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In this paper a comprehensive procedure for the analog modeling of Fractional-Order Elements (FOEs) is presented. Unlike most already proposed techniques, a standard approach from classical circuit theory is applied. It includes the realization of a system function by a mathematical approximation of the desired phase response, and the synthesis procedure for the realization of basic fractional-order (FO) one-port models as passive RC Cauer- and Foster-form canonical circuits. Based on the presented one-ports, simple realizations of two-port differentiator and integrator models are derived. Beside the description of the design procedure, illustrative examples, circuit diagrams, simulation results and practical realizations are presented.

A generalized structure that is capable of implementing power-law filters derived from 1st and 2nd-order mother filter functions is presented in this work. This is achieved thanks to the employment of Operational Transconductance Amplifiers (OTAs) as active elements, because of the electronic tuning capability of their transconductance parameter. Appropriate design examples are provided and the performance of the introduced structure is evaluated through simulation results using the Cadence Integrated Circuits (IC) design suite and Metal Oxide Semiconductor (MOS) transistors models available from the Austria Mikro Systeme (AMS) 0.35 μm Complementary Metal Oxide Semiconductor (CMOS) process.

In this paper we compare two versions of fourth-order band-pass (BP) filter section realized as coupled structure, having negative feedback around two Biquadratic sections in cascade, usually designated as Biquartic section (i.e. it is the fourth-order band-pass section). The frequency response of a filter is subject to vary from the nominal values due to the effects of aging, changing working conditions, the fabrication tolerances of passive and active elements, etc. In order to maintain the filter’s characteristics inside given specifications at the time of manufacture, and as long as possible, the main requirement is to design filters with reduced sensitivity to component tolerances. The Biquartic section has significantly reduced sensitivities to the changes of passive element values, particularly within the pass band, in comparison to common cascade design. For that purpose this structure is very suitable for use as a building block of narrow band, high order BP filters. The sensitivities are further reduced by using three-amplifier biquadratic block instead of more sensitive single-amplifier block. We investigated the sensitivities of two versions of biquartic section with two extreme combinations of second-order sections q-factor values, using Schoeffler’s measure, the dynamic range, and output thermal noise using analysis in PSpice. Biquartic sections used throughout the examples are realized as reduced amplifier version retaining the low sensitivities of biquartic design. They also possess simple tuning features and simple design ; design equations are given.

In this paper the model of a system for measuring small currents, which generates the digital signal proportional to the input current, was designed. The analysed measuring system is the part of a larger microelectronic mixed-signal integrated chip (SoC -System on Chip), and it is located between the input sensor and the external digital device. Since the part that mostly contributes to the noise inside the whole measuring chain is at the input of the chain, the measurement method is analyzed to minimize the impact of noise. Because the noise is not definite, the simulation of measurement by the model was made using statistical principles. Performing multiple measurements of the input value (in our example it is current I d ) it is possible to calculate its mean value and standard deviation. It was shown that the measured quantity can be calculated from the mean value and that the standard deviation represents an amount of noise. Depending on the standard deviation it is needed to do either less or more sequential measurements to determine the reliable measured output digital value.

In this paper we propose a new, programmable, easy-tunable, current-mode general-purpose Biquad (C-GP-B), using second-generation current conveyors (CCIIs), based on the well-known direct form II, which, among other transfer functions, is also suitable to realize band-rejection (BR) or notch transfer function. It is convenient for elliptic transfer function realizations because it can realize finite transfer function zeroes. Since it has low sensitivity and grounded capacitors it is suitable for integrated circuit (IC) design. The new Biquad is then compared in sensitivity performance with other two CCII-based versions of biquadratic sections obtained by different approaches. Since all presented Biquads use CCIIs they operate in the current mode, and are therefore suitable to realize filter circuits operating at higher frequencies, consuming less power, and needing lower supply voltage. As an example, the design procedure, and the Monte Carlo sensitivity analysis using PSpice were performed for a fourth-order elliptic low-pass (LP) filter.

