Equivalent Representations of Lossy Nonuniform Transmission Lines

Equivalent transformations, which were recently derived for mixed lumped and distributed circuits, may be extended to circuits consisting of lumped reactance, resistors, and Iossy transmission lines. It is shown that circuits consisting of a cascade connection of lumped element sections and Iossy uniform transmission lines are equivalent to circuits consisting of a cascade connection of lossy nonuniform transmission lines, lumped elements, and ideal transformers. Furthermore, by considering the limiting case of these transformations, equivalent transformations for circuits consisting of a cascade connection of lumped reactance, resistors, and nonuniform transmission lines are obtained. Exact equivalent circuits of Iossy even-order binomial form transmission lines are derived from these equivalent transformations.     

Equivalent Representations of Nonuniform Transmission Lines Based on the Extended Kuroda's Identity

Kuroda's identity may be extended to circuits consisting of lumped reactance elements and nonuniform transmission lines. It is shown that these circuits are equivalent to circuits consisting of cascade connections of nonuniform transmission lines whose characteristic impedance distributions are different from original ones, lumped reactance elements, and ideal transformers. If a characteristic impedance distribution W(x) of an original nonuniform transmission line is given, a characteristic impedance distribution Z(x) of a transformed nonuniform transmission line may be uniquely obtained using W(x). Moreover, by using these equivalent transformations, network functions of these transformed nonuniform transmission lines can he derived exactly.     

Equivalent transformations for the mixed lumped Brune-type sectionand uniform transmission line

As an analytical method for nonuniform transmission lines (NTLs) equivalent transformations are extended to a more general case, namely a mixed lumped Brune-type section and a uniform transmission line (unit element, UE). Circuits consisting of a cascade connection of a lumped Brune section and a UE are equivalent to one consisting of a cascade connection of a nonuniform transmission line whose characteristic impedance distribution is expressed with a trigonometric function and a lumped Brune section. This equivalent transformation method is easily applied to a circuit consisting of a lumped C section and a UE. The equivalent circuit is a circuit consisting of an NTL and a lumped C section. In this case, the characteristic impedance distribution of the NTL may be expressed in terms of a hyperbolic function. Exact network functions of the NTLs are easily obtained from the equivalent circuits without solving the telegrapher's equation. By considering the limiting case of these equivalent transformations, equivalent transformations for circuits consisting of a cascade connection of a lumped resonance circuit and a circuit and a uniform transmission line are derived     

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     

An equivalent transformation for the mixed lumped lossless two-portand distributed transmission line

It is shown that the circuit consisting of a cascade connection of a lumped type D section and a distributed transmission line is transformed to an equivalent circuit consisting of a cascade connection of a class of a nonuniform transmission line and a lumped type D section. Previously obtained equivalent transformations, i.e., Brune, type C, type A and type B sections, are included in the type D section transformation presented in this paper. The type D section is reduced into the Brune section and both type A and B sections are obtained as a degenerate case of the Brune section. Procedures for obtaining these transformations are given and a general theorem concerned with a mixed lumped lossless two-port and distributed transmission line transformation is established. By using these equivalent transformations, a new analytical method for nonuniform transmission lines is derived without having to solve the telegraph equations     

Two-Port Equivalent Circuits of Two-Wire Parabolic Tapered Coupled Transmission Lines

Two-port equivalent circuits of two-wire parabolic tapered coupled transmission lines (PTCTL) with open or short terminal conditions on the remaining two ports are presented. First, two-port equivalent circuits of PTCTL, whose characteristic admittances increase along the lines, are shown. Second, two-port equivalent circuits of PTCTL, whose characteristic impedances increase along the lines, the dual of the previous circuits, are shown. These two-port circuits of PTCTL are expressed in terms of two equivalent representations, one having mixed, lumped and uniform distributed circuits, and the other consisting of uncoupled nonuniform distributed circuits.     

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.     

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.     

