Consider a channel which contains a logarithmic integrator of the Cooke-Yarborough type with a basic section as shown in Figure 2-1, containing eight sections with the frequency break­points spaced one decade apart and the time constants one decade apart.

Tk

If the ratio of —— is constant over all sections and equal to 0. 0256, and the frequency

Tk. 6

break points are established to provide 0 to 10 volts for a range of 0. 1 to 10 counts/sec input,

Table 2-1 is a tabulation of the fractional standard deviation and average output voltage as a

function of the input count rate obtained by the use of Equations (2-17) and (2-18).

TABLE 2-1

FRACTIONAL STANDARD DEVIATION AND AVERAGE OUTPUT VOLTAGE AS A FUNCTION OF INPUT COUNT RATE____________________________________________

♦Based on the single diode pump relationship

inaccurate at the end point.

If a trip or alarm is provided at 5 x 10° counts/sec then the number of erg’s between the trip or alarm point and the average output voltage is given by

sT — S

cr.

where

n = number of оg’s between ST and S, and S™ = trip or alarm point.

A tabulation of n as a function of count rate is given in Table 2-2.

TABLE 2-2

TABULATION OF n AS A FUNCTION OF COUNT RATE

S Count Rate (sec"*) ffS sT . ■ ; s n 0.5 5.56 9.55 1.00 47. 54 5.0 14.20 9.55 2.40 42.31 50 24.8 9.55 3.85 36. 70 5 x 102 33.3 . 9.55 5.25 27. 27 5 x 103 42. 7. 9. 55 6.70 18. 15 5 x 104 52. 2 . 9.55 8. 10 9. 344 1.0 x 105 . 55. 1 9. 55 8. 53 6. 557 1. 1 x 105 55.5 9.55 8. 59 6. 161 1. 2 x 105 55. 9 9. 55 8.65 5. 814 1.3 x 105 56. 2 9.55 8.70 . 5.491 1.4 x 105 56.5 9. 55 8.74 5. 238 1. 5 x 105 56. 8 9.55 8. 79 4. 913 1.6 x 105 57. 1 9.55 8. 83 4. 654 1. 7 x 105 57. 3 9. 55 8. 87 4.395 1. 8 x 105 57.5 . 9. 55 8. 90 4. 198 1. 9 x 105 57. 8 9. 55 8. 94 • 3. 942 2. 0 x 105 58. 0 9. 55 8. 97 3. 753 2. 2 x 105 58.4 9.55 9.03 3. 364 2.4 x 105 58.7 9. 55 9. 09 2. 970 2. 6 x 105 59. 1 9.55 9. 13 2. 719 2. 8 x 105 59.4 9.55 9. 18 2. 394 3.0 x 105 59. 7 9. 55 9. 22 2. 137 5 x 105 61. 8 9. 55 9. 55 0

can be substituted into Equation (2-25) to produce the tabulation (Table 2-3) of the average rate of crossings over the trip level, ST.

Count Rate (sec"*)	n*	0.09 Tk	r?_ 2	n2 e 2	Average Numb Crossings per Second
0.5	47. 54	0.0012	1130	lxlO”491	1. 2xl0"494
5. 0	42. 31	0.012	895	lxlO"389	1. 2×10"^*
5×10	36.70 .	0. 12	674	lxlO"293	1. 2xl0"294
5X102	27. 27	1. 2	372	lxlO"162	1. 2xlQ"162
5X103	18. 15	12	165	lxlO"72	-71 1.2X10 il
5xl04	9. 344	120	43.66	9×10"20	1.lxlO"17
1. OxlO5	6. 557	280	21. 50	4.675×10"10	1. 3xl0"7
1. lxlO5	6. 161	310	18. 98	5. 61X10"4	1. 7xl0"6
1. 2xl05	5. 814	340	16. 90	4.6X10"8	1.6X10"5
1. 3X105	5.491	370	15. 08	2.81X10"7	1. OxlO’4
1.4X105	5. 238	400	13. 72	1. 12ХЮ"6	4.4ХЮ"4
1.5X105	4. 913	420	12. 07	5. 52ХЮ"6	2. ЗХЮ"3
1. 6X105	4.654	460	10. 83	2. OxlO"5	0. 91×10"2
1. 7xl05	4.395	480	9.658	6. 44xl0"5	3. lxlO’2
1. 8ХЮ5	4. 198	510	8. 812	1. 50ХЮ"4	0. 76×10"1
1. 9ХЮ5	3. 942	550	7. 770	4. 30X10"4	2.4×10"1
2. OxlO5	3. 753	570	7. 043	8. 67X10"4	4.9×10"1
2. 2X105	3.364	630	5. 658	3.5lxlO’3	2. 2
2.4xl05	2. 970	700	4. 411	1. 23ХЮ"2	0. 86ХЮ1
2. 6ХЮ5	2. 719	760	3. 697	2.47X10"2	1. 9X101
2. 8X105	2. 394	820	2. 866	5. 70X10"2	4.6X101
3. OxlO3	2. 137	890	2. 283	1. 02×10"1	0. 9lxlO2
5. OxlO5	0	1200	0	1. 0	1. 2xl03

