Testing Biquad Filters under Parametric Shifts using X-Y Zoning
Testing Biquad Filters under Parametric Shifts using X-Y Zoning Sanahuja R., Barcons V., Balado L., Figueras J. Departament d'Enginyeria Electrònica, Universitat Politècnica de Catalunya Diagonal 647 08006 Barcelona, SPAIN emails: {ricard, victor, balado, figueras}@eel.upc.es
Abstract Testing mixed-signal circuits is a difficult task due to defect modelling challenges, observability and controllability restrictions and ATE bandwidth limitations. In previous works, X-Y Zoning method has been proposed as a BIST technique for mixedsignal and analogue circuits. Some experiments showed its viability in detecting parametric deviations in analog cells. . In this paper the optimal X-Y test of a Biquad filter is addressed in terms of selecting the optimal frequency of the excitation and the best partition of the X-Y plane to obtain the best sensitivity of the BIST scheme to parametric shifts of the parameters defining the filter. The study has been particularized to shifts in the natural frequency ω0 of the Biquad filter. Analytical results on the best input as well as the best partition of the observed X-Y Lissajous plots are obtained. Extensive MATLAB simulations validate the proposal which has been also validated experimentally. For these experiments, multiple implementations of the Biquad with nominal and shifted parameters have been implemented using a commercial Field Programmable Analog Array (FPAA). The experimental measures show good correlation with the analytical expressions and the simulations performed. 1.
Introduction
Testing mixed-signal circuits is a challenging task requiring novel cost effective solutions to overcome the present limitations of the classical analogue test techniques. Analog Automatic Test Equipment (AATE) requires higher bandwidths than the circuit being tested and demanding sampling techniques. In addition the signal processing is performed off-line and requires extensive resources in AATE time and
memory. This makes the cost of the mixed-signal test procedure to become a significant part of the cost of the IC [1], [2]. Available solutions based in the observation and control of the state of internal nodes of the CUT, usually requires the addition of test circuitry loading the analogue signals. The added circuitry to implement the IEEE 1149.4 standard is sometimes not applicable due to the degradation of performance of the CUT in normal operation. The technical challenges and cost of using mixed-signal features in Automatic Test Equipments, have motivated an increasing interest of BIST solutions with different schemas of test resource partitioning. On the other hand the continuous variability of analog signals and the difficulties in modelling defective behaviours in this domain make analog and M-S BIST more demanding and harder to implement than Digital BIST [3]. The “Oscillation Test Method” [4], [5] pioneered work in this domain and triggered interesting solutions implementing variations of the method. The BIST architecture basically produces topological feedback in the CUT which forces the circuit into oscillation. The frequency of these oscillations is used as indicator of Faulty-Non Faulty CUT. The requirements of internal signal generation for M-S Test and also the Digital Signal Processing possibilities to analyze the CUT responses is reported in [6], [7] [8]. A.M. Brosa et al. in [9] and Sanahuja et al. in [10] explore a partial BIST architecture, which uses the information provided by the Lissajous curves generated by the X-Y composition of two internal voltages. In [10] experiments were performed to explore the possibilities of this method using triangular input waveforms [11]. The input and output of the CUT were used as X-Y observables. The experiments performed showed the viability of
the method but the question of selecting the best input frequency and the best partition on the X-Y plane remained unanswered. The purpose of this paper is to design an optimal X-Y test scheme in terms of the selection of the input frequency and the best location of the control line to partition the X-Y plane. The method proposed gives the best excitation and observation solution to maximize the detectability of parametric shifts in the CUT. The method has been applied to detect optimally small shifts in natural frequency of a Biquad filter. The results have been validated experimentally with measures performed in different Biquad (nominal and defective) implementations using a FPAA (Field Programmable Analog Array Anadigm, AN10E40) [12]. The rest of the paper is organized as follows: Section 2 is devoted to briefly explain the X-Y detection method. Section 3 and 4 present the theoretical analysis of the best frequency for the excitation and the best point for the control line position, applying the proposed BIST structure on a Biquad analog filter circuit. Section 5 reports the experimental results. Finally, Section 6 summarizes and presents the conclusions of the work.
Figure 1. X-Y composition curves for a good circuit (left) and with increments (up) and decrements (down) in f0. 2.
