Research Article  Open Access
Hongzhi Hu, Shulin Tian, Qing Guo, "Fault Modeling and Testing for Analog Circuits in Complex Space Based on Supply Current and Output Voltage", Journal of Applied Mathematics, vol. 2015, Article ID 851837, 9 pages, 2015. https://doi.org/10.1155/2015/851837
Fault Modeling and Testing for Analog Circuits in Complex Space Based on Supply Current and Output Voltage
Abstract
This paper deals with the modeling of fault for analog circuits. A twodimensional (2D) fault model is first proposed based on collaborative analysis of supply current and output voltage. This model is a family of circle loci on the complex plane, and it simplifies greatly the algorithms for test point selection and potential fault simulations, which are primary difficulties in fault diagnosis of analog circuits. Furthermore, in order to reduce the difficulty of fault location, an improved fault model in threedimensional (3D) complex space is proposed, which achieves a far better fault detection ratio (FDR) against measurement error and parametric tolerance. To address the problem of fault masking in both 2D and 3D fault models, this paper proposes an effective design for testability (DFT) method. By adding redundant bypassingcomponents in the circuit under test (CUT), this method achieves excellent fault isolation ratio (FIR) in ambiguity group isolation. The efficacy of the proposed model and testing method is validated through experimental results provided in this paper.
1. Introduction
Over the past two decades, fault detection and diagnosis of analog circuits become an important research area where a number of corresponding theories and techniques have been developed. Among the researches in this area, fault dictionary is one of the most important methods that have attracted great interests [1â€“5]. Various circuit variables (e.g., voltage, current, and frequency) have been used in fault dictionary to obtain fault signatures [6â€“8]. In addition to the commonly employed measurement of output voltage, supply current testing has been applied widely in analog and mixed signal integrated circuits (ICs) with excellent fault coverage, especially for the catastrophic fault of CMOS ICs [9â€“11].
Despite the advantages of fault dictionary, there are still persistent challenges, such as test point selection and potential faults simulations, in applications of fault dictionary method for fault diagnosis of analog circuits. Wang and Yang proposed a slope fault mode, which not only reduces the required number of test points to two but also simplifies the potential fault simulation greatly [2]. However, the slope fault model is only limited to dynamic circuits. To achieve detection of parametric faults for analog circuits, Yang et al. proposes a complexcircle based fault model [3]. Although this proposed fault model is improved sequentially by optimal testing frequency selection [12] and fault location [13], the weakness of fault masking still remains. Moreover, a 3D model based on transfer function was proposed in [14]; it extends the distances between fault loci and improves the practicability for applications. Recently, Ma and Wang achieved detecting catastrophic faults and parametric faults, by analyzing the harmonic spectrum of output voltage and supply current [11]. Following the ideas of these researches, this paper proposes two improved fault models with satisfactory FDR based on collaborative analysis of output voltage and supply current. Moreover, to isolate ambiguity groups, a design for testability (DFT) method is proposed to achieve better fault isolation ratio (FIR) against the influence of analog tolerance and measurement errors.
This paper is organized as follows. First, the theory of collaborative fault model on 2D complex plane is introduced in Section 2. An example of a SallenKey filter is also provided to illustrate the theory in this section. In order to reduce the difficulty of fault location and isolation, an improved fault model in 3D complex space is proposed in Section 3. Furthermore, to deal with the ambiguity groups in the fault model, an innovative method of DFT is demonstrated in Section 4 to improve the FIR. Validation of the proposed method is provided by PSPICE. Finally, conclusions and future work are summarized in Section 5.
2. Collaborative Fault Model on 2D Complex Plane
2.1. Theorem of the 2D Collaborative Fault Model
A linear timeinvariant passive CUT is assumed to contain components . is assumed to be stimulated by a direct current (DC) power source and a family of alternating current (AC) signals , where is the number of AC signals. Then a dimension vector is composed as follows:
Furthermore, a single component is assumed to be the potential failure component in , whereThe voltage across is defined as . According to the Substitution Theorem, a voltage source can be used to replace the component in the CUT. Therefore, both voltage and vector are regarded as the excitation of the CUT. Then the collaborative output is defined as follows:where is the output voltage of CUT, is the power supply current, and is the Laplacian operator. and are predefined coefficients. Furthermore, the corresponding transfer function matrices and are defined aswhere , , , , , and are transfer functions with respect to and . Then, can be expressed as
In addition, according to the theories of circuit analysis, can be further defined aswhere is open circuit voltage for the failure component ; is corresponding Thevenin equivalent impedance, while is the impedance of . According to the Theveninâ€™s Theorem, and are independent of . Therefore,
For any determinate frequency , the Laplacian operator is also an imaginary constant, where is imaginary unit vector. In addition, without loss of generality, can be assumed to be a pure resistance or reactance . Therefore, taking as an example,
With the following assumptions,where , , , , , and are all real numbers, thenwhere
If and are all equal to zero, the CUT is equal to an ideal voltage source corresponding to the component . This means that variation in will not affect and . Thus, we have ThenFurthermore, (13) can be transformed into a formula of 2D circle:where and are the real part and imaginary part of the circle center coordinates, respectively. is the radius. It is confirmed that , , and are independent of the parametric change of the failure component . Therefore, fault loci fitting requires a small amount of simulation on potential faults, for both catastrophic faults and parametric faults. Alternatively, if is assumed to be a pure reactance , similar conclusion can be reached.
