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Texas Instruments Analog Engineer's Pocket Reference

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  • 日期: 2015-03-28
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标签: TI模拟电路

一本囊括了50年模拟设计经验的实用电子书,是模拟工程师的随身参考。

关键常量和变换

运算放大器基本配置

传感器概述

模数 (A/D) 和数模 (D/A) 转换

以及更多其它内容

Analog Engineer’s Pocket Reference Art Kay and Tim Green, Editors Download eBook at www.ti.com/analogrefguide Analog Engineer’s Pocket Reference Third 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 © 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: • 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Standard resistor values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Practical capacitor model and specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Practical capacitors vs. frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Capacitor type overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Standard capacitance values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Capacitance marking and tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Diodes and LEDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Capacitor equations (series, parallel, charge, energy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Inductor equations (series, parallel, energy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Capacitor charge and discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 RMS and mean voltage definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 RMS and mean voltage examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Logarithmic mathematical definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 dB definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Pole and zero definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Basic op amp configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Op amp bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Full power bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Small signal step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Noise equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Stability equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Stability open loop SPICE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 PCB and wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 PCB conductor spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Self-heating of PCB traces on inside layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 PCB trace resistance for 1oz and 2oz Cu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Package types and dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 PCB parallel plate capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 PCB via capacitance and inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Common coaxial cable specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Coaxial cable equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Resistance per length for different wire types (AWG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Maximum current for wire types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Temperature sensor overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Thermistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Resistive temperature detector (RTD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Diode temperature characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Thermocouple (J and K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A/D conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Binary/hex conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A/D and D/A transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Quantization error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Signal-to-noise ratio (SNR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Signal-to-noise and distortion (SINAD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Total harmonic distortion (THD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Effective number of bits (ENOB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Noise free resolution and effective resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Texas Instruments Analog Engineer's Pocket Reference 5 6 Texas Instruments Analog Engineer's Pocket Reference Conversions • Standard decimal prefixes • Metric conversions • Temperature scale conversions • Error conversions (ppm and percentage) CoCnovneversrsioionns Texas Instruments Analog Engineer's Pocket Reference 7 Conversions Conversions Table 1: Physical constants Constant Speed of light in a vacuum Permittivity of vacuum Permeability of free space Plank’s constant Boltzmann’s constant Faraday’s constant Avogadro’s constant Unified atomic mass unit Electronic charge Rest mass of electron Mass of proton Gravitational constant Standard gravity Ice point Maximum density of water Density of mercury Gas constant Speed of sound in air (at 273 K) Symbol c εo µo h k F NA mu q me mp G gn Tice ρ ρHg R cair Value 3.00 x 108 8.85 x 10-12 1.26 x 10-6 6.63 x 10-34 1.38 x 10-23 9.65 x 104 6.02 x 1023 1.66 x 10-27 1.60 x 10-19 9.11 x 10-31 1.67 x 10-27 6.67 x 10-11 9.81 273.15 1.00 x 103 1.36 x 104 8.31 3.31 x 102 Units m/s F/m H/m Js J/K C/mol /mol kg C kg kg Nm2/kg m/s2 K kg/m3 kg/m3 J/(K•mol) m/s 8 Texas Instrume8nts Analog Engineer's Pocket Reference Conversions Table 2: Standard decimal prefixes Multiplier 1012 109 106 103 10–3 10–6 10–9 10–12 10–15 10–18 Prefix tera giga mega kilo milli micro nano pico femto atto Abbreviation T G M k m µ n p f a Table 3: English to metric conversions Unit inches mil feet yards miles circular mil square yards pints ounces pounds calories horsepower Symbol in mil ft yd mi cir mil yd2 pt oz lb cal hp Equivalent 25.4 mm/in 0.0254 mm/mil 0.3048 m/ft 0.9144 m/yd 1.6093 km/mi 5.067x10-4 mm2/cir mil 0.8361 m2 0.5682 L/pt 28.35 g/oz 0.4536 kg/lb 4.184 J/cal 745.7 W/hp Unit millimeter millimeter meters meters kilometers square millimeters square meters liters grams kilograms joules watts Symbol mm mm m m km mm2 m2 L g kg J W Table 4: Metric to English conversions Unit millimeter millimeter meters meters kilometers square millimeters square meters liters grams kilograms joules watts Symbol mm mm m m km mm2 m2 L g kg J W Conversion 0.0394 in/mm 39.4 mil/mm 3.2808 ft/m 1.0936 yd/m 0.6214 mi/km 1974 cir mil/mm2 1.1960 yd2/ m2 1.7600 pt/L 0.0353 oz/g 2.2046 lb/kg 0.239 cal/J 1.341x10-3 hp/W Unit inch mil feet yard miles circular mil square yards pints ounces pounds calories horsepower Symbol in mil ft yd mi cir mil yd2 pt oz lb cal hp Example Convert 10 mm to mil. Answer ͳͲ ൈ ͵ͻǤͶ ୫୧୪ ୫୫ ൌ ͵ͺͶ‹Ž Texas Instruments Analog Engineer's P9ocket Reference 9 Conversions Table 5: Temperature conversions �� � 5 9 ��F � �2� �F � 9 5 ���� � �2 Fahrenheit to Celsius Celsius to Fahrenheit � � �� � 2��.15 Celsius to Kelvin �� � � � 2��.15 Kelvin to Celsius Table 6: Error conversions Error�%� � �easured � Ideal Ideal � 100 Error�% FSR� � �easured � Ideal Full‐scale range � 100 %� ppm 10� � 100 m% � ppm 10� � 100 � 1000 ppm � % � 10� ppm � m% � 10 Error in measured value Error in percent of full-scale range Part per million to percent Part per million to milli-percent Percent to part per million Milli-percent to part per million Example Compute the error for a measured value of 0.12V when the ideal value is 0.1V and the range is 5V. Answer Error�%� � 0.12V � 0.1V 0.1V � 100 � 20% Error�% FSR� � 0.12 � 0.1V 5V � 100 � 0.