X-Y Zone detection method
is the output of a Biquad low pass filter. Due to the filtering action of the Biquad the output can be approximated by sinusoidal signal of the same frequency. Then, assuming a triangular input stimuli x(t) with maximum amplitude Vmax and an output sinusoidal y(t) signal of amplitude Vy-max, the composed X-Y curve can be expressed in terms of the phase shift (α) introduced by the CUT as:
y ( x) =
Vy − max 2
(1 + sin( −
xπ π − − α )) Vmax 2
For the [0, T/2] range and,
y ( x) =
Vy − max 2
(1 + sin(
xπ π − − α )) Vmax 2
For the [T/2, T] range. Defects in the filter CUT will affect the phase and amplitude of the output signal changing the shape of the composed signal. In Fig.1 the X-Y composition curves for a nominal (non defective) circuit (internal curve with tangent line) and defective ones are showed. To detect defects in the CUT a control straight line is used to separate the X-Y plane in two zones: a) Defective X-Y points and b) Non-defective X-Y points. Any crossing of this control line indicates a defective behavior. Fig. 2 shows one of these control lines drown tangentially to the non-defective curve. In this case, when a defect changes the shape of the curve; the control line cuts the curve by two points. The objective of this work is the selection of the input frequency and the point of the curve for the control line parameters to optimize the detection of parametric shifts. Next two sections are devoted to 2.5 2
In the X-Y zone testing method the fault detection is based on the X-Y composition of two signals of the CUT circuit, x(t) and y(t), in a similar way that an oscilloscope in X-Y mode represents the evolution of the two signals on the plane. In our case the two signals are the input and output of the CUT. Other alternatives could be used for any (x, y) pair of periodic signals of the circuit and for more than two signals for multidimensional Lissajous curves. The input excitation is chosen among the possibilities of available signals. In this work the use of a triangular signal is explored as stimuli of the circuit [10]. The Y composition signal, in this study,
1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 2. One control line drawn tangentially to the non-defective curve.
search the best solutions to detect changes in parameters of the CUT. The following section is devoted to find the best input frequency for the Biquad filter CUT. 3.
and,
ρ ( w) = − arctg
Best candidate for input frequency
In this section we address the problem of selecting the input frequency or frequencies to detect, with best possible sensibility, parametric variations on the filter natural frequency, quality factor Q of the Biquad and the DC Gain G of the CUT under consideration.
dA dF A
w Q·w0 w 1 − w0
2
being, G the filter gain in the band-pass, A the amplitude of the output signal, Q the quality factor, and w0 and ρ the natural frequency and the phase of the filter response. Supposing that the deviation is infinitesimal, the dF vector represents the vertical distance between the defective and non defective X-Y curves, but because we put the straight line tangentially to the curve, the best sensitivity occurs when the deviation normal to the tangent will be maximized. As can be seen in the Fig.4, we can represent the normal to the tangent vector as a function of dF with the expression:
dN = dF ·cos ξ
A dρ
where ξ is the angle between the dF and dN vectors, then,
ϕ
dN = dF · Figure 3. Phasor representation of non defective and defective X-Y composed signals. dF represents the error vector composed of an amplidude error dA and a phase error dρ. Because the curves obtained with the composition of triangular and sinusoidal signals are sinusoidal segments, we can model in steady state the non defective and the defective signals as two phasors, running at the input frequency f=1/T, see Figure 3. A small infinitesimal parametric shift in the CUT is reflected in the X-Y curves by some deformation in its shape, and this deformation is represented in the phasor diagram by the dF vector. The dF vector is composed by a change in the amplitude of the output signal dA and a change in the phase angle between the input and output signals dρ. Assume the phasor at a certain instant of time determined by ϕ. The amplitude A and the phase ρ are determined by the CUT. We can obtain the dF error vector with the following expression:
dF = A·sin( d ρ )·cos( ϕ ) + dA·sin( ϕ )
1 1 + A ·cos 2 ϕ 2
Maximizing dN/dx, where x can be f0, Q or G respect to w we obtain the best input frequency. MATLAB was used to obtain this maximum because the analytical expression becomes complicated and the closed form solutions unsolvable. Fig.5 represents dN/dx versus f for a defect that changes the natural frequency of the filter (dx = df0), with a quality factor Q=1, from f0=20kHz to f0_def=20.01kHz.
2.5
dF
dN
2
ξ
1.5 1
ξ
0.5 0
ϕ
-0.5 -1 -1.5 -2
Where, for the Biquad filter CUT,
A( w) =
-2.5
G w 1 − w0
2
2
w + Q·w0
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 4.The position of the straight line in the X-Y representation of the signal viewed as an angle from the zero crossing point ϕ.