For a CUT with components , fault modeling is to determine the model parameters , , and for each according to (15), where parameters , , and are corresponding to , , and , respectively. In most actual analog cases, (14) is hard to derive from transfer function directly. However, it is well known that three sets of distinct data are sufficient to determine a circle. In this regard, simulation is a simple method to establish the circle model in (14). Therefore, in the modeling process, three sets of distinct output data are obtained by parametric sweep simulations, corresponding to three distinct values of each component . For the whole CUT, parametric sweep simulations are needed. Three sets of distinct simulation data for each component are assumed as , , and . Since is assumed to be the fault free output, it fits in with all components. The fault modeling process is illustrated in Figure 1. In addition, the following equation is used to calculate the parameters , , and for component :
2.2. Example of a SallenKey Filter
Compared with the difficulty of obtaining explicit mathematical expression, simulation is a relatively easy way for fault modeling. To demonstrate the method, a secondorder SallenKey filter, as shown in Figure 2, is adopted as an example of circuit under test (CUT). By parameter sweeping simulations with respect to the supply current and the output response in PSPICE, the 2D collaborative fault model can be achieved. The simulation results are listed in Table 1, in which parametric ratio means the ratio between failure value and nominal value for each component. Further results are also provided in Figure 3. The stimulation is a 3â€‰kHz, 1â€‰V sine signal, and the parameter sweeping range for each component is assumed to be from to , where is the faultfree value of the th component . In addition, the simulation range of and is reduced, because output power of the filter is limited. As shown in Figure 3, all potential fault statuses compose a family of loci on the complex plane, which is called loci. Every locus corresponds to all the potential fault statuses of a component, not only parametric faults but also catastrophic faults. Furthermore, loci converges in the fault free point A and zero point B.

Note that in some cases, distances between some loci are very small. In extreme cases, some loci may coincide with each other, which results in ambiguity group. Since the 2D collaborative fault model is derived from transfer function of CUT, ambiguity group may be an inevitable problem [15]. Table 2 lists the ambiguity groups of the SallenKey filter, and Figure 3 shows that while the loci of , , , and are obviously different from others, fault status of and are located on the same locus. This means that potential faults of and cannot be isolated based on the fault model in Figure 3; therefore, and fall into the same ambiguity group .

3. Fault Modeling in 3D Complex Space
Based on the measurements of supply current and voltage response at any test point, the 2D collaborative fault model transforms all potential failure statuses into a family of loci. Therefore, both parametric faults and catastrophic faults can all be detected and located. However, due to the limits of component tolerance and measurement error, the 2D fault model is difficult to apply in actual analog circuits. As depicted in Figure 3, between point A and point B, the and loci are too close to be distinguished. Therefore, to increase the practicability for actual applications, the 2D fault model is extended into 3D complex space.
3.1. Theorem of 3D Collaborative Fault Model
The 2D function in (3) is a linear combination of supply current and voltage response , based on the measurements of real part and imaginary part of each complex variable. and are easier to measure incircuit from limited test points than other circuit variables, such as the input admittance. Furthermore, the complex modulus (absolute value) is also employed to compose the 3D function. This helps to increase the distance between loci fault models and improve the FIR.
Similarly, the component in (2) is still assumed to be the failure component. Therefore, by defining , , and as the unit vectors in 3D complex space, a transformation function can be composed as follow [14]:wherewhere and are still adjustment coefficients as in (3). Taking the component pair  as an example, the distance between and loci iswhere , , and are distances corresponding to each axe in 3D complex space:
Although the distance in (20) has been proved to be better than that in 2D loci fault model [14], it is calculated based on voltage and input admittance, whose absolute values are usually far greater than supply current. Therefore, for an appropriate coefficient , the following inequality can be guaranteed:
In addition, a new distance is defined as follows:
Therefore, a new transformation function for the 3D fault model can be defined as
Furthermore, the distance is defined for the new function . Taking and as an example, it can be expressed as
3.2. Simulation Example of SallenKey Filter for the 3D Model
For a given input of 1â€‰V, 3â€‰kHz sine signal, PSPICE results on the SallenKey filter are obtained. The key part of simulation results are listed in Table 3. The original simulation results include and , corresponding to all potential failure status of each component, such as and , and then all the data in Table 3 is obtained according to (18) and (19). Moreover, the simulation results are also illustrated integrally in Figure 4. It shows that all loci converge in the fault free point A, which is similar to the 2D loci. However, due to the definition in (19), some loci tend to be an infinity point B, instead of the determinate point B in 2D model. The associated ambiguity groups are the same as shown in Table 2.