Cascade synthesis of transmission lines and lossless lumped networks

Considerable interest in mixed lumped/distributed networks has arisen in recent years. The use of semiconductor elements, which have essentially lumped equivalent circuits, in microwave networks is an obvious example, and the need to meet even more exacting filter specifications at ever higher frequencies has resulted in many problems which are essentially mixed lumped/distributed. In most applications, the cascade type of structure is required, and it has been claimed that the sufficient condition for a 2-variable bounded real reflection coefficient to be realised by a resistively terminated cascade of lossless transmission lines and lumped 2-ports is that the transmission zeros are the product of the transmission zeros of the basic sections. Primarily by a particular numerical example, it is shown that this result is not generally valid for a canonic cascade realisation. This result is because of the possibility of `dummy? sections existing within the network without violating the bounded-real condition at the terminals. Consideration of the existence of `dummy? sections leads to the conclusion that an additional n(n?1)/2 conditions are required, where n is the number of transmission matrices for both terminated and unterminated networks. Additional conditions for reciprocity etc. are given in the realisability conditions, which are entirely algebraic in nature.     

Design of transformerless quasi-broad-band matching networks forlumped complex loads using nonuniform transmission lines

A simple design method for transformerless and lossless quasi-broadband matching of a lumped RC load is presented by use of a parabolic nonuniform transmission line. The key idea in removing an ideal transformer from the matching network is based on impedance transformation of the nonuniform transmission line, whose mixed lumped and distributed equivalent circuit contains an ideal transformer. Illustrative examples and some design curves are presented     

Impedance Transformation and Matching for Lumped Complex Load with Nonuniform Transmission Line

New nonuniform transmission-linematching networks for a class of lumped complex loads are presented. A parabolic (or reciprocal parabolic) tapered transmission line, whose exact equivalent circuit is represented by a mixed lumped and distributed circuit, can transform the lumped series RC (or parallel RL) loads into different lumped impedances which are more convenient than the original load impedances for ordinary matching network design. Simple design procedures are described and useful design charts are given. Also, numerical examples are shown including experimental verification.     

Analysis and Synthesis of Waveguide Multiaperture Directional Couplers

A precise method of analysis of multiaperture waveguide directional couplers has been developed and used to investigate the performance of couplers designed both by conventional theories and by a new synthesis technique. The analysis is based on the equivalent four-port network of the coupler, the apertures being represented by lumped reactances in series and/or shunt with dispersive transmission lines, representing the waveguides. The effects of finite aperture dimensions and common wall thickness are taken into account. Many couplers designed on the usual basis of a first-order loose-coupling approximation have good directivity, even for tight (3 dB) coupling, but previous theories do not give the predicted directivity and are often far from optimum in design. A new synthesis technique based on the distributed low-pass prototype filter has been devised, and shown to give results close to the predicted performance. It has led to the construction of compact multiaperture directional couplers having directivity greater than 43 dB over complete waveguide operating bands.     

A new multi-valued neural network for the extraction of lumped models of analog circuits

A novel identification technique for lumped models of general distributed circuits (i.e. microwave transmission lines, monolithic integrated circuits and filters) is presented. The approach is based on a hybrid multi-valued neuron neural network with a modified layer and learning process, whose convergence allows the validation of the approximated lumped model. The modified layer is generated by symbolic analysis of the model under exam. The inputs of the neural network are geometrical parameters, while the outputs represent the estimation of the lumped circuit parameters.     

Experimental modelling of high frequency circuits at low audio frequencies part I

At microwave frequencies, frequent use is made in circuit design of the electrical properties of short lengths of resistively terminated transmission line. It will be shown in the following contribution that circuits based on high frequency distributed lines can usefully be modelled at low audio frequencies using lumped element lines.     

