Application Report
SLVA278 – September 2007
How to use TI’s Set of Power Rulers
Juergen Schneider ......................................................................................... High Performance Analog
ABSTRACT
While many experienced designers know all there is to know about power design, a
growing number of designers are confronted daily with unfamiliar power topics that are
either completely new to them, or last encountered years before.
The Power Rulers and this application note provide important rules of thumb and many
approximated values. While they are sufficient for a first estimate, the use of more
accurate numbers and equations should be considered when exact results are
required.
This document describes how to use the four Power Rulers, and gives additional
background information for the topics covered on the rulers.
1
2
Contents
General Information ................................................................................
1.1
Separation of Dedicated Rulers .........................................................
1.2
Scales .......................................................................................
1.3
Triangle Method to Derive Formula Variations ........................................
The Basic Ruler .....................................................................................
2.1
SI Prefixes ..................................................................................
2.2
Unit Conversions: Length, Volume, Speed, Power, Energy, Weight, Area ........
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3
3
3
4
4
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Unit Conversion for Air Flow Velocity ................................................... 4
Physical and Mathematical Constants .................................................. 5
Temperature Conversion (°C ↔ °F)..................................................... 5
Voltage, Current and Power Ratios Expressed in Decibel ........................... 5
Wire Resistance Calculation ............................................................. 6
AWG Conversion into Metric Wire Sizes .............................................. 7
PCB Trace Thickness Conversion (oz ↔ mils ↔ μm) and Resistance
Calculation .................................................................................. 7
The Voltage Regulator/Switching Converter Ruler ............................................. 8
3.1
Calculation of Output Voltage Divider ................................................... 8
3.2
Calculation of Hold-Up Capacitance .................................................... 8
3.3
Most Commonly Used Topologies for Non-Isolated Switching
Regulators................................................................................. 10
3.4
Voltage Transfer Function for the Formerly Mentioned 4 Basic
Topologies ................................................................................ 10
3.5
Basic Time and Frequency Relation for PWM Based Circuits ..................... 10
3.6
Current and Voltage Stress for the 4 Basic Topologies ............................. 11
The Capacitor/Inductor Ruler .................................................................... 11
4.1
Table for the Preferred Values for Inductors / Capacitors .......................... 11
4.2
Graph of Capacitor / Inductor Charge ................................................ 12
4.3
Accuracy and Resolution for Capacitor/Inductor Charging ........................ 12
4.4
Capacitors/Inductors: Symbol, Unit, Basic Laws and Equations................... 13
4.5
Graph of PCB Trace Inductance ....................................................... 13
The Resistor Ruler ................................................................................ 14
5.1
Table for the Preferred Values for Resistors ......................................... 14
5.2
Resistors: Symbol, Unit, Basic Laws and Equations ............................... 15
5.3
Chip Resistors: Basic Data ............................................................. 15
5.4
MELF Resistors: Basic Data ............................................................ 15
References ......................................................................................... 15
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
4
5
6
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
2
mm- and Inch-Scales as Used on All Rulers ................................................... 3
The Triangular Method Allows Easy Derivation of an Equation for Different Items ........ 3
Front Side of the Basic Ruler: Unit Prefixes, Conversions and Constants.................. 4
Rear Side of the Basic Ruler: Conversions, Wire Sizes and PCB Trace Resistance .... 5
Calculation of the Resistance of a Wire ......................................................... 6
PCB Trace Resistance Estimation by Counting Consecutive Squares ..................... 7
The Front Side of the Voltage Regulator/Switching Converter Ruler: Output Voltage
Divider and Hold-Up Capacitance ................................................................ 8
The Rear Side of the Voltage Regulator/Switching Converter Ruler: Switching
Regulator Topologies and Their Basic Properties .............................................. 9
The Duty Cycle D as the Ratio of the ON-Time tON to the Switching Period T ........... 10
The Front Side of the Capacitor/Inductor Ruler: Preferred Capacitors/Inductors
Values and RC-/LC- Charging................................................................... 11
The Rear Side of the Capacitor/Inductor Ruler: Basics of Capacitors/Inductors and
Parasitic Inductance of a PCB Trace ........................................................... 12
PCB Trace Inductance Determined by the Width and Length of the Trace and it's
Distance from Ground ............................................................................ 13
The Front Side of the Resistor Ruler: Preferred Resistor Values (1%–10%) ............. 14
The Rear Side of the Resistor Ruler: Resistor’s Basics, Dimensions and Electrical
Limits................................................................................................ 14
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General Information
1
General Information
1.1
Separation of Dedicated Rulers
The 4 rulers are supplied on a plastic ring that may be opened to separate them from each other.