ϕ between the vector position and x axis. Once selected the best frequency to excite the defect, the dN/dx vector can be maximized to obtain the ϕ angle that specifies the best point to situate the control line. Fig.6 shows the representation of dN/dx versus ϕ for a defect presented below that changes the natural frequency of the filter (dx = df0) from f0=20kHz to f0_def=20.01kHz. The X-Y curves are symmetric then two identical maximum points can be viewed in the graphic. For the example of previous section, these points are 66.9º and 246.9º (66.9º+180º). Fig.7 shows the position of the control line on the X-Y curves for this example. 5.
Figure 5. dN/df0 versus f for f0=20kHz , f0_def=20,01kHz and Q=1. The results show that, instead the natural frequency that could be supposed without analysis, a fin of 22.8kHz is the best input in this case and this result is dependent with Q factor. For Quality factors of 0.7 and 1.41 the results will be 21.6kHz and 22.9kHz respectively. 4.
Best candidate for control line position
Experimental Set-up
To validate experimentally the results, a prototyping platform was designed using the programmable analog device (FPAA) AN10E40 of Anadigm. The prototyping platform consists of an experimental board, a PC, an oscilloscope and a
2.5 out
2 1.5 1 0.5
For each input frequency a straight line need to be selected to obtain the point in the curve of best sensitivity to optimise the change in shape for an infinitesimal parameter deviation. To identify this point, in the temporal representation of the signals, it can be viewed as an angle from the zero crossing point, see ϕ in Fig.4 and 7. In the phasor representation of the signals of Fig.3 the point position over the curve is the angle
in
0 -0.5
ϕ=66.9 ª
-1 -1.5 -2 -2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 7. Control line situated in the position obtained in section 4.
Figure 6. dN/df0 versus ϕ for all possible points in the curve. Two symmetric points are obtained 66.9º and 246.9ª (66.9ª+180º).
signal generator plus the development software supplied by Anadigm and the Test bench with the developed fault injection tool. The experimental validation was based in the use of the BIST proposed architecture and the input frequency and control line position to detect the changes in the Cut-off frequency (f0) and Quality factor (Q) deviations on the Biquad filter. The X-Y zone detector is a simple circuit composed by a weighted adder of the two composed signals plus a reference voltage and a comparator. The output of the circuit will be a DC voltage if the circuit is out of specifications and a repetitive pulse signal if the circuit meets specifications. In this way, the output of the X-Y zoning detector is a digital
signal which can be processed as a digital signature of the CUT. 6.
Experimental results
Using the obtained frequency excitation and control line position experimental results were obtained changing the input frequency (Fig.8) and for the control line position (Fig.9). The figures depicted three different detection ranges of frequencies: assured detection, non-assured detection and non detection. Noise due to switching capacitors in the FPAA, PC computer and instrumentation (240mV pp in the worst case) makes the output have some non repetitive pulses. Non-assured detection zone reflects the interval from first non repetitive pulses to periodic pulse signal.
3% error is found between experimental and theoretical values. The experimental values reflect a very similar results in the range [45º, 85º] of ϕ. After this range, the detection region decreases. Then a 120 100
Assured detection
80
40
ϕ
20 0 -20
Non detection
-40 -60 10
12
14
16
90
Non detection
80
fin (kH z )
60 50
Assured detection
Assured detection
20 10
20
22
24
26
28
30
Figure 9. Ranges of frequencies for assured detection and non detection for shifts in f0 parameter changing the ϕ parameter.
70
30
18
fo (kHz)
100
40
Assured detection
60
17
18
19
20 fo (kH z )
21
22
23
Figure 8. Ranges of frequencies for assured detection, non-assured detection and non detection for shifts in f0 parameter changing the excitation frequency.
shift in the position, if it is in this range, does not change significantly the results. In the theoretical f0 and ϕ values the test bench can detect a shift of f0 from 19.91kHz to 20.12kHz, supposing a resolution for the detection of 0.45% decrement and 0.6% increment of the natural frequency. Fig.10 shows the oscilloscope view of the detector signal output for 20.0 kHz and for 20.12 kHz natural frequencies with the pulsing and DC response respectively. As soon as a change approaches this limit the output of the detector stops oscillating.
