilar lines, the ratio of the coupling coefficients is found to be equat to the ratio of the characteristic impedances. For the cases considered, this is the same as the ratio of the propagation constants of the uncoupled lines, which permits the coupling coefficients to be determined from relatively simple measurements. The adequacy of the theory has been confirmed by a series of experiments.     

Wave Scattering in Nonuniform Waveguides with Large Flare Angles and Near Cutoff

A set of coupled first-order differential equations for the wave amplitudes in nonuniform waveguides is developed. The coupling coefficients are regarded as differentiaI transmission and reflection scattering coefficients between two adjacent elementary radial waveguide sections. The analysis is an extension of an earlier quasi-optical solution. This set of coupled equations is compared with the familiar generalized telegraphist's equations which may be derived by considering the nonuniform waveguide to consist of elementary rectangular waveguide sections. The equations for the wave amplitudes derived in this paper are less coupled than the commonly used telegraphist's equations, and they may also be applied to waveguides with large flare angles and in regions at which the waveguide modes are at their respective cutoff cross sections.     

Ray-optical determination of the coupling coefficients of gratingwaveguide by use of the rigorous coupled-wave theory

The ray-optics approach based on the rigorous coupled wave theory, called rigorous ray-optics method (RROM), is developed for the calculation of backward coupling coefficients of grating waveguide devices. The coupling coefficients of several grating structures, such as rectangular, sinusoidal, triangular, and trapezoidal shapes, are evaluated by the RROM, and they are compared with those obtained by two conventional methods of the ray-optics method (ROM) and the coupled-mode method (CMM). In the case of rectangular gratings, the coupling coefficients are evaluated in more detail by varying grating depth and duty-cycle. We have found that the RROM gives us more exact solutions for the backward coupling coefficients of even arbitrary shapes of diffractive grating waveguides than the other two conventional methods     

Coupled-mode description of a nonuniform transmission line

Coupled-mode analysis provides a conceptually simple technique for the analysis of nonuniform transmission lines and nonuniform coupled systems. The variable series and shunt elements couple backward and forward traveling waves. Reflection coefficients are derived for an exponential and linear taper by this technique.     

Novel microstrip multifunction directional couplers and filters formicrowave and millimeter-wave applications

The design of a particular class of microstrip couplers and filters is presented. The synthesis functions obtained from the solution of first-order nonlinear differential equation of nonuniform lines with a loose coupling assumption are modified and validated for higher coupling values. The design employs a nonuniform coupled line configuration along which a realizable continuous coupling coefficient is obtained by modifying the reflection coefficient distribution function. This modification results in a frequency-selective coupling which minimizes the out-of-band coupling in the specified frequency range. As a result, it is possible to realize -3 dB directional couplers using double-coupled lines without the need for tandem connections or extreme photolithographic techniques. Experimental results for microwave band-pass and periodic couplers are presented together with the computed results. Potential applications of these components are discussed, and the work is extended to include millimeter-wave realization     

Coupled mode analysis of forward and backward coupling in multiconductor transmission lines

The nonorthogonal coupled mode theory is extended to the analysis of multiconductor transmission lines by including backward coupling. Coupling coefficients are expressed as overlap integrals of the eigenfields and currents belonging to individual lines. These eigenmode solutions are calculated using the finite-difference time-domain method, which can provide a broadband solution through a single simulation. General termination conditions are given, and scattering parameters of a multiconductor transmission line can be obtained directly by solving the coupled mode equations subject to these termination conditions. As illustrative examples, several configurations of coupled microstrip lines are analyzed, and numerical results are presented. It is observed that both the forward and backward coupling results agree fairly well with results from Advanced Design System Momentum software.     