1.2
Scales
Each ruler has a 15-cm metric scale and a 6-inch scale to allow easy measurement of length in the Metric
as well as in the Imperial, or English system.
Figure 1. mm- and Inch-Scales as Used on All Rulers
1.3
Triangle Method to Derive Formula Variations
When mathematical equations are shown in a triangle, the value in question can be simply derived by
covering it. The remaining items form the appropriate equation, as shown in Figure 2.
Figure 2. The Triangular Method Allows Easy Derivation of an Equation for Different Items
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The Basic Ruler
2
The Basic Ruler
Figure 3. Front Side of the Basic Ruler: Unit Prefixes, Conversions and Constants
2.1
SI Prefixes
This table lists all the SI prefixes from femto to tera.
2.2
Unit Conversions: Length, Volume, Speed, Power, Energy, Weight, Area
Conversions are given in this section for length, volume, speed, power, energy, weight/mass and area. In
addition to these conversions, the following conversions are also useful:
• Length:
– 1 foot (ft, ‘) = 1/3 yd = 12 in = 0.3048 m
– 1 micron (μm, μ) = 1 micrometer (μm, μ)
Sometimes the symbol "um" is used instead of "μm".
• Area:
– 1 square foot (sq ft, ft2) = 144 in2 = 929.03 cm2
2.3
Unit Conversion for Air Flow Velocity
Air-flow velocity units are often encountered when dealing with SOA (safe operating area) graphs for
DC/DC converters. LFM stands for Linear Feet per Minute.
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The Basic Ruler
2.4
Physical and Mathematical Constants
•
•
Permittivity and permeability of free space
Pi and Euler’s number
Figure 4. Rear Side of the Basic Ruler: Conversions, Wire Sizes and PCB Trace Resistance
2.5
Temperature Conversion (°C ↔ °F)
This diagram relates the 0° and the 100° points for each scale to each other as well as some exemplary
values. To convert other values, Equation 1 (also printed above the temperature diagram on the ruler) can
be used. The term 5/9 in this equation comes from the fact that the freezing and the boiling points of water
are 100° apart on Celsius scale, and 180° apart on the Fahrenheit scale.
5
T°C = (T°F - 32 )
9
(1)
2.6
Voltage, Current and Power Ratios Expressed in Decibel
Ratios such as amplifier gain or filter attenuation are often expressed in decibels (dB). The table on the
ruler lists example values calculated using these equations:
X dB + 20 log10
ǒ Ǔ
X
X0
(2)
for voltage and current ratios, and
X dB + 10 log10
ǒ Ǔ
X
X0
(3)
for power ratios.
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The Basic Ruler
2.7
Wire Resistance Calculation
This section gives Equation 4 for calculating the resistance R for a wire with a length l and a cross
sectional area A, as shown in Figure 5.
re
i
W l
A
d
Figure 5. Calculation of the Resistance of a Wire
R=
r´l
A
(4)
If the diameter of the wire is given instead of it’s cross sectional area, A can be calculated using
Equation 5.
2
A+p d
4
(5)
ρ(rho) in Equation 4 is the electrical resistivity.
For copper, a resistivity value of ρCopper ≈ 1.7μΩcm is given at a temperature of 20°C.
The temperature dependency of the electrical resistivity is expressed by its temperature coefficient TC.
For copper, this is
TC ò
[ 0.0039 K*1
Copper
(6)
The resistance R(T) of a copper wire at a specific temperature T other than 20°C can than be calculated
using Equation 7.
R(T) + R(20 °C)
ƪ1 ) TCò
ǒT * 20 °C)ƫ.