Fig.8 shows the detection zones for shifts in the natural frequency of the filter (20kHz), from 16kHz to 24kHz, for different excitation frequencies from 10kHz to 100kHz. The experimental points have been interpolated with a curve to generate the boundaries of assured detection and non detection zones. As can be seen in the figure, the zone of assured detection for increments (right) and decrements (left) in f0 parameter are not symmetrical. This is because the behaviour of the gain response is different in the two sides of the natural frequency parameter. The wider segment in the zones occurs at an input frequency of 23kHz what supposes an error of 0.9% of the theoretical solution. The experiments were repeated changing the position of the control line. Fig.9 shows the results in a similar graph of previous results. In this case the Y axis is the angle ϕ. The maximum is situated in 64.9 degrees. The theoretical value found is 66.9º then and
Figure 10. Detector signal output for a good circuit (left) and with a 0.7% increment in f0 (right). 7.
Conclusions
The X-Y zoning test method with triangular input has been optimized to detect parametric shifts in a Biquad filter. Using phasors, the analysis of the excitation frequency that maximizes the detection sensitivity has been derived and the best point to situate the control line to partition the X-Y plane has been obtained. An experiment on the use of X-Y zoning detectors has been performed using a programmable analogue device (FPAA). The composition of the input-output signals showed a good detectability to the variations of the natural frequency f0, the quality factor Q, and the DC gain G. These three parameters define uniquely the Biquad filter. The study of the best frequency shows that it is higher than the natural frequency of the Biquad. In fact for the particular CUT studied the optimal frequency is 14% higher than the natural frequency. The experimental results showed a good correlation with the values obtained by the proposed method for the excitation input frequency and the proposed selection of the control line position. The use of a programmable device simplifies the injection of parametric faults to obtain experimental results. In the experience an assured detectability of 0.45% decrement and 0.6% increment of the natural frequency have been achieved. The possibilities of using this method for the design of X-Y BIST in mixed-signal circuits is expected to increase the efficiency of the X-Y method and at the same time help designers in the selection of the architecture of the M-S test scheme. Acknowledgements This work has been partially supported by the Ministerio de Ciencia y Tecnología and FEDER projects TIC2001-2246, TIC2002-03127, the Integrated Action HI2000-0100 and Test D.O.C. project. References [1] M. Burns and G. W. Roberts, An Introduction To Mixed-Signal IC Testing and Measurement, Oxford University Press, 2000. [2] B. Vinnakota (editor), Analog and Mixed-Signal Test, Prentice-Hall PTR, 1998. [3] L.S. Milor, "A Tutorial Introduction to Research on Analog and Mixed-Signal Circuit Testing", IEEE Transactions on Circuits and Systems – II: Analog and Digital Signal Processing, 45 (10); 1389-1407, October 1998. [[4] K. Arabi and B. Kaminska, “Oscillation Test Method for Low Cost Testing of Active Analog Filters” IEEE Trans. on Instrumentation and
Measurement), Vol 48, Number 4, August 1999, pp 798-806 [5] B. Kaminska, K. Arabi, I. Bell, P. Goteti, J.L. Huertas, B. Kim, A. Rueda, and M. Soma., "Analog and Mixed-Signal Benchmark Circuits – First Release", Proceedings of the International Test Conference, November 1-6, 1997. [6] J.W. Lin, C.L. Lee,C.C.Su and J.E. Chen, “Fault Diagnosis for Linear Analog Circuits” Journal of Electronic Testing: Theory and applications (JETTA), Vol 17, Number 6, December 2001, pp 483 – 494. [7] Huertas, G.; Vazquez, D.; Peralias, E.; Rueda, A.; Huertas, J.L. “Testing mixed-signal cores: practical oscillation-based test in an analog macrocell” Proceedings of the Ninth Asian Test Symposium, 2000. (ATS 2000), 2000 , pp 31 –38 [8] M. Hafed, N. Abaskharoun, W. Roberts “A standalone integrated test core for time and frequency domain measurements” Proceedings of the International Test Conference; pp. 1031-1040, 2000. [9] A.M. Brosa, J. Figueras, “Digital Signature Proposal for Mixed-Signal Circuits”, Proc. International Test Conference, Atlantic City, NJ, USA, October, 2000, pp 1041-1050. [10] Sanahuja R., Barcons V., Balado L., Figueras J. “Z-Y Zoning BIST: An FPAA Experiment” IMSTW 2002, June 2002, pp 237-243. [11] F. Azaïs, S. Bernard, Y. Bertrand, X. Michel, M. Renovell, “A Low-Cost Adaptive Ramp Generator for Analog BIST Applications”, 19th VLSI Test Symposium, Marina del Rey, California, April 2001, pp 266-271. [12] Anadigm, http://www.anadigm.com
Abstract Testing mixed-signal circuits is a difficult task due to defect modelling challenges, observability and controllability restrictions and ATE bandwidth limitations. In previous works, X-Y Zoning method has been proposed as a BIST technique for mixedsignal and analogue circuits. Some experiments showed its viability in detecting parametric deviations in analog cells. . In this paper the optimal X-Y test of a Biquad filter is addressed in terms of selecting the optimal frequency of the excitation and the best partition of the X-Y plane to obtain the best sensitivity of the BIST scheme to parametric shifts of the parameters defining the filter. The study has been particularized to shifts in the natural frequency ω0 of the Biquad filter. Analytical results on the best input as well as the best partition of the observed X-Y Lissajous plots are obtained. Extensive MATLAB simulations validate the proposal which has been also validated experimentally. For these experiments, multiple implementations of the Biquad with nominal and shifted parameters have been implemented using a commercial Field Programmable Analog Array (FPAA). The experimental measures show good correlation with the analytical expressions and the simulations performed. 1.