This paper presents the low-sensitivity realization of narrow band-pass analog active filters using second-generation current conveyors. It is known that filters using multiple-feedback structures have reduced sensitivities to passive component tolerances. It was shown that using the follow-the leader (FLF) topology, the sensitivity of current transfer function magnitude was reduced when compared to the common cascade topology (CAS). The sensitivity reduction happens in the current mode in the same way as in the voltage-mode. Besides sensitivity, the output thermal noise is also reduced using the FLF topology. As an example, the design procedure and the Monte Carlo sensitivity, as well as output thermal noise analysis using PSpice were performed for a 4th-order Butterworth narrow BP filter.

This paper presents the realization of third-order low-pass active- RC filters using a new Leap-Frog (LF) topology. New structure is a simplified LF structure with the elements calculated directly from the transfer function coefficients. Several versions of the circuits are presented and compared. The comparison to other common third-order filter sections is done, as well. The new LF filter has the reduced number of components, reduced complexity and straightforward design procedure compared to classical filters. As an illustration of the efficiency of the proposed new LF filter, the sensitivity analysis using Schoeffler sensitivity measure as well as output thermal noise analysis was performed on examples with Butterworth and Chebyshev 0.5dB pass-band ripple transfer functions. Using PSpice with a TL081 opamp model, the filter performance is simulated and the results compared and verified by measurements on a discrete-component breadboard filter. All equations needed for the step-by-step design are given.

In this paper the dynamic-range optimization of a new class of low-pass active-RC filters, based on the leap-frog topology is presented, using the MATLAB Optimization Toolbox. The new structure is a simplified leap-frog structure with a reduced number of components, reduced complexity, and permitting a straightforward design procedure. It is most suitable for the nonbalanced-to-ground version of low-sensitivity filters. As an illustration of the efficiency of the proposed new leap-frog filter, a sensitivity analysis using Monte Carlo runs was performed on examples with Butterworth and Chebyshev 0.5dB pass-band ripple low-pass filters. A comparison with other common filters such as standard leap-frog filters and single-amplifier Biquads demonstrates the sensitivity improvement obtained with the new filters. In this paper we present a numerical method of optimizing these new filters for improved dynamic range.

The application of Matlab, combining its symbolic and numeric calculation capabilities, to calculate noise and sensitivity properties of allpole active-RC filters is shown. Transfer function coefficients calculations, as well as plotting of amplitude-frequency and phase-frequency characteristics (Bode plots) have been performed using Matlab. Using Matlab a comparison of different design strategies of active-RC filters is done. It is shown that active-RC filters can be designed to have low sensitivity to passive components and at the same time possess low output thermal noise. The design procedure of low-noise and low-sensitivity, positive- and negative-feedback, second- and third-order low-pass (LP), high-pass (HP) and band-pass (BP) allpole filters, using impedance tapering, is presented.

One of the problems with on-chip active-RC filters is that in order to satisfy tight specifications accurately, they need to be fine-tuned. This becomes more difficult with increasing filter order. In this paper, we introduce a simplified tuning method which is obtained by a new structure consisting of a filter section of second- or third-order (Biquad or Bitriplet which is referred to as a ‘tuning block’), in cascade with a ladder-RLC filter (passive LC, or simulated-active RC). The cut-off frequency of the resulting filter can be simply tuned by adjusting one component of the tuning block. The new ladder tuning-block (LTB) is compared with two conventional structures, a cascade of Biquads and a single-ladder filter. Fine tuning these two structures is considerably more complicated. It is shown that beside the advantage of ease of tuning, the sensitivity to component tolerances of the new structure is lower than that of the two conventional ones, for the price of a slightly higher filter order.

This paper presents the low-sensitivity and low-power realization of narrow high-order band-pass (BP) active resistance-capacitance (RC) filters. It is known that filters using multiple-feedback structures have reduced sensitivities to passive component tolerances. We present a filter realization using cascaded "two-amplifier biquartic" BP active filter blocks (CBQ), with reduced numbers of operational amplifiers. The "impedance tapering" and the gain-sensitivity-product minimization decrease both passive and active sensitivities of biquartic blocks. As an example, the design procedure and the Schoeffler's sensitivity analysis, were performed for an 8th-order Chebyshev narrow-BP filter.