(7)
2
Example: Copper wire with l = 10 cm, A = 0.14 mm
Wire Resistance at 20°C:
r ´ l 1.7 mW / cm ´ 10 cm
»
» 12 m W
R( 20°C) =
A
0.0014 cm 2
(8)
Wire Resistance at 120°C:
[
]
R(120 °C) = R( 20°C) ´ 1 + TC r ´ (120 °C - 20 °C )
°
R(120 C) [ 12 mW
ƪ1 ) 0.0039 K*1
°
R(120 C) [ 16.7 mW
6
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100 Kƫ
(9)
(10)
(11)
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The Basic Ruler
2.8
AWG Conversion into Metric Wire Sizes
The ruler table allows conversion of the American wire gauge (AWG) sizes into the cross sectional area A
and the diameter d given in metric units.
AWG relates to the diameter d of the wire. The diameter d(n) of a number n AWG wire can be calculated
using Equation 12.
d(n) + 0.005 inch
36 * n
39
92
(12)
Using this equation, the diameter of an AWG26 wire can be calculated.
d(AWG26) + 0.005 inch
2.9
92
36 * 26
39
[ 0.005 inch
3.19 [ 0.016 inch [ 0, 4 mm
(13)
PCB Trace Thickness Conversion (oz ↔ mils ↔ μm) and Resistance Calculation
This table shows the relationship of the different units that describe the thickness of PCB traces. While the
mils and μm are direct units of length and thickness, the use of “ounce per square foot” is based on the
fact that the copper foil with a specific thickness has a specific area weight, expressed in the table as
"copper weight". The copper thickness t (in mils) can then be calculated using Equation 14, by using the
area weight in ounces per square foot.
Copper Thickness [ 1.35 Copper Area Weight
(14)
In common PCB-industry terminology, the unit oz/ft2 is often abbreviated as oz (ounce).
Although Equation 4 can be used to calculate the resistance of a PCB trace, a simple approximation
method is shown in Figure 6 and Equation 15. By counting the number n of squares of a PCB trace and
multiplying it by the sheet resistance RS given in the copper thickness table, a reasonably-accurate
estimate can be calculated.
R + n RS
(15)
3
w
2
t
w
1
w
w
B
C
P
Figure 6. PCB Trace Resistance Estimation by Counting Consecutive Squares
By using this approach, the resistance of a 2 ounce copper trace in the example shown in Figure 6 can be
estimated to be:
R = n ´ R S » 3 sq. ´ 0.25 mW / sq. = 0.75 mW
(16)
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The Voltage Regulator/Switching Converter Ruler
3
The Voltage Regulator/Switching Converter Ruler
Figure 7. The Front Side of the Voltage Regulator/Switching Converter Ruler:
Output Voltage Divider and Hold-Up Capacitance
3.1
Calculation of Output Voltage Divider
These equations are targeted for voltage regulators (linear regulators or switch-mode voltage regulators)
with a ground-referenced voltage reference (VREF), found in almost all modern LDOs, DC/DC controllers
and DC/DC converters. Equation 17 can be used to determine the ratio of the output voltage-divider
resistors. When R1 has been given, Equation 18 is used to calculate the value of the ground-connected
resistor R2. All equations are based on the assumption that the feedback-pin current can be neglected.
The following examples show the use of both equations.
The output voltage VOUT should be 5 V; the internal reference voltage is 1.25 V:
V
R1
+ OUT * 1 + 5 V * 1 + 3.
1.25 V
R2
VREF
(17)
In the second example the output voltage VOUT should be 1.8 V; the internal reference voltage is 0.8 V,
and R1 has been selected to be 1 kΩ.
VREF
0 .8 V
´ R1 =
´ 1kW = 800 W
R2 =
1 .8 V - 0 .8 V
VOUT - VREF
(18)
The nearest resistor value from the E96 series of 1% resistors would be 806 Ω according to the tables on
the resistor ruler.
3.2
Calculation of Hold-Up Capacitance
Designers are confronted with the topic of hold-up capacitance when designing systems that must be able
to operate for a certain time after the input power goes away (for example to store important data or to
switch over to a redundant system). For this so-called hold-up time tHold-Up, the converter output voltage
VOUT must stay in regulation.
