Typical Applications and History of Passive …pec/L 01 Components history... · Web view18-Mar-2017 Typical Applications and History of Passive Components Lecture Plan Introduction

Passive Electronic ComponentsLecture 1 Page 1 of 22

11-Mar-2018Typical Applications and History of Passive Components

Lecture Plan

1. Introduction.2. Passive electronic components. Definition.3. Short history of electronics and passives.4. Typical applications of passive components.5. Why all discrete passive components cannot be integrated in IC or in PCB.

1. Introduction1.1. About Vishay Intertechnology. Vishay Intertechnology is a Fortune 1,000 Company listed on the

New York Stock Exchange (VSH). Vishay has manufacturing plants in Israel, Asia, Europe, and the Americas where it produces discrete semiconductors (diodes, rectifiers, transistors, optoelectronics), selected integrated circuits, passive electronic components (resistors, capacitors, inductors, and transducers). Vishay Intertechnology revenues for 2011 were $2.594 billion. As of December 31, 2011, Vishay Intertechnology had approximately 20,900 full-time employees. Vishay's list of customers include such major manufacturers as AT&T, Alcatel, Bosch, Delco, Ford, IBM, Intel, Motorola, Qualcomm, Samsung, Siemens, Sony. In July 2010 Vishay Precision Group company (approximately 9% of Vishay annual revenue) spun off from Vishay Intertechnology. Vishay Precision Group deals with super precision foil resistors, strain sensors based on resistive foil technology, equipment for stress measurement and industrial weighing.

1.2. About the founder of Vishay Dr. Felix Zandman (1928-2011). In 1962, Dr. Zandman, with the financial help of the late Alfred P. Slaner, founded Vishay to develop and manufacture Dr. Zandman's invention - Bulk Metal Foil resistors. The Company was named after Dr. Zandman's and Mr. Slaner's ancestral village in Lithuania, in memory of family members who perished in the Holocaust. Dr. Felix Zandman was Chairman and CEO of Vishay Intertechnology, Inc. (NYSE: VSH).

Dr. Zandman holded a Ph.D. in Physics from the University of Paris, Sorbonne. He has received numerous honors throughout his life, including

the Musser Award for Excellence in Leadership, the Order of Merit for Research and Invention (France), the Distinguished Contribution Award from the American Society of Stress Analysis, the Franklin Institute Medal for Science, the Legion of Honor (France), EIA Medal of Honor. (The Electronic Industries Alliance is trade organization that represents high

technology community of USA).

Dr. Zandman spoke Russian, Polish, Yiddish, German, Hebrew, French, and English. He has also published three textbooks and held 39 patents. Dr. Zandman´s autobiography "Never the Last Journey", recounts his story from Holocaust victim to Chairman and CEO of a high-tech multinational corporation. (Zandman, Felix, Never the Last Journey. Schocken Books, New York: 1995). It was first issued in English and further was translated in other languages including Hebrew and Russian. The “The Final Victory“ film ("הניצחון הסופי") is the story of Dr. Felix Zandman, an incredible tale of how a small Jewish boy who survived the holocaust in a grave-like shelter, pulled himself together and achieved a life of fame and success in business and scientific achievements. It may be found online in YouTube and many


other sources (see for example https://vimeo.com/47732085 in Hebrew and http://learnmitzvot.com/showsubvideos.php?id=118 in Russian).

1.3. About the lecturer. Dr. Michael Belman is Research and Development Group Leader in Beer-Sheva plant of Vishay Israel Ltd., inventor of 4 patents in passive components field. He graduated with honors from Department of Engineering Physics of Kharkov State Polytechnic University, Ukraine in 1974, and received his Ph.D. from Riga Technical University, Latvia in 1986. Dr. Michael Belman is with Vishay Israel Ltd. from 1991.

1.4. About the course.This course is intended to give the students the understanding of: Passive electronic components constructions, features, modern design, electrical, thermal, and

mechanical properties. Applications (selection, mounting) of passive electronic components in the modern electronics.

2. Characteristics of passive components. Definition: passive component is any electronic device that does not introduce gain or does not have directional functions. Passive components may be linear or non-linear.

Physical parameters of linear (in real life “almost liner”) passive components - resistors, capacitors, and inductors - do not depend significantly on voltage applied to their terminals or electrical current forced across them. It is not so when essentially non-linear components (varistors, thermistors, fuses) are considered.

