Linear Time-invariant Circuits Research Articles

TIME DOMAIN DIAKOPTIC ANALYSIS BASED ON REDUCED-ORDER STATE EQUATIONS

In this paper we present some new tearing techniques to systematically formulate the state equations in symbolic normal-form for linear and/or nonlinear time-invariant large-scale analog circuits. The excess elements of the first and of the second kind are unitarily treated in order to allow a symbolic representation of the circuit with a minimum number of state variables. A procedure to reduce the state equation number of each subcircuit is also presented. The reduced-order is based on an implicit integration algorithm and on the successive elimination of the selected state variables. Examples are given to illustrate the decomposition procedure, the assignment of the connection sources and the reduced-order technique.

A New ATPG Technique (ExpoTan) for Testing Analog Circuits

In analog testing, usage of a single sinusoid as a test signal when compared to multitone signal, and fault detection with digital counting technique, facilitate the test implementation with simple built-in self-test hardware and make testing more cost effective. In this paper, a novel test-set-selection technique known as ExpoTan for testing linear-time-invariant (LTI) circuits is presented. The authors formulate the test generation problem with tan <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ( ) and exponential functions for identification of a test signal with maximum fault coverage. For identification of a test signal the ExpoTan technique combines test generation and test-set-compaction tasks in a single phase and generates an efficient compacted test set. The experimental results show that the testing of LTI circuits using ExpoTan technique for the benchmark circuits achieves the required fault coverage with shorter testing time and test generation time

Application of generalized phasors to eigenvector and natural response computation of LTI circuits

This paper focuses on the natural response computation of linear time-invariant (LTI) circuits, in those cases in which a closed-form solution, rather than numerical integration of the state-variable equations, is preferable. A computationally efficient procedure is presented to individually obtain the eigenvectors of matrix A, both for distinct and repeated eigenvalues. Instead of performing orthogonal matrix transformations, the proposed method relies on the solution of the nodal equations corresponding to a zero-input circuit in the generalized phasor domain. This allows propagation of natural modes to be computed by means of a simple procedure that closely resembles conventional ac analysis.

Recursive calculation of ladder circuit frequency functions

It is shown how the transfer function as well as the input impedance and admittance of any linear time-invariant ladder circuit can be determined in closed form as frequency functions dependent upon the network element parameters. All elements of the ladder circuit must be explicitly specified by their frequency characteristics. The results are achieved with the aid of recursive algorithms. The outcome is demonstrated by means of an example.

A Comparison of Techniques for State-Space Transient Analysis of Transmission Lines

This paper reviews and compares several methods to analytically obtain the transient response of transmission lines in the time domain, in those cases where frequency independent parameters can be assumed. The distributed-parameter line is modeled by the cascaded connection of a number of lumped-parameter /spl pi/ circuits, each one representing a fraction of the line length, leading to a linear time-invariant (LTI) circuit. The associated state-space equations are formulated, allowing explicit expressions for the state variables to be written in the time domain. The solution is then obtained by means of three different approaches, all of them requiring that the natural frequencies be previously computed, namely: eigenvector-based procedure, Vandermonde matrix method, and Lagrange interpolation formula. Numerical integration by the trapezoidal rule is also considered for comparison. Two kinds of test results are presented. First, accuracy of the results provided by the LTI lumped-parameter model are compared with those obtained using the Electromagnetic Transients Program. Second, a comparison is performed in terms of the computational cost involved in each method. Two cases of practical interest are assessed, namely solving from scratch the state equations and updating the solution for a new set of initial conditions.

Efficient Decomposition Techniques for Symbolic Analysis of Large-Scale Analog Circuits by State Variable Method

New tearing techniques for the systematic formulation of the state equations in symbolic normal-form for linear and nonlinear time-invariant large-scale analog circuits are developed. Some examples are given to illustrate the decomposition procedure and the assignment of the connection sources. The partial symbolic state equations of the opamp μA 741 are obtained using the tearing approach, while these equations cannot be written working on the whole circuit.

A New ATPG Technique (MultiDetect) for Testing of Analog Macros in Mixed-Signal Circuits

A new test-set selection technique based on the frequency-domain testing of analog circuits is presented in this paper. We propose a new automatic test pattern generation (ATPG) method known as MultiDetect for testing linear time invariant (LTI) circuits. The proposed technique is best suited for use of existing building blocks in systems-on-chip for implementation of an on-chip test-signal generator and test-response analyzer. The generated test set with the MultiDetect method can effectively detect and diagnose both soft and hard faults and does not require any precision analog signal sources or signal measurement circuits when implemented as built-in self-test (BIST). Testing of analog blocks based on circuit-transfer function makes our ATPG a general purpose method for all kinds of LTI circuits. A new novel test method causing the device under test to saturate or get out of saturation, to detect a fault with simple detection hardware, is also introduced in this paper. In our proposed novel test scheme, usage of a single sinusoid as a test signal when compared to multitone signal, and detection of faults with digital counting technique, facilitate the test implementation with simple BIST hardware and make testing more cost effective. The sinusoid is a very useful waveform for testing and analyzing LTI circuits. In the steady state, both the input and output of a stable LTI circuit are sinusoids of the same frequency. The relationship between the amplitudes and phase angles of the input and output sinusoids is frequency-dependent. Identification of a sinusoid that detects more faults results in an optimized test signal set. The technique used in the MultiDetect method for identification of a sinusoid results in an efficient compacted test set. The search for fault diagnosis is restricted to a limited set of faults, making diagnosis fast in the MultiDetect method. A methodology for test-set compaction of the MultiDetect technique is described and results of experiments on various test circuits are discussed. The proposed method is seen to be efficient for low-power applications. The experimental results show that the testing of LTI circuits using the MultiDetect technique for the benchmark circuits achieves the required fault coverage with much shorter testing time.