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The Voltage Regulator/Switching Converter Ruler
Off-line power supplies, applications of DC/DC converters in telecom/datacom, and embedded computers
often have this requirement. To operate the application during tHold-Up, the energy is taken from the input
bulk capacitor C of the converter. To be used in that way, a converter must be able to operate with a
much lower voltage at it’s input than the nominal input voltage VIN. This lower voltage is defined by the
undervoltage lockout level; the point where the converter switches off (VUVLO_OFF).
The example shown in Equation 19 calculates the minimum required hold-up capacitance CMin for an
off-line power supply with a boost PFC in front of the DC/DC converter. The output capacitor of the boost
PFC acts as the hold-up capacitor for the DC/DC converter in this case. When the mains voltage is
available the capacitor voltage is regulated by the boost PFC to be within a range of 370 V–410 V. It
should be furthermore assumed, that the DC/DC converter has an efficiency η of 85% and is required to
operate over a given hold-up time tHold-Up of 15 ms with an output power POUT of 200W. VUVLO_OFF of the
DC/DC converter is set to 280 V.
POUT ´ t Hold-Up
200 W ´ 15 ms
» 120 mF
C min = 2 ´
= 2´
2
2
85% ´ (370 V ) 2 - (280 V ) 2
h ´ VIN - VUVLO _ OFF
(
)
[
]
(19)
Including the effects of initial tolerance tolInitial and aging tolAging, the final minimum required capacitance
CMin_tol can be calculated according to Equation 20 based on 20% initial tolerance and tolerance for aging
of 20% too.
CMin
120 mF
C Min_tol +
+
[ 188 mF.
(1
*
0.2)
(1 * 0.2)
ǒ1 * tolInitialǓ ǒ1 * tolAgingǓ
(20)
The table of preferred values for capacitors (section 4.1 on the capacitor / inductor ruler) gives 220 μF as
the next fitting capacitance value out of the E12 series.
Figure 8. The Rear Side of the Voltage Regulator/Switching Converter Ruler:
Switching Regulator Topologies and Their Basic Properties
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The Voltage Regulator/Switching Converter Ruler
3.3
Most Commonly Used Topologies for Non-Isolated Switching Regulators
The four figures show the basic topologies for the power stage of the following types of converters:
– Buck Converter
(Step-Down)
VOUT < VIN
– Boost Converter
(Step-Up)
VOUT > VIN
– Buck-Boost Converter (Inverting Step-Up/Down)
VIN ≥ – VOUT ≥ VIN
– SEPIC Converter
VIN ≥ VOUT ≥ VIN
(Step-Up/Down)
All topologies are shown in a non-synchronous mode configuration, for their synchronous-mode
counterparts the diode D1 is replaced by a second active switch.
The term Buck-Boost is inconsistently used in current technical literature. The initial meaning in reference
[1] (See References) followed in this document is that VOUT can be higher (boost) or lower (buck) in
magnitude than VIN and that the polarity of VOUT is inverted with respect to the polarity of VIN. In contrast,
the use of Buck-Boost is also found in relation to circuits not inverting the polarity. Reference [2] in
References is an example. In addition to this, a SEPIC can do the same voltage conversion without
changing the polarity. SEPIC stands for Single-Ended Primary Inductance Converter.
3.4
Voltage Transfer Function for the Formerly Mentioned 4 Basic Topologies
This graph allows to convert a VOUT/VIN ratio directly into the required duty cycle of the DC/DC controller
or converter. This is especially helpful, when limits like maximum duty cycle DMax or minimum controllable
ON-time tON comes into place and in cases where the usability of a dedicated DC/DC controller/converter
for the application needs to be verified. The knowledge of the duty cycle is furthermore required for
estimating the maximum drain current of Q1 as outlined in section 3.6.
The curves and the equations for the duty cycle are valid for the ideal and therefore loss-less case and for
the continuous conduction mode only! Especially under the condition of full output load, the practical
required duty cycle will be larger than that given for the ideal case.
