Analog Engineerâ€™s Pocket Reference Art Kay and Tim Green, Editors

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Analog Engineerâ€™s Pocket Reference Fourth Edition

Edited by: Art Kay and Tim Green

Special thanks for technical contribution and review: Kevin Duke Rafael Ordonez John Caldwell Collin Wells Ian Williams Thomas Kuehl

ÂŠ Copyright 2014, 2015 Texas Instruments Incorporated. All rights reserved.

Texas Instruments Analog Engineer's Pocket Reference

3

Message from the editors: This pocket reference is intended as a valuable quick guide for often used board- and systemlevel design formulae. This collection of formulae is based on a combined 50 years of analog board- and system-level expertise. Much of the material herein was referred to over the years via a folder stuffed full of printouts. Those worn pages have been organized and the information is now available via this guide in a bound and hard-to-lose format! Here is a brief overview of the key areas included: • Key constants and conversions • Discrete components • AC and DC analog equations • Op amp basic configurations • OP amp bandwidth and stability • Overview of sensors • PCB trace R, L, C • Wire L, R, C • Binary, hex and decimal formats • A/D and D/A conversions We hope you find this collection of formulae as useful as we have. Please send any comments and/or ideas you have for the next edition of the Analog Engineer's Pocket Reference to artkay_timgreen@list.ti.com Additional resources: • Browse TI Precision Labs (www.ti.com/precisionlabs), a comprehensive online training curriculum for analog engineers, which applies theory to real-world, hands-on examples. • Search for complete board-and-system level circuits in the TI Designs – Precision reference design library (www.ti.com/precisiondesigns). • Read how-to blogs from TI precision analog experts at the Precision Hub (www.ti.com/thehub). • Find solutions, get help, share knowledge and solve problems with fellow engineers and TI experts in the TI E2E™ Community (www.ti.com/e2e).

4

Texas Instruments Analog Engineer's Pocket Reference

Contents Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Physical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Standard decimal prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Metric conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Temperature conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Error conversions (ppm and percentage) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Discrete components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Resistor color code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard resistor values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical capacitor model and specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical capacitors vs frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitor type overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard capacitance values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitance marking and tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diodes and LEDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 13 14 15 16 17 17 18

Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Capacitor equations (series, parallel, charge, energy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inductor equations (series, parallel, energy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capacitor charge and discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RMS and mean voltage definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RMS and mean voltage examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithmic mathematical definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dB definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Log scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pole and zero definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time to phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20 21 23 24 24 27 28 29 30 34

Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Basic op amp configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Op amp bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Full power bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Small signal step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability open loop SPICE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instrumentation Amp filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 41 42 43 44 48 50 53

PCB and wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 PCB conductor spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-heating of PCB traces on inside layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCB trace resistance for 1oz and 2oz Cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Package types and dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCB parallel plate capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCB microstrip capacitance and inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCB adjacent copper trace capacitanceÂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCB via capacitance and inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common coaxial cable specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coaxial cable equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistance per length for different wire types (AWG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum current for wire types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 57 58 60 61 62 63 64 65 66 67 68

Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Temperature sensor overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resistive temperature detector (RTD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diode temperature characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermocouple (J and K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70 71 72 74 76

A/D conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Binary/hex conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A/D and D/A transfer function (LSB, Data formats, FSR) . . . . . . . . . . . . . . . . . . . . . . . . . . Quantization error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal-to-noise ratio (SNR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total harmonic distortion (THD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal-to-noise and distortion (SINAD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective number of bits (ENOB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noise free resolution and effective resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setting time and conversion accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Texas Instruments Analog Engineer's Pocket Reference

83 84 90 91 92 94 94 95 96

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6

Texas Instruments Analog Engineer's Pocket Reference

Standard decimal prefixes • Metric conversions • Temperature scale conversions • Error conversions (ppm and percentage) •

Texas Instruments Analog Engineer's Pocket Reference

7

Conversions

Conversions Conversions

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Conversions

Conversions

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Table 1: Physical constants Constant Speed of light in a vacuum

Symbol

Value

c

2.997 924 58 x 108

Units m/s -12

εo

8.854 187 817 620 x 10

F/m

Permeability of free space

µo

1.256 637 0614 x 10-6

H/m

Plank’s constant

h

6.626 069 57 x 10-34

J•s

Permittivity of vacuum

-23

Boltzmann’s constant

k

1.380 648 8 x 10

Faraday’s constant

F

9.648 533 99 x 104

C/mol

Avogadro’s constant

NA

6.022 141 29 x 1023

1/mol

mu

1.660 538 921 x 10-27

kg

-19

C

Unified atomic mass unit Electronic charge

q

J/K

1.602 176 565 x 10

-31

Rest mass of electron

me

9.109 382 15 x 10

kg

Mass of proton

mp

1.672 621 777 x 10-27

kg

-11

Nm2/kg2

Gravitational constant

G

Standard gravity

gn

9.806 65

m/s2

Ice point

Tice

273.15

K

Maximum density of water Density of mercury (0°C) Gas constant Speed of sound in air (at 273°K)

6.673 84 x 10

3

ρ

1.00 x 10

kg/m3

ρHg

1.362 8 x 104

kg/m3

R

8.314 462 1

J/(K•mol)

2

cair

3.312 x 10

m/s

Table 2: Standard decimal prefixes Multiplier

Abbreviation

10

tera

T

109

giga

G

106

mega

M

103

kilo

k

10–3

milli

m

10–6

micro

µ

10–9

nano

n

10–12

pico

p

10–15

femto

f

atto

a

–18

10

8

Prefix

12

Texas Instruments Analog Texas Instruments Analog Engineer's Pocket Reference Engineer's Pocket Reference

Conversions

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Table 3: Imperial to metric conversions Unit

Symbol

Equivalent

Unit

Symbol

inches

in

25.4 mm/in

millimeter

mm

mil

mil

0.0254 mm/mil

millimeter

mm

feet

ft

0.3048 m/ft

meters

m

yards

yd

0.9144 m/yd

meters

m

miles

mi

1.6093 km/mi

kilometers

km

circular mil

cir mil

5.067x10-4 mm2/cir mil

square millimeters

mm2

2

2

square yards

yd

0.8361 m

square meters

m2

pints

pt

0.5682 L/pt

liters

L

ounces

oz

28.35 g/oz

grams

g

pounds

lb

0.4536 kg/lb

kilograms

kg

calories

cal

4.184 J/cal

joules

J

horsepower

hp

745.7 W/hp

watts

W

Symbol

Table 4: Metric to imperial conversions Unit

Symbol

Conversion

Unit

millimeter

mm

0.0394 in/mm

inch

in

millimeter

mm

39.4 mil/mm

mil

mil

meters

m

3.2808 ft/m

feet

ft

meters

m

1.0936 yd/m

yard

yd

kilometers

km

0.6214 mi/km

miles

mi

square millimeters

mm2

1974 cir mil/mm2

circular mil

cir mil

square meters

m2

1.1960 yd2/ m2

square yards

yd2

liters

L

1.7600 pt/L

pints

pt

grams

g

0.0353 oz/g

ounces

oz

kilograms

kg

2.2046 lb/kg

pounds

lb

joules

J

0.239 cal/J

calories

cal

watts

W

1.341x10-3 hp/W

horsepower

hp

Example Convert 10 mm to mil. Answer 10 mm x 39.4

mil = 394 mil mm

Texas Instruments Analog Engineer's Pocket Reference

9

Conversions

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Table 5: Temperature conversions Table 5: Temperature conversions Table 5: 5 Temperature conversions Fahrenheit �� � ��F � �2� FahrenheittotoCelsius Celsius 9 5 Fahrenheit to Celsius ��9� ��F � �2� 9 Celsius CelsiustotoFahrenheit Fahrenheit �F � ���� � �2 5 9 Celsius to Fahrenheit �F � ���� � �2 CelsiustotoKelvin Kelvin Celsius � � �� �52��.15 Celsius to Kelvin � � �� � 2��.15 KelvintotoCelsius Celsius Kelvin �� � � � 2��.15 Kelvin to Celsius �� � � � 2��.15

Table 6: Error conversions Table 6: Error conversions Table 6: Error conversions �easured � Ideal Error�%� � � 100 Ideal � Ideal �easured Error�%� � �easured � Ideal� 100 Ideal Error�% FSR� � � 100 Full‐scale range �easured � Ideal Error�% FSR� � ppm � 100 Full‐scale range % � � � 100 10 ppm % � � � 100 ppm 10 � 100 � 1000 m% � 10�ppm � 100 � 1000 m% � 10� ppm � % � 10� ppm � % � 10� ppm � m% � 10 ppm � m% � 10

Error in measured value Errorin inmeasured measuredvalue value Error Error in percent of full-scale range Error Errorin inpercent percent of of full-scale full-scale range range Part per million to percent Part Part per per million million to to percent Part per million to milli-percent Part per per million million to to milli-percent Part Percent to part per million Percent to to part part per per million Percent Milli-percent to part per million Milli-percent to part per million Milli-percent million

Example Example Compute the error for a measured value of 0.12V when the ideal value is 0.1V and theCompute range is 5V. Example the error for a measured value of 0.12V when the ideal value is 0.1V and Compute the range isthe 5V.error for a measured value of 0.12V when the ideal value is 0.1V and the range is 5V. Answer Answer 0.12V � 0.1V Error�%� � � 100 � 20% Answer 0.1V � 0.1V 0.12V V Error�%� � 0.12V 0.12 � 0.1V � 100 � 20% Error % 0.1V Error�% FSR� � 0.1V � 100 � 0.�% 5V � 0.1V 0.12 V � 100 � 0.�% Error�% FSR� � Error % FSR 5V 5V Example

Error in measured value Error in measured value Error in measured value Percent FSR Percent FSR Percent FSR

Example Example Convert 10 ppm to percent and milli-percent. Convert Convert10 10ppm ppmtotopercent percentand andmilli-percent. milli-percent. Answer Answer 10 ppm Answer Part per million to percent � 100 � 0.001% � 10 10 ppm 10 ppm Part Partper permillion milliontotopercent percent � 100 � 0.001% 10 ppm 10 10� � 100 � 1000 � 1 m% Part per million to milli-percent � 10 10 ppm 10 ppm Part Partper permillion milliontotomilli-percent milli-percent � 100 � 1000 � 1 m% 10 10�

10

Texas

Texas Instruments Analog Engineer's Pocket Reference 10

ti.com/precisionlabs Discrete Components

Resistor color code • Standard resistor values • Capacitance specifications • Capacitance type overview • Standard capacitance values • Capacitance marking and tolerance •

11

Texas

Texas Instruments Analog Engineer's Pocket Reference

Discrete

Discrete Components

Discrete Components

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Discrete

Table 7: Resistor color code Color

Digit

Additional Zeros

Black

0

0

Tolerance

Temperature Coefficient

Failure Rate

250

Brown

1

1

1%

100

1

Red

2

2

2%

50

0.1

Orange

3

3

15

0.01

Yellow

4

4

25

0.001

Green

5

5

0.5%

20

Blue

6

6

0.25%

10

Violet

7

7

0.1%

5

0.05%

1

Grey

8

8

White

9

9

Gold

-na-

-1

5%

Silver

-na-

-2

10%

No Band

-na-

-na-

20%

4 Band example: yellow violet orange silver indicate 4, 7, and 3 zeros. i.e. a 47k立, 10% resistor.

Figure 1: Resistor color code 12

Texas

Texas Instruments Analog Engineer's Pocket Reference

10.0 10.1 10.2 10.4 10.5 10.6 10.7 10.9 11.0 11.1 11.3 11.4 11.5 11.7 11.8 12.0 12.1 12.3 12.4 12.6 12.7 12.9 13.0 13.2 13.3 13.5 13.7 13.8 14.0 14.2 14.3 14.5

0.1% 0.25% 0.5%

Texas Instruments Analog Engineer's Pocket Reference

14.3

14.0

13.7

13.3

13.0

12.7

12.4

12.1

11.8

11.5

11.3

11.0

10.7

10.5

10.2

10.0

1%

13

12

11

10

2% 5% 10%

14.7 14.9 15.0 15.2 15.4 15.6 15.8 16.0 16.2 16.4 16.5 16.7 16.9 17.2 17.4 17.6 17.8 18.0 18.2 18.4 18.7 18.9 19.1 19.3 19.6 19.8 20.0 20.3 20.5 20.8 21.0 21.3

0.1% 0.25% 0.5%

21.0

20.5

20.0

19.6

19.1

18.7

18.2

17.8

17.4

16.9

16.5

16.2

15.8

15.4

15.0

14.7

1%

20

18

16

15

2% 5% 10% 21.5 21.8 22.1 22.3 22.6 22.9 23.2 23.4 23.7 24.0 24.3 24.6 24.9 25.2 25.5 25.8 26.1 26.4 26.7 27.1 27.4 27.7 28.0 28.4 28.7 29.1 29.4 29.8 30.1 30.5 30.9 31.2

0.1% 0.25% 0.5%

30.9

30.1

29.4

28.7

28.0

27.4

26.7

26.1

25.5

24.9

24.3

23.7

23.2

22.6

22.1

21.5

1%

30

27

24

22

2% 5% 10% 31.6 32.0 32.4 32.8 33.2 33.6 34.0 34.4 34.8 35.2 35.7 36.1 36.5 37.0 37.4 37.9 38.3 38.8 39.2 39.7 40.2 40.7 41.2 41.7 42.2 42.7 43.2 43.7 44.2 44.8 45.3 45.9

0.1% 0.25% 0.5%

45.3

44.2

43.2

42.2

41.2

40.2

39.2

38.3

37.4

36.5

35.7

34.8

34.0

33.2

32.4

31.6

1%

43

39

36

33

2% 5% 10% 46.4 47.0 47.5 48.1 48.7 49.3 49.9 50.5 51.1 51.7 52.3 53.0 53.6 54.2 54.9 55.6 56.2 56.9 57.6 58.3 59.0 59.7 60.4 61.2 61.9 62.6 63.4 64.2 64.9 65.7 66.5 67.3

0.1% 0.25% 0.5%

Standard resistance values for the 10 to 100 decade

66.5

64.9

63.4

61.9

60.4

59.0

57.6

56.2

54.9

53.6

52.3

51.1

49.9

48.7

47.5

46.4

1%

62

56

51

47

2% 5% 10% 68.1 69.0 69.8 70.6 71.5 72.3 73.2 74.1 75.0 75.9 76.8 77.7 78.7 79.6 80.6 81.6 82.5 83.5 84.5 85.6 86.6 87.6 88.7 89.8 90.9 92.0 93.1 94.2 95.3 96.5 97.6 98.8

0.1% 0.25% 0.5%

97.6

95.3

93.1

90.9

88.7

86.6

84.5

82.5

80.6

78.7

76.8

75.0

73.2

71.5

69.8

68.1

1%

91

82

75

68

2% 5% 10%

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Discrete Components

Table 8: Standard resistor values

13

Discrete Components

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Practical capacitor model and specifications

Rp

ESR

C

ESL

Figure 2: Model of a practical capacitor

Table 9: Capacitor specifications

Parameter

Description

C

The nominal value of the capacitance Table 11 lists standard capacitance values

ESR

Equivalent series resistance Ideally this is zero Ceramic capacitors have the best ESR (typically in milliohms). Tantalum Electrolytic have ESR in the hundreds of milliohms and Aluminum Electrolytic have ESR in the ohms

ESL

Equivalent series inductance Ideally this is zero ESL ranges from 100 pH to 10 nH

Rp

Rp is a parallel leakage resistance (or insulation resistance) Ideally this is infinite This can range from tens of megaohms for some electrolytic capacitors to tens of gigohms for ceramic

Voltage rating

The maximum voltage that can be applied to the capacitor Exceeding this rating damages the capacitor

Voltage coefficient

The change in capacitance with applied voltage in ppm/V A high-voltage coefficient can introduce distortion C0G capacitors have the lowest coefficient The voltage coefficient is most important in applications that use capacitors in signal processing such as filtering

Temperature coefficient

The change in capacitance with across temperature in ppm/째C Ideally, the temperature coefficient is zero The maximum specified drift generally ranges from 10 to 100ppm/째C or greater depending on the capacitor type (See Table 10 for details)

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Discrete Components

Practical capacitors vs. frequency

Impedance (ohms)

Practical capacitors vs. frequency

of ESR ESL on capacitor frequency Figure 3:Figure Effect 3: of Effect ESR and ESL and on capacitor frequency responseresponse

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Table 10: Capacitor type overview Capacitor type

Description

C0G/NP0 (Type 1 ceramic)

Use in signal path, filtering, low distortion, audio, and precision Limited capacitance range: 0.1 pF to 0.47 µF Lowest temperature coefficient: ±30 ppm/°C Low-voltage coefficient Minimal piezoelectric effect Good tolerance: ±1% to ±10% Temperature range: –55°C to 125°C (150°C and higher) Voltage range may be limited for larger capacitance values