�% Error in measured value Percent FSR Example Convert 10 ppm to percent and milli-percent. Answer 10 ppm 10� � 100 � 0.001% 10 ppm 10� � 100 � 1000 � 1 m% Part per million to percent Part per million to milli-percent 10 Texas Instrume1n0ts Analog Engineer's Pocket Reference DisDcirsectreetceocmomppoonneenntts • Resistor color code • Standard resistor values • Capacitance specifications • Capacitance type overview • Standard capacitance values • Capacitance marking and tolerance Discrete Texas Instruments Analog Engineer's Pocket Reference 11 Discrete Discrete components Table 7: Resistor color code Color None Silver Gold Black Brown Red Orange Yellow Green Blue Violet Grey White Digit -na-na-na- 0 1 2 3 4 5 6 7 8 9 Additional Zeros -na-2 -1 0 1 2 3 4 5 6 7 Tolerance 20% 10% 5% 2% Example Yellow, violet, orange and silver indicate 4, 7, and 3 zeros. or a 47 kΩ, 10% resistor. Figure 1: Resistor color code 12 Texas Instrume1n2ts Analog Engineer's Pocket Reference 13 Texas Instruments Analog Engineer's13Pocket Reference 0.1% 0.25% 0.5% 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 1% 10.0 10.2 10.5 10.7 11.0 11.3 11.5 11.8 12.1 12.4 12.7 13.0 13.3 13.7 14.0 14.3 2% 5% 10% 10 11 12 13 0.1% 0.25% 0.5% 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 Standard resistance values for the 10 to 100 decade 1% 14.7 15.0 15.4 15.8 16.2 16.5 16.9 17.4 17.8 18.2 18.7 19.1 19.6 20.0 20.5 21.0 2% 5% 10% 15 16 18 20 0.1% 0.25% 0.5% 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 1% 21.5 22.1 22.6 23.2 23.7 24.3 24.9 25.5 26.1 26.7 27.4 28.0 28.7 29.4 30.1 30.9 2% 5% 10% 22 24 27 30 0.1% 0.25% 0.5% 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 1% 31.6 32.4 33.2 34.0 34.8 35.7 36.5 37.4 38.3 39.2 40.2 41.2 42.2 43.2 44.2 45.3 2% 5% 10% 33 36 39 43 0.1% 0.25% 0.5% 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 1% 46.4 47.5 48.7 49.9 51.1 52.3 53.6 54.9 56.2 57.6 59.0 60.4 61.9 63.4 64.9 66.5 2% 5% 10% 47 51 56 62 0.1% 0.25% 0.5% 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 1% 68.1 2% 5% 10% 68 69.8 71.5 73.2 75.0 75 76.8 78.7 80.6 82.5 82 84.5 86.6 88.7 90.9 91 93.1 95.3 97.6 Table 8: Standard resistor values Discrete components Discrete components Practical capacitor model and specifications Figure 2: Model of a practical capacitor. Table 9: Capacitor specifications Parameter C ESR ESL Rp Voltage rating Voltage coefficient Temperature coefficient Description The nominal value of the capacitance. Table 11 lists standard capacitance values. 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. Equivalent series inductance. Ideally this is zero. ESL ranges from 100 pH to 10 nH. 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. The maximum voltage that can be applied to the capacitor. Exceeding this rating damages the capacitor. 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. The change in capacitance with across temperature in ppm/Ԩ. Ideally, the temperature coefficient is zero. The maximum specified drift generally ranges from 10 to 100ppm/°C depending on the resistor type. 14 Texas Instrume1n4ts Analog Engineer's Pocket Reference Practical capacitors vs. frequency Discrete components Impedance (ohms) Figure 3: Effect of ESR and ESL on capacitor frequency response Texas Instruments Analog Engineer's P15ocket Reference 15 Discrete components Table 10: Capacitor type overview Capacitor type C0G/NP0 (Type 1 ceramic) X7R (Type 2 ceramic) Y5V (Type 2 ceramic) Aluminum oxide electrolytic Tantalum electrolytic Polypropylene film Description 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/Ԩ Low-voltage coefficient Minimal piezoelectric effect Good tolerance: ±1% to ±10% Temperature range: –55Ԩ to 125Ԩ (150Ԩ and higher) Voltage range may be limited for larger capacitance values 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/Ԩ (±15% across temp range) Substantial voltage coefficient Tolerance: ±5% to –20%/+80% Temperature range: –55Ԩ to 125Ԩ Voltage range may be limited for larger capacitance values 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: –30Ԩ to 85Ԩ Other characteristics are similar to X7R and other type 2 ceramic 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/Ԩ Substantial voltage coefficient Tolerance: ±20% Temperature range: –55Ԩ to 125Ԩ (150Ԩ and higher) Higher ESR than other types Capacitance range: 1 µF to 150 µF Similar to aluminum oxide but smaller size 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: –55Ԩ to 100Ԩ 16 Texas Instrume1n6ts Analog Engineer's Pocket Reference Discrete components 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 Example Translate the capacitor marking Figure 4: Capacitor marking code Table 12: Ceramic capacitor tolerance markings Code B C D F G Tolerance ± 0.1 pF ± 0.25 pF ± 0.5 pF ± 1% ± 2% Code J K M Z Tolerance ± 5% ± 10% ± 20% + 80%, –20% Table 13: EIA capacitor tolerance markings (Type 2 capacitors) First letter symbol Z Y X Low temp. limit +10°C –30°C –55°C Second number symbol 2 4 5 6 7 High temp. limit +45°C +65°C +85°C +105°C +125°C Second letter symbol A B C D E F P R S T U V Max. capacitance change over temperature rating ±1.0% ±1.5% ±2.2% ±3.3% ±4.7% ±7.5% ±10.0% ±15.0% ±22.0% ±22% ~ 33% ±22% ~ 56% ±22% ~ 82% Example X7R: –55Ԩ to +125Ԩ, ±15.0% Texas Instruments Analog Engineer's P1o7cket Reference 17 Discrete components Diodes and LEDs Figure 5: Diode and LED pin names Color Infrared Red Orange / yellow Green Blue / white Voltage 1.4 1.7 to 1.9 2 2.1 3.4 Figure 6: LED forward voltage drop by color 18 Texas Instrum1e8nts Analog Engineer's Pocket Reference • 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 AAnnaalloog Analog Texas Instruments Analog Engineer's Pocket Reference 19 Analog Analog Capacitor equations C� � 1 C� � 1 C� 1 � � � 1 C� C� � C�C� C� � C� (1) Series capacitors (2) Two series capacitors C� � C� � C� � � � C� (3) Parallel capacitors Where Ct = equivalent total capacitance C1, C2, C3…CN = component capacitors � � CV (4) Charge storage � � �t (5) Charge defined Where Q = charge in coulombs (C) C = capacitance in farads (F) I = current in amps (A) t = time in seconds (s) � � C dv dt (6) Instantaneous current through a capacitor Where i = instantaneous current through the capacitor C = capacitance in farads (F) �� �� = the instantaneous rate of voltage change � � 1 2 CV� (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) 20 20 Texas Instruments Analog Engineer's Pocket Reference Analog Inductor equations L� � L� � L� � � � L� (8) Series inductors L� � 1 L� � 1 L� 1 � � � 1 L� (9) Parallel inductors L� � L�L� L� � L� (10) Two parallel inductors Where Lt = equivalent total inductance L1, L2, L3…LN = component inductance v � L di dt (11) Instantaneous voltage across an inductor Where v = instantaneous voltage across the inductor L = inductance in Henries (H) � �� = the instantaneous rate of voltage change � � 1 2 LI� (12) Energy stored in an Inductor Where E = energy stored in an inductor in Joules (J) I = current in amps L = inductance in Henries (H) 21 Texas Instruments Analog Engineer's Pocket Reference 21 Analog Equation for charging a capacitor V� � V� �� � ������� (13) General relationship Where VC = voltage across the capacitor at any instant in time (t) VS = the source voltage charging the RC circuit t = time in seconds  = RC, the time constant for charging and discharging capacitors Graphing equation 13 produces