ELECTRONICS Formulas and Concepts.pdf

SEMICONDUCTOR THEORY

The diode current equation kV d

•

I d

Atomic Structure

Diameter of neutron = 10-13 cm Maximum number of electrons per shell or orbit or bit N e = 2n 2 n = 1, 2,3,4 Letter designation K shell – 1 O shell – 5 L shell – 2 P shell – 6 M shell – 3 Q shell – 7 N shell – 4

9.1096 × 10−31 1.6726 × 10−27 1.6726 × 10−27

k =

A = no. of protons + no. of neutrons Z = number of protons or electrons

Where: A = Atomic mass or weight (A) Z = Atomic number (Z) Note: Mass of proton or neutron is 1836 times that of electron. Energy Gap Comparison Element No. of Valence Energy Electrons (Ve) 8 > 5eV Insulator 4 Si = 1.1eV Semiconductor Conductor

1

Ge = .67eV 0eV

At room temperature: there are approximately 10 1.5×10 of free electrons in a cubic centimeter (cm3) for intrinsic silicon and 2.5×1013 for germanium.

•

Diode Theory VthT 1

n

for low levels of diode current n = 1 for Ge and n = 2 for Si for higher levels of diode current n = 1 for both Si S i and Ge

− 1.6022 × 10−19 + 1.6022 × 10−19 No charge

k

Where: Id = diode current Is = reverse saturation current or leakage current Vd = forward voltage across the diode Tk = = room temperature at °K = °C + 273 11600

Mass and Charge of different Particles Particle Mass (kg) Charge (C)

Electron Proton Neutron

= I s (e T − 1)

Temperature effects on Is k ( T −T ) I sT 1 = I sT 0 e 1 0

Where: IsT1 = saturation current at temperature t emperature T1 IsT0 = saturation current at room temperature k = 0.07/°C T1 = new temperature T0 = room temperature (25°C) Reverse Recovery Time (T rr)

T rr = t s

+ t t

Where: Trr = = time elapsed from forward to reverse bias (ranges from a few ns to few hundreds of ps) Tt = transition time Ts = storage tim t imee DC CIRCUITS 1

1 Coulomb = 6.24×1018 electrons By definition: A wire of 1 mil diameter has a cross-

= VthT 0 + k (T 1 − T 0 )

where: VthT1 = threshold voltage at T1 VthT0 = threshold voltage at T0 k = –2.5 mV/°C for Ge k = –2.0 mV/°C for Si

sectional area of 1 Circular Mil (CM) 1 mil = 10-3 in 1 in = 1000 mils Asquare = 1 mil2

PDF created with pdfFactory trial version www.pdffactory.com

Acircle

=

π D 2

1mil 2 =

Quark

Up Down Charm Strange Top Bottom

4 4 π

2

mil

Temperature effects on resistance T + t 2 R2 α1 = R1 T + t 1

CM

Type/Flavors of Quarks Symbol Charge Baryon no.

U D C S T B

+2/3 –1/3 +2/3 –1/3 +2/3 –1/3

1/3 1/3 1/3 1/3 1/3 1/3

R2

Type

D C AA AAA I = =

Q

ρ

1 4 1 1

9

W Q

= RA L

16

3

Ampere( A);

t

V =

2 4 1 34 178 1 34 1

Volt (V );

8

Coulomb(C )

sec ond ( s)

Joule( J ) Coulomb(C )

Ω − m; Ω − cm;

T + t 1

= R1[1 + α 1 (t 2 − t 1 )]

American Wire Gauge (AWG)

AWG #10: A = 5.261 mm2 AWG #12: A = 3.309 mm2 AWG #14: A = 2.081 mm2

Inferred Absolute Temp. for Several Metals Material Inferred absolute zero, °C

Aluminum Copper, annealed Copper, hard-drawn Iron Nickel Silver Steel, soft Tin Tungsten Zinc

-236 -234.5 -242 -180 -147 -243 -218 -218 -202 -250

Temperature-Resistance Coefficients at 20 °C "20 Material

Ω − CM ft

Resistivities of common metals and alloys  (10-8!-m) Material

Aluminum (Al) Brass Carbon Constantan (60% Cu and 40% Ni) Copper (Cu) Manganin (84% Cu, 12%Mn & 4%Ni) Nichrome Silver (Ag) Tungsten (W)

1

where: |T| = inferred absolute temperature, °C R 2 = final resistance at final temp. t 2 R 1 = initial resistance at initial temp. t 1 "1 = temp coefficient of resistance at t 1