X7R (Type 2 ceramic)

Use for decoupling and other applications where accuracy and low distortion are not required X7R is an example of a type 2 ceramic capacitor See EIA capacitor tolerance table for details on other types Capacitance range: 10 pF to 47 µF Temperature coefficient: ±833 ppm/°C (±15% across temp range) Substantial voltage coefficient Tolerance: ±5% to –20%/+80% Temperature range: –55°C to 125°C Voltage range may be limited for larger capacitance values

Y5V (Type 2 ceramic)

Use for decoupling and other applications where accuracy and low distortion are not required Y5V is an example of a type 2 ceramic capacitor See EIA capacitor tolerance table for details on other types Temperature coefficient: –20%/+80% across temp range Temperature range: –30°C to 85°C Other characteristics are similar to X7R and other type 2 ceramic

Aluminum oxide electrolytic

Use for bulk decoupling and other applications where large capacitance is required Note that electrolytic capacitors are polarized and will be damaged, if a reverse polarity connection is made Capacitance range: 1 µF to 68,000 µF Temperature coefficient: ±30 ppm/°C Substantial voltage coefficient Tolerance: ±20% Temperature range: –55°C to 125°C (150°C and higher) Higher ESR than other types

Tantalum electrolytic

Capacitance range: 1 µF to 150 µF Similar to aluminum oxide but smaller size

Polypropylene film

Capacitance range: 100 pF to 10 µF Very low voltage coefficient (low distortion) Higher cost than other types Larger size per capacitance than other types Temperature coefficient: 2% across temp range Temperature range: –55°C to 100°C

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Table 11: Standard capacitance table Standard capacitance table 1

1.1

1.2

1.3

1.5

1.6

1.8

2

2.2

2.4

2.7

3

3.3

3.6

3.9

4.3

4.7

5.1

5.6

6.2

6.8

7.5

8.2

9.1

CK06 223K

Figure 4: Capacitor marking code

Example Translate the capacitor marking 2 2 3 K "K" = ±10% 22 000 pF = 22nF = 0.022µF

Table 12: Ceramic capacitor tolerance markings Code

Tolerance

Code

Tolerance

B

± 0.1 pF

J

± 5%

C

± 0.25 pF

K

± 10%

D

± 0.5 pF

M

± 20%

F

± 1%

Z

+ 80%, –20%

G

± 2%

Table 13: EIA capacitor tolerance markings (Type 2 capacitors) First letter symbol

Low temp limit

Second number symbol

High temp limit

Second letter symbol

Max. capacitance change over temperature rating

Z

+10°C

2

+45°C

A

±1.0%

Y

–30°C

4

+65°C

B

±1.5%

X

–55°C

5

+85°C

C

±2.2%

6

+105°C

D

±3.3%

7

+125°C

E

±4.7%

F

±7.5%

P

±10.0%

R

±15.0%

S

±22.0%

T

±22% ~ 33%

U

±22% ~ 56%

V

±22% ~ 82%

Example X7R: –55°C to +125°C, ±15.0%

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Diodes and LEDs Anode (+)

Cathode (-)

Anode (+)

Cathode (-)

Anode (+)

Anode (+) Long Lead

Cathode (-)

Anode (+)

Cathode (-)

Anode (+) Long Lead

Cathode (-) Short Lead, Flat

Cathode (-)

Figure 5: Diode and LED pin names

Color

Wavelength (nm)

Voltage (approximate range)

Infrared

940-850

1.4 to 1.7

Red

660-620

1.7 to 1.9

Orange / Yellow

620-605

2 to 2.2

Green

570-525

2.1 to 3.0

Blue/White

470-430

3.4 to 3.8

Table 14: LED forward voltage drop by color

Note: The voltages given are approximate, and are intended to show the general trend for forward voltage drop of LED diodes. Consult the manufacturerâ€™s data sheet for more precise values.

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Analog Analog

Analog

Capacitor equations (series, parallel, charge, energy) • Inductor equations (series, parallel, energy) • Capacitor charge and discharge • RMS and mean voltage definition • RMS for common signals • Logarithm laws • dB definitions • Pole and zero definition with examples •

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Capacitor equations Capacitor equations C C

C

1 C C

C C

1 C

1

1 C

(1) Series capacitors

(2) Two series capacitors (3) Parallel capacitors

Analog

Where Ct = equivalent total capacitance C1, C2, C3â€ŚCN = component capacitors V

(4) Charge storage (5) Charge defined

Where Q = charge in coulombs (C) C = capacitance in farads (F) V = voltage in volts (V) I = current in amps (A) t = time in seconds (s) dv dt

(6) Instantaneous current through a capacitor

Where i = instantaneous current through the capacitor C = capacitance in farads (F)

dv = the instantaneous rate of voltage change dt 1 CV 2

(7) Energy stored in a capacitor

Where E = energy stored in an capacitor in Joules (J) V = voltage in volts C = capacitance in farads (F)

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Inductor equations Inductor equations L� � L� � L� � � � L� L� � L� �

1 1 1 1 � � �� L� L� L�

L� L� L� � L�

Series inductors (8)Series (8) inductors Parallel inductors (9)Parallel (9) inductors

Two parallel inductors (10) parallel inductors (10)Two

Where Where total inductance inductance LLtt = equivalent total LL11,, L L33…L …LNN == component componentinductance inductance L22,, L di Instantaneousvoltage voltageacross acrossananinductor inductor (11)Instantaneous v�L (11) dt

Where Where v = instantaneous voltage across the inductor v = instantaneous voltage across the inductor L = inductance in Henries (H) L = inductance in Henries (H) di � = instantaneous rate of current change dt = the instantaneous rate of voltage change ��

1 � � LI� 2

Energystored storedininananinductor Inductor (12)Energy (12)

Where Where EE = energy stored stored in in an an inductor inductorininJoules Joules(J) (J) = energy II ==current currentininamps amps L = inductance in Henries (H) L = inductance in Henries (H)

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Equation for for charging capacitor Equation chargingaan RC circuit ��

V� � V� �� � �� � � �

General relationship (13) (13) General relationship Equation for charging a capacitor Where �� Where relationship V�C =�voltage V� �� � �� � �the � capacitor(13) V across at anyGeneral instant in time (t) across the capacitor VCV ==voltage the source voltage charging at theany RCinstant circuit in time (t) S VSt = =time the source voltage charging the RC circuit in seconds Where t = = time seconds RC,inthe time constant for charging and discharging capacitors VC = voltage across the capacitor at any instant in time (t) τ = RC, the time constant for charging and discharging capacitors VS = the source voltage charging the RC circuit Graphing equation 13 produces the capacitor charging curve below. Note that the t = time in seconds capacitor is 99.3% charged at five time constants. It is common practice to consider = RC, the time constant for charging and discharging capacitors this fully charged. Graphing equation 13 produces the capacitor charging curve below. Note

that the capacitor is 99.3% charged at five time constants. It is common Graphing equation 13 produces the capacitor charging curve below. Note that the practice to consider this fully charged. capacitor is 99.3% charged at five time constants. It is common practice to consider this fully charged.

Figure 7: RC charge curve

Figure 7: RC charge curve

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Figure 6: RC charge curve

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Analog

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Equation for discharging a capacitor

Equation for discharging an RC circuit ��

V� � V� ��� � � �

(14) General Relationship (14) General

relationship

Equation for discharging a capacitor

Where Where V� � V� ��� � � � (14) General relationship == voltage across the capacitor at any instant time (t) in time (t) VV voltage across the capacitor at anyininstant CC == thethe initial voltage of the capacitor at t=0s VV i Where initial voltage of the capacitor at t=0s I t =VCtime in seconds = voltage across the capacitor at any instant in time (t) time intime seconds τ t=V=IRC, constant charging and discharging capacitors = thethe initial voltage of thefor capacitor at t=0s ��

t == time RC,in the time constant for charging and discharging capacitors seconds = RC, the time constant for charging and discharging capacitors

Graphing equation 14 produces the capacitor discharge curve below. Note Graphing equation 14 produces capacitor discharge curve belo that the capacitor is discharged to 0.7%the at five time constants. It is comGraphing equation 14 produces the capacitor discharge curve below. Note that the mon practice to consider this fully discharged. capacitor is 0.7% charged at five time constants. It is common prac capacitor is 0.7% charged at five time constants. It is common practice to consider

this fully this fullydischarged. discharged.

Percentage Discharged vs. Number of Time Constants 100

Percentage Discharged vs. Number of Time Consta

100 90 80

PercentagePercentage Charged Charged

90

70

80

60

70 50 40

60

30

50

20

10 40 0

30 0

1

2

3

4

5

Number of time Constants (τ = RC)

20

Figure Figure 7: RC discharge curve 10 8: RC discharge curve

0 0

1

2

3

4

Number of time Constants (τ = RC)

Figure 8: RC discharge curve

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RMS voltage RMS voltage RMS voltage �� RMS voltage1 (15) General relationship (15) General relationship V��� � � � �V�t��� dt ��� � �� � �� �� 1 �� (15) General relationship � �V�t��� dt V��� � � 1 (15) General relationship V��� �� � ���V�t��� dt � � ��� Where ���� � � �� � �� Where V(t) = continuous function of time V(t) = continuous function of time Where t= time in seconds Where t =V(t) time continuous in seconds function of time T ≤continuous T2 = the time interval that the function is defined over 1 ≤ t= V(t) = function of time t ≤ T the time interval that the function is defined over T1t ≤ = time 2in=seconds t = time in seconds T1 ≤ tvoltage ≤ T2 = the time interval that the function is defined over Mean T1 ≤ t ≤ T2 = the time interval that the function is defined over

Mean voltage Mean voltage 1

��

Mean (16) General relationship � V�t�dt V���� voltage � ��� � �� � �� �� 1 �� (16)(16) General relationship General relationship V���� � 1 � V�t�dt (16) General relationship V � ��V�t�dt ���� � ��� � ��� Where ��� � �� � �� V(t) = continuous function of time Where t= time in seconds Where Where V(t) = continuous function ofthat time T T2 = the time interval 1 ≤=t continuous V(t) function of V(t) = ≤continuous function of time time the function is defined over t =time time inseconds seconds tt == time in in seconds T1 ≤ t ≤ T2 = the time interval that the function is defined over TT11 ≤≤ t ≤ T22 == the the time time interval interval that that the the function functionisisdefined definedover over V���� RMS for full wave rectified V��� � (17) sine wave √2 V���� RMS for full wave rectified RMS for fullsine wave rectified V��� �V���� (17) (17) RMS for full wave rectified wave V��� � 2 √2 (17) sine wave Mean for full wave rectified � V���� (18) sine wave V���� � √2 sine wave π Mean for full wave rectified 2 � V���� for full wave rectified (18) Mean V���� �2 � V���� (18) Mean for full rectified (18)wave sine wave sine wave V���� � π sine wave π

Figure 9: Full wave rectified sine wave

Figure 8: Fullsine wave rectified sine wave Figure 9: Full wave rectified wave Figure 9: Full wave rectified sine wave

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RMS voltage andand mean voltage RMS voltage mean voltage

V

τ 2T

V V π

V

RMS for a half-wave (19) (19) RMS for a half-wave sine wave rectified rectified sine wave τ T

Mean for a half-wave (20) (20) Mean for a half-wave rectified sine wave rectified sine wave

9: sine Half-wave Figure 10: Half-wave Figure rectified wave rectified sine wave V V

V V

τ T

(21) a square wave (21) RMSRMS for afor square wave

τ T

Figure 11: Square wave

(22) Mean for a square wave

Figure 10: Square wave

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RMS voltage and mean voltage RMS voltage and mean voltage

V V

(V τ V 2T

V V 3

V

V

(( T ( τ

(23) RMS for afortrapezoid (23) RMS a trapezoid

(24) Mean for afortrapezoid (24) Mean a trapezoid

Figure 12: Trapezoidal wave Figure 11: Trapezoidal wave

V V

V

τ V 2T

τ 3T

Figure 13: Triangle wave

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(25) RMS a triangle wave wave (25) for RMS for a triangle (26) Mean for a triangle (26) Mean for a triangle wave wave

Figure 12: Triangle wave

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Logarithmic mathematical definitions Logarithmic mathematical definitions A B

A

log AB

log A log

log

ln X

A

log log

A

log log

B

B

(27)ofLog of dividend (27) Log dividend

(28) Log product (28)ofLog of product (29)ofLog of exponent (29) Log exponent (30) Changing the of base log function (30) Changing the base logof function

(31) Example changing to logtobase 2 (31) Example changing log base 2 (32) Natural log is log log is base e (32) Natural log base e (33) Exponential function to 6 digits. (33) Exponential function to 6 digits

Alternative Alternative notations notations exp x

Different notation for exponential (34) Different notation for exponential function (34) function Different notation for scientific (35) Different notation for scientific notation, (35) notation, sometimes confused with sometimes confused with exponential function exponential function

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dB definitions Bode plot basics dB definitions The frequency response for the magnitude or gain plot is the change in voltage gain as frequency changes. This change is specified on a Bode plot, Bode plot basics a plot of frequency versus voltage gain in dB (decibels). Bode plots are The frequency for theplots magnitude or gain ploton is the the change voltage usually plottedresponse as semi-log with frequency x-axis, inlog scale,gain as frequency changes. This change is specified on half a Bode plot,frequency a plot of frequency and gain on the y-axis, linear scale. The other of the versus voltage gain in dB (decibels). Bode plots are usually plotted as semi-log plots response is the phase shift versus frequency and is plotted as frequency with frequency on the x-axis, log scale, and gain on the y-axis, linear scale. The other versus degrees phase shift. Phase plotsshift are versus usuallyfrequency plotted as half of the frequency response is the phase andsemi-log is plotted as plots with frequency on the x-axis, log scale, and phase shift onasthe frequency versus degrees phase shift. Phase plots are usually plotted semi-log y-axis, linear scale.on the x-axis, log scale, and phase shift on the y-axis, linear plots with frequency scale.

Definitions V V

Measured

P P

((36) 36) Voltage Voltagegain gaininindecibels decibels ((37) 37) Power Power gain gain in in decibels decibels

Power Measured (W) 1 mW

(38) Used input or Powerfor gain in decibel (38) output power milliwatt

Table 15: Examples of common gain A (V/V) A (dB) Table 14: Examples of common gain and dB equivalent values and dBvalues equivalent 0.001 –60 A (V/V) A (dB) 0.01 –40–60 0.001 Roll-off rate is the decrease in gain with frequency 0.010.1 –20–40 0.1 –20 Decade is a tenfold increase or decrease in 0 0 frequency (from 10 Hz to 100 Hz is one decade) 1 1 20 20 10 10 Octave is the doubling or halving of frequency 100100 40 40 (from 10 Hz to 20 Hz is one octave) 1,000 60 1,000 60 10,000 80 100,000 10,000 80 100 1,000,000 120 100,000 100 10,000,000 140 1,000,000 120 Roll-off rate is the decrease in gain with frequency 10,000,000 140

Decade is a tenfold increase or decrease in frequency.(from 10 Hz to 100 Hz is one decade) Octave is the doubling or halving of frequency (from 10 Hz to 20 Hz is one octave)

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Analog

Figure 13 illustrates a method to graphically determine values on a logarithmic axis that are not directly on an axis grid line. Figure 14 illustrates a method to graphically determine values on a logarithmic axis that areLnot directly an axis grid line. with a ruler. 1. Given = 1cm; D on = 2cm, measured

2. L/D = log10(fp) 1. (L/D) Given L =(1cm/2cm) 1 cm; D = 2cm, measured with a ruler. 3. fP = 10 = 10 = 3.16 2. L/D = log10(fP) (L/D) (1CM/2CM) (for this example, f = 31.6 Hz) 4. Adjust p 3. for fP =the 10 decade = 10 range = 3.16

A (dB)

4. Adjust for the decade range (for example, 31.6 Hz)

Figure14: 13:Finding Findingvalues valueson onlogarithmic logarithmic axis axis not not directly directly on on aa grid grid line line Figure

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Bode plots: Poles Bode plots: Poles fP 100

0.707*GV/V = –3 dB Actual function

G (dB)

80

Straight-line approximation –20 dB/decade

60 40 20 0 1

10

100

(degrees)

+90 +45

1k

10k

100k

1M

10M

100k

1M

10M

Frequency (Hz)