the capacitor charging curve below. Note that the capacitor is 99.3% charged at five time constants. It is common practice to consider this fully charged. Figure 7: RC charge curve 22 22 Texas Instruments Analog Engineer's Pocket Reference Analog Equation for discharging a capacitor V� � V� �������� (14) General relationship Where VC = voltage across the capacitor at any instant in time (t) VI = the initial voltage of the capacitor at t=0s t = time in seconds  = RC, the time constant for charging and discharging capacitors Graphing equation 14 produces the capacitor discharge curve below. Note that the capacitor is 0.7% charged at five time constants. It is common practice to consider this fully discharged. Percentage Discharged vs. Number of Time Constants 100 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 Number of time Constants (τ = RC) Figure 8: RC discharge curve Percentage Charged 23 Texas Instruments Analog Engineer's Pocket Reference 23 Analog RMS voltage V��� � ���� 1 � ��� �� � �V�t���dt �� (15) General relationship Where V(t) = continuous function of time t = time in seconds T1 ≤ t ≤ T2 = the time interval that the function is defined over Mean voltage V���� � ��� 1 � �� �� � � �� V�t�dt (16) General relationship Where V(t) = continuous function of time t = time in seconds T1 ≤ t ≤ T2 = the time interval that the function is defined over V��� � V���� √2 V���� � 2 � V���� π (17) RMS for full wave rectified sine wave (18) Mean for full wave rectified sine wave Figure 9: Full wave rectified sine wave 24 24 Texas Instruments Analog Engineer's Pocket Reference RMS voltage and mean voltage V��� � V�����2Tτ V���� � 2 � V���� π �Tτ� Analog (19) RMS for a half-wave rectified sine wave (20) Mean for a half-wave rectified sine wave Figure 10: Half-wave rectified sine wave V��� � V�����Tτ V���� � V���� τ T (21) RMS for a square wave (22) Mean for a square wave Figure 11: Square wave 25 Texas Instruments Analog Engineer's Pocket Reference 25 Analog RMS voltage and mean voltage V��� � τ T �V�� � V�V� 3 � V�� V���� � τ 2T �V� � V�� (23) RMS for a trapezoid (24) Mean for a trapezoid Figure 12: Trapezoidal wave V��� � V�����3τT V���� � τ 2T V���� (25) RMS for a triangle wave (26) Mean for a triangle wave Figure 13: Triangle wave 26 Texas Instrume2n6ts Analog Engineer's Pocket Reference Analog Logarithmic mathematical definitions log �AB� � log�A� � log�B� (27) Log of dividend log�AB� � log�A� � log�B� (28) Log of product log�A�� � x�log�A� (29) Log of exponent log��X� � log��X� log���� log��X� � log���X� log����� ln�X� � log��X� e � ��������� (30) Changing the base of log function (31) Example changing to log base 2 (32) Natural log is log base e (33) Exponential function to 6 digits. Alternative notations exp�x� � e� ����� � � � ���� � ���� (34) Different notation function for exponential Different notation for scientific (35) notation, sometimes confused with exponential function 27 Texas Instruments Analog Engineer's Pocket Reference 27 Analog dB definitions Bode plot basics 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, a plot of frequency versus voltage gain in dB (decibels). Bode plots are usually plotted as semi-log plots with frequency on the x-axis, log scale, and gain on the y-axis, linear scale. The other half of the frequency response is the phase shift versus frequency and is plotted as frequency versus degrees phase shift. Phase plots are usually plotted as semi-log plots with frequency on the x-axis, log scale, and phase shift on the y-axis, linear scale. Definitions V���������������� � �� ��� �VV������ (36) Voltage gain in decibels P�������������� � 1� ��� �PP������ (37) Power gain in decibels P�������������m� � 1� ��� �1P�m�W� � (38) Power gain in decibel milliwatt Table 14: Examples of common gain values and dB equivalent A (V/V) 0.001 0.01 0.1 1 10 100 1,000 10,000 100,000 1,000,000 10,000,000 A (dB) –60 –40 –20 0 20 40 60 80 100 120 140 Roll-off rate is the decrease in gain with frequency 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) 28 28 Texas Instruments Analog Engineer's Pocket Reference Analog Figure 14 illustrates a method to graphically determine values on a logarithmic axis that are not directly on an axis grid line. 1. Given L = 1 cm; D = 2cm, measured with a ruler. 2. L/D = log10(fP) 3. fP = 10(L/D) = 10(1CM/2CM)= 3.16 4. Adjust for the decade range (for example, 31.6 Hz) Figure 14: Finding values on logarithmic axis not directly on a grid line A (dB) 29 Texas Instruments Analog Engineer's Pocket Reference 29 Analog Bode plots: Poles fP 100 80 Actual function 60 0.707*GV/V = –3 dB Straight-line approximation –20 dB/decade G (dB) 40 20 0 1 +90 10 100 1k 10k 100k 1M 10M Frequency (Hz) (degrees) +45 0° 10 100 1k 10k 100k 1M 10M 0 –45 –5.7° at fP 10 –45°/decade –90 –45° at fP –84.3° at fP x 10 –90° Figure 15: Pole gain and phase Pole Location = fP (cutoff freq) Magnitude (f < fP) = Gdc (for example, 100 dB) Magnitude (f = fP) = –3 dB Magnitude (f > fP) = –20 dB/decade Phase (f = fP) = –45° Phase (0.1 fP < f < 10 fP) = –45°/decade Phase (f > 10 fP) = –90° Phase (f < 0.1 fP) = 0° 30 Texas Instrume3n0ts Analog Engineer's Pocket Reference Pole (equations) G� � V��� V�� � � G�� �ff�� � � G� � V��� V�� � G�� ��ff��� � � Analog (39) As a complex number (40) Magnitude � � � ����� �ff�� G�� � �������G�� Where Gv = voltage gain in V/V GdB = voltage gain in decibels Gdc = the dc or low frequency voltage gain f = frequency in Hz fP = frequency at which the pole occurs θ = phase shift of the signal from input to output (41) Phase shift (42) Magnitude in dB 31 Texas Instruments Analog Engineer's Pocket Reference 31 Analog Bode plots (zeros) G (dB) 80 60 40 20 0 1 +90 +45 0 –45 Straight-line approximation +20 dB/decade Actual function 10 100 +3 dB 1k 10k 100k 1M 10M +45° at fZ 0° 5.7° at 1fZ0 +90° 84.3° at fZ x 10 +45°/decade 10 100 1k 10k 100k 1M 10M Frequency (Hz) (degrees) –90 Figure 16: Zero gain and phase Zero location = fZ Magnitude (f < fZ) = 0 dB Magnitude (f = fZ) = +3 dB Magnitude (f > fZ) = +20 dB/decade Phase (f = fZ) = +45° Phase (0.1 fZ < f < 10 fZ) = +45°/decade Phase (f > 10 fZ) = +90° Phase (f < 0.1 fZ) = 0° 32 Texas Instrume3n2ts Analog Engineer's Pocket Reference Zero (equations) G� � V��� V�� � G�� �� �ff�� � �� Analog (43) As a complex number G� � V��� V�� � G����ff� � � � � (44) Magnitude � � ����� �ff�� G�� � �������G�� Where GV = voltage gain in V/V GdB = voltage gain in decibels GDC = the dc or low frequency voltage gain f = frequency in Hz fZ = frequency at which the zero occurs θ = phase shift of the signal from input to output (45) Phase shift (46) Magnitude in dB 33 Texas Instruments Analog Engineer's Pocket Reference 33 ATinmae tloopghase shift Figure 17: Time to phase shift � � �� �� ∗ 360° (47) Phase shift from time Where TS = time shift from input to output signal TP = period of signal θ = phase shift of the signal from input to output Example Calculate the phase shift in degrees for Figure 17. Answer � � T� T� ∗ 360° � �0.212m5 sms� ∗ 360° � �1° 34 34 Texas Instruments Analog Engineer's Pocket Reference • Basic op amp configurations • Op amp bandwidth • Full power bandwidth • Small signal step response • Noise equations • Stability equations • Stability open loop SPICE analysis AAmmpplilfifiieer Amplifier Texas Instruments Analog Engineer's Pocket Reference 35 Amplifier Basic op amp configurations G�� � � (48) Gain for buffer configuration Figure 18: Buffer configuration