In this paper two general-purpose (GP) biquadratic filters are analyzed and optimized. The design equations for both filters are given, and the methods used to optimize the dynamic range and reduce the noise are described. On one example of the 2/sup nd/-order Butterworth filter, PSpice analyses were carried out, and a comparison of noise, referred to the input, is given. The dynamic range is optimized such as to have the same signal level at each amplifier output. Both optimization procedures are highly effective and can be applied to the similar filter design problems.

This paper is concerned with a new design method of low-sensitivity current-mode filters, which results from low-sensitivity voltage-mode filter design using impedance tapering. The current– mode filters are obtained by application of a network transposition to their voltage-mode counterparts, in which the passive-RC network remains the same, and both filters are expected to have identical sensitivity properties. However, current– mode filters may be easier to realize in IC form and are expected to have higher bandwidths, greater linearity and wider dynamic range than the voltage-mode filters. In this paper, the 2nd-order class-4 current– mode filters (with positive feedback) will be considered. Design procedures will be given for the design of low-sensitivity low-pass (LP), high-pass (HP), band-pass (BP) and band-rejection (BR) filters, which are all realizable in class-4. A sensitivity analysis is examined using PSpice Monte Carlo runs.

In this paper we introduce a new design procedure for low-sensitivity filter sections, which have a symmetrical passive-RC network in the operational amplifier (opamp) feedback loop. In the design procedure we apply impedance scaling to the symmetrical bridged-T and twin-T networks; they become "potentially symmetrical" (G.S. Moschytz, Linear integrated networks: fundamentals, V. Nostrand Reinhold, 1974). The design of low-sensitivity all-pole active-RC filters, which have an RC-ladder network in the opamp feedback loop, has already been published (G.S. Moschytz, IEEE Trans. on CAS-11, vol. CAS-46, no. 8, pp. 1009-1026, 1999). There, successive I-sections of the ladder structure are impedance scaled upwards, from the driving source to the positive opamp input; we refer to it as "impedance tapering". In both cases we reduce the filter's magnitude sensitivity to variations of the passive components of a circuit. The new design concept is demonstrated by designing two very popular and often used filter sections: a band-pass realized by the Deliyannis SAB, and a band-rejection filter with a twin-T. The sensitivity analysis is examined analytically and double-checked using PSpice Monte Carlo runs.

This paper presents the low-sensitivity realization of band-pass (BP) active resistance-capacitance (RC) filters using "uniform modified leap-frog" (UMLF) structure. Sensitivities with respect to filter passive components are investigated. It is shown how the optimization can be performed in order to achieve minimal sensitivities with relatively small values of Q-factors. Multiple-feedback-structured filters have low sensitivities to passive component tolerances, and additional sensitivity reduction of 2/sup nd/-order building biquads is achieved by the "impedance tapering" design. As an example, the sensitivity analysis, using Shoeffler sensitivity measure, was performed on a Chebyshev 8/sup th/-order BP filter. The filters were realized using UMLF and leap-frog (LF) structures, where sensitivity-optimized single-amplifier (SAB) and Sallen and key (SAK) biquads realize the 2/sup nd/-order biquads.

In this paper we introduce a new design procedure for low-sensitivity and low-noise filter biquads which have a symmetrical bridged-T passive-RC network. They are also low-power circuits, in that they use only one operational amplifier (opamp). The new design concept is based on the recently introduced "impedance tapering" method which was applied to allpole filters in G.S. Moschytz (1999), whereas in this paper it is extended to the design of filters with unite zeros. Thus, it is applied to two commonly used filter sections suitable for the realization of band-rejection and all-pass filters with low and medium pole-Q factors. In the new design procedure, the topology and component count remain the same; we just judiciously select the component values in order to reduce component tolerance sensitivity and improve noise performance. The sensitivity analysis is examined analytically and double-checked using PSpice Monte Carlo runs. In the PSpice noise analysis, a macro-model of the u uA741 opamp is used. It is found that the minimum-sensitivity and minimum-noise filters coincide.