Characteristics of Coupled Microstrip Transmission Lines-I: Coupled-Mode Formulation of Inhomogeneous Lines

This paper consists of two parts. In Part I, coupled-mode theory is employed to determine the effects of reflection at the various ports and unequal inductive and capacitive coupling coefficients on the coupling and directivity of two coupled lines. Since couplers utilizing microstrip lines generally have unequal inductive and capacitive coupling coefficients, the results presented here should be useful in explaining the behavior of microstrip coupled lines. It is shown how the difference in the coupling coefficients leads to finite directivity and, under certain conditions, to "codirectional" instead of "contradirectional coupling." In Part II, the coupling coefficients and other parameters of various microstrip-line geometries are presented. Using these parameters in the results obtained here leads to an improved understanding of and design criteria for coupled microstrip lines.     

Field coupling to nonuniform and uniform transmission lines

We study time-domain and frequency-domain responses of nonuniform and uniform transmission lines excited by incident electromagnetic waves. Externally excited uniform transmission lines permit closed-form solutions in terms of inverse chain matrix, whereas nonuniform lines cannot be analytically solved, in general. We adopt a method of equivalent cascaded network chain as the method of solving the latter situation. Useful and compact expressions for the load currents induced at terminal loads are derived. To confirm the validity of this method and the forcing terms, theoretical and experimental results of coupling calculations for a few typical (uniform/nonuniform) line geometries, relevant in the EMC field, are presented and discussed     

Coupled Nonuniform Transmission Line and Its Applications

Theory and applications of coupled nonuniform transmission lines are described. Matrix representations of a general coupled nonuniform transmission line are presented, by means of which the behavior of any coupled nonuniform transmission line maybe completely described. Among a wide variety of applications of coupled nonuniform transmission lines, two typical networks, one the coupled nonuniform transmission-line folded all-pass network and the other the coupled nonuniform transmission-line directional coupler, are treated in detail. Equivalent circuit representatious of these two networks are presented, which enable the designer to synthesize them in a greatly simplified manner by making use of the theories now available for more conventional single nonuniform transmission lines. In addition, the properties of these two networks using coupled exponential line are investigated. Design procedure is also given for asymmetrical coupled exponential-line directional couplers having excellent characteristics.     

Equivalent transformations for the mixed lumped Richards sectionand distributed transmission line

The authors introduce an analysis method for nonuniform transmission lines. Equivalent transformations between a circuit consisting of a cascade connection of a lumped Richards section, an ideal transformer, and a distributed transmission line and one consisting of a cascade connection of a class of a nonuniform transmission line, a lumped Richards section, and an ideal transformer, are given. Characteristic impedance distributions of these nonuniform transmission lines are expressed as hyperbolic or trigonometric functions. It is quite difficult to find the exact network functions of nonuniform transmission lines from the telegraph equation, but by using the equivalent transformation described it becomes possible to obtain exact network functions of a class of nonuniform transmission lines     

Equivalent Transformations for Mixed-Lumped and Multiconductor Coupled Circuits

Distributed circuits consisting of a cascade connection of m -port stab circuits and multiconductor coupled transmission lines are equivalent to ones consisting of cascade connections of multiconductor coupled transmission lines whose characteristic impedances are different from original ones, m-port stub circuits, and an m-port ideal transformer bank. Because of the reciprocity of the circuit, values of transformer ratio must be identified. In the special case of a one conductor transmission line, these equivalent transformations are equivalent to Kuroda's identities. These extended equivalent transformations may be applied to mixed-lumped and multiconductor coupled circuits. By using these equivalent transformations, equivalent circuits and exact network functions of multiconductor nonuniform coupled transmission lines can be obtained.     