G�� � R� R� � � (49) Gain for non-inverting configuration Amplifier + + Figure 19: Non-inverting configuration 36 Texas Instrume3n6ts Analog Engineer's Pocket Reference Amplifier Basic op amp configurations (cont.) G�� � � R� R� (50) Gain for inverting configuration + Figure 20: Inverting configuration V��� � �R��RV�� � V� R� � � � V� R� � V��� � � R� R� �V� � V� � � � V�� (51) Transfer function for inverting summing amplifier Transfer function for inverting (52) summing amplifier, assuming R1 = R2 = …=RN RN VN V2 V1 R2 R1 Rf Vcc -+ + Vee Figure 21: Inverting summing configuration Vout Texas Instruments Analog Engineer's P37ocket Reference 37 Amplifier Basic op amp configurations (cont.) V��� � �RR��� � �� �VN� � V� N � � � VN�� Where R1 = R2 = … = RN N = number of input resistors Transfer function for non(53) inverting summing amplifier for equal input resistors Figure 22: Non-inverting summing configuration 38 Texas Instrume3n8ts Analog Engineer's Pocket Reference Amplifier Simple non-inverting amp with Cf filter G�� � R� R� � 1 G�� � 1 f� � 2π 1 R� C� (54) Gain for non-inverting configuration for f < fc (55) Gain for non-inverting configuration for f >> fc (56) Cut off frequency for non-inverting configuration Figure 23: Non-inverting amplifier with Cf filter Figure 24: Frequency response for non-inverting op amp with Cf filter Texas Instruments Analog Engineer's P39ocket Reference 39 Amplifier Simple inverting amp with Cf filter G�� � � R� R� G�� � 1 f� � 2π 1 R� C� (57) Gain for inverting configuration for f < fC (58) Gain for inverting configuration for f >> fC (59) Cutoff frequency for inverting configuration Cf R1 Rf Vin Vcc -+ + Vout Vee Figure 25: Inverting amplifier with Cf filter Figure 26: Frequency response for inverting op amp with Cf filter 40 Texas Instrume4n0ts Analog Engineer's Pocket Reference Amplifier Op amp bandwidth GBW � Gain���BW (60) Gain bandwidth product defined Where GBW = gain bandwidth product, listed in op amp data sheet specification table Gain = closed loop gain, set by op amp gain configuration BW = the bandwidth limitation of the amplifier Example Determine bandwidth using equation 60 Gain � �100 (from amplifier configuration) GBW � �22MHz (from data sheet) BW � GBW Gain � � 22MHz 100 � 220�Hz Note that the same result can be graphically determined using the AOL curve as shown below. Open-loop gain and phase vs. frequency Figure 27: Using AOL to find closed-loop bandwidth Texas Instruments Analog Engineer's P41ocket Reference 41 Amplifier Full power bandwidth V� � SR 2πf (61) Maximum output without slew-rate induced distortion Where VP = maximum peak output voltage before slew induced distortion occurs SR = slew rate f = frequency of applied signal Maximum output voltage vs. frequency �� � �� ��� � �. ��/�� ��������� � �. ����� �� �. ����� Figure 28: Maximum output without slew-rate induced distortion Notice that the above figure is graphed using equation 61 for the OPA188. The example calculation shows the peak voltage for the OPA277 at 40kHz. This can be determined graphically or with the equation. Example V� � SR 2πf � 0.8V/μs 2π��0k��� � 3.�8Vpk or 6.37Vpp 42 Texas Instrume4n2ts Analog Engineer's Pocket Reference Amplifier Small signal step response τ� � 0.35 f� (62) Rise time for a small signal step Where 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 Figure 29: Small signal step response Texas Instruments Analog Engineer's Pocket Reference 43 43 Amplifier Op amp noise model Figure 30: Op amp noise model Op amp intrinsic noise includes:  Noise caused by op amp (current noise + voltage noise)  Resistor noise 44 Texas Instrume4n4ts Analog Engineer's Pocket Reference Noise bandwidth calculation BW� � ��f� (63) Noise bandwidth Where BWN = noise bandwidth of the system KN = the brick wall correction factor for different filter order fC = –3 dB bandwidth of the system Amplifier Figure 31: Op amp bandwidth for three different filters orders Table 15: Brick wall correction factors for noise bandwidth Number of poles 1 2 3 4 KN brick wall correction factor 1.57 1.22 1.13 1.12 Broadband total noise calculation E� � ����BW� (64) Total rms noise from broadband Where EN = total rms noise from broadband noise eBB = broadband noise spectral density (nV/rtHz) BWN = noise bandwidth (Hz) Texas Instruments Analog Engineer's P4o5cket Reference 45 Amplifier 1/f total noise calculation E�_������ � � ����f� (65) Normalized 1/f noise at 1 Hz Where EN_NORMAL = 1/f noise normalized to 1 Hz eBF = noise spectral density measured in the 1/f region fO = the frequency that the 1/f noise eBF is measured at E�_������� � � E�_��������� �ff�� � (66) 1/f total noise calculation Where EN_FLICKER = total rms noise from flicker EN_NORMAL = 1/f noise normalized to 1 Hz fH = upper cutoff frequency or noise bandwidth fL = lower cutoff frequency, normally set to 0.1 Hz Table 16: Peak-to-peak conversion Number of standard deviations 2σ (same as ±1σ) 3σ (same as ±1.5σ) 4σ (same as ±2σ) 5σ (same as ±2.5σ) 6σ (same as ±3σ) 6.6σ (same as ±3.3σ) Percent chance reading is in range 68.3% 86.6% 95.4% 98.8% 99.7% 99.9% 46 Texas Instruments Analog Engineer's Pocket Reference 46 Amplifier Thermal noise calculation E�_� � √4 kTRΔf (67) Total rms thermal noise Where EN_R = total rms noise from resistance, also called thermal noise k = Boltzmann’s constant 1.38 x 10-23 J/K T = temperature in Kelvin ∆f = noise bandwidth in Hz 1000 Noise Spectral Density (nV/rtHz) 100 10 ‐55C 1 25C 125C 0.1 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Resistance (Ω) 1.E+06 1.E+07 Figure 32: Noise spectral density vs. resistance Texas Instruments Analog Engineer's Pocket Reference 47 47 Amplifier Ac response versus frequency Figure 33 illustrates a bode plot with four different examples of ac peaking. Figure 33: Stability – ac peaking relationship example Phase margin versus ac peaking This graph illustrates the phase margin for any given level of ac peaking. Note that 45° of phase margin or greater is required for stable operation. Figure 34: Stability – phase margin vs. peaking for a two-pole system 48 Texas Instrume4n8ts Analog Engineer's Pocket Reference Amplifier Transient overshoot Figure 35 illustrates a transient response with two different examples of percentage overshoot. Figure 35: Stability – transient overshoot example Phase margin versus percentage overshoot This graph illustrates the phase margin for any given level of transient overshoot. Note that 45° of phase margin or greater is required for stable operation. Figure 36: Stability – phase margin vs. percentage overshoot Note: The curves assume a two-pole system. Texas Instruments Analog Engineer's P49ocket Reference 49 Amplifier Figure 37: Common spice test circuit used for stability A��_������ �� V� V�� (68) Loaded open-loop gain β � � V�� (69) Feedback factor 1 β � � 1 V�� (70) Closed-loop noise gain A��_������ � β � � V� (71) Loop gain Where VO = the voltage at the output of the op amp. VOUT = the voltage output delivered to the load, which may be important to the application but is not considered in stability analysis. VFB = feedback voltage RF , R1, RISO and CL = the op amp feedback network and load. Other op amp topologies will have different feedback networks; however, the test circuit will be the same for most cases. Figure 38 shows the exception to the rule (multiple feedback). 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. 