Proton – 2 Up and 1 Down Neutron – 1 Up and 2 Down Types of Battery Height (in) Diameter (in)

=

2.6 6 350 50 1.7 44 100 1.5 5.6

Absolute zero = 0 K = –273°C Cu = 10.37 ! –CM/ft PDF created with pdfFactory trial version www.pdffactory.com

Nickel Iron, commercial Tungsten Copper, annealed Aluminum Lead Copper, hard-drawn Silver Zinc Gold, pure Platinum Bras Nichrome German Silver Nichrome II Manganin Advance Constantan

0.006 0.0055 0.0045 0.00393 0.0039 0.0039 0.00382 0.0038 0.0037 0.0034 0.003 0.002 0.00044 0.0004 0.00016 0.00003 0.000018 0.000008

G

=

1 R

=

A ρ L

=σ

Gold Silver None

A L

where: # = specific conductance or conductivity of the material in siemens/m or mho/m.

P =

t

=

Q t

E = IE =

E 2

= I 2 R

R

Voltage Division Theorem 2 resistors in series with one R1 V 1 = E V 2 R1 + R 2 Current Division Theorem R 2 I 1 = I T I 2 R1 + R2

=

=

R1 + R2

E

I T

Transformations or Conversations: Delta (#) to Wye (Y) Pr oduct _ of _ adjacent _ R _ in _ ∆ RY = of _ all _ R _ in _ ∆

∑

Wye (Y) to Delta (#) of _ cross _ products _ in _ Y R∆ = Opposite _ R _ in _ Y

∑

Color Coding Table Color 1 2nd Toler- Temp signifi- signifi- Multiplier rance Coef cant cant (±%) ppm/°C Black 0 0 100 20 0 Brown 1 1 101 1 -33 Red 2 2 102 2 -75 Orange 3 3 103 3 -150 Yellow 4 4 104 GMV -220 Green 5 5 105 5 -330 Blue 6 6 106 -470 Violet 7 7 107 -750 Gray 8 8 108 +30 White 9 9 109 +500 st

+100 Bypass -

1.0% 0.1% 0.01% 0.001%

Ampere − hour _ rating ( Ah) Amperes _ drawn ( A)

Cell Types and Open-Circuit Voltage Cell Name Type Nominal OpenCircuit Voltage

R 2

R1 + R2

5 10 20

Brown Red Orange Yellow

Battery life =

R1

0.1 0.01 -

Fifth band reliability color code Color Failures during 1000 hours of operation

Batteries

where: W = work in Joules (J) t = time in seconds (s) Q = charge in Coulomb (C)

-

GMV = Guaranteed Minimum Value: -0%, +100%

Note: The best is silver with 1.68×10 24 free electrons per in3. Next is copper with 1.64×1024 free electrons per in3 and then aluminum with 1.6×1024 free electrons per in3. W

-

Carbon-zinc Zinc-chloride Manganese dioxide (alkaline) Mercuric oxide Silver oxide Lead-acid Nickel-cadmium Nickel-iron (Edison cell) Silver-zinc Silver-cadmium Nickel metal hydride (NiMH)

Primary Primary Primary or Secondary Primary Primary Secondary Secondary Secondary

1.5 1.5 1.5 1.35 1.5 2.1 1.25 1.2

Secondary Secondary Secondary

1.2 1.1 1.2

DIODES

•

Diode Applications Half–wave Rectification V V DC = m = 0.318V m π PIV rating  V m Full–wave Rectification


V rms

=

1

t

V (t ) dt T ∫ 2

0

V DC PIV rating  V m

= 0.36V m for bridge-type

PIV rating  2V m

for center-tapped

•

Other Semiconductor Devices Zener Diode ∆V Z T CC = V Z (T 1 − T 0 )

1 +    V T 

where: C(0) = capacitance at zero-bias condition

where: TCC = temperature coefficient T1 –T0 = change in temperature VZ = Zener Voltage at T0

Also,

TC C

Basic Zener Regulator I. Vi and R L fixed

(a) Determine the state of the Zener diode by removing it from the network and calculating the voltage across the resulting open circuit. (b) Substitute the appropriate equivalent circuit and solve for the desired unknown. II. Fixed R L, variable Vi ( R + RS )V Z V i min = L R L III. Fixed Vi, variable R L RV Z R L min = V i − V Z