0°

10

100

1k

10k

0 –45

–5.7° at

fP 10

–45° at fP

–90

–45°/decade –84.3° at fP x 10

–90°

Figure 14: Pole gain and phase Figure 15: Pole gain and phase Pole Location fP (cutoff freq) Pole=Location = fP (cutoff freq) Magnitude (f < fP) = Gdc (for example, 100 dB) Magnitude (f < fP) = GDC (for example, 100 dB) Magnitude (f = fP) = –3 dB (f dB/decade = fP) = –3 dB MagnitudeMagnitude (f > fP) = –20 Phase (f =Magnitude fP) = –45° (f > fP) = –20 dB/decade Phase (0.1 fP < f <(f10 Phase = ffPP) == –45°/decade –45° Phase (f > 10 fP) = –90° Phase (0.1 f < f < 10 fP) = –45°/decade Phase (f < 0.1 fP) = 0° P

Phase (f > 10 fP) = –90° Phase (f < 0.1 fP) = 0°

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Pole (equations) Pole (equations) G

V V

G

V V

G

G j f f

(39) As (39) a complex number number As a complex

G

f f

(40) Magnitude (40) Magnitude

f f

(41)shift Phase shift (41) Phase (42) Magnitude (42) Magnitude in dB in dB

Where Gv = voltage gain in V/V Where GdB = voltage gain in decibels GG voltage gain in V/V v= dc = the dc or low frequency voltage gain Hz in decibels GfdB= frequency = voltageingain frequency which the pole occurs P == GfDC the dc oratlow frequency voltage gain θ = phase shift of the signal from input to output

f = frequency in Hz fP = frequency at which the pole occurs θ = phase shift of the signal from input to output j = indicates imaginary number or √ –1

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BodeBode plotsplots (zeros) (zeros) 80 Straight-line approximation +20 dB/decade

G (dB)

60

40 Actual function

20

+3 dB

0 1

10

100

+90

1k

10k

100k

1M

10M

+90°

+45° at fZ

(degrees)

84.3° at fZ x 10 +45 0°

f 5.7° at Z 10

+45°/decade

0 10

100

1k

10k

100k

1M

10M

Frequency (Hz)

–45

–90

Figure 15: Zero gain and phase Figure 16: Zero gain and phase Zero location fZ Zero =location = fZ Magnitude (f < fZ) = 0 dB Magnitude (f < fZ) = 0 dB Magnitude (f = fZ) = +3 dB Magnitude (f = fZ) = +3 dB Magnitude (f > fZ) = +20 dB/decade Phase (f Magnitude = fZ) = +45°(f > fZ) = +20 dB/decade Phase fZ)fZ=) +45° = +45°/decade Phase (0.1 fZ < f(f<=10 Phase (0.1 fZ < f < 10 fZ) = +45°/decade Phase (f > 10 fZ) = +90° Phase (f Phase < 0.1 fZ(f) = > 0° 10 fZ) = +90° Phase (f < 0.1 fZ) = 0°

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Zero (equations) Zero (equations) G� �

V��� f � G�� �j � � � �� V�� f�

G� �

f � V��� � G�� �� � � � f� V��

f � � ����� � � f�

G�� � �������G� �

As a complex (43) As(43) a complex number number

(44) Magnitude (44) Magnitude

(45) shift Phase shift (45) Phase

(46) Magnitude in dB in dB (46) Magnitude

Where GV = voltage gain in V/V Where voltage gain decibels GG == voltage gain in in V/V V dB the dc or lowinfrequency voltage gain DC==voltage GG gain decibels dB f = frequency in Hz GDC = the dc or low frequency voltage gain fZ = frequency at which the zero occurs f = frequency in Hz θ = phase shift of the signal from input to output fZ = frequency at which the zero occurs θ = phase shift of the signal from input to output j = indicates imaginary number or √ –1

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Time to phase shift

P

S

Figure 17: Time to phaseFigure shift 16: Time to phase shift

θ

(47) Phase fromshift timefrom time (47) shift Phase

• 360°

Where Where T S = time shift from input to output signal TPS==period time shift from input to output signal of signal T θTP= = phase shiftofofsignal the signal from input to output period

θ = phase shift of the signal from input to output Example Calculate the phase shift in degrees for Figure 17.

Example

Answer Calculate the phase shift in degrees for Figure 16. T Answer

θ=

34

T Ts

Tp

0.225 1

• 360° =

(

Texas

ms

0.225ms ms 1 ms

) • 360° = 81°

34 Texas Instruments Analog Engineer's Pocket Reference

Amplifier Amplifier

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Amplifier

Basic op amp configurations • Op amp bandwidth • Full power bandwidth • Small signal step response • Noise equations • Stability equations • Stability open loop SPICE analysis •

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Amplifier

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Basic op amp configurations Basic op amp configurations Basic op amp configurations

G�� � �

Gain for buffer (48) Gain (48) for buffer configuration

configuration

(48) Gain for buffer configuration

G�� � �

VCC VOUT

VIN

+

+

VEE

Figure 17: Buffer configuration Figure 18: Buffer configuration

Amplifier

FigureR18: Buffer configuration � (49) forfor non-inverting configuration Gain non-inverting configuration G�� � �� (49)Gain R�

G�� �

R� �� R�

R1

Rf

(49) Gain for non-inverting configuration

VCC VOUT

+

VIN

VEE

+

Figure 18: Non-inverting configuration Figure 19: Non-inverting configuration

Figure 19: Non-inverting configuration ti.com/amplifiers 36

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Basic op amp configurations (cont.) ti.com/precisionlabs R�

G�� � �

Amplifier

Gain for inverting configuration

(50)

R� Basic op amp configurations (cont.)

Basic op amp configurations (cont.)

+

G�� � �

R� R�

Gain for inverting (50) Gain(50) for inverting configuration R1

configura

Rf VCC

VIN

+

VOUT

VEE Figure 19: Inverting configuration Figure 20: Inverting configuration

V� V� V� Transfer function for i V��� � �R � � � � � � � (51) Transfer function (51) for inverting R R R Figure 20: configuration summing amplifier summing amplifier � Inverting � �

Transfer function for i R� VV� �V� Transfer function � �V � V � Transfer function forsumming V��� V � � � �V � amplifier, as (52) (52) � � � � � �� � (51)inverting summing ��� R �R � � � summing R� R � R � amplifier, assuming R R = …=R R1 = 2 = …=R N amplifi 1 = R 2 N V��� � � VN

RN R� �V� � V� � � � V� � R� RN R 2

V2

VN

VN

(52)

R2 V1

V2

V2

V1

R1

R RN1

Transfer function summing amplifi R1 = R2 = …=RN

Rf VCC Rf

R2

VOUT

Vcc R1

- + VEE

Rf

+

Vcc

Vee

-+

Vout

V1 Inverting summing configuration Figure 20: ti.com/amplifiers Texas Instruments Analog Engineer's Pocket Reference

+

Figure 21: Inverting summing configuration

Vout 37

Amplifier

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Basic op amp configurations (cont.) Basic op amp configurations (cont.)

V��� � �

R� V� V� V� � �� � � � � � � R �� N N N

Transfer function (53) Transfer function forfor noninverting summing amplifier (53)noninverting summing amplifier equalinput inputresistors resistors forforequal

Where R1 = R2 = … = RN Where of input resistors R1N==Rnumber 2 = … = RN N = number of input resistors

Rin R1 V1

Rf VCC VOUT

R2 V2

RN

VEE

VN Figure 22: Non-inverting summing configuration Figure 21: Non-inverting summing configuration

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Simple amp Cf filter Simplenon-inverting non-inverting amp withwith Cf filter R � non-inverting amp with Cf filter Simple Gain for non-inverting configuration G�� � �1 (54) Gain(54) for non-inverting configuration for f < fc for f < fc R� R� Gain non-inverting configuration Gain forfor non-inverting configuration G�� �1 (54) (55) Gain(55) for non-inverting configuration for f >> fc �� 1 G�� f <fcfc R� forfor f >>

Gain frequency for non-inverting configuration for non-inverting (55) Cut off for (56) Cut off configuration (56)frequency for f >> fnon-inverting c configuration Cut off frequency for non-inverting (56) Cf configuration

G �11 f� ��� 2π R � C� 1 f� � 2π R � C�

R1

Rf VCC VOUT

VIN

VEE

Figure 23: Non-inverting amplifier with Cf filter

Figure 22: Non-inverting amplifier with Cf filter Figure 23: Non-inverting amplifier with Cf filter

Figure 24: Frequency response for non-inverting op amp with Cf filter Figure 23: 24: Frequency Frequency response response for for non-inverting non-inverting op op amp amp with with C Cff filter ti.com/amplifiers

Texas Instruments Analog Engineer's Pocket Reference 39

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Simple inverting amp with Cf filter SimpleRinverting amp with Cf filter (57) Gain for inverting configuration for f < fC G R R Gain for inverting configuration (57) −20dB/decade after f C GG < fC R (58) Gainfor forf inverting configuration for f > fC f

G f

until op amp bandwidth limitation

−20dB/decade after fC 1until op amp bandwidth (59) Cutoff frequency for inverting configuration 2π Rlimitation C 1 frequency for inverting configuration (59) Cutoff Cf 2π R C Cf

R1 R1

Rf

VIN

Rf VCC

Vcc

Vin

VOUT

-+ +

Vout

VEE

Vee

Figure 24: Inverting amplifier with Cf filter Figure 25: Inverting amplifier with Cf filter

Figure 25: Frequency response for inverting op amp with Cf filter Figure 26: Frequency response for inverting op amp with C f filter ti.com/amplifiers 40

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Pocketreference_6815_final.indd 40

Texas Instruments Analog Engineer's Pocket Reference

40

6/11/15 3:06 PM

Amplifier

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Op amp bandwidth Op amp bandwidth GBW = Gain • BW GBW � Gain���BW

(60) Gain product defined Gain bandwidth product defined (60) bandwidth

Where Where = bandwidth gain bandwidth product, op amp sheet specification table GBWGBW = gain product, listedlisted in opinamp datadata sheet Gain = closed loop gain, set by op amp gain configuration specification table = the loop bandwidth limitation of thegain amplifier Gain BW = closed gain, set by op amp configuration BW = the bandwidth limitation of the amplifier Example Determine bandwidth using equation 60

Example Gain � �100 Determine bandwidth using equation 60(from amplifier configuration) Gain = 100 (from amplifier configuration) (from data sheet) GBW � �22MHz GBW = 22MHz (from data sheet) GBW 22MHz 22MHz GBW = 220�Hz 220 kHz BW BW = � Gain=� � 100 � 100 Gain

Note that the same result can be graphically determined using the AOL curve as Note shown that thebelow. same result can be graphically determined using the AOL curve as shown below. Open-loop gain and phase vs. frequency

Open-loop gain and phase vs. frequency

Figure 27: Using AOL to find closed-loop bandwidth Figure 26: Using AOL to find closed-loop bandwidth

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Full power Full powerbandwidth bandwidth Full power bandwidth SR Maximum output without slew-rate induced (61) Maximum (61) output without slew-rate induced distortion V� � distortion 2πfSR Maximum output without slew-rate induced (61) V� � distortion 2πf Where VP = maximum peak output voltage before slew induced distortion occurs Where Where = slew ratepeak output voltage before slew induced distortion occurs VSR P= = maximum peak output voltage before slew induced distortion occurs VPmaximum f = =frequency of applied signal SR raterate SRslew = slew f = ffrequency of applied signal = frequency of applied signal Maximum output voltage vs. frequency Maximum output voltage vs. frequency Maximum output voltage vs. frequency

�. ��/�� �� � �. ����� �� �. ����� � ��� ��������� �� �. ��/�� �� � � � �. ����� �� �. ����� ��� ���������

�� �

Figure 28: Maximum output without slew-rate induced distortion Figure 27: Maximum output without slew-rate induced distortion Figure output withoutusing slew-rate induced Notice that28: theMaximum above figure is graphed equation 61 for distortion the OPA188. The example calculation shows the peak voltage for the OPA277 at 40kHz. This can be Notice thatthat thethe above figure is graphed using equation 6161 forfor thethe OPA277. Notice above figure is graphed using equation OPA188. The determined graphically or with the equation. The example calculation shows the peak voltage for the OPA277 at 40kHz. example calculation shows the peak voltage for the OPA277 at 40kHz. This can be Thisdetermined can be determined graphically with the equation. graphically or with theorequation. Example Example

SR Example V� � �

0.8V/μs � 3.�8Vpk or 6.37Vpp 2πfSR 2π��0k��� 0.8V/μs V� � � SR 0.8V/µs � 3.�8Vpk or 6.37Vpp VP = 2πf= 2π��0k��� = 3.18Vpk or 6.37Vpp

2πf

2π(40kHz)

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ti.com/precisionlabs Small signal signal step response Small step response τ� �

0.35 f�

a small signal (62) (62) RiseRise timetime for afor small signal stepstep Small signal step response

Where 0.35 τ� �of a small signal step response (62) Rise time for a small signal step R = the rise time f� Where C = the closed-loop bandwidth of the op amp circuit t f= the rise time of a small signal step response R

Wherebandwidth of the op amp circuit fC = the closed-loop Small signal step response waveform R = the rise time of a small signal step response fC = the closed-loop bandwidth of the op amp circuit Small signal step response waveform Small signal step response waveform

Figure 29: Small signal step response

Figure 29: Small signal step response Figure 28: Maximum output without slew-rate induced distortion

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Op amp noise model Op amp noise model

Figure 30: Op amp noiseFigure model29: Op amp noise model Op amp intrinsic noise includes: Noise caused by op amp (current noise + voltage noise) Op amp intrinsic noise includes: Resistor noise • Noise caused by op amp (current noise + voltage noise) • Resistor noise

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ti.com/precisionlabs Noise bandwidth calculation Noise bandwidth calculation Noise bandwidth calculation (63) Noise bandwidth NoiseBW bandwidth � � � � f� calculation Noise bandwidth BW� � � � f� (63)Noise (63) bandwidth

BW Where �� f (63) Noise bandwidth Where� BW � =� noise bandwidth of the system N Where BWN = noise of the systemfactor for different filter order KN = bandwidth the brick wall correction Where BW noise bandwidth of the factor system KNN==the brick wall correction for different filter order –3 dB bandwidth system noise bandwidth of of thethe system BWN f=C = brick wall correction factor for different filter order KNfC == the –3 dB bandwidth of the system KN = the brick wall correction factor for different filter order fC = –3 dB bandwidth of the system fC = –3 dB bandwidth of the system

Figure 31: Op amp bandwidth for three different filters orders Figure 31: Op amp bandwidth for three different filters orders Figure 30: Op amp bandwidth for three different filters orders Figure 31: Op amp bandwidth for three different filters orders Table 15: Brick wall correction factors for noise bandwidth Table 15: Brick wall correction factors for noise bandwidth wall for noise bandwidth KN brickfactors 16:ofBrick wall correction Number poles Table Table 15: Brick wall correction factors for noise bandwidth wall KN brick correction factor Number of poles correction factorKN brick wall correction factor Number of poles KN brick wall 1 1.57 Number 1 of poles1 1.57 correction factor 1.57 2 1.22 2 1 1.221.57 2 1.22 3 1.13 3 2 1.13 1.22 3 1.13 4 1.12 4 3 1.121.13 4 1.12 4 1.12 Broadband total noise calculation Broadband totalcalculation noise calculation Broadband total noise Broadband total noise calculation Totalfrom rmsbroadband noise from broadband E� � ��� (64) noise �BW (64) Total rms � E� � ��� �BW� (64) Total rms noise from broadband E � Where ��� �BW� (64) Total rms noise from broadband � Where Where EN = total rms noise from broadband noise ENrms = total rms noise from broadband noise EN = total noise from broadband noise eBB = broadband noise spectral density (nV/rtHz) Where eBB = broadband noise spectral density (nV/rtHz) e = broadband noise spectral density BB = noise bandwidth (Hz) EN = BW totalN rms noise from broadband noise (nV/rtHz) BWN = noise bandwidth (Hz) bandwidth (Hz) N = noisenoise eBB =BW broadband spectral density (nV/rtHz) BWN = noise bandwidth (Hz)

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1/f total noise calculation 1/f total noise calculation 1/f total noise calculation

Normalized1/f 1/fnoise noiseatat1 1Hz Hz (65) Normalized E�_������ � � ��� �f� (65) E�_������ � � ��� �f� (65) Normalized 1/f noise at 1 Hz Where Where E Where = 1/f noise normalized to 1 Hz ENN_NORMAL _NORMAL = 1/f noise normalized to 1 Hz EN_NORMAL = 1/f noise normalized to 1 in Hzthe 1/f region eBF = noise spectral density measured = =noise spectral density measured in the eBF noise spectral the1/f 1/fregion region frequency thatdensity the 1/f measured noise eBF isinmeasured at fOe=BFthe fOfO= =the frequency that the 1/f noise e is measured BF the frequency that the 1/f noise eBF is measuredat at f� E�_������� � � E�_������ ��� � f� f � E�_������� � � E�_������ ��� �� � f�

1/f total noise calculation (66) 1/fnoise total noise calculation (66) (66) 1/f total calculation