50 Texas Instruments Analog Engineer's Pocket Reference 50 VFB R1 Cin RF CF Amplifier Vin C1 1T V- ++ V+ Riso Vo Vout CL Figure 38: Alternative (multiple feedback) SPICE test circuit used for stability A��_������ � V� (72) Loaded open loop gain β� V�� V� (73) Feedback factor 1 β � V� V�� (74) Closed-loop noise gain A��_������ � β � V�� (75) Loop gain Where VO = the voltage at the output of the op amp. VOUT = the voltage output delivered to the load. This may be important to the application but is not considered in stability analysis. VFB = feedback voltage RF, R1, RISO and CF = the op amp feedback network. Because there are two paths for feedback, the loop is broken at the input. 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. CIN = the equivalent input capacitance taken from the op amp datasheet. 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. Texas Instruments Analog Engineer's Pocket Reference 51 51 Amplifier R1 RF +Vs - Vout + Voffset -Vs + Vin Volts Voffset 50mVpp Figure 39: 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).  User 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 52 Texas Instruments Analog Engineer's Pocket Reference 52 Amplifier Figure 40: Input filter for instrumentation amplifier Select C��� � 1�C��� R��� � R��� C��� � C��� f�� � 1 2πR���C��� (76) Differential filter is sized 10 times the common-mode filter (77) Input resistors must be equal (78) Common-mode capacitors must be equal (79) Differential filter cutoff f��� � 1 2π�2R�����C��� � 1 2 C���� (80) Common-mode filter cutoff Where fDIF = differential cutoff frequency fCM = common-mode cutoff frequency RIN = input resistance CCM = common-mode filter capacitance CDIF = differential filter capacitance Note: Selecting CDIF ≥ 10 CCM sets the differential mode cutoff frequency 10 times lower than the common-mode cutoff frequency. This prevents common-mode noise from being converted into differential noise due to component tolerances. Texas Instruments Analog Engineer's Pocket Reference 53 53 Amplifier Notes 54 Texas Instruments Analog Engineer's Pocket Reference PCPCBBaannddwwiirre • 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 PCB and wire Texas Instruments Analog Engineer's Pocket Reference 55 PCB and wire Table 17: Printed circuit board conductor spacing Voltage between conductors (dc or ac peaks) B1 Minimum spacing Bare board B2 B3 B4 Assembly A5 A6 A7 0-15 0.05 mm 0.1 mm 0.1 mm 0.05 mm 0.13 mm 0.13 mm 0.13 mm [0.00197 in] [0.0039 in] [0.0039 in] [0.00197 in] [0.00512 in] [0.00512 in] [0.00512 in] 16-30 0.05 mm 0.1 mm 0.1 mm 0.05 mm 0.13 mm 0.25 mm 0.13 mm [0.00197 in] [0.0039 in] [0.0039 in] [0.00197 in] [0.00512 in] [0.00984 in] [0.00512 in] 31-50 0.1 mm 0.6 mm 0.6 mm 0.13 mm 0.13 mm 0.4 mm 0.13 mm [0.0039 in] [0.024 in] [0.024 in] [0.00512 in] [0.00512 in] [0.016 in] [0.00512 in] 51-100 0.1 mm 0.6 mm 1.5 mm 0.13 mm 0.13 mm 0.5 mm 0.13 mm [0.0039 in] [0.024 in] [0.0591 in] [0.00512 in] [0.00512 in] [0.020 in] [0.00512 in] 101-150 0.2 mm 0.6 mm 3.2 mm 0.4 mm 0.4 mm 0.8 mm 0.4 mm [0.0079 in] [0.024 in] [0.126 in] [0.016 in] [0.016 in] [0.031 in] [0.016 in] 151-170 0.2 mm 1.25 mm 3.2 mm 0.4 mm 0.4 mm 0.8 mm 0.4 mm [0.0079 in] [0.0492 in] [0.126 in] [0.016 in] [0.016 in] [0.031 in] [0.016 in] 171-250 0.2 mm 1.25 mm 6.4 mm 0.4 mm 0.4 mm 0.8 mm 0.4 mm [0.0079 in] [0.0492 in] [0.252 in] [0.016 in] [0.016 in] [0.031 in] [0.016 in] 251-300 0.2 mm 1.25 mm 12.5 mm 0.4 mm 0.4 mm 0.8 mm 0.8 mm [0.0079 in] [0.0492 in] [0.492 in] [0.016 in] [0.016 in] [0.031 in] [0.031 in] 301-500 0.25 mm 2.5 mm 12.5 mm 0.8 mm 0.8 mm 1.5 mm 0.8 mm [0.00984 in] [0.0984 in] [0.492 in] [0.031 in] [0.031 in] [0.0591 in] [0.031 in] B1 Internal conductors B2 External conductors uncoated sea level to 3050 m B3 External conductors uncoated above 3050 m 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 3050 m 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 PCB and wire 56 Texas Instrum5e6nts Analog Engineer's Pocket Reference PCB and wire Figure 41: Self heating of PCB traces on inside layer Example Find the current that will cause a 20Ԩ temperature rise in a PCB trace that is 0.1 inch wide and uses 2 oz/ft2 copper. (Assume traces on outside of PCB.) Answer First translate 0.1 inch to 250 sq. mils. using bottom chart. Next find the current associated with 10Ԩ and 250 sq. mils. using top chart (Answer = 5A). Extracted with permission from IPC-2152, Figure 5-1. For additional information the entire specification can be downloaded at www.ipc.org Texas Instruments Analog Engineer's P57ocket Reference 57 PCB and wire PCB trace resistance for 1 oz Cu 1 100m 10m 5mil 10mil 25mil 50mil 100mil 1m 100µ 10µ 1µ 1 10 100 1000 10000 Trace length (mils) Figure 42: PCB trace resistance vs. length and width for 1 oz-Cu, 25°C Figure 43: PCB trace resistance vs. length and width for 1 oz-Cu, 125°C Example What is the resistance of a 20 mil long, 5 mil wide trace for a 1 oz-Cu thickness at 25°C and 125°C? Answer R25C = 2 mΩ, R125C = 3 mΩ. The points are circled on the curves. 58 Texas Instrume5n8ts Analog Engineer's Pocket Reference PCB and wire PCB trace resistance for 2 oz Cu 1 100m 10m 5mil 10mil 25mil 50mil 100mil 1m 100µ 10µ 1µ 1 10 100 1000 Trace length (mils) 10000 Figure 44: PCB trace resistance vs. length and width for 2 oz-Cu, 25°C Figure 45: PCB trace resistance vs. length and width for 2 oz-Cu, 125°C Example What is the resistance of a 200 mil long, 25 mil wide trace for a 2 oz-Cu thickness at 25°C and 125°C? Answer R25C = 2 mΩ, R125C = 3 mΩ. The points are circled on the curves. Texas Instruments Analog Engineer's5P9 ocket Reference 59 PCB and wire Common package type and dimensions 60 Texas Instrume6n0ts Analog Engineer's Pocket Reference PCB parallel plate capacitance ����� � 2.249 ∗ 10�� ∗ �� ∗ � ∗ � � (81) Where r = PCB dielectric constant (r ≈ 4.2 for FR-4) l = common length of copper planes (mils) w = common width of copper planes (mils) h = separation between copper planes (mils) Example r = 4.2 l = 400 mils w = 400 mils h = 63 mils Answer C = 2.4 pF PCB and wire Capacitance for parallel copper planes w l A h r Figure 46: PCB parallel plate capacitance Texas Instruments Analog Engineer's P61ocket Reference 61 PCB and wire PCB via capacitance and inductance L�nH� � � 197 �1 � �n �4 ∗ � ��� (82) ����� � 1.41 ∗ 10�� ∗ �� ∗ � ∗ �� � �� �� (83) Where h = separation between planes (mils) d = diameter of via hole (mils) r = PCB dielectric constant (r ≈ 4.2 for FR-4) d1 = diameter of the pad surrounding the via (mils) d2 = diameter of the clearance hole in the plane (mils) Example h = 63 mils d = 16 mils r = 4.2 d1 = 32 mils d2 = 63 mils Answer L = 1.20 nH C = 0.39 pF Inductance for via Capacitance for via Figure 47: Inductance and capacitance of via 62 Texas Instru6m2ents Analog Engineer's Pocket Reference PCB and wire Adjacent copper trace capacitance ����� � 2.249 ∗ 10�� � ∗ � ∗ � ����� � 2.249 ∗ 10�� � ∗ ε� ∗ � ∗ � (58) Same layer (59) Different layers Where l = length of copper trace (mils) t = thickness of copper trace (mils) copper thickness (mils) = 1.37 * (number of ounces) ex: 1 oz. copper thickness = 1.37 mils ex: ½ oz. copper thickness = 0.685 mils d = distance between traces (mils) r = PCB dielectric constant (r ≈ 4.2 for FR-4) w = width of copper trace (mils) h = separation between planes (mils) Example l = 100 mils t = 1.37 mils (1 oz. copper) d = 10 mils εr = 4.2 w = 25 mils h = 63 mils Answer C (same layer) = 0.003 pF C (different layers) = 0.037 pF Figure 48: Capacitance for adjacent copper