V i max

In terms of the applied reverse bias voltage: C (0) C T = n  V R 

= I R max RS + V Z

R L max

=

V Z I L min

Varactor diode or Varicap diode A C T = ε W d

where: CT = transition capacitance which is due to the established covered charges on either side of the junction A = pn junction area Wd = depletion width In terms of the applied reverse bias voltage: k C T = (V T + V R ) n

where: CT = transition capacitance which is due to the established covered charges on either side of the junction k = constant determined by the semiconductor material and construction technique VT = knee voltage VR = reverse voltage n = ½ for alloy junctions and $ for diffused junctions

=

∆C C O (T 1 − T 0 )

where: TCC = temperature coefficient T1 – T0 = change in temperature C0 = capacitance at T0 Photodiode

W = hf = h

c λ

; Joules

where: W = energy associated with incident light waves h = Planck’s constant (6.624×10-34 J-sec) f = frequency 1eV = 1.6×10-9 J 1 Angstrom (Å) = 10-10 m Solar Cell

η

=

P O P i

P max

=

 1W    cm 2 

( Area)

where: ( = efficiency P0 = electrical power output Pi = power provided by the light source Pmax = maximum power rating of the device Area = in cubic centimeters Note: The power density received from the sun at

sea level is about 1000 mW/cm2

BIPOLAR JUNCTION TRANSISTOR Ratio =

•

widthtotal width base

= 0.150 = 150 0.001

Basic Operation Relationship between I E, IB and IC: I E = I B + I C


h21 = forward transfer current ratio, h f h22 = output conductance, ho

IC is composed of two components: I C = I majority + I min ority

H-Parameters typical values CE CB CC hi 1k ! 20! 1k ! -4 hr 2.5×10-4 3×10 *1

DC Transistor Parameters

α

 ∆ I  =  C    ∆ I E  Vcb=cons tan t

α

=

I C I E

β

 ∆ I  =  C    ∆ I B  Vce= cons tan t

β

=

hf ho

I C I B

where: IE = emitter current IB = base current IC = collector current " = CB short-circuit amplification factor ) = CE forward-current amplification factor Relationship between " and $: β α= β β +1

=

Zi Zo Ai Av A p shift

1−α

Unitless Siemens Ampere

FIELD EFFECT TRANSISTORS

•

JFET 2

Small Signal Analysis A. Hybrid Model V i = h11 I in

+ h12V 0 = h21 I in + h22V 0

If Vo = 0 V i I in

ohms

If Iin = 0 h12

=

V i V 0

unitless

If Vo = 0 h21 =

low moderate high high moderate low low high moderate high high low moderate high low none 180° none

Note: Common Base : hib = r e ; hfb = –1 Common Emitter: ) = hfe ; )r e = hie

I D

=

I 0 I in

unitless

 V  2 I DSS g = I DSS 1 − GS  = mo  V P  V P  2 I  V  ∆ I d = g m = DSS 1 − GS  V P  V P   ∆V gs V =0 0 ≤ V GS ≤ 5 ds

where: Id = drain current Idss = drain-source saturation current Vgs = gate source voltage V p = Vgs (off), pinch-off voltage gm = gfs, device transconductance gmo = the maximum ac gain parameter of the JFET

•

MOSFET

= k (V GS − V TH ) 2 k = 0.3mA / V 2

I DS

If Iin = 0 h11

=

-50 25+S

B. R e Model

∆ I C S ( I CO ) = ∆ I CO ∆ I C S ( I CO ) = ∆V BE ∆ I S ( I CO ) = C ∆β

h11

-0.98 0.5+S

Comparison between 3 transistor configurations CB CE CC

α

Stability Factor (S):

I o

50 25+S

I 0 V 0

siemens

•

FET biasing

where: h11 = input-impedance, hi h12 = reverse transfer voltage ratio, hr PDF created with pdfFactory trial version www.pdffactory.com

DC bias of a FET requires setting the gate-source voltage, which results in a desired drain current. Vgg is used to reverse bias the gate so that Ig = 0.