Where Where = total rms noise from flicker EN_FLICKER rmsnormalized noise fromtoflicker EN_FLICKER= =1/ftotal noise 1 Hz EN_NORMAL Where 1/f frequency noise normalized tobandwidth 1 Hz E E ==total rms noise flicker fHN_FLICKER =N_NORMAL upper cutoff orfrom noise f=H = upper frequency or noise bandwidth fLN_NORMAL lower cutoff normally to 0.1 Hz E =cutoff 1/f frequency, noise normalized to set 1Hz f = lower cutoff frequency, normally set to 0.1 Hz L f = upper cutoff frequency or noise bandwidth H

fL = lower cutoff frequency, normally set to 0.1Hz Table 16: Peak-to-peak conversion Table 16: Peak-to-peak conversion Number of Percent chance Table 17:Number Peak-to-peak conversion standard deviations reading is inchance range of Percent reading is in range 2σstandard (same as deviations ±1σ) 68.3% Number of standard deviations Percent chance reading is in range 2σ (same as ±1σ) 3σ (same as 2σ ±1.5σ) (same as ±1σ) 3σ (same as ±1.5σ) (same as ±1.5σ) 4σ (same as3σ ±2σ) 4σ (same as ±2σ) 4σ (same as ±2σ) 5σ (same as ±2.5σ) 5σ (same as ±2.5σ) 5σ (same as ±2.5σ) 6σ (same as ±3σ) 6σ (same as ±3σ) 6σ (same as 6.6σ (same as ±3σ) ±3.3σ) 6.6σ (same as(same ±3.3σ) 6.6σ as ±3.3σ)

68.3% 86.6% 86.6% 95.4% 95.4% 98.8% 98.8% 99.7% 99.7% 99.9% 99.9%

68.3% 86.6% 95.4% 98.8% 99.7% 99.9%

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Thermal noise calculation En_R =

√ 4kTR�f √ 4kTR

Thermal noise calculation en_R =

(67) Total rms Thermal Noise (68) Thermal Noise Spectral Density

(67)

E�_� � √4 kTRΔf

Total rms thermal nois

Where En_R = Total rms noise from resistance, also called thermal noise (V rms) Where en_R = Noise spectral density from resistance, also called thermal noise (V/√Hz ) EkN_R = total rms noise from resistance, also called thermal noise = Boltzmann’s constant 1.38 x 10-23J/K -23 kT== Boltzmann’s constant 1.38 x 10 J/K Temperature in Kelvin T∆f==temperature ininKelvin Noise bandwidth Hz

∆f = noise bandwidth in Hz

Noise Spectral Density (nV/rtHz)

1000

100

10 ‐55C 25C

1

0.1 1.E+01

125C

1.E+02

1.E+03 1.E+04 1.E+05 Resistance (Ω)

1.E+06

1.E+07

Figure 32: Noise spectral vs. resistance Figure 31: Noise spectral densitydensity vs. resistance

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Ac response frequency (Dominant 2-Pole System) Ac responseversus versus frequency Ac response versus frequency

Figure 32 illustrates a bode four different differentexamples examples of peaking. ac peaking. Figure 33 illustrates a bodeplot plotwith with four of ac Figure 33 illustrates a bode plot with four different examples of ac peaking.

Figure 32: Stability – ac peaking relationship Figure 33: Stability – ac peaking relationship exampleexample Figure 33: Stability – ac peaking relationship example

Phase margin versus ac peaking

Phase marginversus versus ac Phase margin acpeaking peaking

This graph illustrates the phase margin for any given level of ac peaking.

graph illustrates thephase phasemargin margin for of of ac ac peaking. This This graph illustrates the for any anygiven givenlevel level peaking. Note that 45° of phase margin or greater is required for stable operation. NoteNote thatthat 45°45° of phase isrequired requiredforfor stable operation. of phasemargin marginor or greater greater is stable operation.

Figure 34: Stability – phase margin vs. peaking for a two-pole system Figure 34: Stability – phase margin vs. peaking for a two-pole system Figure 33: Stability – phase margin vs. peaking for a two-pole system

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Transient overshoot (Dominant 2-Pole System) Transient overshoot

Figure 34 35 illustrates withtwo two different examples of Figure illustratesa atransient transient response response with different examples percentage overshoot. of percentage overshoot. Transient overshoot Figure 35 illustrates a transient response with two different examples of percentage overshoot.

Figure 35:Figure Stability transient overshootovershoot example example 34:–Stability – transient Phase margin versus percentage overshoot Figure 35: Stability – transient overshoot example Phase margin versus overshoot This graph illustrates the percentage phase margin for any given level of transient overshoot. Note that 45° of overshoot phase margin or greater is required Phase margin versus percentage Thisforgraph stableillustrates operation. the phase margin for any given level of transient This graph illustrates theof phase margin for any given level of overshoot. Note that 45° phase margin or greater is required for transient overshoot. Note that 45° of phase margin or greater is required stable operation. for stable operation.

Figure 36: Stability – phase margin vs. percentage overshoot Note: assume a two-pole system. Figure 35: Stability ––phase margin FigureThe 36:curves Stability phase marginvs. vs.percentage percentageovershoot overshoot Note: The curves assume a two-pole system.

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VFB R1

C1 1T

RF

L1 1T

VIN V+

VO Riso

VOUT CL

V– Figure 36: Common spice test circuit used for stability Figure 37: Common spice test circuit used for stability A��_������ � � β � � V��

V� V��

1 1 �� β V��

Loaded open-loop gain (68) (69) Loaded open-loop gain

(70) Feedback Feedback factor (69) factor (71) Closed-loop gain noise gain Closed-loop (70) noise

A��_������ � β � � V�

(72) Loop gain (71)

Loop gain

Where VO = the voltage at the output of the op amp. Where = the voltage delivered the load, which may be important to the VVOOUT = the voltage at output the output of thetoop amp. application but is not considered in stability analysis. VOUT = the voltage output delivered to the load, which may be important to feedback voltage VFB =the application but is not considered in stability analysis. RF , R1, RISO and CL = the op amp feedback network and load. Other op amp VFB = feedback voltage topologies will have different feedback networks; however, the test circuit will be the Rsame , RiS0 andcases. CL = the op amp feedback network and load. most Figure 38 shows the exception to the rule (multiple feedback). F , R1for C1 and Other op components amp topologies have different feedbackThey networks; L1 are thatwill facilitate SPICE analysis. are large (1TF, 1TH) to make however, the test circuit will the same most the circuit closed-loop forbe dc, but openfor loop for cases. ac frequencies. SPICE requires Figure 37 showsoperation the exception rule (multiple feedback). closed-loop at dc to forthe convergence. C1 and L1 are components that facilitate SPICE analysis. They are large (1TF, 1TH) to make the circuit closed-loop for dc, but open loop for ac frequencies. SPICE requires closed-loop operation at dc for convergence.

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R1

R1

VFB

CIN

CF

V-

VIN Vin

C1 1T

C1 1T

RF

RF

L1 1T

Cin

VFB

CF

V+ -

Riso

+

+

+

V+

V–

Vo

Riso VO

Vout C

CL

VOUT

L

Figure 37: Alternative (multiple feedback) SPICE test circuit used for stability Figure 38: Alternative (multiple feedback) SPICE test circuit used for stability A��_������ � V� β �

V�� V�

V� 1 � V�� β

A��_������ � β � V��

open loop gain (73) (72) LoadedLoaded open loop gain Feedback (74) (73) Feedback factor factor Closed-loop noise gain (75) (74) Closed-loop noise gain Loop gain (76) (75) Loop gain

Where Where VO = the voltage at the output of the op amp. VO = the voltage at the output of the op amp. VOUT = the voltage output delivered to the load. This may be important to the V output delivered to the load. This may be important to application but is not considered in stability analysis. OUT = the voltage VFB =the application but is not considered in stability analysis. feedback voltage

, RISO and voltage CF = the op amp feedback network. Because there are two paths for RFB F, R V =1feedback feedback, the loop is broken at the input. RF, R1, Riso and CF = the op amp feedback network. Because there are two C and L1 are components that facilitate SPICE analysis. They are large (1TF, 1TH) 1 paths for feedback, the loop is broken at the input. to make the circuit closed loop for dc, but open loop for ac frequencies. SPICE C are components thatatfacilitate SPICE analysis. They are large 1 and L1 requires closed-loop operation dc for convergence. C =(1TF, 1TH) to make circuit closed loop forthe dc,op but open loop for This the equivalent inputthe capacitance taken from amp datasheet. IN ac frequencies. SPICE requires closed-loop operation at dc for capacitance normally does not need to be added because the model includes it. convergence. However, when using this simulation method the capacitance is isolated by C = theinductor. equivalent input capacitance taken from the op amp datasheet. IN 1TH the This capacitance normally does not need to be added because the model includes it. However, when using this simulation method the capacitance is isolated by the 1TH inductor.

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R1

RF +Vs VOUT

+

Voffset +

-Vs

VIN

Volts

VOUT

Voffset

50mVpp

Figure 38: Transient real world stability test

Test tips • Choose test frequency << fcl • Small signal (Vpp ≤ 50 mV) ac output square wave (for example, 1 kHz) • Adjust VIN amplitude to yield output ≤ 50 mVpp • Worst cases is usually when Voffset = 0 (Largest RO, for IOUT = 0A). • Use Voffset as desired to check all output operating points for stability • Set scope = ac couple and expand vertical scope scale to look for amount of overshoot, undershoot, and ringing on VOUT • Use 1x attenuation scope probe on VOUT for best resolution

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+15V RIN1 VIN-

CCM1

1nF

CDIF

10nF

1kΩ RIN2

VIN+

Rg 1kΩ

RG RG

Ref

Out

VOUT

U1 INA333

1kΩ CCM2

1nF

-15V

Figure filter for instrumentation amplifier Figure 40: 40: Input Input filter39: forInput instrumentation amplifier Figure filter for instrumentation amplifier Figure 40: Input filter for instrumentation amplifier Differential filter is sized sized 10 (77) Differential filter is sized 10 is times the filter 10 (76) Differential Select C��� � � 1�C 1�C��� (76) Select C times the common-mode filter ��� ��� common-mode filter Differential filter is sized 10 times the common-mode filter (76) Input resistors must be equal Select C � 1�C��� (77) R �R R��� ��� � ��� times the common-mode filter (77) Input resistors must be equal R ��� ��� (78) Input resistors must be equal Common-mode capacitors (77) Input resistors must be equal R��� RC��� Common-mode capacitors ��� � (78) C � ��� (78) must be equal C��� � C��� Common-mode must be equal capacitors (79) Common-mode capacitors must be equal (78) C��� � C���1 1 must be equal ff �� � � (79) Differential filter cutoff cutoff (79) Differential filter �� 2πR ��� 1 C��� 2πR ��� C��� f�� � (79) Differential filter cutoff (80) Differential filter cutoff 2πR ��� C��� 1 1 � ��� � (80) Common-mode Common-mode filter filter cutoff cutoff ff��� 1 (80) ��C1��� � �1C C����� 2π�2R �����C 2 f��� � 2π�2R ��� ��� ��� (81) Common-mode filter cutoff (80) Common-mode filter cutoff 21 2π�2R ��� ��C��� � C��� � 2 Where Where fWhere DIF = differential cutoff frequency f DIF = differential cutoff frequency Where ff CM = = common-mode common-mode cutoff frequency differentialcutoff cutoff frequency cutoff frequency DIF==differential CM frequency fDIF = common-mode input resistance R IN = fRCM cutoff frequency input resistance IN fCM ==common-mode cutoff frequency C common-mode filter capacitance capacitance CM resistance filter RCM C =input common-mode IN = C = differential filter capacitance DIF RC = =input resistance common-mode capacitance differential filter filter capacitance CM INDIF

= common-mode differential filter filter capacitance DIF= CCCM capacitance Note: Selecting Selecting C CDIF ≥ ≥ 10 10 C CCM sets sets the the differential differential mode mode cutoff cutoff frequency frequency 10 10 times times Note: DIF CM Clower = differential filter capacitance than the common-mode cutoff frequency. This prevents common-mode noise DIF Note: than Selecting CDIF ≥ 10 CCM sets differential mode cutoff common-mode frequency 10 times lower the common-mode cutoffthe frequency. This prevents noise from being converted into differential differential noise due due to to component tolerances. lowerbeing than the common-mode cutoff frequency. This prevents tolerances. common-mode noise from converted into noise component 10 Cdifferential differential cutoff tolerances. frequency 10 times Note: CDIF ≥ into from Selecting being converted due to mode component CM sets thenoise lower than the common-mode cutoff frequency. This prevents common-mode noise from being converted into differential noise due to component tolerances.

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Notes

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Texas Instruments Analog Engineer's Pocket Reference

PCB andWire Wire PCB and

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PCB and wire

PCB trace resistance for 1oz and 2oz Cu • Conductor spacing in a PCB for safe operation • Current carrying capacity of copper conductors • Package types and dimensions • PCB trace capacitance and inductance • PCB via capacitance and inductance • Common coaxial cable specifications • Coaxial cable equations • Resistance per length for wire types • Maximum current for wire types •

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PCB and Wire

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Table 18: Printed circuit board conductor spacing Minimum spacing

PCB and wire

Voltage between conductors (dc or ac peaks)

Bare board

Assembly

B1

B2

B3

B4

A5

A6

A7

0-15

0.05 mm [0.00197 in]

0.1 mm [0.0039 in]

0.1 mm [0.0039 in]

0.05 mm [0.00197 in]

0.13 mm [0.00512 in]

0.13 mm [0.00512 in]

0.13 mm [0.00512 in]

16-30

0.05 mm [0.00197 in]

0.1 mm [0.0039 in]

0.1 mm [0.0039 in]

0.05 mm [0.00197 in]

0.13 mm [0.00512 in]

0.25 mm [0.00984 in]

0.13 mm [0.00512 in]

31-50

0.1 mm [0.0039 in]

0.6 mm [0.024 in]

0.6 mm [0.024 in]

0.13 mm [0.00512 in]

0.13 mm [0.00512 in]

0.4 mm [0.016 in]

0.13 mm [0.00512 in]

51-100

0.1 mm [0.0039 in]

0.6 mm [0.024 in]

1.5 mm [0.0591 in]

0.13 mm [0.00512 in]

0.13 mm [0.00512 in]

0.5 mm [0.020 in]

0.13 mm [0.00512 in]

101-150

0.2 mm [0.0079 in]

0.6 mm [0.024 in]

3.2 mm [0.126 in]

0.4 mm [0.016 in]

0.4 mm [0.016 in]

0.8 mm [0.031 in]

0.4 mm [0.016 in]

151-170

0.2 mm [0.0079 in]

1.25 mm [0.0492 in]

3.2 mm [0.126 in]

0.4 mm [0.016 in]

0.4 mm [0.016 in]

0.8 mm [0.031 in]

0.4 mm [0.016 in]

171-250

0.2 mm [0.0079 in]

1.25 mm [0.0492 in]

6.4 mm [0.252 in]

0.4 mm [0.016 in]

0.4 mm [0.016 in]

0.8 mm [0.031 in]

0.4 mm [0.016 in]

251-300

0.2 mm [0.0079 in]

1.25 mm [0.0492 in]

12.5 mm [0.492 in]

0.4 mm [0.016 in]

0.4 mm [0.016 in]

0.8 mm [0.031 in]

0.8 mm [0.031 in]

301-500

0.25 mm [0.00984 in]

2.5 mm [0.0984 in]

12.5 mm [0.492 in]

0.8 mm [0.031 in]

0.8 mm [0.031 in]

1.5 mm [0.0591 in]

0.8 mm [0.031 in]

B1 Internal conductors B2 External conductors uncoated sea level to 3050m B3 External conductors uncoated above 3050m B4 External conductors coated with permanent polymer coating (any elevation) A5 External conductors with conformal coating over assembly (any elevation) A6 External component lead/termination, uncoated, sea level to 3050m A7 External component lead termination, with conformal coating (any elevation) Extracted with permission from IPC-2221B, Table 6-1. For additional information, the entire specification can be downloaded at www.ipc.org

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PCB and Wire

Figure Self of heating of PCB on inside layer Figure 41: Self 40: heating PCB traces on traces inside layer Example

cause a 20Ԩ temperature rise in a PCB trace that is 0.1 inch Example Find the current that will 2 wide and useswill 2 oz/ft copper. (Assume traces on outside Find the current that cause a 20°C temperature rise in of a PCB.) PCB trace 2 that is 0.1 Answer inch wide and uses 2 oz/ft copper. (Assume traces on outside of PCB.) First translate 0.1 inch to 250 sq. mils. using bottom chart. Next find the current

Answer associated with 10Ԩ and 250 sq. mils. using top chart (Answer = 5A). First translate 0.1 inch to 250 sq. mils. using bottom chart. Next find with permission from IPC-2152, Figure 5-1. the currentExtracted associated with 10°C and 250 sq. mils. using top chart For additional information the entire specification can be downloaded at www.ipc.org (Answer = 5A). Extracted with permission from IPC-2152, Figure 5-1. For additional information the entire specification can be downloaded at www.ipc.org 57 Texas Instruments Analog Engineer's Pocket Reference

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PCBtrace trace resistance for 1 PCB PCB traceresistance resistancefor for11oz ozCu Cu

oz-Cu 5mil 5mil 10mil 10mil 25mil 25mil 50mil 50mil 100mil 100mil

11

100m 100m 10m 10m 1m 1m 100µ 100µ 10µ 10µ 1µ 1µ 11

10 10

100 1000 100 1000 Trace Tracelength length(mils) (mils)

10000 10000

Figure 42: trace resistance vs. and width for oz-Cu, 25°C Figure 41: trace resistance vs. length width for 1 oz-Cu, 25°C Figure 42:PCB PCBPCB trace resistance vs.length length andand width for11 oz-Cu, 25°C

Figure 43: PCB trace resistance vs. and width for 11oz-Cu, 125°C Figure PCB trace resistance vs. length width 1 oz-Cu, 125°C Figure 43:42: PCB trace resistance vs.length length andand width forfor oz-Cu, 125°C Example Example What Example Whatisisthe theresistance resistanceof ofaa20 20mil millong, long,55mil milwide widetrace tracefor foraa11oz-Cu oz-Cuthickness thicknessat at 25°C and 125°C? What is the resistance of a 20 mil long, 5 mil wide trace for a 25°C and 125°C? 1 oz-Cu thickness at 25°C and 125°C? Answer Answer

Answer R25C ==33mΩ. The points are on curves. R25C==222mΩ, mΩ, R125C TheThe points arecircled circled onthe theon curves. R25C mΩ,R125C R125C = mΩ. 3 mΩ. points are circled the curves.

