COUNTING STATISTICS

This section contains a discussion on some of the properties of a counting channel com­prising a detector, logarithmic count-rate meter, and period meter. The properties considered are:

a. Variance of the count-rate indication,

‘ b. Probability of false trip from the count-rate indication,

c. Variance of the period indication, and

d. Probability of false trip from the period indication.

The derivations are based on the assumption of constant average counting rate (infinite period), but the results should be valid for finite periods greater than the longest time-constant in the system.

2. 1 VARIANCE OF THE COUNT-RATE INDICATION

The logarithmic count-rate meter considered is of the multiple-diode pump type (Cooke^ Yarborough), of which one section is delineated in Figure 2-1, and in which R is the same for

image007

Figure 2-1. One Section of Multiple-Diode Pump Type of Log Count-Rate Meter

each section, C^/C ^ is the same for each section, and of any section is — Jq ^f °* preceding section. That is,

(Rcf)k = 10(RC()(k+D

and

Подпись: (2-2)(RCT)k — 10(RCT)(k,

where к is the section number and starts at 1 for the section with the lowest frequency break­point. All sections of the multiple pump are driven from the output of a scale-of-two circuit.

Let the average counting rate, r, at the input of the scale-of-two be such that the output of the k— diode pump is one-half of its saturation value; i. e.,

r

2

 

(2-3)

 

(RCt)i

 

Then individual pulses from the (k-1)-^, k—, and (k +-l)^ pumps are

(2-4)

 

‘(к — 1)

 

image009
image010

(2-5)

 

image011

and

image012

(2-6)

 

‘(k+’l)

 

(k+ 1)

 

image013

respectively, where Tk ^ ^RCf]k, тк = (RCjk , and V is the voltage swing of the output of the scale-of-two. To obtain these expressions, note that the charge delivered to C, each time the scale-of-two output swings in the positive direction, is

VT

R (1 + 0. 5 rT)

and this charge leaks off through R with a timerconstant, r, hence,

Подпись: (2-7)Подпись: (2-8) (2-3a) (2-9) » . VT—e-Vr.

t (1 + 0.5 ГТ) ‘

By combining Equation (2-7) with the relations,

. 0-5 r T(k_ 1} » 1 ,

0. 5 r Tk. = 1 ,

, . 0 5 rT(k+ 1) <<1 ’

Equations (2-4), (2-5), and (2-6) are obtained.

image016 image017

Now rewrite Equations (2-4), (2-5), and (2-6) as

An individual pulse from the log count-rate meter is a fraction, A, of the sum of these three pulses:

Подпись: A V /Tk 10 I T,. AvfeV (2-13)

(Actually, the total pulse is a fraction of the sum of the outputs of all of the individual sections; however, the terms below (к — 1) and above (k+ 1) contribute an insignificant amount of energy to the total pulse and can be neglected. )

2

Now the variance-, a„ , of a voltage, S, that consists of a linear superposition of randomly •Э (11

image019 Подпись: 0.5 Подпись: dt . Подпись: (2-14)
image023

arriving pulses of the form v (t) and an average arrival rate of 0. 5 r is given by’ ‘

image024 Подпись: 0. 5 r Подпись: 0. 43 T.
image027

(Equation (2-14) is true for pulses whose arrival satisfies Poisson’s distribution; pulses from a scale-of-two do not. However, it is shown in Appendix A that the variance obtained by the use of Equation (2-14) is in error by less than a factor of 2 and is on the large side.) Substitution of Equation (2-13) into Equation (2-14) yields the variance,

image028 Подпись: 0. 43 A2 V2 Подпись: (2-15)

and this can be combined with Equation (2-3a) to obtain

The average voltage from one diode pump is given by

0. 5 r VTt 1+ 0. 5 r Tt ’

image031
Подпись: (2-17)

It should be remembered that the manner in which Equation (2-18) was obtained imposes the following restrictions:

a. It is valid when the counting rate is such that the k— pump is half saturated,

b. It is approximately valid for noninteger values of k,

c. It is too large by a factor less than /2 because of the slight regularity in the output of the scale-of-two, and

d. It is invalid for к = 1 or к = N (where N is the number of pumps in the circuit).

For к = N (high count rate, N— pump half saturated, all other pumps saturated), the

image033 image034 Подпись: (2-19)

standard deviation is

The average output voltage is

image036S

Подпись: S image038 Подпись: (2-21)
image040

and the fractional standard deviation is

This expression is subject to restriction c. of the preceding paragraph.

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