R

*Trip point at 5×10 counts/sec

5

If the counting channel is to be operated in the region near lxlO cps with a trip setting at

c

5×10 cps, the digest of Table 2-3 (shown in Table 2-4) illustrates the average number of spurious trips possible in this region.

AVERAGE TIME BETWEEN TRIPS AS A FUNCTION OF COUNT RATE Count Rate (sec’*) Average Time Between Trips

lxlOb.	2. 9	months
1. lxlO5	6.6	days
1.2xl05	18	hours
1. 3xl05	2. 7	hours
1.4X105	44	minutes
1. 5xl05	7. 1	minutes
1.6ХЮ5	1. 8	minutes
1. 7xl05	32-	seconds
1. 8xl05	13	seconds
1. 9xl05	4. 1	seconds
2. 0ХЮ5	2.0	seconds

It is shown in Table 2-4 that the average time to trip changes very rapidly for a very small increment of change in the reading, especially since the scale is logarithmic and covers seven decades.

In order that a smaller operating range not be required, the response time of the

logarithmic integrator may be made slower and the average time between trips at the region of

5 . .

lxlO counts/sec can be made much more tolerable.

In Equation (2-25) the part of the function which varies rapidly is the exponential, which is

a function of the number of root mean-square fluctuations between the operating point and the trip

‘ °S

or alarm point. From Equation (2-18) it is seen that the fractional standard deviation — is

s

directly proportional to. ‘ where

Tk is the frequency break, and is the time constant break.

For the counting channel considered in the previous analysis, the time constant breaks and response time (measured) as a function of к and rate are listed in Table 2-5.

Count Rate-(sec_1)	к	7k	Measured (0-63%) тк
0. 05		781 sec
0. 5	1	78.1 sec	34 sec
5. 0	2	7.81 sec	5.4 sec
5×10і	3	0. 781 sec	0. 54 sec
5xl02	4	78. 1 msec	54 msec
5xl03	5	7. 81 msec	5. 4 msec
5X104	6	0. 781 msec	0. 54 msec
5xl05	7	78. 1 Msec	54 psec
If the measured response time at	4 5×10 counts/sec is	established to be 40 msec, then the
Tk 1 ratio of — will change from —	to approximately —-— :	therefore, the standard deviation
Tk 39		3900
will be decreased by a factor of 10.	and Table 2-1 is amended as shown in Table 2-la.

TABLE 2-la

FRACTIONAL STANDARD DEVIATION AND AVERAGE OUTPUT VOLTAGE
AS A FUNCTION OF INPUT COUNT RATE

к	<V S	CTs. ■	S
1	0.0180	0.0180	1.00
2	0.00705	0.0169	2.40
3	0.00423	0. 0163	3. 85
4	0.00300	0. 0158	5. 25
5	0.00234	0. 0157	6. 70
6	0.00192	0.0156	8. 10
7	0.00162	0. 0155	9. 55

Table 2-2 is then amended as shown in Table 2-2a.