There is some "grey" area of interchangeability between active and passive components. For example:

The Field Effect Transistor (active component) may be used as a Voltage Controlled Resistor. Source and Drain will be “resistor” terminals, control voltage should be applied between Gate and Source terminals.

Variable Capacitance Diode or Varicap (active component) may replace metal plate variable

capacitor.

http://learnmitzvot.com/showsubvideos.php?id=118

https://vimeo.com/47732085


Incandescent bulb that may be regarded as passive component may serve as a current stabilizer (baretter). It may be used as Automatic Gain Control (AGC) element too. The classic example is Wien bridge circuit in Audio Oscillator HP 200A developed by Bill Hewlett and Dave Packard in 1938.

Varistor (passive component) is commonly used as surge suppressor (Voltage Clamping Device).

It acts very similar to symmetrical Zener diode.

Electrolytic capacitor (passive component) is semi-conductive and therefore may rectify

alternating current the same as a diode. Anyhow, the reverse voltage mode without current limitation may result in capacitor damage and commonly is not allowed.

Let us note that the equations that express conductance and capacitance of solid electrically conductive bodies through electric field, and their inductance through magnetic field look very similar:

G= 1R

= iV

=∫S

σ E⃗ ∙ d S⃗

∫a

b

E⃗ ∙ d l⃗

G=qV =

∮S+¿¿

ε E⃗ ∙ d S⃗

∫S+¿¿

S−¿ E⃗ ∙ d l⃗ ¿(1)

Baretter


L= N ∙Φi

=N∫

Sμ H⃗ ∙d S⃗

∮C

H⃗ ∙d l⃗

G – conductance;R – resistance;i – current;V – voltage; – conductivity;E⃗ – electrical field;S – cross-section of resistive element;a, b – terminal points.

C – capacitance;q – charge; – permittivity;S+ – positive electrode surface;S− – negative electrode surface.

L – coil inductance;Φ – flux;N – number of turns - permeability;H⃗ - magnetic field,(B⃗=μ H⃗ – magnetic induction);S – internal area of each turn;C – closed field line.

Each of the equations (1) includes only geometric parameters and respective physical constant of the medium.

3. Historical excursus.

Inventions of principal passive components: Capacitor – 1745; Resistor – 1827; Inductor – 1831.

Evolution of electronic components (including passive components).It follows three directions:

Smaller size. Lower cost. Higher performance.

Let us compare for example radio sets manufactured in different times.

Crosley TRIRDYN REGULAR. Regenerative 3-tubes Reflex Receiver.　 The Crosley Radio Corporation,　(1925), 54cm length, $50, (approximately $677 in 2016).


Smaller sizes and higher performance of newer radios is a commonplace. Price comparison is shown in the graph below.

1922 1954 20160

100

200

300

400

500

600

700

800677

440

3.7

Radio Set Price

USD

(201

6)

Regency TR-1

4 transistors Super-Heterodyne. Texas Instruments of Dallas, Texas and Industrial Development Engineering Associates (I.D.E.A.) of Indianapolis, Indiana. (1954), 12.7 cm length, $50 (approximately $440 in 2016).

XIBOMAN AS-875

Mini FM Scan Radio + headphones. (2016), 8.8 cm length, $3.7.


This progress has become possible as the result of perfection of electronic components (both active and passive). Let us consider a 1W resistors for example.

Applications of passive components .

22.5

6.5

6.3

0.6

1W resistors:

Axial leaded Chip


Passive components in 2003 constituted are about 80% of BOM (Bill of Materials) count, occupied about 60% of PCB area and constituted about 20% of BOM cost 6).

1984 1987 1990 19930

20

40

60

80

100

Passive components part count on PCB, %

DDR4-2133 SDRAM (see picture below) comprises 8 active components and 90 passive components visible outside the labels that constitutes 92%.

Typical count of passive components in some products. Car electronics – 3400. PC – 2200. LCD TV – 2100. iPhone 6 Plus – 1100. Game console – 1020. Digital camera – 840. iPod – 230.


486 Pentium Pentium II Pentium III Pentium IV0

100200300400500600700

124

252345

440

600

Typical passive components count in PC moth-erboard

Processor TypeTransistors

on chip

External passives

count486 1.2M 124Pentium 3.2M 252Pentium II 7.5M 345Pentium III 28M 440Pentium 4 (Willamette) 42M 600


There are about 1100 passive components in iPhone 6 PCB (see picture below).Passive components occupy over 40% of the iPhone's PCB area 7).


