Parametric Fault Testing of Non-Linear Analog Circuits Based on Polynomial and V-Transform Coefficients

Abstract

This paper is an exposition of recent advances made in polynomial coefficient and V-transform coefficient based testing of parametric faults in linear and non-linear analog circuits. V-transform is a non-linear transform that increases the sensitivity of polynomial coefficients with respect to circuit component variations by three to five times. In addition, it makes the original polynomial coefficients monotonic. Using simulation, the proposed test method is shown to uncover most parametric faults in the range of 5–15 % on a low noise amplifier (LNA) and an elliptic filter benchmark. Diagnosis of parametric faults clearly illustrates the effect of enhanced sensitivity through V-transform. Finally, we report an experimental validation of the polynomial coefficient based test scheme, with and without V-transform, using the National Instruments’ ELVIS bench-top testbed. The result demonstrates the benefit of V-transform.

Appendix

Theorem 1

If coefficient a _i is a monotonic function of all parameters, then a _i takes its limit (maximum and minimum) values when at least one or more of the parameters are at the boundaries of their individual hypercube.

Proof

Let a _i be a function of three parameters say x, y and z. Let a _i reach its maximum value for (x₀, y₀, z₀). Further let $\rm{x_0,~y_0} \ne \alpha$. Now if we can show that the maximum value of the coefficient a _i occurs at the z₀ = α we have proved the theorem. From definition of monotonic dependence of a _i on circuit parameters, it follows that a _i (x₀, y₀, α) ≥ a_i(x₀, y₀, z₀), ∀ z₀ ≤ α. Because the maximum value taken by z is α, it follows that z₀ = α. With similar arguments we can show that the minimum value for the coefficient occurs when z₀ = − α. Hence, the statement of theorem follows. □

Theorem 2

In polynomial expansion of a non-linear analog circuit there exists at least one coefficient that is a monotonic function of all of the circuit parameters.

Proof

Consider the block diagram in Fig. 14, which models an 2n^th order non-linear analog circuit. It has an input x and an output y. Constants a₁ ⋯ a_n are added at the input of each stage. The coefficient corresponding to input x raised to the 2n^th power is given by G, as follows:

$$ \rm G=\prod\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{g}}_{\rm{i}}^{\rm{2i}} } $$

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where g_i  ∀ i = 1 ...n are the monotonic gains of individual stages in the cascaded blocks. As the product of two or more monotonic functions is also monotonic we have G to be a monotonic function. G constitutes the coefficient of the n^th power of x in this expansion, as it lines in the main signal flow path from input to output. Thus, it is proved that there is at least one monotonically varying coefficient in a polynomial expansion of a Non-Linear analog circuit. Further, in general the coefficient of 2n^th power of such a polynomial expansion is monotonic.□

Fig. 14

A system model for a non-linear circuit