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Statistical evaluation of switching circuit characteristic parameters
NUCLEAR INSTRUMENTS AND METHODS 64 (1968) 205-207;
STATISTICAL
EVALUATION
OF SWITCHING
N. ABBATTISTA*,
CIRCUIT
M. COLIf, G. GIANNELLI* *Istituto
+Laboratori
0 NORTH-HOLLAND
CHARACTERISTIC
PUBLISHING
co.
PARAMETERS+
and V. L. PLANTAMURA*
di Fisica, Bari, Italy
Nazionali
de1 CNEN,
Frascati,
Italy
Received 28 June 1968 The problem of correlating
the threshold
and switching delay with the statistical characteristics statistical way.
In the last few years the evaluation of the characteristics of regenerative circuits, namely of the threshold and of the switching delay, has been the subject of many studiesiP4). In these studies the evaluation was of a deterministic nature. We propose here a different approach to lhis problem, of statistical nature, that allows to correlate the threshold and switching delay with the statistical characteristics of the noise (both of the circuit and of the signal). Let us consider the tunnel diode monostable circuit shown in fig. 1 and an input signal of the step type with added noise (fig. lb). The analysis has been carried out on a digital computer. The intrinsic noise of the circuit has been simulated with noise added to the input signal, and was represented with a square-wave with random (halfwave) amplitudes and durations. t Work supported by Consiglio Nazionale delle Ricerche.
RP=Vp K
= I.
/
I
of the noise are approached
The amplitude distribution was uniform and limited between A,,, and -A,,,. The duration distribution was uniform and limited between rmin and z,,; A,,,, Zmin,T,,_ and the probabilities distribution define the statistical characteristics of the noise. Fig. 2 shows the result of the analysis. It gives, for a fixed noise power, the percentage of times that the circuit triggers, as a function of the step amplitudeji,. Let us now suppose of experimenting with an input signal jin = 0.3. Fig. 2 shows that the result of the experiment will be that the circuit will trigger always and we will deduce from this experiment that the circuit’s threshold is lower than 0.3. But we can express the result in a different way. We can say that the threshold has a probability of 100% of being lower than 0.3. We can now extend this point of view and, if in an experiment with a certain value of jin we find that the circuit triggers n times percent, we can say that n% is
lp
RpC
b)
a) Fig. 1. a. Scheme of the monostable
circuit; b. Typical input signal wave shape.
205
in a
206
et cd.
N. ABBATTISTA
rmin= 0.5
Switt
101
Tma=
‘O
= jbias k = 100
0.9
0 =2
50
.15
.l
.05
.3
.25
.2
. 'in
Fig. 2. Switch percentage of the circuit vs the input current amplitude.
the probability for the threshold to be lower than the given ji,. This is a cumulative probability and we can interprete the diagram of fig. 2 as the distribution function of the cumuiative probability that the threshold of the circuit has to have a value 5 ji,.
The probability density function can be obtained from that diagram and this function is shown in fig. 3; we can define the most probable value and the standard deviation of the threshold. The same approach can be used for the switching delay. Fig. 4 shows the most probable value and the
5
. 1
* 0
05
.I
.I5
.2
.25
Fig. 3. Probability density function of the threshold.
.3
Threnkold
value
[Q,]
SWlTCHING
CIRCUIT
CHARACTERISTIC
207
PARAMETERS
Tim 8
6
t Fig. 4. Most probable
0.2
0.3
0.4
0.5
0.6
values and its variance vs input current amplitude. Most probable value (-----delay density probability function.
standard deviation of the switching delay as a function ofji, for two values of the noise power. These definitions are quite general and can be applied to all types of switching circuits and they define the behaviour of switching circuits in presence of noise in a quantitative way. In particulair we think to apply this point of view to the study of a cascade of switching circuits.
0.7
‘in
) and fwhm (- - - -) of the
References I) I. De Lotto and L. Stanchi, Rapport Euratom, EUR 430. a) I. De Lotto, P. F. Manfredi and L. Stanchi, Alta Frequenza 35 (1966) 830. 3, F. Pandarese, IEEE Trans. NS-11 (1964) 16. 4, N. Abbattista,
M. Coli and V. L. Plantamura,
Meth. 45 (1966) 157.