traces Texas Instruments Analog Engineer's P6o3 cket Reference 63 PCB and wire Table 18: Coaxial cable information Type ZO Capacitance / length (pF/feet) Outside diameter (inches) dB attenuation /100 ft at 750 MHz Dielectric type RG-58 RG-8 RG-214/U 9914 RG-6 RG-59/U RG-11/U RG-62/U RG-174 RG-178/U 53.5 Ω 52 Ω 50 Ω 50 Ω 75 Ω 73 Ω 75 Ω 93 Ω 50 Ω 50 Ω 28.8 29.6 30.8 26.0 20 29 17 13.5 31 29 0.195 0.405 0.425 0.405 0.270 0.242 0.412 0.242 0.100 0.071 13.1 5.96 6.7 4.0 5.6 9.7 3.65 7.1 23.5 42.7 Application Test equipment and RF power PE to a few hundred watts, and a couple hundred MHz PE PE RF power to a few kW, up to several hundred MHz PE PF Video and CATV applications. RF to a few hundred watts, up to a few hundred MHz, sometimes to higher PE frequencies if losses can be tolerated PE RF power to a few kW, up to several hundred MHz ASP Used in some test equipment and 100 Ω video applications PE Miniature coax used primarily for test equipment ST interconnection. Usually short runs due to higher loss. 64 Texas Instrume6n4ts Analog Engineer's Pocket Reference Coaxial cable equations C l � 2πε l� �Dd� L l � μ 2π l� �Dd� �� � �CL � 1 2π �με PCB and wire (84) Capacitance per length (85) Inductance per length (86) Characteristic impedance Where L = inductance in henries (H) C = capacitance in farads (F) Z = impedance in ohms (Ω) d = diameter of inner conductor D = inside diameter of shield, or diameter of dielectric insulator ε = dielectric constant of insulator (ε = εr εo ) µ = magnetic permeability (µ = µr µo ) l = length of the cable Figure 49: Coaxial cable cutaway Texas Instruments Analog Engineer's P6o5 cket Reference 65 PCB and wire Table 19: Resistance per length for different wire types (AWG) AWG 36 36 34 34 32 32 30 30 28 28 26 26 24 24 22 22 20 20 18 18 16 16 14 14 Stds Solid 7/44 Solid 7/42 Solid 7/40 Solid 7/38 Solid 7/36 Solid 10/36 Solid 7/32 Solid 7/30 Solid 7/28 Solid 7/26 Solid 7/24 Solid 7/22 Outside diameter in mm 0.005 0.006 0.0063 0.0075 0.008 0.008 0.010 0.012 0.013 0.015 0.016 0.021 0.020 0.024 0.025 0.030 0.032 0.038 0.040 0.048 0.051 0.060 0.064 0.073 0.127 0.152 0.160 0.192 0.203 0.203 0.254 0.305 0.330 0.381 0.409 0.533 0.511 0.610 0.643 0.762 0.813 0.965 1.020 1.219 1.290 1.524 1.630 1.854 Area circular mils mm2 25 0.013 28 0.014 39.7 0.020 43.8 0.022 67.3 0.032 67.3 0.034 100 0.051 112 0.057 159 0.080 175 0.090 256 0.128 250 0.128 404 0.205 448 0.229 640 0.324 700 0.357 1020 0.519 1111 0.562 1620 0.823 1770 0.902 2580 1.310 2828 1.442 4110 2.080 4480 2.285 dc resistance Ω / 1000 ft Ω / km 445 1460 371 1271 280 918 237 777 174 571 164 538 113 365 103 339 70.8 232 64.9 213 43.6 143 41.5 137 27.3 89.4 23.3 76.4 16.8 55.3 14.7 48.4 10.5 34.6 10.3 33.8 6.6 21.8 5.9 19.2 4.2 13.7 3.7 12.0 2.6 8.6 2.3 7.6 66 Texas Instrume6n6ts Analog Engineer's Pocket Reference Table 20: Maximum current vs. AWG PCB and wire Wire gauge Polyethylene Neoprene Polyvinylchloride (semi-ridged) at 80°C Polypropylene Polyethylene (high density) at 90°C Polyvinylchloride Nylon at 105°C Kynar Polyethylene Thermoplastic at 125°C Kapton Teflo n Silicon at 200°C AWG Imax (A) Imax (A) 30 2 3 28 3 4 26 4 5 24 6 7 22 8 9 20 10 12 18 15 17 16 19 22 14 27 30 12 36 40 10 47 55 Note: Wire is in free air at 25°C Imax (A) 3 4 5 7 10 13 18 24 33 45 58 Imax (A) 3 5 6 8 11 14 20 26 40 50 70 Imax (A) 4 6 7 10 13 17 24 32 45 55 75 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 Texas Instruments Analog Engineer's P67ocket Reference 67 PCB and wire Notes 68 Texas Instruments Analog Engineer's Pocket Reference • Thermistor • Resistive temperature detector (RTD) • Diode temperature characteristics • Thermocouple (J and K) SSeennssoor Sensor Texas Instruments Analog Engineer's Pocket Reference 69 Sensor Thermistor RTD Diode Thermocouple Sensor Table 21: Temperature sensor overview 70 70 Texas Instruments Analog Engineer's Pocket Reference Temp range Cost Accuracy Linearity –55°C < T < 150°C Low Good accuracy at one temperature. Less accurate over full range. Very nonlinear. Follows reciprocal of logarithmic function. Construction Less rugged Output range Applications Typically 10s to 100s of kΩ full scale. Very wide variation in resistance. General purpose General Requires excitation –200°C < T < 850°C High Excellent accuracy Fairly linear Nonlinearity < 4.5% of full scale. Relatively simple quadratic function. Depends on Type (can be rugged) 18 to 390 Ω for PT100 180 to 3.9 kΩ for PT1000 Scientific and industrial Requires excitation –55°C < T < 150°C Low Poor accuracy without calibration. –250°C < T < 1800°C Low Good accuracy with polynomial correction. Fairly linear Slope ≈ -2mV/C Slope varies according to current excitation, diode type, and diode processing. Rugged Fairly linear Nonlinearity < 10% of full scale Complex 10th order polynomial Most rugged 0.4 to 0.8V 10s of millivolts Low cost temperature monitor Low cost linear response Requires excitation Industrial temperature measurement Self-powered Requires cold junction comp Sensor Thermistor: Resistance to temperature, Steinhart-Hart equation 1 T � � � b ���R� � c ����R��� (87) Where T = temperature in Kelvin a, b, c = Steinhart-Hart equation constants R = resistance in ohms Convert resistance to temperature for a thermistor Thermistor: Temperature to resistance, Steinhart-Hart equation R � ��� �� � y 2 � �� � ��� �� � y 2 � �� y � � � c 1 T (88) Convert temperature to resistance for a thermistor (89) Factor used in Equation 88 � � ��3bc�� � y� 4 (90) Factor used in Equation 88 Where R = resistance in ohms T = temperature in Kelvin a, b, c = Steinhart-Hart equation constants x, y = Steinhart-Hart factors used in temperature to resistance equation Texas Instruments Analog Engineer's P7o1 cket Reference 71 Sensor RTD equation temperature to resistance R��� � R��1 � A�T � B�T� � ���T � 100�T�� (91) RTD resistance for T<0°C R��� � R��1 � A�T � B�T�� (92) RTD resistance for T>0°C Where Rrtd = resistance of RTD over temperature range of (–200°C < T < 850°C) Ro = 100 Ω for PT-100, 1000 Ω for PT-1000 AO, BO, CO = Callendar-Van Dusen coefficients T = temperature in degrees Celsius (�) RTD equation resistance to temperature (T>0°C) T � �A� � �A�� � �B� 2B� �1 � RR����� (93) RTD resistance for T>0°C Where RRTD = resistance of RTD over temperature range of (–200°C < T < 850°C) Ro = 100 Ω AO, BO, CO = Callendar-Van Dusen coefficients T = temperature in degrees Celsius (�) Table 22: Callendar-Van Dusen coefficients for different RTD standards IEC-751 DIN 43760 BS 1904 ASTM-E1137 EN-60751 JISC 1604 US Industrial Standard D-100 American US Industrial Standard American ITS-90 A0 +3.9083E-3 +3.9739E-3 +3.9787E-3 +3.9692E-3 +3.9888E-3 B0 –5.775E-7 –5.870E-7 –5.8686E-7 –5.8495E-7 –5.915E-7 C0 –4.183E-12 –4.4E-12 –4.167E-12 –4.233E-12 –3.85E-12 Example What is the temperature given an ITS-90 PT100 resistance of 120 Ω? Answer T � ��3.9888 ∙ 10��� � ��3.9888 ∙ 10���� � ����.91� 2���.91� ∙ 10��� ∙ 10��� �1 � 112000� � �0.