Efficient DDD-based symbolic analysis of linear analog circuits

The general framework of a symbolic analysis environment for linear time-invariant circuits is presented. Such a framework consists of two major parts: a simplification at circuit level and a simplifying expression-generation. An overview of the known techniques for both circuit simplification and expression generation is given, and in addition a new nonhierarchical expression generation technique based on the concept of determinant decision diagrams (DDDs) is presented. The DDD-based technique is evaluated on a set of benchmark circuits together with a known fast symbolic analysis algorithm, and turns out to be the most efficient. The DDD-based expression generation algorithm is, therefore, the best solution for nonhierarchical linear symbolic expression generation to the authors' knowledge.

Time-domain thermal noise simulation of switched capacitor circuits and delta-sigma modulators

This paper presents an accurate and efficient algorithm for simulating the effects of thermal noise in time-domain. The algorithm presented in this paper is based on Monte Carlo methods and is applicable to linear time invariant circuits. In addition to linear time invariant elements, the algorithm can easily be extended to include clock controlled switches and single or multibit quantizers. This expands the thermal noise analysis capability to switched capacitor circuits and oversampled delta-sigma modulators. Thermal noise generated by different elements is modeled, in the time-domain, by a random pulse waveform having a desired power spectral density. Typically the noise power level is much smaller than that of other desired signals in the circuit and accurate simulation is needed to obtain correct results. In addition, random noise waveforms require the computation of many time points in each transient analysis and efficient simulation methods are needed. In this paper, a new method for computing the transient response of linear time invariant circuits is used. This method accurately and efficiently computes the response to the random pulse waveforms. In addition to thermal noise, the method can also be used to simulate the effect of random dither signals in delta-sigma modulators. Examples of noise simulation are given and, when possible, comparison with measurements, previously published results, or analytical expressions is done.

Limits to the fluctuation-dissipation theorem for nonlinear circuits

It is well known that the equilibrium thermal noise behavior at the terminals of any linear time-invariant (LTI) RLC circuit can be predicted from knowledge of the driving-point impedance and temperature alone. This paper examines the conjecture that similar results hold if the capacitors and inductors are nonlinear. We refine the conjecture by analyzing the behavior of an RLC bridge circuit with the nonlinear inductor and capacitor carefully matched so the terminal behavior reduces to that of a linear resistor R. We show that the terminal noise current is not that predicted by the Nyquist-Johnson model for R if the driving voltage is time dependent or the inductor and capacitor are time varying. This counterexample disproves the conjecture, which does hold, however, for the bridge circuit with nonlinear (but time invariant) devices if the driving voltage is zero or constant. This paper makes exact calculations using techniques from stochastic differential equations and using reversibility arguments.

The transition matrix for linear circuits

The state transition matrix /spl Phi/(t)=e/sup At/ plays an important role in the state variable analysis of linear time-invariant circuits. In this paper, we give a method to compute the equivalent matrix for a set of differential-algebraic equations. Specifically, the algorithm is illustrated for the modified nodal analysis (MNA). A benefit of using MNA formulation is that equation formulation is straightforward and computer-aided analysis of large circuits is simplified. Another benefit is that inconsistent initial conditions in the analysis of switched circuits is automatically and correctly handled by the algorithm. Comparison with the state variable approach is made, and application to the simulation of linear circuits is given. The method is based on circuit theory concepts accessible to all electrical engineers. In addition to the transition matrix, a numerical method to compute the zero state response of linear circuits for a restricted set of inputs is given. The transition matrix along with the zero state response results in a special algorithm that is used to compute the time response of lumped linear time-invariant circuits at equally spaced intervals of time. This method is compared with the solution of linear circuits by SPICE-like simulators.

Analysis of stochastic behaviour of linear circuitsusing first-order second-moment and adjoint network techniques

An efficient computational method is proposed to analyse the stochastic behaviour of large-scale linear time-invariant circuits. The method is based on the first-order second-moment and adjoint network techniques. Numerical results obtained using the proposed technique are compared with those from Monte Carlo simulation.