3.5
Basic Time and Frequency Relation for PWM Based Circuits
This section shows how the duty cycle D, the switching frequency f, the switching period T and the times
during which the switch Q1 (see section 3.3) is switched ON (ON-time tON) respectively OFF (OFF-time
tOFF ) are related to each other.
tON
tOFF
T
Figure 9. The Duty Cycle D as the Ratio of the ON-Time tON to the Switching Period T
The basic relations are:
T + t ON ) t OFF + 1
ƒ
(21)
D+
t ON
+ t ON
T
ƒ
(22)
In addition to that, the following relations should be kept in mind:
10
D' = 1 - D
(23)
t OFF = (1 - D) ´ T = D'´T
(25)
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t ON = D ´ T
(24)
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The Capacitor/Inductor Ruler
3.6
Current and Voltage Stress for the 4 Basic Topologies
The given equations are rules of thumb and allow a first estimation of the component’s stress. As already
mentioned in section 3.4, the effect of losses are neglected. The inductor ripple current and peaks caused
by parasitics needs to be considered additionally.
4
The Capacitor/Inductor Ruler
Figure 10. The Front Side of the Capacitor/Inductor Ruler:
Preferred Capacitors/Inductors Values and RC-/LC- Charging
4.1
Table for the Preferred Values for Inductors / Capacitors
The capacitance and inductance values are very often taken out off the E3, E6 or E12 series. The number
following the E specifies the number of logarithmic steps per decade. Each E-series is related to a
dedicated tolerance of the component:
• E3 — 50%; included within the E6 and the E12 series,
• E6 — 20%, included within the E12 series,
• E12 —10%.
Exceptions from this assignment can be found for instance for some types of electrolytic capacitors which
are offered with values out off the E12-series (which means usually a tolerance of 10%), although the
devices have a tolerance of 20%.
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The Capacitor/Inductor Ruler
4.2
Graph of Capacitor / Inductor Charge
The graph represents the charging curves when applying a unit step function (from 0 to V0) to an RCrespectively an RL- circuit.
The maximum rate of change for the capacitor voltage vC(t) respectively for the inductor current IL(t)
happens at the beginning of the charging cycle at t = 0 and equals to
V0
t for the capacitor charging, respectively to
•
V0
•
4.3
R
t for the inductor charging.
Accuracy and Resolution for Capacitor/Inductor Charging
The table lists the:
• Ratio (%) between the capacitor voltage vC(t) or inductor current IL(t) at a dedicated time t and their
maximum achievable value at t = ∞.
• Accuracy (dB or bit) of vC(t) or IL(t) at a dedicated time t referred to their maximum achievable value at
t = ∞.
Figure 11. The Rear Side of the Capacitor/Inductor Ruler:
Basics of Capacitors/Inductors and Parasitic Inductance of a PCB Trace
12
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The Capacitor/Inductor Ruler
4.4
Capacitors/Inductors: Symbol, Unit, Basic Laws and Equations
Especially the given units and basic laws help to remember the basic behavior of a capacitor and an
inductor.
Equation 26 and Equation 27 allows for example to derive easily, how much the capacitor voltage
changes during a given time:
1 F + 1 As
1V
(26)
means simply that a constant current of 1 A applied to a capacitor with 1 F changes the capacitor voltage
by 1 V in 1 s. Because this seems to be kind of unrealistic case, a better example can be achieved by
multiplying the left and the right side of the equation with μ:
1 A ms
1 mF +
,
1V
(27)
representing now the same 1-V change, but based on applying 1 A to a capacitor of 1 μF for a time of 1
μs.
The same way of using the unit of a device can be applied for inductors.
1 H + 1 Vs
1A
1 V ms
1 mH +
1A
(28)
(29)
where Equation 29 can be translated into the fact that applying a constant voltage of 1V across an
inductor of 1 μH over a time of 1 μs causes the inductor current to change by 1 A.
4.5
Graph of PCB Trace Inductance
This graph should remind all designers dealing with fast changing voltages or currents, that each PCB
trace has a parasitic inductance associated to it. Switching regulators are a good example for this. While it
is clear, that the power path where the large currents flow needs to be designed by keeping the rules of
minimizing parasitics, this is very often overlooked for parts of the circuit where no fast currents changes
or no high currents are (expected). The connection of a MOSFET driver output to the gate of a MOSFET
is a good example for this. Unwanted inductance can cause significant ringing and can slow-down the
switching of the MOSFET. The curves for the trace inductance per cm of trace length show, that the
differences between thin and wide traces (w) as well as between ground planes placed near of far away
(h) from the trace are significant.
h
w
B
C
P
Figure 12. PCB Trace Inductance Determined by the Width and Length of the Trace and it's Distance from
Ground
There are a number of different methods for estimating/calculating trace inductance as outlined in
references [3] and [4] (See References). They all are suffering from the fact that they are valid under
certain conditions only.
