Passive Ladder Research Articles

Exact discrete time synthesis

A procedure for the exact synthesis of LDI ladder filters is developed. Starting from any realisable discrete-time z-plane transducer function H(z), irrespective of the s-z transformation used, a discrete-time loss-less passive ladder prototype is derived whose complex frequency variable is γ, the LDI variable, and whose terminations are resistive and functions of μ, where =(z 1/2 + z -l/2 )/2,γ =(z 1/2 - z -1/2 )/2 and z is the discrete-time complex frequency variable. The proposed synthesis procedure is carried out by introducing a discrete-time frequency transformation which maps the discrete-time z-plane into another discrete-time plane called the λ-plane, where λ = α + jβ. The discrete-time physical frequency variable s describes a straight line axis rather than a unit circle axis. The frequency characteristics of the synthesised discrete-time transducer function H(z) are exactly preserved in the discrete-time /lambda/-plane.

A Graduate Course on Active, Digital, and Switched Capacitor Filters

This paper presents a solution to the problem of accommodating the important new subject of switched capacitor networks in already crowded electrical engineering curricula. This is achieved by combining in a one-semester course three different, but related, subjects. This is made possible 1) by placing a strong emphasis on the underlying principles that are common to active, digital, and switched capacitor filters, and 2) by concentrating on the realization of these three types of filters by simulation of passive ladder networks.

The Gauss Hypergeometric Ratio As a Positive Real Function

The Gauss continued fraction for the ratio of two hypergeometric functions is converted into an ordinary fraction (all partial numerators are 1) and simplifications occurring for particular relations between the parameters are discussed. In particular, a very simple expansion is obtained for the ratio ${E /K}$ of the complete elliptic integrals. For the argument $ - z$ and for certain ranges of the parameters, the Gauss expansion is a Stieltjes fraction and represents the input impedance of a passive ladder network. The Stieltjes integral representations of the corresponding positive real functions are established and yield many new definite integrals. A general method for obtaining indefinite integrals involving independent solutions of a self-adjoins differential equation, by means of the Wronskian, is also mentioned. Finally, some continued fractions originating from other contiguity relations for hypergeometric functions are discussed.

Analysis and synthesis of switched-capacitor circuits using switched-capacitor immittance converters

Switched-capacitor immittance converters (SCIC's) which are the switched-capacitor (SC) counterparts of GIC's are proposed. In this paper the definition and analysis of the SCIC's are given. Their fundamental properties are derived from the definition. Three methods for LC ladder simulation using the concept of SCIC's are presented. In the proposed bottom-plate stray-insensitive SCIC filter structure, only one operational amplifier is needed per node (excluding the grounded node) in the passive ladder network. This saving in the number of active elements is an advantage of this method especially in case of bandpass filter simulation.

Realization of Matrix Transfer Functions via a State-Space Approach

A state-space technique is presented for minimal realization of an impedance matrix or a voltage transfer function matrix without using integrators or the combination of integrators and multiwinding transformers. The Cauer II type passive RC ladder multiport and state-space concepts are employed for the realization. A new similarity transformation matrix is derived to transform a state-space equation in a RC-type Cauer second form into a state-space equation in a block companion form.

Exact analysis of inhomogeneous ladder networks having a tridiagonal interaction matrix

AbstractA projection‐recurrence method is applied to active and passive ladder networks with a tridiagonal interaction matrix to obtain a factorization of the response functions. The factors are partial chain parameters, determined by third order recurrences. For a ladder with N nodes, the number of arithmetic operations to calculate response functions ranges from 4N to 6N. A diagrammatic representation is developed for passive ladders which makes it possible to obtain the partial chain parameters by inspection, and a compact formula for the node‐voltage response is presented for active doubly‐terminated ladders.