58

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Texas Instruments Analog Engineer's Pocket Reference 58 58

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PCB trace resistance for 2 oz Cu

PCB trace 1 resistance for 2 oz-Cu PCB trace resistance for 2 oz Cu

1 100m 100m 10m 10m 1m

5mil 10mil 25mil 5mil 10mil 50mil 25mil 100mil 50mil 100mil

1m 100µ 100µ 10µ 10µ 1µ

1µ 1 1

10

100 1000 10000 Trace length 100 (mils) 1000 10000 Trace length (mils) Figure 44: PCB trace resistance vs. length and width for 2 oz-Cu, 25°C 10

Figure PCB trace resistance length and width 2 oz-Cu, 25°C Figure 44:43: PCB trace resistance vs.vs. length and width forfor 2 oz-Cu, 25°C

Figure 44: PCBtrace traceresistance resistancevs. vs.length length and and width width for for 2 2 oz-Cu, Figure 45: PCB oz-Cu, 125°C 125°C Figure 45: PCB trace resistance vs. length and width for 2 oz-Cu, 125°C

Example Example What is the resistance of a 200 mil long, 25 mil wide trace for a 2 oz-Cu thickness at Example What is the resistance of a 200 mil long, 25 mil wide trace for a 2 oz-Cu thickness at 25°Cisand What the125°C? resistance of a 200 mil long, 25 mil wide trace for a 25°C and 125°C?

2 oz-Cu thickness at 25°C and 125°C? Answer Answer Answer

R25C 22mΩ, == 3 mΩ. The Thepoints pointsare arecircled circled on R25C= = mΩ,R125C R125C = mΩ. thecurves. curves. =2 mΩ, R125C 33 mΩ. The points are circled ononthe theR25C curves. Texas Instruments Analog Engineer's Pocket Reference

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Common package typetype and dimensions Common package and dimensions

120.2mil 3.05mm

60

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Texas Instruments Analog Engineer's Pocket Reference 60

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PCB parallel plate capacitance PCB parallel plate capacitance Capacitance for parallel copper (82) Capacitance for parallel copper planes (81) planes

k ∙ ℓ ∙ w ∙ εr

PCB parallel plate capacitance Where Where of free space. kk ==Permittivity Permittivity of free space. Both the metric and imperial Capacitance for parallel copper k ∙ ℓ version ∙ w ∙ εofr the constant are given. k = Both 8.854·10-3 pF/mm, or 2.247·10-4 pF/mil the metric and imperial version of (81) the constant are given. planes ℓ =PCB length (metric in mm, or imperial in mil) parallel plate capacitance -4 w = width (metric in mm, or-3 imperial in mil) k = 8.854∙10 pF/mm, or 2.247∙10 pF/mil h = separation between planes (metric in mm, or imperial in mil) PCB relative dielectric constant (εr ≈ 4.5 for FR-4) Where Capacitance for parallel copper ℓε == length (metric k ∙ in ℓ ∙ mm, w ∙ εr or imperial in mil) (81) planes k = Permittivity of free space. w = width (metric in mm, or imperial in mil) r

Both the metric and imperial version of the constant are given.

k = 8.854·10-3 pF/mm, or 2.247·10-4 pF/mil w h= between (metric in mm, or imperial in mil) ℓ = separation length (metric in mm, or imperial inplanes mil) Where w = width (metric in mm, or imperial in mil) εrhk === Permittivity PCB relative dielectric constant ≈ 4.5 for FR-4) separation between planes (metric in mm, or imperial in(ε mil) r of free space. εr = PCB relative dielectric constant (εr ≈ 4.5 for FR-4) Both the metric and imperial version of the constant are given. k = 8.854·10-3 pF/mm, or 2.247·10-4 pF/mil ℓ = length (metric in mm, or imperial in mil) l or imperial in mil) w = width (metric in mm, h = separation between planes (metric in mm, or imperial in mil) εr = PCB relative dielectric constant (εr ≈ 4.5 for FR-4)

h

A

w

w

A

l

εr

A Figure 45: PCBl parallel plate capacitance h Example Calculate the total capacitance for ℓ=5.08mm, w=12.7mm, h=1.575mm, h Figure 45: PCB parallel plate capacitance

εr = 4.5

εr εr

(8.854 ∙ 10–3 pF⁄mm) ∙ (5.08mm) ∙ (12.7mm) ∙ (4.5) Example Calculate the total capacitance for ℓ=5.08mm, C(pF) = 45: PCB parallel plate capacitance = 1.63pF Figure 1.575mm w=12.7mm, h=1.575mm, ε = 4.5

r

Example Calculate –3 the total capacitance for ℓ=5.08mm, (8.854 ∙ 10

pF⁄mm) ∙ (5.08mm) ∙ (12.7mm) ∙ (4.5)

C(pF) = Calculate = 1.63pF w=12.7mm, h=1.575mm, εr = 4.5 Example the1.575mm total capacitance for ℓ=200mil,

w=500mil, h=62mil, εr = 4.5

(8.854 ∙ 10–3 pF⁄mm) ∙ (5.08mm) ∙ (12.7mm) ∙ (4.5)

C(pF) = = 1.63pF –4 C(pF) = (2.247 ∙ 10 pF⁄mil) ∙ (200mil) ∙ (500mil) (4.5) Example Calculate the1.575mm total capacitance for ∙ℓ=200mil, = 1.63pF 62mil ε = 4.5 w=500mil, h=62mil,

r

Example C(pF) = (2.247 ∙ 10

–4 Calculate

the total capacitance for∙ (4.5) ℓ=200mil, pF⁄mil) ∙ (200mil) ∙ (500mil) = 1.63pF

w=500mil, h=62mil, 62mil εr = 4.5 –4

C(pF) = (2.247 ∙ 10

pF⁄mil) ∙ (200mil) ∙ (500mil) ∙ (4.5) 62mil

Texas Instruments Analog Engineer's Pocket Reference 61

= 1.63pF

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Microstrip capacitance and inductance L(nH) = kL ∙ ℓ ∙ ln C(pF) =

( 0.85.98∙ w∙ +h t (

kC ∙ ℓ ∙ (εr + 1.41) ln

5.98 ∙ h ( 0.8 ∙w+t (

(83) Inductance for microstrip

(84) Capacitance for microstrip

Where kL = PCB inductance per unit length. Both the metric and imperial version of the constant are given. kL = 2nH/cm, or 5.071nH/in kC = PCB capacitance per unit length. Both the metric and imperial version of the constant are given. kC = 0.264pF/cm, or 0.67056pF/in ℓ = length of microstrip (metric in cm, or imperial in inches) w = width of microstrip (metric in mm, or imperial in mil) t = thickness of copper (metric in mm, or imperial in mil) h = separation between planes (metric in mm, or imperial in mil)

εr = relative permittivity, approximately 4.5 for FR-4 PCB

For imperial: Copper thickness (mils) = 1.37 • (number of ounces) i.e. 1oz Cu = 1.37mils i.e. ½oz Cu = 0.684mils

ℓ t

W

h Figure 46: PCB Microstrip capacitance and inductance

Example Calculate the total inductance and capacitance for ℓ=2.54cm, w=0.254mm, t=0.0356mm, h=0.8mm, εr = 4.5 for FR-4 L(pF) = (2 nH⁄cm) ∙ (2.54cm) ∙ ln

(

5.98 ∙ 0.8mm

0.8 ∙ 0.254mm + 0.0356mm

C(pF) = (0.264pF/cm) ∙ (2.54cm)(4.5 + 1.41) = 1.3pF 5.98 ∙ 0.8mm ln ( ) 0.8 ∙ 0.254mm + 0.0356mm

) = 15.2nH

Example Calculate the total inductance and capacitance for ℓ=1in, w=10mil, t=1.4mil, h=31.5mil, εr = 4.5 for FR-4 L = 15.2nH, C=1.3pF. Note: this is the same problem as above with imperial units. 62

Texas

Texas Instruments Analog Engineer's Pocket Reference

Where l = length of copper trace (mils) t = thickness of copper trace (mils) copper thickness (mils) = 1.37 * (number of ounces) PCB ti.com/precisionlabs ex: 1 oz. copper thickness = 1.37 mils ex: ½ oz. copper thickness = 0.685 mils d = distance Adjacent copperbetween tracestraces (mils)

and Wire

r =∙ tPCB r ≈ 4.2 for FR-4) k ∙ ℓ dielectric constant ((85) Same layer C(pF) ≈ w = width of copper trace (mils) d h = separation between planes (mils)

εr ∙ w ∙ ℓ C(pF) ≈ Example h k∙

(86) Different layers

l = 100 mils

Where t = 1.37 mils (1 oz. copper) ℓ = length themils copper trace (mil, or mm) d of = 10 -3 εr = 4.2 k = 8.854*10 pF/mm, or k=2.247*10-4 pF/mil w = 25 t = thickness of mils trace (in mil, or mm) h = 63 mils d = distance between traces if on same layer (mil, or mm)

For imperial: Copper thickness (mils) = 1.37 • (number of ounces)

w = widthAnswer of trace. (mil, or mm)

h = separation between planes. (mil, or mm) C (same layer) = 0.003 pF

i.e. 1oz Cu = 1.37mils i.e. ½oz Cu = 0.684mils

εr = PCBCdielectric constant (εr = 4.5 for FR-4) (different layers) = 0.037 pF

Figure 47: Capacitance for adjacent copper traces Figure 48: Capacitance for adjacent copper traces

Example: Calculate the total capacitance for both cases: ℓ=2.54mm, t=0.0348mm, d=0.254mm, w=0.635mm, h=1.6mm, εr = 4.5 for FR-4 C(pF) ≈

(8.854 ∙ 10–3 pF/mm) (0.0348mm) (2.54mm) 63 0.254mm

= 0.0031pF Same layer

(8.854 ∙ 10–3 pF/mm) (4.5mm) (0.635mm) (2.54mm) = 0.04pF 1.6mm Adjacent layers Example: Calculate the total capacitance for both cases: ℓ=100mil, t=1.37mil, d=10mil, w=25mil, h=63mil, εr = 4.5 for FR-4 C(pF) ≈

C = 0.0031pF (Same layer), C=0.4pF (Adjacent layers). Note: this is the same problem as above with imperial units. Texas Instruments Analog Engineer's Pocket Reference

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PCB via capacitance and inductance

[

L(nH) ≈ kL ∙ h 1 + ln C(pF) ≈

(4hd )]

(87) Inductance for via

kC ∙ εr ∙ h ∙ d1 d2 — d1

(88) Capacitance for via

Where kL = PCB inductance per unit length. Both the metric and imperial version of the constant are given. kL = 0.2nH/mm, or 5.076∙10-3nH/mil kC = PCB capacitance per unit length. Both the metric and imperial version of the constant are given. kC = 0.0555pF/mm, or 1.41∙10-3pF/mil h=

separation between planes

d=

diameter of via hole

d1 = diameter of the pad surrounding the via d2 = distance to inner layer ground plane.

εr =

PCB dielectric constant (εr = 4.5 for FR-4) d1 d

Top Layer

Trace Middle Layer GND Plane

h

d2 Bottom Layer Trace

Figure 48: Inductance and capacitance of via

Example: Calculate the total inductance and capacitance for h=1.6mm, d=0.4mm, d1=0.8mm, d2=1.5mm

[

L(nH) ≈ (0.2 nH⁄mm) ∙ (1.6mm) 1 + ln C(pF) ≈

∙ 1.6mm (40.4mm )] = 1.2nH

(0.0555pF/mm) ∙ (4.5) ∙ (1.6mm) ∙ (0.8mm) 1.5mm — 0.8mm

= 0.46pF

Example: Calculate the total inductance and capacitance for h=63mil, d=15.8mil, d1=31.5mil, d2=59mil L=1.2nH, C=0.46pF. Note: this is the same problem as above with imperial units.

64

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Texas Instruments Analog Engineer's Pocket Reference

PCB and Wire

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Type

ZO

Capacitance / length (pF/feet)

Outside diameter (inches)

dB attenuation /100 ft at 750 MHz

Dielectric type

Table 19: Coaxial cable information

RG-58

53.5Ω

28.8

0.195

13.1

PE

Application Test equipment and RF power to a few hundred watts, and a couple hundred MHz

RG-8

52Ω

29.6

0.405

5.96

PE

RG-214/U

50Ω

30.8

0.425

6.7

PE

9914

50Ω

26.0

0.405

4.0

PE

RG-6

75Ω

20

0.270

5.6

PF

RG-59/U

73Ω

29

0.242

9.7

PE

RG-11/U

75Ω

17

0.412

3.65

PE

RF power to a few kW, up to several hundred MHz

RG-62/U

93Ω

13.5

0.242

7.1

ASP

Used in some test equipment and 100Ω video applications

RG-174

50Ω

31

0.100

23.5

PE

RG-178/U

50Ω

29

0.071

42.7

ST

Texas Instruments Analog Engineer's Pocket Reference

RF power to a few kW, up to several hundred MHz Video and CATV applications. RF to a few hundred watts, up to a few hundred MHz, sometimes to higher frequencies if losses can be tolerated

Miniature coax used primarily for test equipment interconnection. Usually short runs due to higher loss.

