Low-pass Transfer Function Research Articles

Explicit design equations for an active distributed RC network

Explicit design and sensitivity equations are given for an active distributed RC network. This network is used to approximate a low-pass transfer function with a pair of complex poles. The expression for the transient response shows that the dominant poles alone yield a good approximation.

Active Distributed RC Realization of Low-Pass Magnitude Specifications

This paper introduces a design method that allows one to find easily an active distributed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> network transfer function whose magnitude approximates the magnitude of a specified lumped low-pass transfer function. The resulting distributed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC(\overline{RC})</tex> transfer function is a product of simple transfer functions rational in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P = \cosh \sqrt{sRC}</tex> , each using a different <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> product. The overall transfer function is then realized as a cascade of the realizations of the simple transfer functions using a variety of available synthesis techniques. In addition, the characterization of a distributed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RC</tex> transfer function as a lumped transfer function times a delay factor, as presented herein, permits easy calculation of the transient response of the network. An example of the magnitude approximation is given together with the phase correction made by adding a delay term.

Synthesis of Rational Transfer Function Approximations Using a Tapped Distributed RC Line With Feedback

This paper describes a simple procedure for synthesizing an active distributed RC network which, by using dominant poles and zeros, realizes a very accurate approximation of an arbitrary stable rational transfer function. The network uses a single uniformly distributed RC line with taps spaced along its length. A linear combination of tap voltages is added to the input signal to form the driving voltage for the RC line; the output signal is also a linear combination of the tap voltages. The network offers a number of significant advantages. Since it realizes a nearly rational transfer function, the approximation problem can be conveniently solved using readily available results on rational function approximation. Also, the network uses only one uniform RC line, the transfer function can be changed simply by changing resistor values, and the frequency can be scaled by minor connection changes. Thus one standard network with minor modification is useful for a wide variety of applications. This paper develops the design procedure and derives the various sensitivity functions of importance. Two example designs are carried out: an approximation to a second-order low-pass transfer function and an approximation to a second-order band-pass transfer function with a Q of 100. The sensitivities for the examples are very reasonable and the measurements made on laboratory models indicate excellent agreement with theoretical predictions.

Synthesis of delay networks for an analogue computer

The merits and limitations of simulating a time delay by means of active RC networks are considered. The various methods of determining transfer functions which approximate to that of a pure delay, are discussed. Optimum all-pass and low-pass transfer functions based on a minimum-mean-square-error criterion are derived and compared to the Padé functions and equal-ripple envelope delay functions. The relative merits of the low-pass and all-pass approximations are discussed. Some methods of implementing the optimum transfer functions on an analogue computer are described.

Sensitivity in some simple RC active networks

Five different RC active networks are considered, each of which produces a second-order low-pass transfer function. Two of the networks use positive feedback (one from an amplifier of nominally unity gain); two use negative feedback (one from an amplifier of nominally infinite gain); and the fifth uses a negative-impedance convertor. Equations are developed for the sensitivity of each circuit to its active and passive components, and these are compared with those for the sensitivity of a passive LCR circuit. The four active circuits using conventional amplifiers are also compared in terms of the total error in the transfer-function amplitude caused by given errors in the gain and feedback of the amplifier used.

The Symmetrical Transfer Characteristics of the Narrow-Bandwidth Four-Crystal Lattice Filter

The conventional four-crystal lattice filter is analyzed, without recourse to image parameter theory, in order to obtain an (approximate) expression for a general symmetrical insertion-loss characteristic. The resulting expression is given as an equivalent low-pass transfer function which is the ratio of two polynomials of the second degree in complex frequency. These in turn have coefficients which are functions of three parameters: 1) shunt capacitance ratio; 2) fractional-bandwidth, crystal-Q products; 3) fractional stagger of the inner critical frequencies of the crystal. Numerous curves illustrate the effect on the insertion-loss characteristic of the three design parameters. Where comparison is possible, the approximate equations reached in this paper are shown to agree with the corresponding approximate image parameter equations of Kosowsky. Also, a variety of responses are shown to result which were not discovered by previous image parameter techniques. The approximations involved are equivalent to ignoring critical frequencies of the impedance elements (and hence of the transfer function) when these are far removed from the filter center frequency. Thus the representation is only valid near the filter center frequency. However, the error involved in the approximation is typically less than 0.1 db to the 40-db attenuation points, with narrow-band filters.