In this paper transformations, which are very often used in design of active filters, are summarized, and used for revealing the relations between different classes of active filter sections. Among various transformations we distinguish: frequency transformations and network transformations. As frequency transformation examples we consider the well known "low-pass to band-pass" (LP-BP) and less known "lossy low-pass to band-pass" (LLP-BP) frequency transformation. As examples of network transformations, which are performed by network elements substitution we consider (i) "lossy" LP-BP network transformation: to obtain BP filter it substitutes each resistor of the LP prototype filter by a series RC circuit, and each capacitor by a parallel RC circuit ; and (ii) complementary transformation: passive network in negative feedback loop is transformed into passive network in positive feedback loop of operational amplifier.

To reduce the low-power allpole active-RC filter's output magnitude sensitivity to passive components, the "impedance tapering" design method was used. This design procedure has been applied to the design of 2nd- and 3rd-order low-pass (LP) (class 4) Sallen-and-Key filters and has already been published in G. S. Moschytz (1999). In this paper the active sensitivity analysis using Mathematica was performed on the same "impedance tapered" filter sections. The analysis was performed by calculation of the real filter's transfer-function magnitude which was denormalized to various cut-off frequencies from low to high, using a single-pole model of a real operational amplifier. The amount of active and passive sensitivity reduction was compared using the same degree of impedance tapering. It was demonstrated that both sensitivities, i.e. active and passive, were corresponding reduced for the same filter. Thus, a judicious selection of component values accounts for a considerable decrease in both passive and active sensitivities.

The design procedure of 2/sup nd/- and 3/sup rd/-order low-sensitivity low-power allpole active resistance-capacitance (RC) filters, using the impedance tapering design method has already been published. Beside sensitivity minimization the impedance tapering was used for output noise minimization and output dynamic range maximization. In this paper it is shown that active resistance-capacitance (RC) filters can also be designed for low input thermal noise. The design procedure of filters with minimum input noise is very similar to the design of minimum sensitivity filters in its most important steps. Thus, the judicious selection of component values accounts for the considerable decrease in sensitivity and in input thermal noise. The minimum noise at the filters input is obtained by minimizing the total resistance of the filter circuit. The noise analysis was performed on the 2/sup nd/- and 3/sup rd/-order (class 4) Sallen and Key low-pass filter sections using MATLAB.

The design procedure for high-order single amplifier BP filters is presented. A method for the design of 2nd- and 4th-order band-pass (BP) active-RC filters using a modified low-pass to band-pass (LP-BP) frequency transformation, was already presented in previous works [1], [2]. It was shown that a BP filter could be realized simply by the substitution of resistors and capacitors of the ladder in a low-pass (LP) prototype filter, by serial and parallel RC circuits. Such a substitution results from a so-called, "lossy" LP-BP transformation. In this paper, the design procedure is extended to higher-order BP filters, such as 6th- and 8th-order. The design procedure is simple, and the closed-form design equations, are presented. Furthermore, it is shown that "impedance tapering" decreases sensitivities to component tolerances for the LP prototype, as well as for the resulting BP filter. The Schoeffler’ s sensitivity measure is used for the sensitivity analysis.

This paper presents fault location technique for testing analog filters using a fault dictionary. The single hard faults of the filter passive elements can be located and identified. To enable the efficient testing, an initial fault dictionary is optimised in order to get the minimal size of the dictionary with maximum number of uniquely recognised faults. After a simulation, the procedure tests the circuit and locates the failed element using adequate fault isolation criterion. The performances of the described method are illustrated with an analog filter fault diagnosis example performed in the frequency domain. The fault detection problem has also been considered.

Third order low-pass filters are analyzed. Two different configuration of General Impedance Converter (GIC) based filters are compared to two Sallen and Key (SAK) based structures. For GIC based third-order filters realization procedures are given. The transfer functions and the component values for the third-order filters with Butterworth response are presented. The sensitivity analysis is done, and the results of Monte Carlo runs, as well as Schoeffler sensitivities are shown. The best results are obtained with the GIC based filter, which use four capacitances for the realization.