Equivalent circuit modeling of losses and dispersion in single andcoupled lines for microwave and millimeter-wave integrated circuits

Losses and dispersion in open inhomogeneous guided-wave structures such as microstrips and other planar structures at microwave and millimeter-wave frequencies and in MMICs (monolithic microwave integrated circuits) have been modeled with circuits consisting of ideal lumped elements and lossless TEM (transverse electromagnetic) lines. It is shown that, given a propagation structure for which numerical techniques to compute the propagation characteristics are available, an equivalent circuit whose terminal frequency and time-domain properties are the same as the structure can be synthesized. This is accomplished by equating the network functions of the given single or coupled line multiport with that of the model and extracting all the parameters of the equivalent circuit model by using standard parameters identification procedures. This model is valid over a desired frequency range and can be used to help design both analog and digital circuits consisting of these structures and other active and passive elements utilizing standard CAD (computer-aided design) programs. To validate the accuracy and usefulness of the models, results for a mismatched 50-Ω line in alumina and a high-impedance MMIC line stub are included     

Understanding Soliton Wave Propagation in Nonlinear Transmission Lines for Millimeter Wave Multiplication

In this paper, soliton propagation in nonlinear transmission lines (NLTLs) periodically loaded with symmetric voltage dependent capacitances is studied. From the lumped element equivalent circuit of the line we have analyzed the influence of nonlinear shunt reactances on soliton propagation characteristics. It is shown that by increasing the non linearity of the C–V characteristic, a faster separation of input signal into solitons is achieved. The fact that frequency multiplication in NLTLs is governed by soliton formation makes the results of this work relevant to understand the influence of nonlinear loading devices on multiplier performance. Since a heterostructure barrier varactor (HBV)-like voltage dependent capacitance has been considered for the nonlinear devices, this study can be of interest for the design of millimeter wave frequency multipliers loaded with HBVs.     

On the synthesis of equivalent-circuit models for multiports characterized by frequency-dependent parameters

The synthesis of lumped-element equivalent circuits for time-domain analysis of problems with frequency-dependent parameters is of great interest in microwave theory. This paper presents a systematic approach to generate minimal-order realizations for passive microwave circuits characterized by either admittance, impedance, or scattering-parameter data. In addition, a very efficient method to ensure inherent system properties such as stability and passivity is described. The proposed method is suitable for simulating frequency-response-based linear subnetworks in a general circuit environment consisting of lumped/distributed elements and nonlinear devices. Modeling examples for a two-, four-, and ten-port system with nonlinear and linear terminations, respectively, are given.     

Synthesis of Reduced Equivalent Circuits for Transmission Lines

This paper presents a method for generating lumped models for symmetrical transmission-line two-ports. These models consist of an ideal transformer and frequency-domain approximations for two physical driving-point impedances. The lumped element values are obtained directly from the distributed parameters or propagation constant and characteristic impedance. The method is applied to dispersive transmission lines, skin effect and waveguides. It is shown that the equivalent circuit is superior in accuracy and number of elements compared to spatial discretizations like ladder approximation.     

Time-domain solution of field-excited multiconductor transmissionline equations

Analytical solution of lossless field-excited multiconductor transmission lines is presented. The equivalent circuit of a multiconductor transmission line with distributed sources is reduced to a simple lumped parameter circuit with independent voltage sources at both the ends of the transmission line. The transient source waveforms are analytically estimated for exponential time dependence of the external field, as EMP, ESD, and lightning. The method is suitable for a direct implementation in computer-aided circuit analysis codes and enables a very fast analysis for any load condition. Some numerical results are presented for single conductor and multiconductor lines excited by all EMP plane-wave field     

A New Class of Distributed Prototype Filters with Applications to Mixed Lumped/Distributed Component Design

The distributed prototype filter consists of a cascade of shunt stubs of equal length alternating with uniform transmission lines, each of twice the stub length. If the stubs are open circuited, they may be replaced by lumped capacitors to synthesize a mixed lumped/distributed (L/D) filter having near optimum Chebyshev or Zolotarev characteristics. The rate of cutoff and the general character of the stopband region is predictable. The use of Zolotarev prototypes enables the impedance level within the filter to be controlled and gives greater selectivity. Designs suitable for either transverse electromagnetic (TEM) line or waveguide low-pass filters are presented. If the stubs are short-circuited, the prototype may be used to design quasi-high-pass or bandpass filters of very large bandwidth. An example is given of an inductive-iris-type filter of approximately 100-percent bandwidth for which previous theories have been unsuitable. The theory is capable of extension to more complicated mixed L/D structures containing both series and shunt elements.