Introduction
Testing mixed-signal circuits is a challenging task requiring novel cost effective solutions to overcome the present limitations of the classical analogue test techniques. Analog Automatic Test Equipment (AATE) requires higher bandwidths than the circuit being tested and demanding sampling techniques. In addition the signal processing is performed off-line and requires extensive resources in AATE time and
memory. This makes the cost of the mixed-signal test procedure to become a significant part of the cost of the IC [1], [2]. Available solutions based in the observation and control of the state of internal nodes of the CUT, usually requires the addition of test circuitry loading the analogue signals. The added circuitry to implement the IEEE 1149.4 standard is sometimes not applicable due to the degradation of performance of the CUT in normal operation. The technical challenges and cost of using mixed-signal features in Automatic Test Equipments, have motivated an increasing interest of BIST solutions with different schemas of test resource partitioning. On the other hand the continuous variability of analog signals and the difficulties in modelling defective behaviours in this domain make analog and M-S BIST more demanding and harder to implement than Digital BIST [3]. The “Oscillation Test Method” [4], [5] pioneered work in this domain and triggered interesting solutions implementing variations of the method. The BIST architecture basically produces topological feedback in the CUT which forces the circuit into oscillation. The frequency of these oscillations is used as indicator of Faulty-Non Faulty CUT. The requirements of internal signal generation for M-S Test and also the Digital Signal Processing possibilities to analyze the CUT responses is reported in [6], [7] [8]. A.M. Brosa et al. in [9] and Sanahuja et al. in [10] explore a partial BIST architecture, which uses the information provided by the Lissajous curves generated by the X-Y composition of two internal voltages. In [10] experiments were performed to explore the possibilities of this method using triangular input waveforms [11]. The input and output of the CUT were used as X-Y observables. The experiments performed showed the viability of
the method but the question of selecting the best input frequency and the best partition on the X-Y plane remained unanswered. The purpose of this paper is to design an optimal X-Y test scheme in terms of the selection of the input frequency and the best location of the control line to partition the X-Y plane. The method proposed gives the best excitation and observation solution to maximize the detectability of parametric shifts in the CUT. The method has been applied to detect optimally small shifts in natural frequency of a Biquad filter. The results have been validated experimentally with measures performed in different Biquad (nominal and defective) implementations using a FPAA (Field Programmable Analog Array Anadigm, AN10E40) [12]. The rest of the paper is organized as follows: Section 2 is devoted to briefly explain the X-Y detection method. Section 3 and 4 present the theoretical analysis of the best frequency for the excitation and the best point for the control line position, applying the proposed BIST structure on a Biquad analog filter circuit. Section 5 reports the experimental results. Finally, Section 6 summarizes and presents the conclusions of the work.
Figure 1. X-Y composition curves for a good circuit (left) and with increments (up) and decrements (down) in f0. 2.