S Count Rate (sec" *) CTs ST, S i. n ‘ 0. 5 55.6 9.55 .1.00 475. 3 5.0 142. 0 9. 55 2. 40 423. 2 50 ’ 248 9. 55 3. 85 367. 0 5xl02 333 . 9.55 5.25 272. 7 5xl03 427 • 9.55 6. 70 181. 5 5xl04 522 9.55 8. 10 93. 43 1. 0X105 551 9.55 8.53 65.57 1. lxlO5 555 9.55 8. 59 61.61 1. 2X105 559 9. 55 8. 65 58. 14 1.3X105 562 9.55 8. 70 54. 91 1.4ХЮ5 565 9.55 . 8. 74 52. 38 1.5ХІ05 ‘ 568 9. 55 8. 79 49. 13 1. 6X105 571 9. 55 8. 83 46.54 1. 7X105 573 9.55 8. 87 43. 95 1. 8X105 575 9.55 8. 90 41. 98 1. 9xl05 578 9. 55 8. 94 39.42 2. OxlO5 580 9.55 8. 97 37.53 2. 2X105 .584 . 9.55 9.03 . 33.64 2. 4ХЮ5 587 9.55 9. 09 29. 70 2. 6ХЮ5 591 9. 55 9. 13 27. 19 2.8ХІ05 594 9. 55 9. 18 23. 94 . 3. 0X105 597 9. 55 9. 22 21.37 5X105 ■ 618 9.55 9. 55 0

From Table 2-2a and the proposed log integrator circuit constants, the average number of crossing-per-unit time over the trip level is tabulated in Table 2-‘3a. . , •

0,09

Count Rate (sec"1) n Tk

0. 5	475. 3	1. 2×10"
5.0	423. 2	1. 2X10*
5X10	367. 0	1. 2X10*
5ХІ02	272. 7	1. 2X10"
5X103	181. 5	to X h—* О 1
1 О H X	93.43	1. 2
1. OxlO[1]	65. 57	2. 8
Lrt 1 О X 1-4 i-H	61.61	3. 1
1. 2X105	58. 14	3.4
1.3ХЮ5	54. 91	3.7
1.4X105	52. 38	4.0
1. 5xl05	49. 13	4. 2
1. 6xl05	46. 54	4.6
1. 7xl05	43. 95	4. 8
1. 8xl05	41. 98	5. 1
1. 9xl05	39. 42	5. 5
2. OxlO5	37. 53	5. 7
2. 2ХЮ5	33.64	6.3
2.4xl05	29. 70	7.0
2. 6X105	27. 19	7. 6
2. 8ХІ05	23. 94	8. 2
3. OxlO5	21. 37	8. 9
5. OxlO5	0	12

n^ 2	-n! e 2	Average Number Crossings per Second
1. 13X105	lxlO"49000	1. 2xl0"49005
8.95X104	lxlO"38840	1. 2xl0"38844
6. 73ХЮ4	lxlO’29210	1. 2ХЮ"29213
3.72X104	. 1×10"16150	1.2×10"16152
1.65X104	1ХЮ"7161	1. 2ХЮ"7162
4.37X103	lxlO"1897	1.2xl0"1897
2. 15ХЮ3	7.94×10"934	2.2xl0"933
1.90xl03	3.96X10"824	1. 2X10"823
1.69ХЮ3	3.16×10"733	1.lxlO-732
1.51X103	2. OxlO"655	0.74xl0"654
1.37X103	3. 96X10"594	1. 6xl0"593
1. 2ІХЮ3	1.2xl0"525	0.54X10"524
1. 08ХІ03	6.3ХІ0"468	2. 9ХЮ"467
9. 66ХЮ2	1.59X10"419	0. 72X10"418
8.81X102	2.5xl0"382	1. 3xl0"381
7.77X102	1. 59X10"337	0.88X10"336
7.04X102	3. 14ХІ0’305	1.8X10"304
5. 66ХЮ2	4. OxlO"245	2.5X10"244
4. 4ІХЮ2	2. 5ХЮ"191	1.8×10"190
3.7OxlO2	3.94×10"160	3.OxlO"159
2.87X102	3. 94×10"124	3. 2×10"123
2. 27ХЮ2	9ХЮ"98	0.80xl0"97
0	1	12