Nucl. Instr. and
STATISTICAL
EVALUATION
OF SWITCHING
N. ABBATTISTA*,
CIRCUIT
M. COLIf, G. GIANNELLI* *Istituto
+Laboratori
0 NORTH-HOLLAND
CHARACTERISTIC
PUBLISHING
co.
PARAMETERS+
and V. L. PLANTAMURA*
di Fisica, Bari, Italy
Nazionali
de1 CNEN,
Frascati,
Italy
Received 28 June 1968 The problem of correlating
the threshold
and switching delay with the statistical characteristics statistical way.
In the last few years the evaluation of the characteristics of regenerative circuits, namely of the threshold and of the switching delay, has been the subject of many studiesiP4). In these studies the evaluation was of a deterministic nature. We propose here a different approach to lhis problem, of statistical nature, that allows to correlate the threshold and switching delay with the statistical characteristics of the noise (both of the circuit and of the signal). Let us consider the tunnel diode monostable circuit shown in fig. 1 and an input signal of the step type with added noise (fig. lb). The analysis has been carried out on a digital computer. The intrinsic noise of the circuit has been simulated with noise added to the input signal, and was represented with a square-wave with random (halfwave) amplitudes and durations. t Work supported by Consiglio Nazionale delle Ricerche.
RP=Vp K
= I.
/
I
of the noise are approached
The amplitude distribution was uniform and limited between A,,, and -A,,,. The duration distribution was uniform and limited between rmin and z,,; A,,,, Zmin,T,,_ and the probabilities distribution define the statistical characteristics of the noise. Fig. 2 shows the result of the analysis. It gives, for a fixed noise power, the percentage of times that the circuit triggers, as a function of the step amplitudeji,. Let us now suppose of experimenting with an input signal jin = 0.3. Fig. 2 shows that the result of the experiment will be that the circuit will trigger always and we will deduce from this experiment that the circuit’s threshold is lower than 0.3. But we can express the result in a different way. We can say that the threshold has a probability of 100% of being lower than 0.3. We can now extend this point of view and, if in an experiment with a certain value of jin we find that the circuit triggers n times percent, we can say that n% is
lp
RpC
b)
a) Fig. 1. a. Scheme of the monostable
circuit; b. Typical input signal wave shape.
205
in a
206
et cd.
N. ABBATTISTA
rmin= 0.5
Switt
101
Tma=
‘O
= jbias k = 100
0.9
0 =2
50
.15
.l
.05
.3
.25
.2
. 'in
Fig. 2. Switch percentage of the circuit vs the input current amplitude.
the probability for the threshold to be lower than the given ji,. This is a cumulative probability and we can interprete the diagram of fig. 2 as the distribution function of the cumuiative probability that the threshold of the circuit has to have a value 5 ji,.
The probability density function can be obtained from that diagram and this function is shown in fig. 3; we can define the most probable value and the standard deviation of the threshold. The same approach can be used for the switching delay. Fig. 4 shows the most probable value and the
5
. 1
* 0
05
.I
.I5
.2
.25
Fig. 3. Probability density function of the threshold.
.3
Threnkold
value
[Q,]
SWlTCHING
CIRCUIT
CHARACTERISTIC
207
PARAMETERS
Tim 8
6
t Fig. 4. Most probable
0.2
0.3
0.4
0.5
0.6
values and its variance vs input current amplitude. Most probable value (-----delay density probability function.
standard deviation of the switching delay as a function ofji, for two values of the noise power. These definitions are quite general and can be applied to all types of switching circuits and they define the behaviour of switching circuits in presence of noise in a quantitative way. In particulair we think to apply this point of view to the study of a cascade of switching circuits.
0.7
‘in
) and fwhm (- - - -) of the
References I) I. De Lotto and L. Stanchi, Rapport Euratom, EUR 430. a) I. De Lotto, P. F. Manfredi and L. Stanchi, Alta Frequenza 35 (1966) 830. 3, F. Pandarese, IEEE Trans. NS-11 (1964) 16. 4, N. Abbattista,
M. Coli and V. L. Plantamura,
Meth. 45 (1966) 157.
Nucl. Instr. and