Passive components in analog circuit. There are: 1 active component (IC), 15 passive components (or 94%) in the typical amplifier circuit below:

Example of the modern mixed-mode integrated circuit: Bluetooth - Radio Modem based on Silicon Wave SiW1711 (see picture below).

Passive components Active components16 capacitors6 inductors1 resistor1 ceramic filter1 quartz resonator1 antenna

1 IC

Total: 26 (96%) Total: 1 (4%)


Some reasons of growth of passive components count: Higher clock frequencies require impedance matching at the ends of transmission lines. Lower operating voltages require lower level of ripple voltage in power line and therefore

better filtering. Higher operating currents require higher capacitance of filtering capacitor. Commonly several

smaller capacitors distributed on PCB are used instead of single capacitor of highr capacitance.

Increasing number of analogue applications.

New issues related to passive components: Surface-Mount Technology (SMT) is a method for constructing electronic circuits

in which the components are mounted directly onto the surface of PCB. SMT changed the appearance of passive electronic components, reduced their dimensions, and enhanced their HF performance.

Environmental concerns. Lead-free components are being introduced. CFC cleaning substances elimination in assembly process.

Special passive components. Special resistors: current sense, sulfuration-proof, pulse-proof. Special capacitors: X2Y, Z-chip, feed-through, double layer. Special inductors: common mode choke, ferrite bead, coupler, balun, power divider, antenna.

4. Why all discrete passive components cannot be integrated in IC or in PCB?


Passive components count in PCB is growing despite the intuitive feeling that it has to drop as far as IC integration increases. Why cannot resistors, capacitors, and inductors be integrated into silicon of ICs and put on the Moore's Law?

Regardless of high cost of semiconductor chip surface some modern ICs have on-die termination resistors which improve signal integrity at higher clock rates. Examples: DDR2 computer memory, Pentium 4 processor.

Pentium 4 processor P4 Celeron-D 310 Prescott processor comprises 125 millions of transistors and dissipates 73 W of thermal power. Termination resistors are provided on the processor silicon and are terminated to its core voltage (VCC) (but not for all signals). Let us evaluate how many additional transistors may be placed on chip instead of single termination resistor. (We shall take into consideration dissipated power only).Suppose that:

Core voltage is 1.3 V; Transmission line and termination resistor impedance is 60 ; Digital signal in transmission line may be regarded as a periodic pulse train with a duty factor

t=1/2.

Impossibility to integrate all

passives in ICs

IC universality

OA (gain setting, offset nulling, muster bias current setting)

Digital IC (open-collector output, pull-up, pull-down)

Timer (time constant setting)

Integrated passives capabilities

Narrow span of nominal values

High tolerance, TCR, voltage coefficient

Low power dissipation

Parameters of passives are functions of their

dimensions but silicon chip area is costly


R = 60 U = 1.3 VPtotal = 73 Wt = 1/2N = 125106

P=t⋅U 2

R;

N1

N=P

Ptotal

N1=tNU 2

RPtotal

N1=125⋅106⋅1 .32

2⋅60⋅73=24(115 )≈24⋅103

V2//W = V A/W = W/W = 1

N1 - ?

5. Integration in PCB. (Integrated passives). There is possibility to integrate passive components in PCB (see picture below). It is used more and more because saves PCB area. But only limited quantity of passives may be integrated because of: poor performance of integrated passives, limited range of their values, higher PCB price.

References.

1. R. Klaiber, C. Lassen, “Critical issues in electronic packaging assembly. Part 1,” Circuits Assembly, December 1995, pp.30-33.

2. G. Smith, “Resistors and capacitors: a renaissance,” in Proc. CARTS 2000: 20th Capacitor and Resistor Technology Symposium, Huntington Beach, CA, March 2000, pp.1-6.