Signal flow graph approach to active RC simulation of LC ladder filters

Abstract A simple method is reported for the simulation of doubly resistively terminated lossless ladders by means of active RC networks. The method is based on a proper matrix description of the two-port subnetworks, into which the passive ladder is partitioned. Through the use of signal flow graph concepts, known ladder simulating active networks as well as new ones can be obtained. An example of an all integrator canonic ‘active bandpass ladder’ is also given.

Minimum sensitivity active (leapfrog) and passive ladder bandpass filters

It is shown that the sensitivities of resistively terminated <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LC</tex> ladder networks can be significantly reduced by inserting losses into the internal sections of the overall structure. Sensitivity improvements on the order of 20 percent are obtained for many of the filters considered. This observation is particularly useful in active ladder (leapfrog) synthesis where the additional losses are readily compensated, if necessary, by the gains of the active elements. To achieve optimum overall sensitivity, a previously introduced statistical multiparameter sensitivity measure is minimized.

Design and Evaluation of Shifted-Companion-Form Active Filters

Two techniques for designing a class of low-sensitivity, follow-the-leader, feedback-type active filters have been introduced by Hurtig and Laker-Ghausi. The FLF configuration consists of a cascade of second-and/or first-order sections, with feedback from each section back to the first. This paper presents an approach for designing FLF-type realization for all classes of filter functions. The technique is based on a shifted-companion form of the associated-state equations. Some salient features of Hurtig's primary resonator block, Laker-Ghausi's follow-the-leader feedback, and the shifted-companion-form techniques are presented below. (i) Hurtig's PRB realizes any all-pole (no finite transmission zeros) filter function. This includes the low-pass, high-pass, and symmetrical bandpass filters without finite zeros. Explicit design equations are available, and the individual sections in the array are identical. (ii) Laker-Ghausi's FLF realizes any symmetrical (including finite transmission zeros) bandpass filter function. The sections are not constrained to be identical, which allows optimization using this degree of freedom. Finite zeros are realized by a summation technique. (iii) The SCF realizes all types of filter functions, i.e., low-pass, high-pass, bandpass, all-pass, or band-reject filters. Explicit design equations are available. The first section can differ from the rest, thus allowing some optimization with standardization. Feed-forward as well as summation techniques can be used to realise the finite zeros. Two bandpass design examples using SCF, PRB, and/or Laker-Ghausi FLF techniques are given and compared with the low-sensitivity coupled (leapfrog) biquad, the conventional cascade biquad, and the passive ladder filter designs. The comparison shows that the passive filter gives

Design and predistortion of passive filters by optimization

AbstractA method has been developed for design of passive filters based on optimization in the complex frequency domain. On the basis of the observation that a network response takes on very large values near the poles and very small values near the zeros it is possible to select an error function with the value zero when the network under consideration has the wanted pole‐zero positions. Application of the damped least square algorithm for minimization of that error function has produced excellent results in design and predistortion of passive ladder filters.

Synthesis of Multiple-Feedback Active Filters

A synthesis technique is developed for the active RC realization of transfer functions that have all their transmission zeros on the imaginary (jω) axis. The method leads to a realization using biquadratic blocks in a multiple-feedback arrangement that is the generalization of the structure obtained from the passive, double-terminated reactive equivalents. The realization combines the easy independent tuning properties of the cascade, with the low-sensitivity characteristics of passive ladders.

The recurrent-continuant method of transfer function synthesis

It is shown that the ratio of input voltage to output voltage for any passive ladder type low-pass filter network is always a polynomial in the complex frequency s. This polynomial can always be written by inspection as a single determinant in the form of a recurrent. The recurrent determinant is then converted by elementary manipulation into an alternate form known as a continuant. The transfer function of any passive ladder type low-pass filter network can also be written by inspection as a single determinant in continuant form. The synthesis of such networks is thus reduced to the simple procedure of writing the given transfer function in recurrent form by inspection, use of simple arithmetic operations to change the recurrent into continuant form, and reading the numerical values of the network elements directly from the continuant.

Active filters using a predistortion technique

A method is described of converting passive predistorted ladder networks to active equivalents.