Los fluidos supercríticos

In this chapter, a new methodology for the testability measure and ambiguity group determina- tion was presented based on the well-known pole and zero analysis and on pole-zero sensitiv- ity. Testability measure at a certain node of a circuit can be computed from the number of the poles and zero in addition to the DC gain of the transfer function. Also, the testability measure can be computed directly from the circuit matrix as given in Eq. (4-43). The ambiguity groups can be determined using the pole and zero sensitivity. Thus, the ambiguity group can be inter- preted as a group of circuit elements which affect the poles and zeros (and the DC gain if the

DC sensitivities are taken into account) of a circuit in the same way.

The main advantages of this methodology can be summarized as follows:

1) Our methodology is based on the pole-zero analysis and on the pole-zero sensitivity which can be employed in many applications such as stability of the control system, design and optimization of filters, compensation circuits in feedback amplifiers, simplification of the symbolic transfer function, behavioral modeling, and model order reduction.

2) The proposed methodology is independent of the frequency, unlike the sensitivity analysis where the sensitivity must be evaluated at each frequency point for the arbitrary range of the frequency. Furthermore, the normalized sensitivity is undefined at the zeros of the transfer function (the zero frequencies), and is equal to zero at the poles of the transfer

S_x j H s( ) s x, _j ( ) S_x j K_{0 s} s–z ---S_x j z_{i s} s– p_i ---S_x j p_i i=1 m

∑

∑

∑

function (the pole frequencies), thus no information is provided at these frequency points (cf. Eq. 2-5).

3) This methodology provides a new interpretation of the ambiguity groups based on the cir- cuit theory, unlike the matrix manipulation techniques which provide the mathematical interpretation given by the linearly dependent columns of the testability matrix.

4) The proposed method provides the relationship between the concept of the controllability and observability and testability measure. The controllability and observability essentially govers the existence of a solution to control problem, this is similar to the testability which also govers the solution of the diagnosis problem of the analog circuits.

5) The ambiguity groups, which are determined using numerical methods, depend on circuit specifications. Thus, the ambiguity groups are different from one specification to another. In contrast, our methodology provides the ambiguity groups which are the same for all cir- cuit specifications in the time domain or in the frequency domain.

6) The sensitivity analysis can be achieved based on the pole-zero sensitivity analysis (cf. Eq (4-47) and Eq. (4-48)). Thus, the applications of the sensitivity analysis in an analog test such as fault diagnosis, and test signal generation can also be achieved based on the pole- zero sensitivity analysis.

The analog and mixed-signal circuits have a large number of specifications. Checking all spec- ifications is a very time-consuming task. For this reason, an algorithm for selecting relevant specifications is required in order to reduce the test cost without affecting the quality of the test in terms of fault coverage.

In the following chapters, we will address the measurement selection problem based on the pole-zero analysis. The selection of specifications that have to be measured depends on the element testability concept which can be obtained based on the pole sensitivity.

Chapter 5

Element Testability and Measurement Selection for

Second-Order Circuits

5.1. Introduction

Analog and mixed-signal circuits have large numbers of specifications. Checking all these specifications can result in prohibitive testing times. Therefore, minimizing the specifications that need to be measured is required to reduce the test cost. The reduction of the test cost can be performed by ordering the tests to achieve high fault coverage or by dropping some specifi- cations to reduce test time [Milo98]. The shortcoming of these algorithms is that they do not provide the complete link between the circuit elements and circuit specifications.

Selecting a subset of specifications that have to be measured can be performed using the sensi- tivity analysis which provides the relation between the circuit elements and the performances. This relationship can be obtained by the ratio of the relative deviation of the circuit perfor- mance to the relative deviation of the circuit elements as a function of the frequency [Slam96] or the time [Dai90]. The sensitivity analysis ensures both the structural and functional test, where the circuit elements are tested by verifying the circuit functionality.

The measurement selection methods based on the sensitivity analysis are proposed in several previous works such as in [Sten87, Sten89, Dai90, Hami93, Slam96, Spaa96b]. However, since the sensitivity analysis is dependent on the frequency or the time, there is a need for han- dling a large matrix which is constructed by circuit sensitivities. The objective of the measure-

ment selection based on the sensitivity analysis is to improve the fault coverage, but the reduction of the test cost is not considered.

In order to overcome the shortcomings of the above-mentioned methods, we will present an algorithm for measurement selection which is independent of the frequency or the time. The aim of this algorithm is to reduce test cost by reducing the number of specifications that need to be measured. The selected specifications can serve as a signature in fault simulators to improve the fault coverage and to evaluate the test vectors applied at the inputs of the circuit under test. Also, this algorithm provides maximum information about the fault identification for fault diagnosis by breaking up the ambiguity groups.

The proposed measurement selection algorithm is based on the element testability concept which can be computed based on the sensitivity of the circuit poles. The element testability will provide the information about the difficulty in testing the circuit elements as well as the effect of the changes of the circuit elements on the circuit specifications.

We will devote this chapter to the discussion of the element testability and measurement selec- tion for second-order circuits. Higher-order circuits will be addressed in Chapter 6.

The algorithm is valid for parametric faults which caused by manufacturing process variations and do not affect the circuit topology. This algorithm can be classified under the specification- based test.