SPECIFIC RESULTS: COUNT-RATE INDICATION

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

Подпись: Vs

Подпись: S (Average output voltage. at rate specified)
Подпись: Count Rate (sec-1) . _k

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

Подпись: 0.5 1 0. 180* 1. 0 volts 5. 0 2 0. 0705 2.40 50 3 0. 0423 3. 85 5 x 102 4 0.0300 5. 25 5 x 103 5 0.0234 6. 70 5 x 104 6 0.0192 8. 10 5 x 105 7 0.0162 9. 55

Подпись: since Equation (2-18) isimage069♦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

Подпись: sT - S
Подпись: and the values of for the log integrator being considered
Подпись: These values of n

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