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Coaxial cable equations Coaxial cable equations C ℓ

L ℓ

2πε D Coaxial cable equations d

μC ℓ 2π

D2πε d D d

LL μ1 μD 2π εd ℓ C 2π

Capacitance per length (89) per length (84)Capacitance

per length (90) per length (85)Inductance Capacitance per length (84) Inductance

impedance (91) impedance (86)Characteristic Inductance per length (85) Characteristic

L 1 μ Characteristic impedance (86) Where Where C 2π ε inductance in in henries henries (H) (H) LL==inductance C = capacitance in farads (F) C = capacitance in farads (F) Z = impedance in ohms (Ω) Where Zd == impedance ohms (Ω) diameter of in inner conductor L = inductance in henries (H) D = inside diameter of shield, or diameter of dielectric insulator d = diameter of inner conductor C = capacitance in farads (F) ε = dielectric constant of insulator (ε = εr εo ) D = inside diameter ofinshield, Z = impedance ohms or (Ω)diameter of dielectric insulator µ = magnetic permeability (µ = µr µo ) d = diameter of inner conductor εl = length dielectric constant of the cable of insulator (ε = εr εo ) D = inside diameter of shield, or diameter of dielectric insulator μ = magnetic permeability (μof = insulator μr μo ) (ε = εr εo ) ε = dielectric constant µ = magnetic permeability (µ = µr µo ) ℓ = length of the cable l = length of the cable

Insulation

Figure 49: Coaxial cable cutaway

Figure 49: Coaxial cable cutaway Figure 49: Coaxial cable cutaway

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Texas Instruments Analog Engineer's Pocket Reference 65

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Table 20: Resistance per length for different wire types (AWG) AWG

Stds

36 36

Outside diameter

Area

dc resistance

in

mm

circular mils

mm2

â„Ś / 1000 ft

â„Ś / km

Solid

0.005

0.127

25

0.013

445

1460

7/44

0.006

0.152

28

0.014

371

1271

34

Solid

0.0063

0.160

39.7

0.020

280

918

34

7/42

0.0075

0.192

43.8

0.022

237

777

32

Solid

0.008

0.203

67.3

0.032

174

571

32

7/40

0.008

0.203

67.3

0.034

164

538

30

Solid

0.010

0.254

100

0.051

113

365

30

7/38

0.012

0.305

112

0.057

103

339

28

Solid

0.013

0.330

159

0.080

70.8

232

28

7/36

0.015

0.381

175

0.090

64.9

213

26

Solid

0.016

0.409

256

0.128

43.6

143

26

10/36

0.021

0.533

250

0.128

41.5

137

24

Solid

0.020

0.511

404

0.205

27.3

89.4

24

7/32

0.024

0.610

448

0.229

23.3

76.4

22

Solid

0.025

0.643

640

0.324

16.8

55.3

22

7/30

0.030

0.762

700

0.357

14.7

48.4

20

Solid

0.032

0.813

1020

0.519

10.5

34.6

20

7/28

0.038

0.965

1111

0.562

10.3

33.8

18

Solid

0.040

1.020

1620

0.823

6.6

21.8

18

7/26

0.048

1.219

1770

0.902

5.9

19.2

16

Solid

0.051

1.290

2580

1.310

4.2

13.7

16

7/24

0.060

1.524

2828

1.442

3.7

12.0

14

Solid

0.064

1.630

4110

2.080

2.6

8.6

14

7/22

0.073

1.854

4480

2.285

2.3

7.6

Texas Instruments Analog Engineer's Pocket Reference

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Polypropylene Polyethylene (high density) at 90°C

Polyvinylchloride Nylon at 105°C

Imax (A)

Imax (A)

Imax (A)

Imax (A)

Kapton Teflon Silicon at 200°C

Polyethylene Neoprene Polyvinylchloride (semi-ridged) at 80°C

AWG

Kynar Polyethylene Thermoplastic at 125°C

Wire gauge

Table 21: Maximum current vs. AWG

Imax (A)

30

2

3

3

3

4

28

3

4

4

5

6

26

4

5

5

6

7

24

6

7

7

8

10

22

8

9

10

11

13

20

10

12

13

14

17

18

15

17

18

20

24

16

19

22

24

26

32

14

27

30

33

40

45

12

36

40

45

50

55

10

47

55

58

70

75

Note: Wire is in free air at 25°C

Example What is the maximum current that can be applied to a 30 gauge Teflon wire in a room temperature environment? What will the self-heating be? Answer Imax = 4A Wire temperature = 200°C

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Texas Instruments Analog Engineer's Pocket Reference

Sensor

Sensor

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Sensor

Thermistor • Resistive temperature detector (RTD) • Diode temperature characteristics• Thermocouple (J and K) •

69

Texas Instruments Analog Engineer's Pocket Reference

70

Pocketreference_6815_final.indd 70

–55°C < T < 150°C

Low LowLow Low

Accuracy

Less accurate over full range

Very nonlinear. Follows reciprocal |of logarithmic function

Less rugged

General

Applications General General General General

Requires excitation

General purpose Requires excitation Requires excitation Requires excitation Requires excitation

Very wide variation in resistance Applications General purpose General purpose Applications General purpose Applications General purpose Applications

range

Low LowLow Low

Low

polynomial correction

180 to 3.9 kΩ for PT1000

Requires excitation

Scientific and industrial Requires excitation Requires excitation Requires excitation Requires excitation

Scientific industrial Scientific andindustrial industrial Scientific and and industrial Scientific and

Industrial temperature

Requires excitation

Self-powered Requires cold junction comp

Requires Self-powered Requires excitation Self-powered Requires excitation Requires excitation Self-powered Lowexcitation cost linear response Self-powered measurement Requires junction comp Requires cold junction comp Requires coldcold junction comp Requires cold junction comp

Low cost temperature monitor

10s of millivolts

0.4 to 0.8V

cost temperature monitor Industrial temperature Low cost temperature monitor Industrial Industrial temperature LowLow cost temperature monitor temperature Low cost temperature monitor Industrial temperature cost linear response measurement Low cost linear response measurement LowLow cost linear response measurement Low cost linear response measurement

of millivolts 10s millivolts 10s 10s of millivolts 10s ofofmillivolts

Most rugged

0.40.8V to0.4 0.8V 0.8V 0.4 to 0.4 toto0.8V

Complex 10th order polynomial

rugged Most rugged MostMost rugged Most rugged

Rugged

diode processing

Rugged Rugged Rugged Rugged

Depends on Type (can be rugged)

Depends on on Type (can be bebe Depends on Type (can Depends on Type (can be(can Depends Type rugged) rugged) rugged) rugged)

Relatively simple quadratic function

Poor accuracy without calibration

accuracy without calibration. Good accuracy polynomial Poor accuracy without calibration. Good accuracy with polynomial PoorPoor accuracy without calibration. Good accuracy withwith polynomial Poor accuracy without calibration. Good accuracy with polynomial correction. correction. correction. correction. Good accuracy with

Low

–250°C T< <1800°C –250°C <–250°C T<T 1800°C –250°C <<<T1800°C –250°C <1800°C T < 1800°C

Thermocouple Thermocouple Thermocouple Thermocouple Thermocouple

Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Fairly linear Nonlinearity < 4.5% of scale. full scale. Slope ≈ -2mV/C Nonlinearity < 10% of scale full scale Slope -2mV/C Nonlinearity 10% fullscale scale Nonlinearity 4.5% fullscale. scale. Slope ≈Slope -2mV/C < 10% full Nonlinearity < 4.5% full Nonlinearity <of<4.5% ofoffull ≈ ≈linear -2mV/C Nonlinearity <of<10% ofoffull Fairly Slope ≈ -2mV/CNonlinearity Fairly linear Fairly linear Relatively simple quadratic Slope varies according to current Complex 10th order polynomial Slope variesaccording according tocurrent current Complex 10th order polynomial Relatively simplequadratic quadratic Slope varies according to current Complex 10th order polynomial Relatively simple simple quadratic Relatively Slope varies toto Complex 10th order polynomial Slope varies according current function. excitation, diode type, and diode function. excitation, diode type, and diode excitation, diode type, and diode function. function. excitation, diode type, and diode Nonlinearity < 10% of full scale Nonlinearity < 4.5% of full scale excitation, diode type, and processing. processing. processing. processing.

Output range Typically to 100s of full kΩ full to18 390 Ω for PT100 Typically 10s to100s 100s kΩfull fullto18390 390 forPT100 PT100 Output range 18 Ωtofor PT100 Typically 10s 10s to 100s kΩ Output range Typically 10s toof ofofkΩ 18 to 390 ΩΩfor Output range Typically 10s toVery 100s ofvariation kΩ scale. Very wide variation to 3.9 kΩ for 180 3.9 kΩfor forPT1000 PT1000 scale. wide variation scale. Very wide variation in in inin180 180 to 180 3.9 kΩ for PT1000 to kΩ scale. Very wide Output 18 toto3.9 390 ΩPT1000 for PT100 resistance. resistance. resistance. resistance. full scale

Construction

High

Low LowLow Low

Diode

–55°C T< <150°C –55°C < –55°C T<T 150°C –55°C <<<T150°C –55°C <150°C T < 150°C

Diode Diode Diode Diode

Table 22: Temperature sensor overview Table 21: Temperature sensor overview Table 21: Temperature sensor overview Table 21: Temperature sensor overview Table 21: Temperature sensor overview

rugged Construction Less rugged Construction LessLess rugged Construction Less rugged Construction

Linearity

nonlinear. Follows Linearity Very nonlinear. Follows Linearity VeryVery nonlinear. Follows Linearity Very nonlinear. Follows Linearity reciprocal of logarithmic reciprocal logarithmic reciprocal of logarithmic reciprocal ofoflogarithmic function. function. function. function.

Accuracy

High HighHigh High

RTD

–200°C 850°C –200°C <–200°C T<T 850°C –200°C <<<T850°C –200°C <T<T<850°C < 850°C

RTD RTDRTD RTD

Excellent accuracy Excellent accuracy Excellent accuracy Excellent accuracy Good accuracy at one temperature. temperature. temperature. Goodtemperature. accuracy ataccurate one temperature accurate full full range. Less over fullrange. range. LessLess accurate overover fullover range. Less accurate Excellent accuracy

CostAccuracy Low Good accuracy at one Accuracy Good accuracy at one AccuracyGood accuracy at one

Cost CostCost Cost

<150°C 150°C Temprange range –55°C –55°C < –55°C T < 150°C Temp range T T<150°C Temp Temp range –55°C < T< <<

Temp range

Thermistor Thermistor Thermistor Thermistor Thermistor

Sensors

Sensor ti.com/precisionlabs

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6/11/15 3:08 PM

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Thermistor: Resistance to temperature, Steinhart-Hart equation Thermistor: Resistance to temperature, Steinhart-Hart equation 1 Convert resistance to (87) R to temperature, (92) Convert Steinhart-Hart resistance to temperature a thermistor Thermistor: RResistance equation temperature for for a thermistor T

1 Convert resistance to Where (87) R R Where temperature for a thermistor T T = temperature in Kelvin T = temperature in Kelvin a, b, c = Steinhart-Hart equation constants a,Where b, c = Steinhart-Hart equation constants R = resistance in ohms temperature in Kelvin RT==resistance in ohms a, b, c = Steinhart-Hart equation constants Steinhart-Hart equation Thermistor: Temperature to resistance, R = resistance in ohms

[ [

[ [

Thermistor:x Temperature to resistance, Steinhart-Hart equation Convert temperature to x (88) y Temperature y+ Thermistor: to resistance, Steinhart-Hart equation resistance for a thermistor 2 2 x x y y

c

1y T

1 T cb 3c b 3c

x 2

y+

x 4 x 4

x 2

Convert temperature to (88) (93) Convert to resistance resistance forinaEquation thermistor Factor used 88 (89)temperature for a thermistor Factor used93 in Equation 88 (94) Factor(89) used in Equation Factor used in Equation 88 (90)

(95) Factor(90) used in Equation Factor used93 in Equation 88

Where R = resistance in ohms T = temperature in Kelvin Where Where a, b, c = Steinhart-Hart equation constants R== resistance Rx, inin立ohms yresistance = Steinhart-Hart factors used in temperature to resistance equation T = temperature Kelvin T = temperature ininKelvin a, b, c = Steinhart-Hart equation constants a, b, c = Steinhart-Hart equation constants x, y = Steinhart-Hart factors used in temperature to resistance equation x, y = Steinhart-Hart factors used in temperature to resistance equation

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RTD equation temperature to resistance RTD equation temperature to resistance T R 0 0 0T 0 RTD equation temperature to resistance R

0

0

0T

0

T

RTD resistance for (96) RTD resistance for T<0°C (91) T<0°C RTD resistance for (97) RTD resistance for T>0°C (92) (91) T>0°C T<0°C

Where RTD resistance for Where R 0 0 of RTD 0 T over temperature range of (–200°C (92) Rrtd = resistance < T < 850°C) T>0°C Rrtd = resistance of RTD over temperature range of (–200°C < T < 850°C) Ro = 100 Ω for PT-100, 1000Ω for PT-1000 RWhere for PT-100, 1000Ω for PT-1000 A0O= , B100Ω O, CO = Callendar-Van Dusen coefficients Rrtd = resistance of RTD over temperature range of (–200°C < T < 850°C) in degreesDusen Celsius ( ) AT0,=Btemperature coefficients 0, C0 = Callendar-Van Ro = 100 Ω for PT-100, 1000Ω for PT-1000 TA=O,temperature in degrees Celsius (°C) BO, CO = Callendar-Van Dusen coefficients RTD equation resistance to temperature (T>0°C) T = temperature in degrees Celsius ( ) R RTD resistance A RTD equation resistance to temperature RTD equation resistance toRtemperature (T>0°C) (T>0°C) (93) 0 for T>0°C 2B R RTD A (98) RTD resistance for resistance T>0°C Where R0 (93) for T>0°C RRTD = resistance2B of RTD over temperature range of (–200°C < T < 850°C) Ro = 100 Ω Where AO, BO, CO = Callendar-Van Dusen coefficients Where RRTD = resistance of RTD over temperature range of (–200°C < T < 850°C) = temperature Celsius ( ) RTRTD = resistanceinofdegrees RTD over temperature range of (–200°C < T < 850°C) Ro = 100 Ω RA0O= , B100Ω , C = Callendar-Van Dusen coefficients TableO22:OCallendar-Van Dusen coefficients for different RTD standards T = temperature in degreesDusen Celsius ( ) A0, B0, C0 = Callendar-Van coefficients IEC-751 T = temperature in degrees Celsius (°C) US Industrial 43760 Table 22: DIN Callendar-Van Dusen coefficients for different RTD standards US Industrial Standard BS 1904 IEC-751 Standard D-100 ASTM-E1137 US Industrial Table 23: Callendar-Van Dusen coefficients for different RTD standardsITS-90 DIN 43760 American EN-60751 JISC 1604 American US Industrial Standard BS 1904 A0 +3.9083E-3 +3.9787E-3 IEC-751 DIN 43760 +3.9739E-3 US Industrial +3.9692E-3 US Industrial +3.9888E-3 Standard D-100 ASTM-E1137 1904 ASTM-E1137 –5.870E-7 Standard –5.8495E-7 Standard –5.915E-7 B0 BS –5.775E-7 –5.8686E-7 American EN-60751 JISC 1604 American ITS-90 EN-60751 JISC 1604 D-100 American American ITS-90 C0 –4.183E-12 –4.4E-12 –4.167E-12 –4.233E-12 –3.85E-12 A0 +3.9083E-3 +3.9739E-3 +3.9787E-3 +3.9692E-3 +3.9888E-3 +3.9083E-3 +3.9739E-3 +3.9787E-3 +3.9692E-3 +3.9888E-3 A0 B0 –5.775E-7 –5.870E-7 –5.8686E-7 –5.8495E-7 –5.915E-7 –5.775E-7 –5.870E-7 –5.8686E-7 –5.8495E-7 –5.915E-7 B0 C0 –4.183E-12 –4.4E-12 –4.167E-12 –4.233E-12 –3.85E-12 Example –4.183E-12 –4.4E-12 –4.167E-12 –4.233E-12 –3.85E-12 C 0

What is the temperature given an ITS-90 PT100 resistance of 120 Ω? Example Answer What is the temperature given an ITS-90 PT100 resistance of 120 Ω? Example 120 What is the temperature3.9888 given∙ an 10 ITS-90 PT100 resistance of 120Ω? 100 Answer

Answer

3.9888 ∙ 10

72

120 120 100 100

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RTD equation resistance to temperature RTD equation resistance to temperature (T<0°C) (T<0°C) �

� � � �� �� ��� ��

(99) RTD resistance for T<0°C (94)

���

RTD resistance for T<0�

Where Where (�) T = temperature in degrees Celsius (°C) RRTD = resistance of RTD over temperature range of (T<0�) RRTD = resistance of RTD over temperature range of (T<0°C) αI = polynomial coefficients for converting RTD resistance to temperature for T<0� αi = polynomial coefficients for converting RTD resistance to temperature for T<0°C th

Table 23: Coefficients for 5 order RTD resistance to temperature IEC-751 Table 24:DIN Coefficients for 5th orderUS RTD resistance to temperature Industrial 43760 US Industrial Standard BS 1904 IEC-751 Standard D-100 ASTM-E1137 DIN 43760 American EN-60751 JISC 1604 American ITS-90 BS 1904 US Industrial α0 –2.4202E+02 –2.3864E+02 ASTM-E1137 –2.3820E+02 –2.3818E+02 Standard US Industrial –2.3791E+02 EN-60751 JISC 1604 D-100 American Standard American ITS-90 α1 2.2228E+00 2.1898E+00 2.1956E+00 2.1973E+00 2.2011E+00

αα2 0

–2.4202E+02 2.5857E-03 –2.3820E+02 2.5226E-03

–2.3818E+02 2.4413E-03

–2.3864E+02 2.3223E-03 –2.3791E+02 2.4802E-03

αα3 1

–4.8266E-06 –4.7825E-06 2.2228E+00 2.1898E+00

–4.7517E-06 2.1956E+00

–4.7791E-06 2.1973E+00 –4.6280E-06 2.2011E+00

αα4 2 αα5

–2.8152E-08 –2.7009E-08 2.5857E-03 2.5226E-03 1.5224E-10 1.4719E-10 –4.8266E-06 –4.7825E-06

–2.3831E-08 2.4413E-03 1.3492E-10 –4.7517E-06

–2.5157E-08 2.4802E-03 –1.9702E-08 2.3223E-03 1.4020E-10 1.1831E-10 –4.7791E-06 –4.6280E-06

α4

–2.8152E-08

–2.7009E-08

–2.3831E-08

–2.5157E-08

–1.9702E-08

1.5224E-10 α5 Example

1.4719E-10

1.3492E-10

1.4020E-10

1.1831E-10

3

Find the temperature given an ITS-90 PT100 resistance of 60 Ω. Answer

Example � � ��������� � 0�� ∗ �60�� � ����0��� � 00� ∗ �60�� � �������� � 0�� ∗ �60�� � � � Find the temperature given an ITS-90 resistance of 60 Ω. � �������� � 0�� ∗ �60�PT100 � ����6� Answer