3. “Flat-chip resistors: a booming market for 2000,” Passive Component Industry, March/April 2000, pp.25-26.4. D. Zogbi, “Letter from the publisher,” Passive Component Industry, November/December 1999, p.4.5. “Tantalum capacitors: global trends in 2000,” Passive Component Industry, January/February 2000, pp.24-27.6. Passive components: Capacitors & Resistors,

http://ecee.colorado.edu/~mcclurel/passivecomponents_lhm_handouts2.pdf7. Jim Stratigos , Shrink the iPhone, http://www.electroiq.com/articles/ap/2008/08/executive-

viewpoint.html

http://www.electroiq.com/articles/ap/2008/08/executive-viewpoint.html

http://www.electroiq.com/articles/ap/2008/08/executive-viewpoint.html

http://ecee.colorado.edu/~mcclurel/passivecomponents_lhm_handouts2.pdf


10-Mar-2013Solving of Engineering Problem

1. Stages of a solution

Solution of the problem should be performed in 3 steps:

A. Brief record of problem situation.B. Derivation of necessary formulas. C. Calculations and units check.

Example. Voltage drop on 2.4 k resistor is 230 V. Calculate the power dissipated by the resistor.

A B CR = 2.4 k = 2.4103 U = 230 V P=U2

RP=2302

2 .4⋅103=22 .(04 )≈22 (W)

V2/ = V A = W

P = ?

1.1. Brief record of problem situation.

Brief record of problem situation has to provide a compact view of all given data. It may be regarded as translation of the problem situation from common language to the language of formulas and drawings. The given physical magnitudes have to be represented as the following: Numbers. Digits in each number represent both quantity and precision of a physical magnitude.

Therefore, notation of the number should be kept as it is. At least a count of significant digits has not to be changed. At that, a leading zeros (any 0 digits that are in the left of the first nonzero digit) do not represent precision of a physical magnitude. Trailing zeros that are to the right of the last non-zero digit indicate precision of a physical magnitude. For example, 2.4 k may be represented as 24102 , 2.4103 , 0.24104 , 0.024105 but not as 240101 or 2400 .

Units. Each given physical magnitude must be represented as a number accompanied by respective physical unit. If a given unit does not belong to selected system of units (preferably to SI system of units) it has to be converted. Prefixes as for example k (kilo), M (mega), (micro) have to be substituted by respective 103, 106, 10-6 multipliers.

Illustrations. All the necessary sketches, circuit diagrams, etc. have to be drawn.

1.2. Derivation of necessary formulas. Selection of mathematical model. “Essentially, all models are wrong, but some are useful”. (George

E. P. Box (1919-2013), a statistician).

U

R


Derivation of necessary formulas has to result in general view of solution, i.e. expression of the desired value in terms of the given literal constants and variables. The worth of general view is the possibility to analyze it using algebra and calculus.

No numbers are admissible in general view of solution. The exceptions are dimensionless summands, multipliers, exponents, bases of logarithm, etc. They may be integer, rational or irrational (, e) numbers.

Substitution of any physical magnitude by its numerical value is forbidden at this stage.

1.3. Calculation stage includes the following steps: (a) substitution of literal constants and variables by their numerical values, (b) calculations with monitoring of significant digits, (c) rounding of final result, (d) checking the physical units of final result.1.3.1. Substitution. Each literal values has to be substituted by its numerical value accompanied by respective physical unit (if applicable).1.3.1.1. Numerical values. Numerical value of each literal constant or variable has to be represented as common fraction (1/2,

10/3), decimal fraction (0.5, 0.079), or mixed decimal numbers (1.47, 2.54). It is suggested to use scientific notation in the following three cases: (a) a magnitude of a numerical

value is too big (6.25108), (b) a magnitude is too small (1.610-12), (c) to display correctly the number of significant digits (1.2101 instead of 120 if only 2 significant digits are available).

Do not use literal prefixes like k (kilo), M (mega), (micro) with numerical values instead of respective power of 10 multiplier: 103, 106, 10-6. For example, do not use notation 1.5k in place of 1.5103.

All literal values have to be substituted by their numerical equivalents at once. Partial substitution is unacceptable.

1.3.1.2. Physical units. Each physical magnitude has to be represented by numerical value accompanied by respective

physical unit. (It may be regarded as multiplication of numerical value by physical unit). No literal prefix shall forgo a physical unit. All physical units have to belong to the same system of units, preferably SI (see Appendix). The arguments of elementary and special functions (trigonometric, exponential, logarithmic,

hyperbolic, error, gamma, Bessel, etc.) must be dimensionless. All computations have to be performed both with numbers and their physical units in parallel. It is

acceptable to treat the units separately like in Example given in paragraph 1. Computation of physical unit related to final result is called “checking of units”. The computed unit

has to agree with expected in advance physical unit that is specific for the found physical value.