Sampled data simulation of linear and nonlinear circuits

In this paper, an efficient, accurate, stable, and explicit method for transient analysis of lumped linear time invariant circuits is given. The method formulates a set of finite difference equations in the analog domain for sampled data simulation of the circuit. The solution of these equations gives the network response at fixed and equally spaced discrete instants of time. The fixed-time interval between each solution can be chosen arbitrarily and does not depend on the circuit time constants. The transient solution at each time point requires one matrix/vector multiplication and vector addition only. The algorithm is a general computer oriented formulation method that can be applied to any linear circuit. In addition to linear time invariant circuits, the method can be applied to a restricted class of time varying and/or nonlinear circuits. It is directly applicable to linear networks containing periodically clocked ideal switches, for example, switched capacitor and switched current circuits. In these networks, the circuit is linear inside each phase and changes its topology at fixed discrete instants of time. All nonlinear elements allowed in analysis must be clocked, such that their characteristics change only at discrete instants of time. For example, clocked digital circuits, such as, ideal comparators, A/D and D/A blocks are allowed. In particular, the method is applied to the simulation of oversampled delta-sigma modulators. The method can be used to study the effect of clock feed through, finite switch resistance, and finite gain bandwidth of operational amplifiers on the performance of these modulators. In addition, the algorithm can be used for analysis of continuous time delta-sigma modulators. Examples of simulation results are given.

Fault-based ATPG for linear analog circuits with minimal size multifrequency test sets

An automatic test pattern generation (ATPG) procedure for linear analog circuits is presented in this work. A fault-based multifrequency test approach is considered. The procedure selects a minimal set of test measures and generates the minimal set of frequency tests which guarantee maximum fault coverage and, if required, maximal fault diagnosis, of circuit AC hard/soft faults. The procedure is most suitable for linear time-invariant circuits which present significant frequency-dependent fault effects. For test generation, the approach is applicable once parametric tests have determined DC behaviour. The advantage of this procedure with respect to previous works is that it guarantees a minimal size test set. For fault diagnosis, a fault dictionary containing a signature of the effects of each fault in the frequency domain is used. Fault location and fault identification can be achieved without the need of analog test points, and just in-circuit checkers with an observable go/no-go digital output are required for diagnosis. The procedure is exemplified for the case of an analog biquadratic filter. Three different self-test approaches for this circuit are considered. For each self-test strategy, a set of several test measures is possible. The procedure selects, in each case, the minimal set of test measures and the minimal set of frequency tests which guarantee maximum fault coverage and maximal diagnosis. With this, the self-test approaches are compared in terms of the fault coverage and the fault diagnosability achieved.

Minimum dimensional state-space formulation of linear systems

The state-space modelling of linear time-invariant RLC circuits with independent sources is reconsidered. The principal idea presented is two-fold. In the first part of the paper it is shown that, if the circuits considered have capacitor and current source-only cut sets and/or inductor and voltage source-only loops, a new set of state equations can be obtained which will have a lower dimension compared with the state equations obtained using the standard textbook approach. A simple algorithm is presented to derive the reduced-order state equations and initial conditions. In the second part of the paper, it is demonstrated that further reductions are possible if the state equations are written at other coordinate systems rather than the usual Cartesian co-ordinates.

New approach to the design of thyristor and transistor switching circuits

The paper gives a new approach to the design of power electronic circuits. Time periodic functions of excitation and response are required as input data for the design. A sequence of the linear time-invariant circuits is found on the basis of these data. As a result of the synthesis method new switching structures are obtained. The synthesis method presented can be applied to create new thyristor and transistor arrangements, particularly thyristor compensators.

DC and AC Analysis of Thyristor Circuits by Coordinate Transformation and Describing-Function Method

A new and practical method is developed for evaluating approximately both dc and fundamental ac components of waveforms in steady-state thyristor circuits. The process is purely algebraic and is not based on simulation techniques, which utilizes a modified form of describing-function method for each thyristor. The method can be applied, at least in principle, to any configuration of linear time-invariant circuits containing several thyristors. Two simple but fundamental examples are included to show the limit and utility of the proposed method.

Efficient Calculation of Exact Group Delay Sensitivities

This paper presents, explicitly, an approach to the exact calculation of group delay and its sensitivities with respect to component parameters based on the adjoint network concept and applicable to linear time-invariant circuits. In general, no more than four analyses are required and the computational effort is only moderately more than is necessary for a single analysis. The results presented are in a form particularly suited to the computer-aided design of microwave circuits and include useful tables of sensitivity expressions.

Modes in Linear Circuits

The purpose of this review paper is to show that the concept of modes does apply to any linear time invariant circuit. This is accomplished by reducing the network equations to the standard vector form \dot{y} = Ay + f . In particular, it is shown that 1) any free oscillation of a linear circuit can be thought of as a superposition of non-interacting modes, 2) in the case of free oscillations, the amount of excitation of each mode can easily be expressed in terms of the initial conditions, 3) any forcing function excites each mode independently, and finally, 4) the resonance phenomenon is easily interpreted and the importance of the proper type of excitation is made obvious. Vector notation is used throughout. Examples of RC, RLC and active circuits are included.