For the purpose of this document the 2D Field Solver software Si8000m/Si9000e from Polar Instruments
(www.polarinstruments.com) was therefore used. The Si8000m provides an additional EXCEL-Interface
for integration into an EXCEL Spreadsheet to simulate various geometry scenarios. Further background
information can be found in [5].
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The Resistor Ruler
5
The Resistor Ruler
Figure 13. The Front Side of the Resistor Ruler: Preferred Resistor Values (1%–10%)
5.1
Table for the Preferred Values for Resistors
Standard resistor values are often taken from the E12, E24, E48 and E96 series. The number following
the "E" specifies the number of logarithmic steps per decade. Each E-series is related to a dedicated
tolerance of the resistor:
• E12 — 10%; included within the E24 series,
• E24 — 5%
• E48 — 2%; included within the E96 series,
• E96 — 1%
Figure 14. The Rear Side of the Resistor Ruler: Resistor’s Basics, Dimensions and Electrical Limits
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References
5.2
Resistors: Symbol, Unit, Basic Laws and Equations
The well-known basic Ohm’s Law and its derivatives are used to calculate the power P, when the
resistance R and the voltage V respectively the current I are known.
5.3
Chip Resistors: Basic Data
The table lists the basic specification found for chip resistors as a starting point for design activities. Data
sheets needs to be used for ultimate values.
• Type: related to inch-size
– The first two digits represent the length
– The remaining digits stand for the width
– 2-digit numbers are measured in hundredths of inch
– 3-digit numbers are measured in thousands of inch (theoretical value)
• Dimensions: length and width given in mm
– Values are based on the maximum values found in Data sheets from different manufacturers
• Power P: maximum power dissipation at an ambient temperature of 70°C
– Given values in the Data sheets of the manufacturers differ from case to case.
• Max RCWV: maximum rated continuous working voltage
– Different values are found in different data sheets
5.4
MELF Resistors: Basic Data
•
•
•
6
Type
Dimensions: length and diameter given in mm
– Values are based on the maximum values found in data sheets
Power and Max RCWV (See Section 5.3
References
1. Everett Rogers; Understanding Buck-Boost Power Stages in Switch Mode Power Supplies, Texas
Instruments Application Report (SLVA059A) – November 2002
2. TPS63000/01/02 High Efficient Single Inductor BUCK-BOOST Converter with 1.8-A Switches, Texas
Instruments Data sheet (SLVS520A) – April 2006
3. Robert Kollman, Constructing Your Power Supply – Layout Considerations, 2004-5 TI Unitrode Design
Seminar Series - SEM1600; Topic 4 (SLUP230)
4. Howard Johnson, Martin Graham, High-Speed Digital Design – A Handbook of Black Magic, Prentice
Hall
5. http://www.polarinstruments.com/support/cits/cits_index.html
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Following are URLs where you can obtain information on other Texas Instruments products and application solutions:
Products
Applications
Amplifiers
amplifier.ti.com
Audio
www.ti.com/audio
Data Converters
dataconverter.ti.com
Automotive
www.ti.com/automotive
DSP
dsp.ti.com
Broadband
www.ti.com/broadband
Interface
interface.ti.com
Digital Control
www.ti.com/digitalcontrol
Logic
logic.ti.com
Military
www.ti.com/military
Power Mgmt
power.ti.com
Optical Networking
www.ti.com/opticalnetwork
Microcontrollers
microcontroller.ti.com
Security
www.ti.com/security
RFID
www.ti-rfid.com
Telephony
www.ti.com/telephony
Low Power
Wireless
www.ti.com/lpw
Video & Imaging
www.ti.com/video
Wireless
www.ti.com/wireless
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