•

60

• •

60

Texas Instruments Analog Engineer's Pocket Reference 73

60

•

60

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Diode equation vs. temperature Diode equation vs. temperature V� �

nkT I nkT I �n � � �� � �n � � q I� q I�

(100) Diode(95) voltage Diode voltage

Diode equation vs. temperature Where Where vs. nkT temperature and current VD = diode nkT voltage I I VV diode�n voltage vs.�temperature and1 current � � �� �n � �from D�= n =�diode factor (ranges to 2) q ideality I-23 q I � � 1.38 xideality 10 J/K, Boltzmann’s constant n k==diode factor (ranges from 1 to 2) T = temperature in Kelvin k Where = 1.38 x 10-23 -19 J/K, Boltzmann’s constant q x 10 C, charge of an electron diode voltage vs. temperature and current VD= =1.60 TI==temperature in current Kelvin in amps forward diode n = diode ideality factor (ranges from 1 to 2) -19 -23 C, charge saturation of an electron qkIS===1.60 xx10 1.38 10 current J/K, Boltzmann’s constant

(95)

Diode voltage

T forward = temperature in Kelvinin amps I= diode current qV� -19 ⁄�� ��x q =�1.60 10 C, charge � of an electron (96) Saturation current �T ��� �� I � IS = saturation current nkT I = forward diode current in amps IS = saturation current Where IS = saturation current qV� Saturation current �T ��⁄�� ��� �� to�the cross sectional area (96) current I� =�constant α related of the junction (101) Saturation nkT VG = diode voltage vs. temperature and current n = diode ideality factor (ranges from 1 to 2) Where -23 x 10 current J/K, Boltzmann’s constant IkS == 1.38 saturation Where T= = constant temperature in Kelvin α related IS = saturation-19 currentto the cross sectional area of the junction q = =1.60 x 10 C, charge of an electron V diode voltage vs. temperature and current G α = constant related to the cross sectional area of the junction n = diode ideality factor (ranges from 1 to 2) -23 VkG == 1.38 diodex voltage vs.Boltzmann’s temperatureconstant and current 10 J/K, temperature in Kelvin n T= =diode ideality factor (ranges from 1 to 2) -19 q = 1.60 x 10-23 C, charge of an electron k = 1.38 x 10 J/K, Boltzmann’s constant

T = temperature in Kelvin q = 1.60 x 10-19 C, charge of an electron

74

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Diode voltage versus temperature Figure 50 shows an example of the temperature drift for a diode. Depending on the characteristics of the diode and the forward current versus temperature the Diode slopevoltage and offset of this curve will change. However, typical diode drift is about –2mV/°C. A forward drop of about 0.6V is typical for Figure 50 shows an example of the temperature drift for a diode. Depending on the room temperature. characteristics of the diode and the forward current the slope and offset of this curve will change. However, typical diode drift is about –2mV/°C. A forward drop of about 0.6V is typical for room temperature.

Figure 50: Diode voltage vs. temperature Figuredrop 50: Diode voltage drop vs. temperature

75

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Type J thermocouples translating temperature to voltage (ITS-90 standard)

Type J thermocouples translating temperature to voltage (ITS-90 standard) �

V� � � �� ����

(102) Thermoelectric voltage

���

(97)

Thermoelectric voltage

Where Where T = thermoelectric voltage VV T = thermoelectric voltage T = temperature in degrees Celsius T = temperature in degrees Celsius ci = translation coefficients ci = translation coefficients Table 24: Type J thermocouple temperature to voltage coefficients Type J thermocouple temperature to voltage Table 25: Type J thermocouple temperature to voltage coefficients –219� to–219°C 760�to 760°C 760� to 1,200� c0 c0 c1 c1

c2 c2 c3 c3 c4 c4 c5 c5 c6 c c76 c c87

c8

760°C to 1,200°C

0.0000000000E+00 2.9645625681E+05 2.9645625681E+05 0.0000000000E+00 5.0381187815E+01 –1.4976127786E+03 5.0381187815E+01 –1.4976127786E+03 3.0475836930E-02 3.1787103924E+00 3.1787103924E+00 3.0475836930E-02

–8.5681065720E-05 –3.1847686701E-03 –8.5681065720E-05 1.3228195295E-07 1.5720819004E-06 1.3228195295E-07 –1.7052958337E-10 –3.0691369056E-10 –1.7052958337E-10 2.0948090697E-13 -2.0948090697E-13 –1.2538395336E-16 -–1.2538395336E-16 1.5631725697E-20 --

1.5631725697E-20

76

–3.1847686701E-03 1.5720819004E-06 –3.0691369056E-10 — — —

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Type J thermocouples translating voltage to temperature (ITS-90 standard) Type J thermocouples translating voltage to temperature (ITS-90 standard) �

� � � �� �V� ��

(103) Temperature

(98)

Temperature

���

Table 25: Type J thermocouple voltage to temperature coefficients Table 26: Type J thermocouple voltage to temperature coefficients Type J thermocouple voltage to temperature –219°C to 0°C 0°C to 760°C 760°C to 1,200°C –219°C to 0°C 0°C to 760°C 760°C to 1,200°C 0.000000000E+00 0.000000000E+00 –3.113581870E+03 c0 c0 0.000000000E+00 0.000000000E+00 –3.113581870E+03 1.952826800E-02 1.978425000E-02 3.005436840E-01 c1 c1 1.952826800E-02 1.978425000E-02 3.005436840E-01 –1.228618500E-06 –2.001204000E-07 –9.947732300E-06 c2 c2 –1.228618500E-06 –2.001204000E-07 –9.947732300E-06 c3 –1.075217800E-09 1.036969000E-11 1.702766300E-10 c3 –1.075217800E-09 1.036969000E-11 1.702766300E-10 –5.908693300E-13 –2.549687000E-16 –1.430334680E-15 c4 c4 –5.908693300E-13 –2.549687000E-16 –1.430334680E-15 –1.725671300E-16 3.585153000E-21 4.738860840E-21 cc5 5 –1.725671300E-16 3.585153000E-21 4.738860840E-21

cc6 6 cc7 7

–2.813151300E-20 –2.813151300E-20 –2.396337000E-24 –2.396337000E-24

cc8 8

–8.382332100E-29 –8.382332100E-29

–5.344285000E-26 –5.344285000E-26 5.099890000E-31 5.099890000E-31 -- —

Texas Instruments Analog Engineer's Pocket Reference

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—

--

—

--

—

77

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Type K thermocouples translating temperature to voltage (ITS-90 standard) translating temperature to voltage (ITS-90 standard) Type K thermocouples �

Thermoelectric voltage (99) (104) Thermoelectric voltage for T<0°C for T<0�

V� � � �� ���� ��� �

�

V� � �� �� ���� � � �� e��� ������������� ���

(105) Thermoelectric voltage forT>0°C voltage (100) Thermoelectric forT>0�

Where VT = thermoelectric voltage Where in degrees VTT==temperature thermoelectric voltage Celsius translation coefficients Tci == temperature in degrees Celsius α0, α1 = translation coefficients ci = translation coefficients 26: Type K thermocouple temperature to voltage coefficients αTable 0, α1 = translation coefficients –219°C to 760°C

760°C to 1,200°C

0.0000000000E+00 –1.7600413686E+01 c0 27: Type Table K thermocouple temperature to voltage coefficients 3.9450128025E+01 3.8921204975E+01 c1 –219°C to 760°C 760°C to 1,200°C 2.3622373598E-02 1.8558770032E-02 c2 0.0000000000E+00 –1.7600413686E+01 c0 –3.2858906784E-04 –9.9457592874E-05 c 3

cc41

3.9450128025E+01 3.1840945719E-07 3.8921204975E+01 –4.9904828777E-06 2.3622373598E-02 –5.6072844889E-10 1.8558770032E-02 –6.7509059173E-08

cc6

–5.7410327428E-10 –3.2858906784E-04 5.6075059059E-13 –9.9457592874E-05 –3.1088872894E-12 –3.2020720003E-16 –4.9904828777E-06 3.1840945719E-07 –1.0451609365E-14 9.7151147152E-20 –6.7509059173E-08 –5.6072844889E-10 –1.9889266878E-17 –1.2104721275E-23 –5.7410327428E-10 5.6075059059E-13 –1.6322697486E-20 --

cc52

3

c7

c4

c8

c

c9 5

cc 106 αc07 αc1 8

78

–3.1088872894E-12 -1.1859760000E+02 –3.2020720003E-16 -–1.1834320000E-04 9.7151147152E-20 –1.0451609365E-14

c9

–1.9889266878E-17

–1.2104721275E-23

c10

–1.6322697486E-20

—

α0

—

1.1859760000E+02

α1

—

–1.1834320000E-04

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Type K thermocouples translating voltage to temperature (ITS-90 standard)

Type K thermocouples translating voltage to temperature (ITS-90 standard) �

(106) Temperature

� � � �� �V� ��

(101)

Temperature

���

Table 27: Type K thermocouple voltage to temperature coefficients

Table 28: Type K thermocouple voltage to temperature coefficients

–219°C to 0°C –219°C to 0°C

0°C to 760°C

760°C to 1,200°C

0°C to 760°C

760°C to 1,200°C

0.0000000E+00

0.0000000E+00

–1.3180580E+02

2.5173462E-02

2.5083550E-02

4.8302220E-02

7.8601060E-08

–1.6460310E-06

–1.0833638E-09 –8.9773540E-13

–2.5031310E-10 8.3152700E-14

5.4647310E-11 –9.6507150E-16

–8.9773540E-13 –3.7342377E-16

8.3152700E-14 –1.2280340E-17

–9.6507150E-16 8.8021930E-21

c5c6

–3.7342377E-16 –8.6632643E-20

–1.2280340E-17 9.8040360E-22

8.8021930E-21 –3.1108100E-26

c6c7

–1.0450598E-23 –8.6632643E-20

–4.4130300E-26 9.8040360E-22

-–3.1108100E-26

–5.1920577E-28 –1.0450598E-23

1.0577340E-30 –4.4130300E-26

— --

–1.0527550E-35 1.0577340E-30

—

–1.0527550E-35

—

c0

0.0000000E+00

c1

2.5173462E-02

c2

–1.1662878E-06

c0 c1

c2c3 c3c4 c4c5

c7c8 c9

–1.1662878E-06

–1.0833638E-09

c8

-–5.1920577E-28

c9

—

0.0000000E+00 2.5083550E-02

7.8601060E-08

–2.5031310E-10

Texas Instruments Analog Engineer's Pocket Reference

–1.3180580E+02 4.8302220E-02

–1.6460310E-06 5.4647310E-11

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Table 29: Seebeck coefficients for different material Material

Seebeck coefficient

Material

Seebeck coefficient

Material

Seebeck coefficient

Aluminum

4

Gold

6.5

Rhodium

6

Antimony

47

Iron

19

Selenium

900

Bismuth

–72

Lead

4

Silicon

440

Cadmium

7.5

Mercury

0.6

Silver

6.5

Carbon

3

Nichrome

25

Sodium

–2.0

Constantan

–35

Nickel

–15

Tantalum

4.5

Copper

6.5

Platinum

0

Tellurium

500

Germanium

300

Potassium

–9.0

Tungsten

7.5

Note: Units are μV/°C. All data at temperature of 0°C

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A/D Conversion

A/D conversion

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Texas Instruments Analog Engineer's Pocket Reference

A/D conversion

Binary/hex conversions • A/D and D/A transfer function • Quantization error • Signal-to-noise ratio (SNR) • Signal-to-noise and distortion (SINAD) • Total harmonic distortion (THD) • Effective number of bits (ENOB) • Noise-free resolution and effective resolution •

A/D Conversion

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Numbering systems: Binary, decimal, and hexadecimal Numbering systems: Binary, decimal, and hexadecimal Numbering systems: Binary, decimal, and hexadecimal

Binary (Base-2)

0 1 0 1 2 3 4 5 6 7 8 9 01 2 3 4 5 6 7 8 9 A B C D E F

Decimal (Base-10) Hexadecimal (Base-16)

2(1000) + 3(100) + 4(10) + 1(1) = 2,341

MSD = Most significant digit 2(1000) + 3(100) + 4(10) + 1(1) = 2,341

MSD = Most significant digit Example conversion: todecimal decimal Example conversion: Binary Binary to Example conversion: Binary to decimal Binary

Decimal Decimal

Binary

= =

LSD

LSD

8 + 4 + 0 + 1 8 + 4 + 0 + 1 Example conversion: Decimal to binary Example conversion: Decimal to binary Example conversion: Decimal to binary Decimal

Binary Binary

Decimal

LSD

= =

A/D conversion

LSD 128 + 64 + 32 + 8 + 4 = 236 MSD

128 + 64 + 32 + 8 + 4 = 236 LSD = Least Significant Digit MSD = Most Significant Digit ti.com/adcs

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A/D Conversion

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Example conversion: Binary to hexadecimal Example conversion: Binary to hexadecimal

Binary

Example conversion: Binary MSD to hexadecimal

LSD

128 + 64 + 16 + 8 + 1 = 217

Hexadecimal Conversion

128 + 64 + 16 + 8 + 1 = 217 8 + 4 + 1 = 13 (D) 8 8++14=+91 = 13 (D)

161 160

161 160 Hexadecimal

8+1=9

D 9

D 9

MSD LSD

MSD

208 + 9 = 217

208 + 9 = 217

Example Conversion: Hexadecimal to decimal Example Conversion: Hexadecimal binary and decimal totohexadecimal Decimal (Base-10) Hexadecimal (Base-16)

0

1

2

3

4

5

6

8

9 10 11 12 13 14 15

0 1 2 3 4 5 6 7 8 9 A BCDE F Decimal

Hexadecimal

x16 3 x16 2 x16 1 x16 0

16 9903 R = 15 (F) =

2 6 A F MSD

7

LSD LSD

16 618 R = 10 (A) 16 38 R = 6 (6) 16 38 R = 2 (2)

LSD

MSD

2(4096) + 6(256) + 10(16) + 16(1) = 9903

LSD = Least Significant Digit MSD = Most Significant Digit83 ti.com/adcs Texas Instruments Analog Engineer's Pocket Reference

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A/D Conversion

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A/D Converter with PGA 5V

FSR 0 to 2.5V

VREF

PGA x2

ADC 12 bits

ADC in 0 to 5V

Digital I/O

Figure 51: ADC full-scale range (FSR) unipolar

Full Scale Range (FSR) Unipolar VREF FSR = PGA 1LSB =

FSR

2n Example calculation for the circuit above. FSR =

VREF

1LSB =

PGA

FSR 2n

= =

5V = 2.5V 2

2.5V = 610.35ÂľV 212

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A/D Conversion

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A/D Converter with PGA 2.5V

FSR 0 to ±1.25V

VREF

PGA x2

ADC 12 bits

ADC in 0 to ± 2.5V

Digital I/O

Figure 52: ADC full-scale range (FSR) Bipolar

Full Scale Range (FSR) Bipolar FSR =

VREF PGA

1LSB =

FSR 2n

Example calculation for the circuit above. FSR =

±VREF ±2.5V = = ±1.25V ⇒ 2.5V PGA 2

1LSB =

FSR 2n

=

2.5V = 610.35µV 212

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A/D Conversion

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Table 30: Different data formats Code

Straight binary

Offset binary

2’s complement Decimal value

Binary

Decimal value

Decimal value

11111111

255

127

11000000 Code

192 Straight binary

Table 29: Different data formats

Binary 10000000

128Decimal value Table 29: Different data formats 11111111 255 127 11000000 192 Code Straight binary 01000000 64Decimal 10000000 128value Binary 01111111 127 11111111 255 00000000 0 01000000 64 11000000 192 00000000 0 10000000 128 01111111

01111111

–1

64 Offset binary

–64 2’s complement

Decimal 0 value 127 –1 Offset 64 binary –640value Decimal –1 127 –128 –64 64 –128 0

Decimal –128 value –1 127 –64 2’s complement 64 –128 Decimal value 127 0 –1 64 –64 0 –128

–1

127 64 0

127

Converting two’s complement to decimal: 01000000 64 –64 Converting two’s complement to decimal: Negative number example 00000000 0 –128

Negative number example

Converting two’s complement to decimal: SIGN x4 x2 x1 Negative number example Step 1: Check sign bit This case is negative