X-Y Zone detection method
is the output of a Biquad low pass filter. Due to the filtering action of the Biquad the output can be approximated by sinusoidal signal of the same frequency. Then, assuming a triangular input stimuli x(t) with maximum amplitude Vmax and an output sinusoidal y(t) signal of amplitude Vy-max, the composed X-Y curve can be expressed in terms of the phase shift (α) introduced by the CUT as:
y ( x) =
Vy − max 2
(1 + sin( −
xπ π − − α )) Vmax 2
For the [0, T/2] range and,
y ( x) =
Vy − max 2
(1 + sin(
xπ π − − α )) Vmax 2
For the [T/2, T] range. Defects in the filter CUT will affect the phase and amplitude of the output signal changing the shape of the composed signal. In Fig.1 the X-Y composition curves for a nominal (non defective) circuit (internal curve with tangent line) and defective ones are showed. To detect defects in the CUT a control straight line is used to separate the X-Y plane in two zones: a) Defective X-Y points and b) Non-defective X-Y points. Any crossing of this control line indicates a defective behavior. Fig. 2 shows one of these control lines drown tangentially to the non-defective curve. In this case, when a defect changes the shape of the curve; the control line cuts the curve by two points. The objective of this work is the selection of the input frequency and the point of the curve for the control line parameters to optimize the detection of parametric shifts. Next two sections are devoted to 2.5 2
In the X-Y zone testing method the fault detection is based on the X-Y composition of two signals of the CUT circuit, x(t) and y(t), in a similar way that an oscilloscope in X-Y mode represents the evolution of the two signals on the plane. In our case the two signals are the input and output of the CUT. Other alternatives could be used for any (x, y) pair of periodic signals of the circuit and for more than two signals for multidimensional Lissajous curves. The input excitation is chosen among the possibilities of available signals. In this work the use of a triangular signal is explored as stimuli of the circuit [10]. The Y composition signal, in this study,
1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 2. One control line drawn tangentially to the non-defective curve.
search the best solutions to detect changes in parameters of the CUT. The following section is devoted to find the best input frequency for the Biquad filter CUT. 3.
and,
ρ ( w) = − arctg
Best candidate for input frequency
In this section we address the problem of selecting the input frequency or frequencies to detect, with best possible sensibility, parametric variations on the filter natural frequency, quality factor Q of the Biquad and the DC Gain G of the CUT under consideration.
dA dF A
w Q·w0 w 1 − w0
2
being, G the filter gain in the band-pass, A the amplitude of the output signal, Q the quality factor, and w0 and ρ the natural frequency and the phase of the filter response. Supposing that the deviation is infinitesimal, the dF vector represents the vertical distance between the defective and non defective X-Y curves, but because we put the straight line tangentially to the curve, the best sensitivity occurs when the deviation normal to the tangent will be maximized. As can be seen in the Fig.4, we can represent the normal to the tangent vector as a function of dF with the expression:
dN = dF ·cos ξ
A dρ
where ξ is the angle between the dF and dN vectors, then,
ϕ
dN = dF · Figure 3. Phasor representation of non defective and defective X-Y composed signals. dF represents the error vector composed of an amplidude error dA and a phase error dρ. Because the curves obtained with the composition of triangular and sinusoidal signals are sinusoidal segments, we can model in steady state the non defective and the defective signals as two phasors, running at the input frequency f=1/T, see Figure 3. A small infinitesimal parametric shift in the CUT is reflected in the X-Y curves by some deformation in its shape, and this deformation is represented in the phasor diagram by the dF vector. The dF vector is composed by a change in the amplitude of the output signal dA and a change in the phase angle between the input and output signals dρ. Assume the phasor at a certain instant of time determined by ϕ. The amplitude A and the phase ρ are determined by the CUT. We can obtain the dF error vector with the following expression:
dF = A·sin( d ρ )·cos( ϕ ) + dA·sin( ϕ )
1 1 + A ·cos 2 ϕ 2
Maximizing dN/dx, where x can be f0, Q or G respect to w we obtain the best input frequency. MATLAB was used to obtain this maximum because the analytical expression becomes complicated and the closed form solutions unsolvable. Fig.5 represents dN/dx versus f for a defect that changes the natural frequency of the filter (dx = df0), with a quality factor Q=1, from f0=20kHz to f0_def=20.01kHz.
2.5
dF
dN
2
ξ
1.5 1
ξ
0.5 0
ϕ
-0.5 -1 -1.5 -2
Where, for the Biquad filter CUT,
A( w) =
-2.5
G w 1 − w0
2
2
w + Q·w0
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 4.The position of the straight line in the X-Y representation of the signal viewed as an angle from the zero crossing point ϕ.