�� 72 72 Texas Instruments Analog Engineer's Pocket Reference RTD equation resistance to temperature (T<0°C) � � � � �� ������� ��� Sensor (94) RTD resistance for T<0� Where T = temperature in degrees Celsius (�) RRTD = resistance of RTD over temperature range of (T<0�) αI = polynomial coefficients for converting RTD resistance to temperature for T<0� Table 23: Coefficients for 5th order RTD resistance to temperature IEC-751 DIN 43760 BS 1904 ASTM-E1137 EN-60751 JISC 1604 US Industrial Standard US Industrial D-100 Standard American American ITS-90 α0 –2.4202E+02 –2.3820E+02 –2.3818E+02 –2.3864E+02 –2.3791E+02 α1 2.2228E+00 2.1898E+00 2.1956E+00 2.1973E+00 2.2011E+00 α2 2.5857E-03 2.5226E-03 2.4413E-03 2.4802E-03 2.3223E-03 α3 –4.8266E-06 –4.7825E-06 –4.7517E-06 –4.7791E-06 –4.6280E-06 α4 –2.8152E-08 –2.7009E-08 –2.3831E-08 –2.5157E-08 –1.9702E-08 α5 1.5224E-10 1.4719E-10 1.3492E-10 1.4020E-10 1.1831E-10 Example Find the temperature given an ITS-90 PT100 resistance of 60 Ω. Answer � � ��������� � 0�� ∗ �60�� � ����0��� � 00� ∗ �60�� � �������� � 0�� ∗ �60�� � � � �������� � 0�� ∗ �60�� � ����6� 73 Texas Instruments Analog Engineer's Pocket Reference 73 Sensor Diode equation vs. temperature V� � nkT q �n �II� � �� � nkT q �n �II�� (95) Diode voltage Where VD = diode voltage vs. temperature and current n = diode ideality factor (ranges from 1 to 2) k = 1.38 x 10-23 J/K, Boltzmann’s constant T = temperature in Kelvin q = 1.60 x 10-19 C, charge of an electron I = forward diode current in amps IS = saturation current I� � �T��⁄����� �� nqkVT� � (96) Saturation current Where IS = saturation current α = constant related to the cross sectional area of the junction VG = diode voltage vs. temperature and current n = diode ideality factor (ranges from 1 to 2) k = 1.38 x 10-23 J/K, Boltzmann’s constant T = temperature in Kelvin q = 1.60 x 10-19 C, charge of an electron 74 74 Texas Instruments Analog Engineer's Pocket Reference Sensor 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 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 drop vs. temperature 75 Texas Instruments Analog Engineer's Pocket Reference 75 Sensor Type J thermocouples translating temperature to voltage (ITS-90 standard) � V� � � �� ���� ��� (97) Thermoelectric voltage Where VT = thermoelectric voltage T = temperature in degrees Celsius ci = translation coefficients Table 24: Type J thermocouple temperature to voltage coefficients Type J thermocouple temperature to voltage –219� to 760� 760� to 1,200� c0 0.0000000000E+00 2.9645625681E+05 c1 5.0381187815E+01 –1.4976127786E+03 c2 3.0475836930E-02 3.1787103924E+00 c3 –8.5681065720E-05 –3.1847686701E-03 c4 1.3228195295E-07 1.5720819004E-06 c5 –1.7052958337E-10 –3.0691369056E-10 c6 2.0948090697E-13 -- c7 –1.2538395336E-16 -- c8 1.5631725697E-20 -- 76 76 Texas Instruments Analog Engineer's Pocket Reference Sensor Type J thermocouples translating voltage to temperature (ITS-90 standard) � � � � �� �V��� ��� (98) Temperature Table 25: 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 c0 0.000000000E+00 0.000000000E+00 –3.113581870E+03 c1 1.952826800E-02 1.978425000E-02 3.005436840E-01 c2 –1.228618500E-06 –2.001204000E-07 –9.947732300E-06 c3 –1.075217800E-09 1.036969000E-11 1.702766300E-10 c4 –5.908693300E-13 –2.549687000E-16 –1.430334680E-15 c5 –1.725671300E-16 3.585153000E-21 4.738860840E-21 c6 –2.813151300E-20 –5.344285000E-26 -- c7 –2.396337000E-24 5.099890000E-31 -- c8 –8.382332100E-29 -- -- 77 Texas Instruments Analog Engineer's Pocket Reference 77 Sensor Type K thermocouples translating temperature to voltage (ITS-90 standard) � V� � � �� ���� ��� � V� � �� �� ����� � ��e����������������� ��� (99) Thermoelectric voltage for T<0� (100) Thermoelectric voltage forT>0� Where VT = thermoelectric voltage T = temperature in degrees Celsius ci = translation coefficients α0, α1 = translation coefficients Table 26: Type K thermocouple temperature to voltage coefficients –219°C to 760°C 760°C to 1,200°C c0 0.0000000000E+00 –1.7600413686E+01 c1 3.9450128025E+01 3.8921204975E+01 c2 2.3622373598E-02 1.8558770032E-02 c3 –3.2858906784E-04 –9.9457592874E-05 c4 –4.9904828777E-06 3.1840945719E-07 c5 –6.7509059173E-08 –5.6072844889E-10 c6 –5.7410327428E-10 5.6075059059E-13 c7 –3.1088872894E-12 –3.2020720003E-16 c8 –1.0451609365E-14 9.7151147152E-20 c9 –1.9889266878E-17 –1.2104721275E-23 c10 –1.6322697486E-20 -- α0 -- 1.1859760000E+02 α1 -- –1.1834320000E-04 78 78 Texas Instruments Analog Engineer's Pocket Reference Sensor Type K thermocouples translating voltage to temperature (ITS-90 standard) � � � � �� �V��� ��� (101) Temperature Table 27: Type K thermocouple voltage to temperature coefficients –219°C to 0°C c0 0.0000000E+00 c1 2.5173462E-02 c2 –1.1662878E-06 c3 –1.0833638E-09 c4 –8.9773540E-13 c5 –3.7342377E-16 c6 –8.6632643E-20 c7 –1.0450598E-23 c8 –5.1920577E-28 c9 -- 0°C to 760°C 0.0000000E+00 2.5083550E-02 7.8601060E-08 –2.5031310E-10 8.3152700E-14 –1.2280340E-17 9.8040360E-22 –4.4130300E-26 1.0577340E-30 –1.0527550E-35 760°C to 1,200°C –1.3180580E+02 4.8302220E-02 –1.6460310E-06 5.4647310E-11 –9.6507150E-16 8.8021930E-21 –3.1108100E-26 ---- 79 Texas Instruments Analog Engineer's Pocket Reference 79 Sensor Table 28: Seebeck coefficients for different material Material Aluminum Antimony Bismuth Cadmium Carbon Constantan Copper Germanium Seebeck coefficient 3.5 47 –72 7.5 3.0 –35 6.5 300 Material Gold Iron Lead Mercury Nichrome Nickel Platinum Potassium Seebeck coefficient 6.5 19 4.0 0.6 25 –15 0 –9.0 Material Rhodium Selenium Silicon Silver Sodium Tantalum Tellurium Tungsten Seebeck coefficient 6.0 900 440 6.5 –2.0 4.5 500 7.5 Note: Units are µV/°C. All data at temperature of 0°C 80 80 Texas Instruments Analog Engineer's Pocket Reference A/AD/Dcocnonvveerrssiioon • 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 Texas Instruments Analog Engineer's Pocket Reference 81 A/D conversion Numbering systems: Binary, decimal, and hexadecimal 2(1000) + 3(100) + 4(10) + 1(1) = 2,341 MSD = Most significant digit Example conversion: Binary to decimal Binary Decimal = 8+4 +0 +1 Example conversion: Decimal to binary Decimal Binary = 128 + 64 + 32 + 8 + 4 = 236 A/D conversion 82 82 Texas Instruments Analog Engineer's Pocket Reference Example conversion: Binary to hexadecimal A/D conversion 128 + 64 + 16 + 8 + 1 = 217 8 + 4 + 1 = 13 (D) 8 + 1 = 9 161 160 D9 MSD 208 + 9 = 217 Example Conversion: Hexadecimal to binary Decimal (Base-10) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Hexadecimal (Base-16) 0 1 2 3 4 5 6 7 8 9 A B C D E F Hexadecimal x163 x162 x161x160 2 6 AF = MSD 2(4096) + 6(256) + 10(16) + 16(1) = 9903 Decimal 16 9903 R = 15 (F) 16 618 R = 10 (A) 16 38 R = 6 (6) 16 38 R = 2 (2) 83 Texas Instruments Analog Engineer's Pocket Reference 83 A/D conversion Table 29: Different data formats Code Binary 11111111 11000000 10000000 01111111 01000000 00000000 Straight binary Decimal value 255 192 128 127 64 0 Offset binary Decimal value 127 64 0 –1 –64 –128 Converting two’s complement to decimal: Negative number example SIGN x4 x2 x1 Step 1: Check sign bit This case is negative 10 11 MSD 2’s complement Decimal value –1 –64 –128 127 64 0 Step 2: Invert all bits 01 00 Step 3: Add 1 01 01 Final result –(4+1) = –5 Converting two’s complement to decimal: Positive number example SIGN x4 x2 x1 Just add bit weights Final result 01 01 MSD 4+1 = 5 84 84 Texas Instruments Analog Engineer's Pocket Reference A/D conversion Table 30: LSB voltage vs. resolution and reference voltage 1.024 8 4 mV 10 1 mV Reference voltage 1.25 2.048 4.88 mV 8 mV 1.22 mV 2 mV 2.5 9.76 mV 2.44 mV Resolution 12 250 µV 14 52.5 µV 16 15.6 µV 18 3.91 µV 305 µV 76.3 µV 19.1 µV 4.77 µV 500 µV 125 µV 31.2 µV 7.81 µV 610 µV 152.5 µV 38.14 µV 9.53 µV 20 0.98 µV 22 244 nV 24 61 nV 1.19 µV 299 nV 74.5 nV 1.95 µV 488 nV 122 nV 2.384 µV 596 nV 149 nV Table 31: LSB voltage vs. resolution and reference voltage Reference voltage Resolution 3 8 11.7 mV 10 2.93 mV 12 732 µV 14 183 µV 16 45.77 µV 