High-frequency electromagnetic field coupling to long terminatedlines

We present a theory for the EMC problem of electromagnetic field coupling to a long line with arbitrary terminations. The theory is applicable for the high-frequency plane wave electromagnetic field excitations, when the transmission line approximation is no longer valid. Analytical expressions are derived for the induced current along the line, and at the two-line terminals. The coefficients of these expressions are determined using a procedure based on the exact solutions of the integral equation for two similar line configurations, but having a significantly shorter length. The method is, therefore, particularly efficient when considering the electromagnetic field coupling to very long lines. The advantage of the proposed approach is that, in contrast with transmission line approximation, it takes into account high-frequency radiation effects. Furthermore, it allows a considerable reduction in computation time and storage requirements with respect to conventional numerical solutions based on the thin-wire approximation     

Circuit-Based Analysis of Electromagnetic Field Coupling With Nonuniform Transmission Lines

This paper presents a new algorithm for simulating electromagnetic (EM) field coupling with nonuniform multiconductor transmission lines in a circuit simulation environment. The proposed algorithm is based on the concept of passive model-order reduction, whereby an algorithmically developed passive reduced-order model, coupled with a set of equivalent sources representing the incident filed, are shown to accurately capture the behavior of the transmission line under EM excitation. The reduced-order model is developed independently from the particular shape of the incident field pulse, in the sense that, in constructing the model, one does not need prior knowledge about the waveform of the incident pulse of the EM field. In addition, it is also shown that the model developed can be used to simulate the transmission line in the absence of the EM field. The derived equivalent sources, representing the field coupling, are given directly in the time domain, thereby making simulation under nonlinear circuit terminations an easy task. Although the proposed work is aimed mainly at simulating nonuniform transmission lines, it can be applied to uniform lines as a special case. The proposed algorithm has been validated numerically with several examples.     

Plane wave scattering by a material coated parabolic cylinder

An eigenfunction solution to the problem of transverse magnetic (TM) or transverse electric (TE) scattering by a coated parabolic cylinder is presented. Paralleling the well-known solution for the coated circular cylinder, eigenfunction expansions involving parabolic cylinder functions are obtained for the fields in the exterior and coating regions. Next, boundary conditions are enforced to obtain a pair of coupled equations for the unknown coefficients in the eigenfunction expansions for the fields. Unlike the corresponding solution for the coated circular cylinder, the eigenfunctions in the exterior and coating regions are not orthogonal, and an exact term-by-term solutions of these equations is not possible. Instead, the equations are solved by the method of moments. For thin coatings both an uncoupled-mode approximation and a surface-impedance model are described. In particular, for the TM polarization it is shown that a thin coating can be modeled by a specific nonuniform surface impedance for which an exact term-by-term solution is possible. Numerical data are presented, showing the convergence of the solution and comparing the solutions for the uncoupled-mode and surface-impedance models     

Forward and Backward Scattered Modes in Multimode Nonuniform Transitions

Analyses previously published on the subject of mode conversion consider only the forward scattered modes. The present paper investigates both the forward and backward scattered modes at frequencies from cutoff to far away from cutoff in a multimode nonuniform waveguide. The four coupled telegraphist's equations with varying coefficients are transformed into the form of coupled Volterra integral equations of the second kind and these integral equations are solved by an iteration method. The solutions are valid for all frequencies from cutoff to far away from cutoff. For uniform waveguides the solutions correctly reduce to those of the original forward traveling launching mode. The solutions also show the characteristics of "propagation" in the tapered cutoff region of the waveguide. The accuracy of the series solution is discussed, and possible wide applications of the results to a variety of problems are mentioned.     

Characteristics of Coupled Microstrip Transmission Lines-II: Evaluation of Coupled-Line Parameters

This part of the paper presents the parameters of coupled microstrip lines which are required in the equations and results derived in Part I for determining the characteristics of coupled lines. Several geometries are considered and the inductive and capacitive coupling coefficients, the effective dielectric constant, and the characteristic impedance for various dimensions of these geometries are presented.     

Equivalent Circuits of Binomial Form Nonuniform Coupled Transmission Lines

Equivalent circuits of nonuniform coupled transmission lines whose self and mutual characteristic admittance distributions obey binomial form are presented. Telegrapher's equations of these nonuniform coupled transmission lines can he solved exactly rising Bessel functions of fractional order. By decomposing the chain matrix, it is shown that equivalent circuits of these nonuniform conpled transmission lines consist of cascade connections of lumped reactance elements, uncoupled uniform transmission lines and ideal transformers.