ϕ between the vector position and x axis. Once selected the best frequency to excite the defect, the dN/dx vector can be maximized to obtain the ϕ angle that specifies the best point to situate the control line. Fig.6 shows the representation of dN/dx versus ϕ for a defect presented below that changes the natural frequency of the filter (dx = df0) from f0=20kHz to f0_def=20.01kHz. The X-Y curves are symmetric then two identical maximum points can be viewed in the graphic. For the example of previous section, these points are 66.9º and 246.9º (66.9º+180º). Fig.7 shows the position of the control line on the X-Y curves for this example. 5.
Figure 5. dN/df0 versus f for f0=20kHz , f0_def=20,01kHz and Q=1. The results show that, instead the natural frequency that could be supposed without analysis, a fin of 22.8kHz is the best input in this case and this result is dependent with Q factor. For Quality factors of 0.7 and 1.41 the results will be 21.6kHz and 22.9kHz respectively. 4.
Best candidate for control line position
Experimental Set-up
To validate experimentally the results, a prototyping platform was designed using the programmable analog device (FPAA) AN10E40 of Anadigm. The prototyping platform consists of an experimental board, a PC, an oscilloscope and a
2.5 out
2 1.5 1 0.5
For each input frequency a straight line need to be selected to obtain the point in the curve of best sensitivity to optimise the change in shape for an infinitesimal parameter deviation. To identify this point, in the temporal representation of the signals, it can be viewed as an angle from the zero crossing point, see ϕ in Fig.4 and 7. In the phasor representation of the signals of Fig.3 the point position over the curve is the angle
in
0 -0.5
ϕ=66.9 ª
-1 -1.5 -2 -2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure 7. Control line situated in the position obtained in section 4.
Figure 6. dN/df0 versus ϕ for all possible points in the curve. Two symmetric points are obtained 66.9º and 246.9ª (66.9ª+180º).
signal generator plus the development software supplied by Anadigm and the Test bench with the developed fault injection tool. The experimental validation was based in the use of the BIST proposed architecture and the input frequency and control line position to detect the changes in the Cut-off frequency (f0) and Quality factor (Q) deviations on the Biquad filter. The X-Y zone detector is a simple circuit composed by a weighted adder of the two composed signals plus a reference voltage and a comparator. The output of the circuit will be a DC voltage if the circuit is out of specifications and a repetitive pulse signal if the circuit meets specifications. In this way, the output of the X-Y zoning detector is a digital
signal which can be processed as a digital signature of the CUT. 6.
Experimental results
Using the obtained frequency excitation and control line position experimental results were obtained changing the input frequency (Fig.8) and for the control line position (Fig.9). The figures depicted three different detection ranges of frequencies: assured detection, non-assured detection and non detection. Noise due to switching capacitors in the FPAA, PC computer and instrumentation (240mV pp in the worst case) makes the output have some non repetitive pulses. Non-assured detection zone reflects the interval from first non repetitive pulses to periodic pulse signal.
3% error is found between experimental and theoretical values. The experimental values reflect a very similar results in the range [45º, 85º] of ϕ. After this range, the detection region decreases. Then a 120 100
Assured detection
80
40
ϕ
20 0 -20
Non detection
-40 -60 10
12
14
16
90
Non detection
80
fin (kH z )
60 50
Assured detection
Assured detection
20 10
20
22
24
26
28
30
Figure 9. Ranges of frequencies for assured detection and non detection for shifts in f0 parameter changing the ϕ parameter.
70
30
18
fo (kHz)
100
40
Assured detection
60
17
18
19
20 fo (kH z )
21
22
23
Figure 8. Ranges of frequencies for assured detection, non-assured detection and non detection for shifts in f0 parameter changing the excitation frequency.
shift in the position, if it is in this range, does not change significantly the results. In the theoretical f0 and ϕ values the test bench can detect a shift of f0 from 19.91kHz to 20.12kHz, supposing a resolution for the detection of 0.45% decrement and 0.6% increment of the natural frequency. Fig.10 shows the oscilloscope view of the detector signal output for 20.0 kHz and for 20.12 kHz natural frequencies with the pulsing and DC response respectively. As soon as a change approaches this limit the output of the detector stops oscillating.