1.3.2. Calculations (Operations with approximate numbers).1.3.2.1. Addition and subtraction of approximate numbers. Align the numbers with respect to the decimal point as shown in the example below. In the case when

scientific notation of any of the numbers is preferable all the numbers have to be presented in scientific notation with the same exponent.

Perform addition (subtraction) as usually. Define significant digits in the sum. For this purpose compare positions of the rightmost significant

digits in all the summands. The leftmost of them defines position of the last significant digit in the sum. Close in parenthesis all non-significant digits in the sum.


12. 5+ 0. 023

31. 2343. 7(53)

1.3.2.2. Multiplication and division of approximate numbers. The number of significant digits in product has to be the same as in the multiplier with the least number of significant digits. Close in parenthesis all non-significant digits in the product.

2.156 1.2 = 2.5(872)

1.3.2.3. Keep non-significant digits in all intermediate calculations to minimize the round-off error. At that, all non-significant digits have to be closed in parenthesis as shown in paragraphs 1.3.2.1 and 1.3.2.2.1.3.2.4. Perform round-off in final result only. The　typical　rounding　rule　is　rounding　up　if　the　leftmost　non-significant　digit　is　five　or　more　and　rounding　down　if　it　is　four　or　less.

2.5(572) 2.62.5(472) 2.5

2. General rules of paperwork.

Use ruled paper (2 lines per centimeter is preferable). Write on one side of paper only (leave the backside blank). The written part should be neatly lettered. Do not use pencil and eraser. Do not blacken the wrong text. Cross out it by thin line. Crossed out text

must be legible. Do not crowd your work. Leave one or more blank lines above and below equations. Leave margins

for drafts and remarks. The pages must be numbered, dated, and marked with a problem identifier.

Appendix 1Physical Quantity

A physical quantity is always the product of two quantities:

Physical quantity = Number Unit

Examples are T = 297 K, R = 82 . Physical quantity that is presented as number only (dimensionless value) has no meaning. Dimensionless value may be used as: integer number that represent a quantity of identical objects, ratio of two physical quantities having the same units.

Numerical part of physical quantity is characterized by its accuracy and precision.


Accuracy is a measure of deviation of measured quantity from its unknown exact value. By other words accuracy is a measure of rightness. It is not represented in notation of a number.

Precision characterizes repeatability of measurement result. By other words precision is a measure of exactness. It is represented by quantity of significant digits in notation of a number.

Accuracy Precision10 NO NO

5.7236914 NO YES3.1 YES NO

3.1415926 YES YES

Unit. There are seven physical quantities having dimensionally independent (base) units.

SI Base Units

Physical quantity Name of SI unit

Symbol for SIunit

Length meter mMass kilogram kgTime second sElectric current ampere ATemperature kelvin KAmount of substance mole molLuminous intensity candela cd

In addition to the seven base units there are so called derived units. Some of them are presented in the table below. They can be expressed in terms of the above SI base units.


Some Derived SI Unites

Name Symbol Dimensions QuantityCoulomb C A·s Electric charge

Volt V J/C = kg·m2·s−3·A−1 Potential differenceOhm Ω V/A = kg·m2·s−3·A−2 Resistance, Impedance, ReactanceFarad F C/V = kg−1·m−2·A2·s4 CapacitanceSiemens S Ω−1 = kg−1·m−2·s3·A2 Conductance, Admittance, SusceptanceWeber Wb V·s = kg·m2·s−2·A−1 Magnetic fluxTesla T Wb/m2 = kg·s−2·A−1 Magnetic flux densityHenry H V·s/A = kg·m2·s−2·A−2 Inductance

Ω·m = kg·m3·s−3·A−2 Electric resistivityS/m = kg−1·m−3·s3·A2 Electric conductivityF/m = kg−1·m−3·A2·s4 PermittivityH/m = kg·m·s−2·A−2 PermeabilityWm-2 = kgs-3 Heat flux density, IrradianceJK-1 = kgm2s-2K-1 Heat capacity, EntropyWm-1K-1= kgm2s-

3K-1Thermal conductivity

Newton N N= kgms-2 ForceJoule J J = kgm2s-2 Work, EnergyWatt W W = kgm2s-3 Power

Symbol of unit that is named after scientist’s name starts from capital letter (A, V, W, Wb, Pa, etc.).