Step 1: Check sign bit This case is negative

Step 2: Invert all bits

1 x40 SIGN MSD

1 x11 x2 LSD

1 0 1 1 MSD 0 1 0 0

Step 2: Invert all bits Step 3: Add 1

0 1 0 0 0 1 0 1

Step : Add 1 Final3result

0–(4+1) = 1 0 –51

Converting two’s complement to decimal: –(4+1) = –5 Positive number example Converting two’s complement to decimal: Converting two’s complement to decimal: Positive number example Positive number exampleSIGN x4 x2 x1 Final result

Just add bit weights

SIGN 0 x41

x2 0 x11

MSD

Just add bit weights Final result Final result

0 4+1 = 5 1 0 1

LSD

MSD

4+1 = 5 84

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Table 31: LSB voltage vs. resolution and reference voltage voltage FSR Reference (Full-Scale Range) 1.024V

1.25V

2.048V

2.5V

4 mV

4.88 mV

8 mV

9.76 mV

10

1 mV

1.22 mV

2 mV

2.44 mV

12

250 µV

305 µV

500 µV

610 µV

14

52.5 µV

76.3 µV

125 µV

152.5 µV

16

15.6 µV

19.1 µV

31.2 µV

38.14 µV

18

3.91 µV

4.77 µV

7.81 µV

9.53 µV

20

0.98 µV

1.19 µV

1.95 µV

2.384 µV

22

244 nV

299 nV

488 nV

596 nV

24

61 nV

74.5 nV

122 nV

149 nV

Resolution

8

Table 32: LSB voltage vs. resolution and reference voltage

Resolution

voltage FSR Reference (Full-Scale Range) 3V

3.3V

4.096V

5V

8

11.7 mV

12.9 mV

16 mV

19.5 mV

10

2.93 mV

3.222 mV

4 mV

4.882 mV

12

732 µV

806 µV

1 mV

1.221 mV

14

183 µV

201 µV

250 µV

305 µV

16

45.77 µV

50.35 µV

62.5 µV

76.29 µV

18

11.44 µV

12.58 µV

15.6 µV

19.07 µV

20

2.861 µV

3.147 µV

3.91 µV

4.768 µV

22

715 nV

787 nV

976 nV

1.192 µV

24

179 nV

196 nV

244 nV

298 nV

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DAC definitions

DAC definitions

Resolution = n The number of bits used to quantify the output n Resolution The number of bits used quantify the output The number of input codetocombinations Codes == 2n Number of Codes = 2n The number of input code combinations Sets the LSB voltage or current size and Reference voltage = V Full-Scale Range output = FSRREF Sets the converter output range and the LSB voltage converter rangestep size of each LSB n The voltage LSB = FSR / 2 n 2 The output voltage or current LSB = V REF / voltage Full-scale output voltage of thestep DAC size of each Full-scale output = (2n – 1) • 1LSB code Largest code that can be written Full-scale input code = 2n – 1 n n ) largest Relationship between output and input code Transfer Function: Vout==2Number Full-scale code – 1 of Codes • (FSR/2 The code that can bevoltage written Full-scale voltage = VREF – 1LSB

Full-scale output voltage of the DAC n

Transfer function = VREF x (code/ 2 ) Relationship between input code and output voltage or current

FSR = 5V

Output voltage (V)

Full-scale voltage = 4.98V

Resolution 1LSB = 19mV

Full-scale code = 255

Number of codes = 2n

Resolution = 8bits

Figure 53: DAC transfer function Figure 51: DAC transfer function

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Resolution = n

The number of bits used to quantify the output

n ADC Codesdefinitions =2

The number of input code combinations

Sets theThe LSB voltage current sizethe and Reference Resolution = nvoltage = VREF number of bitsorused to quantify input converter Therange number of output code combinations Number of Codes = 2n n Full-Scale input = FSR Sets the converter input rangecode. and the LSB voltage – 1) LSB = VRange The voltage step size of each Note that REF / (2 n n The voltage step of 2 each LSB = FSR / 2 some topologies maysize use asLSB opposed to input voltage of the ADC Full-scale input voltage = (2n – 1) • 1LSB 2n – 1 inFull-scale the denominator. n Largest code that can be read Full-scale output code = n 2 –1 The largest code that can be written. Full-scale code = 2 – 1 Transfer Function: Number of Codes = Vin / (FSR/2n) Relationship between input voltage and output code Full-scale output voltage of the DAC. Note that Full-scale voltage = VREF the full-scale voltage will differ if the alternative definition for resolution is used. n

Transfer function = VREF x (code/ 2 ) Relationship between input code and output voltage or current

Full-scale code=255

Input voltage (V)

Figure 54: ADC transfer function Figure 52: ADC transfer function

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Full-scale Range FSR = 5V

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Quantization error of ADC Quantization error of ADC

Quantization error Figure 53: Quantization error of an A/D converter

Figure 55: Quantization error of an A/D converter

Quantization error

The error introduced as a result of the quantization process. The amount of this error is a function of the resolution of the converter. The quantization error of an A/D Quantization error converter is ½ LSB. The quantization error signal the difference between the actual The error introduced result of the quantization The amount voltage applied andas theaADC output (Figure 53). The rmsprocess. of the quantization signal of is this error is a function of the resolution of the converter. The quantization 1LSB⁄√12

error of an A/D converter is ½ LSB. The quantization error signal is the difference between the actual voltage applied and the ADC output (Figure 55). The rms of the quantization signal is 1LSB ⁄√12

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Signal-to-noise ratio (SNR) from quantization Signal-to-noise ratio (SNR) from quantization noise only noise only MaxRMSSignal � RMSNoise � SNR �

FSR/2

1LSB √12

√2

�

1LSB � 2���

(102) (107)

√2

(103) (108)

from quantization only

MaxRMSSignal 1LSB � 2��� /√2 � � 2��� √6 RMSNoise 1LSB⁄√12

SNR�dB� � 2�log�SNR� � �2� log�2��N � 2�log � SNR�dB� � 6��2N � 1��6

√6 � 2

(109) (104) (110) (105) (111) (106)

Where FSR = full-scale range of the A/D converter Where n 1LSB = the voltage of 1LSB, VREF/2 FSR = full-scale range of the A/D converter N = the resolution of the A/D converter MaxRMSSignal = the rms equivalent 1LSB = the voltage of 1LSB, VREF/2n of the ADC’s full-scale input RMSNoise = the rms noise from quantization NSNR = the= resolution thesignal A/D converter the ratio ofofrms to rms noise MaxRMSSignal = the rms equivalent of the ADC’s full-scale input RMSNoise = the rms noise from quantization Example SNR = the ratio of rms signal to rms noise What is the SNR for an 8-bit A/D converter with 5V reference, assuming only quantization noise?

Answer

Example SNR �is2��� � 2� �� 314 A/D converter with 5V reference, What the√6 SNR for√6 an�8-bit assuming only quantization noise? SNR�dB� � 2�log�314� � 4��� dB

Answer SNR�dB� � 6��2�8� � 1��6 4��� dB SNR = 2N-1 √6 = 28-1 √6 =�314 SNR(dB) = 20log(314) = 49.9 dB SNR(dB) = 6.02(8) + 1.76 = 49.9 dB

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Total harmonic distortion (Vrms) Total harmonic distortion (Vrms)

%

THD dB

RMSDistortion MaxRMSSignal

•

100

RMSDistortion MaxRMSSignal

V

V

V V

V

•

100

(112) (107) (113) (108)

Total harmonic distortion (Vrms)

Where RMSDistortion V V V V • 100rms(107) % THD = total harmonic distortion, the• 100 ratio of the rmsVdistortion to the signal MaxRMSSignal Where RMSDistortion = the rms sum of all harmonic components THD = total harmonic the ratio of the rms distortion to the rms signal RMSDistortion MaxRMSSignal = thedistortion, rms value of the input signal THD dB (108) MaxRMSSignal generally input signal V1 = the fundamental, RMSDistortion = the rms sum of the all harmonic components V2, V3, V4, …Vn = harmonics of the fundamental MaxRMSSignal = the rms value of the input signal Where THD = total harmonic distortion, the ratio of the rms distortion to the rms signal

V1 = the fundamental, generally signal components RMSDistortion = the rmsthe suminput of all harmonic

MaxRMSSignal = the rms value of the input signal V2, V3, V4, …V = harmonics of the fundamental V n= the fundamental, generally the input signal 1

V2, V3, V4, …Vn = harmonics of the fundamental

Figure 56:and Fundamental harmonics in Vrms Figure 54: Fundamental harmonicsand in Vrms Figure 54: Fundamental and harmonics in Vrms

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Total harmonic distortion (dBc) Total harmonic distortion (dBc) Total harmonic distortion (dBc) ୈమ ୈయ ୈర ୈ THD(dBc) ൌ ͳͲ ͳͲቀ ଵ ቁ ͳͲቀ ଵ ቁ ͳͲቀ ଵ ቁ ڮ ͳͲቀ ଵ ቁ ൨ ൌ

ୈమ ͳͲ ͳͲቀ ଵ ቁ

ୈయ ͳͲቀ ଵ ቁ

ୈర ͳͲቀ ଵ ቁ

ڮ

(114)

ୈ ͳͲቀ ଵ ቁ ൨

(109) (109)

Where Where THDWhere = total harmonic distortion. The ratio of the rms distortion to the rms signal THD D = total harmonic distortion. The ratio of the rms distortion to the rms signal 1 = the fundamental, generally the input signal. This is normalized to 0 dBc THD = total harmonic distortion. The ratio of the rms distortion to the rms signal D , D , D of the fundamental relative the 2 3 4, …Dn = harmonics D1 = the D fundamental, generally the input signal. Thismeasured is normalized to 0to dBc 1 = the fundamental, generally the input signal. This is normalized to 0 dBc fundamental D 2, D3, D4, …Dn = harmonics of the fundamental measured relative to the D2, D3, D4, …Dn = harmonics of the fundamental measured relative to fundamental the fundamental

Figure 57: Fundamental andin harmonics in dBc Figure 55: 55: Fundamental andand harmonics dBc Figure Fundamental harmonics in dBc

Example Example Determine THDTHD for the example above. Determine for the example above.

Example

Answer Answer

Determine THD for the example above. ିଽଶ ିଽଶ

ିହ ିହ

ିଽହିଽହ

ିଵଵ ିଵଵ

ቀ ቁ ቀ ቀ ቁ ቀ ቁ ቀ ଵ ቀ ଵ ቁ ቁ ଵ ቁ ଵ ଵ ቁଵ ଵ ቁଵ ڮ ൌ ͳͲ ͳͲ ቀͳͲ Ͳͳ ڮ ͳͲ ൌ ͳͲ ͳͲቀͳͲ ͳͲ ͳͲ ൨ ൨

-75

)

)

)

-95

)

-92

)

-110

)

)

Answer

10 10 10 ൌ െͶǤ10 ൌ= െͶǤ THD(dBc) 10 log 10 +10 +10 + ... +10

)

THD(dBc) = -74.76 dB

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Ac signals Ac signals Signal-to-noise and distortion (SINAD) and effective number of bits (ENOB) SINAD�dB� � 20 log �

SINAD�dB� � �20log ��10 �N�B �

MaxRMSSignal

√RMSNoise� � RMSDis�or�ion� �������� � � ��

� 10

�

������� � � �� �

SINAD�dB� � 1.76dB 6.02

(115) (110)

(116) (111)

(117) (112)

Where MaxRMSSignal = the rms equivalent of the ADC’s full-scale input Where RMSNoise = the=rms integratedofacross the A/D converters MaxRMSSignal the noise rms equivalent the ADC’s full-scale input RMSDistortion = the rms sum of all harmonic components RMSNoise = the noise integrated across the A/D SINAD = the ratiorms of the full-scale signal-to-noise ratioconverters and distortion THD = total harmonic distortion. ratio of the rms distortion to the rms signal. RMSDistortion = the rms sum of The all harmonic components SNR = the ratio of rms signal to rms noise SINAD = the ratio of the full-scale signal-to-noise ratio and distortion THD = total harmonic distortion. The ratio of the rms distortion to the rms signal. SNR = the ratio of rms signal to rms noise Example

Calculate the SNR, THD, SINAD and ENOB given the following information: MaxRMSSignal = 1.76 Vrms Example RMSDistortion 50 µVrms Calculate the = SNR, THD, SINAD and ENOB given the following RMSNoise = 100 µVrms information:

MaxRMSSignal = 1.76 Vrms Answer RMSDistortion = 50 μVrms RMSNoise = 100 1.76 Vrms μVrms

SNR�dB� � 20 log � � � ��.� dB 100 μVrms Answer THD�dB� SNR dB � 20 log �

50 μVrms 1.76 Vrms � � � �0.� dB 1.76 Vrms

1.76V rms SINAD�dB� � � ��.� dB THD dB � 20 log � � � �50 μVrms�� ��100 μVrms� 1.76 Vrms ���.�1.76V �� ���.� �� � rms� �� � �� � 10 �

SINAD dB � �20 log ��10� SINAD�dB�

��.�dB � 1.76dB �N�B � 1�.65 SINAD�dB 10 6.02 6.02

94

� ��.� dB

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Dcsignals signals Dc Noise free resolution and effective resolution Noise�ree�eso��tion � �o� � �

2� � PeaktoPeakNoiseinLSB

2� ���e�ti�e�eso��tion � �o� � � � rmsNoiseinLSB

PeaktoPeakNoiseinLSB � 6.6 � rmsNoiseinLSB ���e�ti�e�eso��tion � Noise�ree�eso��tion � 2.7

(118)(113) (119)(114) (120)(115) (121)(116)

Note: The maximum effective resolution is never greater than the ADC resolution. Note: The maximum resolution is never greater than the ADC resolution. For example, a 24-biteffective converter cannot have an effective resolution greater than Forbits. example, a 24-bit converter cannot have an effective resolution greater 24 than 24 bits.

Example Example What is the noise-free resolution and effective resolution for a 24-bit converter What is the resolution effective resolution for a assuming the noise-free peak-to-peak noise is 7and LSBs?

24-bit converter assuming the peak-to-peak noise is 7 LSBs?

Answer ��

2 Answer Noise�ree�eso��tion � �o� � � � � 2�.2 72 7 2�� ���e�ti�e�eso��tion � �o� � � � � 2�.� 7 2 6.6 7 6.6 ���e�ti�e�eso��tion � 2�.2 � 2.7 � 2�.�

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Time Constant

R

VIN A/D

VIN

C

Figure 58: Settling time for RC circuit-related to A/D converters

Figure 56: Settling time for RC circuit-related to A/D converters Table 33: Conversion accuracy achieved after a specified time Table 32: Conversion accuracy achieved after a specified time Settling time in time Settling time in time Settling time time in (NTC) Accuracy in bits (N)Settling constants Accuracy in bits constants (NTC)in time constants Accuracy in time constants Accuracy in 1 ) 1.44 10 14.43 (N bits (NTC) bits TC 21 2.89 11 15.87 10 14.43 1.44 11 15.87 2.89 32 4.33 12 17.31 3 12 17.31 4.33 4 5.77 13 18.76 4 13 18.76 5.77 55 7.21 14 20.20 14 20.20 7.21 66 8.66 15 21.64 15 21.64 8.66 77 88 9 9

ൌ ଶ ሺିి ሻ

16 17 18

10.10 10.10 11.54 11.54 12.98 12.98 (122)

16 17 18

23.08 24.53 25.97

23.08 24.53 25.97

(117)

Where Where N = the number of bits of accuracy the RC circuit has settled to after NTC number of N = the number of bits of accuracy the RC circuit has settled to after NTC number of time constants. time constants. NTC = the number of RC time constants NTC = the number of RC time constants Note: For a FSR step. For single-ended input ADC with no PGA front end FSR (Full Scale Range) = VREF

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Table 34: Time required to settle to a specified conversion accuracy Table in 33: Time required to settle accuracy Accuracyconversion in bits Settling time in time Accuracy bits Settling time in timeto a specified ) (N) (NTC) (N) constants (N Settling timeconstants in Settling time inTC

Accuracy 8 in bits (N) 9 8 10 9 11 10 11 12 12 13 13 14 14 15 15 16 16

time constants 5.5 (NTC) 6.24 5.55 6.93 6.24 6.93 7.62 7.62 8.32 8.32 9.01 9.01 9.70 9.70 10.40 10.40 11.09 11.04

N�� � ����� �

Accuracy in bits (N) 17 18 19 20 21 22 23 24 25

17 time constants (NTC) 18 11.78 19 12.48 13.17 20 13.86 21 14.56 22 15.25 23 15.94 16.64 24 17.33 25

11.78 12.48 13.17 13.86 14.56 15.25 15.94 16.64 17.33

(118)

(123)

Where NTC = the number of time constants required to achieve N bits of settling Where N the = the number bitsconstants of accuracy NTC = number ofof time required to achieve N bits of settling N = the number of bits of accuracy Note: For a FSR step. For single-ended input ADC with no PGA front end FSR (Full Scale Range) = VREF

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