Fig.8 shows the detection zones for shifts in the natural frequency of the filter (20kHz), from 16kHz to 24kHz, for different excitation frequencies from 10kHz to 100kHz. The experimental points have been interpolated with a curve to generate the boundaries of assured detection and non detection zones. As can be seen in the figure, the zone of assured detection for increments (right) and decrements (left) in f0 parameter are not symmetrical. This is because the behaviour of the gain response is different in the two sides of the natural frequency parameter. The wider segment in the zones occurs at an input frequency of 23kHz what supposes an error of 0.9% of the theoretical solution. The experiments were repeated changing the position of the control line. Fig.9 shows the results in a similar graph of previous results. In this case the Y axis is the angle ϕ. The maximum is situated in 64.9 degrees. The theoretical value found is 66.9º then and
Figure 10. Detector signal output for a good circuit (left) and with a 0.7% increment in f0 (right). 7.
Conclusions
The X-Y zoning test method with triangular input has been optimized to detect parametric shifts in a Biquad filter. Using phasors, the analysis of the excitation frequency that maximizes the detection sensitivity has been derived and the best point to situate the control line to partition the X-Y plane has been obtained. An experiment on the use of X-Y zoning detectors has been performed using a programmable analogue device (FPAA). The composition of the input-output signals showed a good detectability to the variations of the natural frequency f0, the quality factor Q, and the DC gain G. These three parameters define uniquely the Biquad filter. The study of the best frequency shows that it is higher than the natural frequency of the Biquad. In fact for the particular CUT studied the optimal frequency is 14% higher than the natural frequency. The experimental results showed a good correlation with the values obtained by the proposed method for the excitation input frequency and the proposed selection of the control line position. The use of a programmable device simplifies the injection of parametric faults to obtain experimental results. In the experience an assured detectability of 0.45% decrement and 0.6% increment of the natural frequency have been achieved. The possibilities of using this method for the design of X-Y BIST in mixed-signal circuits is expected to increase the efficiency of the X-Y method and at the same time help designers in the selection of the architecture of the M-S test scheme. Acknowledgements This work has been partially supported by the Ministerio de Ciencia y Tecnología and FEDER projects TIC2001-2246, TIC2002-03127, the Integrated Action HI2000-0100 and Test D.O.C. project. References [1] M. Burns and G. W. Roberts, An Introduction To Mixed-Signal IC Testing and Measurement, Oxford University Press, 2000. [2] B. Vinnakota (editor), Analog and Mixed-Signal Test, Prentice-Hall PTR, 1998. [3] L.S. Milor, "A Tutorial Introduction to Research on Analog and Mixed-Signal Circuit Testing", IEEE Transactions on Circuits and Systems – II: Analog and Digital Signal Processing, 45 (10); 1389-1407, October 1998. [[4] K. Arabi and B. Kaminska, “Oscillation Test Method for Low Cost Testing of Active Analog Filters” IEEE Trans. on Instrumentation and
Measurement), Vol 48, Number 4, August 1999, pp 798-806 [5] B. Kaminska, K. Arabi, I. Bell, P. Goteti, J.L. Huertas, B. Kim, A. Rueda, and M. Soma., "Analog and Mixed-Signal Benchmark Circuits – First Release", Proceedings of the International Test Conference, November 1-6, 1997. [6] J.W. Lin, C.L. Lee,C.C.Su and J.E. Chen, “Fault Diagnosis for Linear Analog Circuits” Journal of Electronic Testing: Theory and applications (JETTA), Vol 17, Number 6, December 2001, pp 483 – 494. [7] Huertas, G.; Vazquez, D.; Peralias, E.; Rueda, A.; Huertas, J.L. “Testing mixed-signal cores: practical oscillation-based test in an analog macrocell” Proceedings of the Ninth Asian Test Symposium, 2000. (ATS 2000), 2000 , pp 31 –38 [8] M. Hafed, N. Abaskharoun, W. Roberts “A standalone integrated test core for time and frequency domain measurements” Proceedings of the International Test Conference; pp. 1031-1040, 2000. [9] A.M. Brosa, J. Figueras, “Digital Signature Proposal for Mixed-Signal Circuits”, Proc. International Test Conference, Atlantic City, NJ, USA, October, 2000, pp 1041-1050. [10] Sanahuja R., Barcons V., Balado L., Figueras J. “Z-Y Zoning BIST: An FPAA Experiment” IMSTW 2002, June 2002, pp 237-243. [11] F. Azaïs, S. Bernard, Y. Bertrand, X. Michel, M. Renovell, “A Low-Cost Adaptive Ramp Generator for Analog BIST Applications”, 19th VLSI Test Symposium, Marina del Rey, California, April 2001, pp 266-271. [12] Anadigm, http://www.anadigm.com