Quantities that are characterized by different dimensions (units) cannot be added, subtracted, or even compared.

SI Prefixes

Multiple Prefix Symbol Multiple Prefix Symbol100 deca da 10-1 deci d102 hecto h 10-2 centi c103 kilo k 10-3 milli m106 mega M 10-6 micro 109 giga G 10-9 nano n1012 tera T 10-12 pico p1015 peta P 10-15 femto f1018 exa E 10-18 atto a


Appendix 2Subtraction of two floating-point numbers

If the magnitudes of the numbers are similar and signs are the same a LARGE PRECISION LOSS may occur.

Example. Derivation of the function y=x2.

Analytical derivation gives precise result:

y=x2 ;y ' ( x )=2 x ;y ' (1)=2.

Let us evaluate approximate value of the derivative by simple calculations based on definition of a derivative:

y ' ( x )≈~y '( x , Δx )=y ( x+ Δx)− y ( x )Δx

;

~y '(1 , Δx )=(1+ Δx )2−1Δx

Δx ~y ' (1 , Δx )

~y ' (1 , Δx )− y ' (1)y ' (1 )

1.0E+00

3.000000000000000E+00 0.5

1.0E-01

2.100000000000000E+00 0.05

1.0E-02

2.010000000000000E+00 0.005

1.0E-03

2.000999999999700E+00 0.0005

1.0E-04

2.000099999999170E+00 5E-05

1.0E-05

2.000010000013930E+00 5E-06

1.0E-06

2.000000999924370E+00 5E-07

1.0E-07

2.000000101087810E+00 5.05E-08

1.0E-08

1.999999987845060E+00 -6.1E-09

1.0E-09

2.000000165480740E+00 8.27E-08

1.0E-10

2.000000165480740E+00 8.27E-08

1.0E-11

2.000000165480740E+00 8.27E-08

1.0E- 2.000177801164680 8.89E-05


12 E+001.0E-

131.998401444325280

E+00 -0.00081.0E-

141.998401444325280

E+00 -0.00081.0E-

152.220446049250310

E+00 0.1102231.0E-

160.000000000000000

E+00 -11.0E-

170.000000000000000

E+00 -1


Appendix 3

Wheatstone bridge(Example)

Four resistors R1; R’1; R’2; R2 form a Wheatstone bridge. Its adjacent arms are:R1 R1', R1

' R2'

, R2' R2 , R1 R2 .

The adjacent arms R1 R2 and R1' R2

' form the branches between power corners. Input voltage U is applied

across the power corners.

The resistances of the bridge arms were measured using 3-1/2 digits ohmmeter. The measurement results are: R1=1013 ; R’1=1051 ; R’2=1057 ; R2=1019 .

Calculate the ratio of output to input voltages.

A B C

R1=1013 ;R2=1019 ;R’1=1051 ;R’2=1057 .

Uout=|UR 2

R1+R2−

UR2'

R1' +R2

'|=

¿U|R2 R1

' −R1 R2' |

(R1+R2) (R1' +R2

' ).

Uout

U=|R2 R1

' −R1 R2'|

( R1+R2 ) (R1' +R2

' ).

Uout

U=

¿|1019⋅1051−1013⋅1057|(1013+1019 ) (1051+1057 )

=

¿|1070(969 )−1070 (741 )|4283 (456 )

=

¿(228 )4283 (456 )

=0 . 0000(532) .

Uout /U = ?

The calculation results in the number with no significant digit.Input data have to originate from more precise measurements in order to receive a meaningful result.

U

Uout


Triangle(Example)

A B C

a = 1.2 cm;c = 1.8 cm; b=√c2−a2

b=√ (c−a ) ( c+a )

b=√1. 82−1 .22=√3 .2( 4 )−1. 4( 4 )=¿√1. 8(0 )=1 . 3(4 )≈1. 3

b=√ (1.8−1.2 ) (1 . 8+1. 2 )=√0 .6⋅3 . 0=√1.( 8)=1 .(34 )≈1

b = ?

The first sequence of calculations gives more precise result than the second one.

a

b c

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Typical Applications and History of Passive …pec/L 01 Components history... · Web view18-Mar-2017 Typical Applications and History of Passive Components Lecture Plan Introduction