The design procedure of low-sensitivity second-and third-order class-4 Sallen and Key active resistance-capacitance (RC) allpole filters, using impedance tapering, has already been published. There a ladder structure in the positive feedback loop of an operational amplifier was used. In this paper desensitization using impedance tapering is applied to class-3 circuits with negative feedback referred to as a “single-amplifier biquad” (SAB). The RC-section in the negative feedback loop is impedance scaled upwards, from the driving source to the negative amplifier input. Second-order band-pass filters are considered. The improvement comes free of charge; component count and topology remain unchanged, whereas the component values, selected for impedance tapering, account for the significant decrease in sensitivity to component tolerances.

The design of second-order band-pass (BP) active RC filters using a modified low-pass to band-pass (LP-BP) frequency transformation is presented. The transformation is applied to a first-order low-pass (LP) filter as the (odd-order-)prototype, from which a single-amplifier second-order BP filter is constructed. The operational amplifier is added to the first-order LP circuit in order to provide a low output impedance and supply a positive feedback loop to enable a pole shifting process needed in the realization. It is shown that a BP filter can be realized by substitution of resistor and capacitor in the low-pass prototype filter, by serial parallel RC circuits in the resulting band-pass structure. Schoeffler sensitivity is used as a measure of the magnitude sensitivity to component tolerances. A step-by-step design procedure is verified for several second-order band-pass filter circuits, using different impedance scaling ratios, resulting by different sensitivities. It is shown that the circuit with equal impedance scaling ratios yields the best results. Obtained results are double-checked using PSPICE.

The design procedure of low-sensitivity active resistance-capacitance (RC) allpole filters, using impedance tapering, has already been published. The low-sensitivity filter sections already described in publications are class 4 (TT-SABB) Sallen and Key (1955) sections. In this paper class 3 (SAB) sections for low pole-Q realization are considered. Impedance tapering is applied on L-sections, which are situated in the negative-feedback of the operational amplifier in open-loop mode. L-sections are impedance scaled upwards, from the driving source to the negative amplifier input. A second-order band-pass filter is considered. Pole-Q factors for low-Q building blocks take their values up to, say, 5. The sensitivity to component tolerances of the circuit is shown to be small for any type of impedance tapering regardless of the gain-sensitivity product (GSP) value.

The design of low sensitivity band-pass (BP) active resistance-capacitance low (RC) filters using impedance tapering is presented. 4/sup th/-order band-pass filters are considered. The design procedure for low-sensitivity low-pass (LP) prototype second-order class 4 Sallen and Key active-RC allpole filters, using impedance tapering, has already been published. In this paper a low-pass to band-pass (LP-BP) transformation is applied to an impedance tapered 2/sup nd/-order LP filter, and a 4/sup th/-order BP filter is constructed. The component values, selected for impedance tapering, account for the considerable decrease in sensitivity to component tolerances for the LP as well as for the BP filter. As an example, a Chebyshev 4/sup th/-order BP filter is realized, and the sensitivities are analysed. Schoeffler sensitivity measure is used as a basis for comparison, and Monte Carlo runs are performed as a double-check. A considerable improvement in sensitivity is achieved, both for the BP filter as well as well as for the original LP prototype.

It is shown that low-sensitivity active resistance-capacitance (RC) filters are also low in output thermal noise. The design procedure of second- and third-order low-sensitivity allpole filters, using impedance tapering, has already been published. The component values selected for impedance tapering account for the considerable decrease in output thermal noise. The method of Zurada and Bialko (1975) was used to determine output noise spectral density and total RMS output noise of filters. Passive elements and operational amplifiers are represented by substitute noise models. The noise contribution of each device to the output node is calculated using noise transfer functions. The noise analysis was performed on the second-order (class 4) Sallen and Key low-pass, high-pass and band-pass filter sections using MATLAB. The extension to third- and higher-order filters follows the same principles. It was found that the filters with minimum noise coincide with the filters with minimum sensitivity to component tolerances.