18 11.44 µV 20 2.861 µV 22 715 nV 24 179 nV 3.3 12.9 mV 3.222 mV 806 µV 201 µV 50.35 µV 12.58 µV 3.147 µV 787 nV 196 nV 4.096 16 mV 4 mV 1 mV 250 µV 62.5 µV 15.6 µV 3.91 µV 976 nV 244 nV 5 19.5 mV 4.882 mV 1.221 mV 305 µV 76.29 µV 19.07 µV 4.768 µV 1.192 µV 298 nV Texas Instruments Analog Engineer's P8o5cket Reference 85 A/D conversion DAC definitions Resolution = n The number of bits used to quantify the output Codes = 2n The number of input code combinations Reference voltage = VREF Sets the LSB voltage or current size and converter range LSB = VREF / 2n The output voltage or current step size of each code Full-scale code = 2n – 1 The largest code that can be written Full-scale voltage = VREF – 1LSB Full-scale output voltage of the DAC Transfer function = VREF x (code/ 2n) Relationship between input code and output voltage or current Reference voltage = 5V Full‐scale voltage = 4.98V Output voltage (V) Resolution = 19mV Number of codes = 2n Figure 51: DAC transfer function Full‐scale code = 255 Resolution = 8bits 86 Texas Instrume8n6ts Analog Engineer's Pocket Reference A/D conversion ADC definitions Resolution = n The number of bits used to quantify the output Codes = 2n The number of input code combinations Reference voltage = VREF Sets the LSB voltage or current size and converter range LSB = VREF / (2n – 1) The voltage step size of each code. Note that some topologies may use 2n as opposed to 2n – 1 in the denominator. Full-scale code = 2n – 1 The largest code that can be written. Full-scale voltage = VREF Full-scale output voltage of the DAC. Note that the full-scale voltage will differ if the alternative definition for resolution is used. Transfer function = VREF x (code/ 2n) Relationship between input code and output voltage or current Figure 52: ADC transfer function 87 Texas Instruments Analog Engineer's Pocket Reference 87 A/D conversion Quantization error of ADC Figure 53: 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 converter is ½ LSB. The quantization error signal the difference between the actual voltage applied and the ADC output (Figure 53). The rms of the quantization signal is 1LSB⁄√12 88 88 Texas Instruments Analog Engineer's Pocket Reference A/D conversion Signal-to-noise ratio (SNR) from quantization noise only MaxRMSSignal � FSR/2 √2 � 1LSB � 2��� √2 RMSNoise � 1LSB √12 from quantization only SNR � MaxRMSSignal RMSNoise � 1LSB � 2���/√2 1LSB⁄√12 � 2���√6 SNR�dB� � 2�log�SNR� � �2� log�2��N � 2�log �√26� (102) (103) (104) (105) SNR�dB� � 6��2N � 1��6 (106) Where FSR = full-scale range of the A/D converter 1LSB = the voltage of 1LSB, VREF/2n N = the resolution of the A/D converter MaxRMSSignal = the rms equivalent of the ADC’s full-scale input RMSNoise = the rms noise from quantization SNR = the ratio of rms signal to rms noise Example What is the SNR for an 8-bit A/D converter with 5V reference, assuming only quantization noise? Answer SNR � 2���√6 � 2� ��√6 � 314 SNR�dB� � 2�log�314� � 4��� dB SNR�dB� � 6��2�8� � 1��6 � 4��� dB 89 Texas Instruments Analog Engineer's Pocket Reference 89 A/D conversion Total harmonic distortion (Vrms) THD � �MRMaxSRDMisStoSrigtinoanl� � �V�� � V�� � V�� V� � � � V�� THD�dB� � ��log �MRMaxSRDMisStoSrigtinoanl� (107) (108) Where THD = total harmonic distortion, the ratio of the rms distortion to the rms signal RMSDistortion = the rms sum of all harmonic components MaxRMSSignal = the rms value of the input signal V1 = the fundamental, generally the input signal V2, V3, V4, …Vn = harmonics of the fundamental Figure 54: Fundamental and harmonics in Vrms 90 90 Texas Instruments Analog Engineer's Pocket Reference Total harmonic distortion (dBc)   ൌ ͳͲŽ‘‰ ൤ͳͲቀୈଵ଴మቁ ൅ ͳͲቀୈଵ଴యቁ ൅ ͳͲቀୈଵ଴రቁ ൅ ‫ ڮ‬൅ ͳͲቀୈଵ଴౤ቁ൨ A/D conversion (109) Where THD = total harmonic distortion. The ratio of the rms distortion to the rms signal D1 = the fundamental, generally the input signal. This is normalized to 0 dBc D2, D3, D4, …Dn = harmonics of the fundamental measured relative to the fundamental Figure 55: Fundamental and harmonics in dBc Example Determine THD for the example above. Answer   ൌ ͳͲŽ‘‰ ൤ͳͲቀିଵଽ଴ଶቁ ൅ ͳͲቀିଵ଻଴ହቁ ൅ ͳͲቀିଵଽ଴ହቁ ൅ ‫ ڮ‬൅ ͳͲቀିଵଵ଴ଵ଴ቁ൨   ൌ െ͹ͶǤ͹͸† 91 Texas Instruments Analog Engineer's Pocket Reference 91 A/D conversion Ac signals Signal-to-noise and distortion (SINAD) and effective number of bits (ENOB) SINAD�dB� � 20 log �√RMSNoMisaex�R�MRSSMigSnDaisl �or�ion�� (110) SINAD�dB� � �20log ��10������������ � 10������������ (111) �N�B � SINAD�dB� � 6.02 1.76dB (112) Where MaxRMSSignal = the rms equivalent of the ADC’s full-scale input RMSNoise = the rms noise integrated across the A/D converters RMSDistortion = the rms sum of all harmonic components 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 RMSDistortion = 50 µVrms RMSNoise = 100 µVrms Answer SNR�dB� � 20 log �110.706μVVrrmmss� � ��.� dB THD�dB� � 20 log �15.076μVVrrmmss� � � �0.� dB SINAD�dB� � 20 log ���100 1.76V rms μVrms�� � �50 μVrms��� � ��.� dB SINAD�dB� � �20 log ��10�����.�� ��� � 10�����.�� ���� � ��.� dB �N�B � ��.�dB � 1.76dB 6.02 � 1�.65 92 92 Texas Instruments Analog Engineer's Pocket Reference Dc signals Noise free resolution and effective resolution Noise�ree�eso��tion � �o�� �PeaktoPea2k�NoiseinLSB� ���e�ti�e�eso��tion � �o�� �rmsNo2is�einLSB� PeaktoPeakNoiseinLSB � 6.6 � rmsNoiseinLSB A/D conversion (113) (114) (115) ���e�ti�e�eso��tion � Noise�ree�eso��tion � 2.7 (116) Note: The maximum effective resolution is never greater than the ADC resolution. For example, a 24-bit converter cannot have an effective resolution greater than 24 bits. Example What is the noise-free resolution and effective resolution for a 24-bit converter assuming the peak-to-peak noise is 7 LSBs? Answer Noise�ree�eso��tion � �o�� �27��� � 2�.2 ���e�ti�e�eso��tion � �o�� � 2�� 7 � � 2�.� 6.6 ���e�ti�e�eso��tion � 2�.2 � 2.7 � 2�.� 93 Texas Instruments Analog Engineer's Pocket Reference 93 A/D conversion Figure 56: Settling time for RC circuit-related to A/D converters Table 32: Conversion accuracy achieved after a specified time Settling time in time constants (NTC) 1 2 3 4 5 6 7 8 9 Accuracy in bits 1.44 2.89 4.33 5.77 7.21 8.66 10.10 11.54 12.98 Settling time in time constants (NTC) 10 11 12 13 14 15 16 17 18 Accuracy in bits 14.43 15.87 17.31 18.76 20.20 21.64 23.08 24.53 25.97  ൌ Ž‘‰ଶሺ‡ି୒౐ిሻ (117) Where N = the number of bits of accuracy the RC circuit has settled to after NTC number of time constants. NTC = the number of RC time constants 94 94 Texas Instruments Analog Engineer's Pocket Reference A/D conversion Table 33: Time required to settle to a specified conversion accuracy Accuracy in bits (N) 8 9 10 11 12 13 14 15 16 Settling time in time constants (NTC) 5.55 6.24 6.93 7.62 8.32 9.01 9.70 10.40 11.09 Accuracy in bits (N) 17 18 19 20 21 22 23 24 25 Settling time in time constants (NTC) 11.78 12.48 13.17 13.86 14.56 15.25 15.94 16.64 17.33 N�� � ������ (118) Where NTC = the number of time constants required to achieve N bits of settling N = the number of bits of accuracy 95 Texas Instruments Analog Engineer's Pocket Reference 95 Notes 96 Texas Instruments Analog Engineer's Pocket Reference IMPORTANT NOTICE Texas Instruments Incorporated and its subsidiaries (TI) reserve the right to make corrections, enhancements, improvements and other changes to its semiconductor products and services per JESD46, latest issue, and to discontinue any product or service per JESD48, latest issue. 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