Vdc = average value of the filter’s output voltage

POWER SUPPLY

•

Transformer I p a= I s

=

V s V p

=

N s N p

=

•

Z s

Rectifier Half–wave signal

V dc

V rms

= 0.636V rms

=

V m

2

PIV = 2V rms

Ripple frequency = AC input frequency Full–wave rectified signal (bridge type) V V dc = 0.636V m V rms = m

2

V dc

= 0.9V rms

PIV =

2V rms

Ripple frequency = 2×AC input frequency Full–wave with center-tapped transformer V dc = 0.9V rms of the half the secondary = 0.45V rms of the full secondary = 0.637V pk of half of the secondary = 0.637V pk of the full secondary PIV = 1.414V rms of full secondary

r =

AC DC

=

V r ( rms ) V dc

V r ( rms) = V rms

2

V r ( rms) = 0.308V m

full–wave rectified signal

Filter

Z p

•

= 0.318V m

half–wave rectified signal

V r ( rms ) =

where: a = turns ratio Vs = secondary induce voltage V p = primary voltage Ns = no. of turns on the secondary windings N p = no. of turns on the primary windings I p = current in the primary windings Is = current in the secondary windings Zs = impedance of the load connected to the secondary winding Z p = impedance looking into the primary from source

V dc

V r ( rms) = 0.385V m

− V dc 2

where: r = ripple factor Vr (rms) = rms value of the ripple voltage

V r (rms) =

V r ( p )

3

I dc

=

=

V r ( p − p)

2 3

2.4 I dc

=

2.4V dc

C R L C 4 3 fC I 4.17 I dc V ( p − p) = V m − dc = V m − V dc = V m − r 2 4 fC C 2.4 I dc V (rms ) 2.4 r = r × 100% = × 100% = × 100% V dc

CV dc

R L C

where: Idc = the load current in mA C = filter capacitor in +F R L = load resistance at the filter stage in k ! Vm = the peak rectified voltage Idc = the load current in mA C = filter capacitor in +F f = frequency at 60 Hz

•

Regulator Voltage Regulation V − V V R . . = noload fload × 100% V fload Stability factor (S) ∆V out S = (constant output current) ∆V in Improved series regulation R + R2 V o = 1 (V Z + V BE 2 ) R2

INSTRUMENTATION

•

DC Ammeter Relationship between current without the ammeter and current with the ammeter I wm Ro

I wom

=

Ro + Rm

where: Iwm = current with meter Iwom = current without meter R o = equivalent resistance


R m = internal resistance of ammeter Accuracy Equation of an ammeter I accuracy = wm I wom Percent of loading error %error = (1 − accuracy) × 100%

D =

Rin sh

=

Rin sh

=

V in I in

S ac

Rm R sh Rm + R sh

I t

where: R sh = shunt resistance Ifs = full scale current R m = meter resistance It = total current Rinsh = input resistance of the shunted meter Vin = voltage input Iin = current input Voltmeter For full scale current V fs = (R s + Rm )I fs V fs R s = − Rm I fs

where: Vfs = full scale voltage R s = series resistor R in = input resistance

S ac

•

Ohmmeter V I fs = oc Ro

Sensitivity for a full-wave rectifier

I fs

DC Bridges Wheatstone bridge ohmmeter

Bridge is balance if R1 R 2

•

=

R3 R 4

Attenuators

=

Rins Rino

L type or the voltage divider V R2 gain = attenuation = in R1 + R2 V out

Rin

=

= 0.9

R1

Voltmeter Loading Error V accuracy = wm V wom

V wm

Sensitivity for a half-wave rectifier

I fs

•

Sensitivity of Voltmeter

I fs

+ Ru

where: R o = characteristic resistance R ins = input resistance with output terminals shorted R ino = input resistance with output terminals open

Rin = R s + Rm

1

Ro

= 0.45

Ro

•

S =

I fs

Ro

AC Detection

I fs Rm

=

=

where: Ifs = full scale current Voc = open circuit voltage R o = internal resistance of ohmmeter D = meter deflection R u = unknown resistance

• Ammeter Shunt I fs Rm R sh = I t − I fs

I

=

=

V fs I fs

X C 1

C 1

X C 2

Symmetrical Attenuator R m = 2 ; R 2 R1

Rin Rin + Ro

RinV wom Rin + Ro

I =

R 2

=

Ro + Ru

1 gain

R2 C 2 R1

= mR1

Symmetrical T Analysis

R0

V oc

=

=

= R1 1 + 2m

a

=

V in V out

=

1 + m + 1 + 2m m

Symmetrical Pi Analysis

R0


=

R2

1 + 2m

a

=

V in V out

=

1 + m + 1 + 2m m

6 7 8 9

Design Formulas for T Attenuator a2 −1 a +1 R1 = Ro R2 = Ro 2a a −1

Gray Code (Reflected Code) DECIMAL DIGIT Gray Code

Design Formulas for T Attenuator a −1 2a R1 = Ro R2 = 2 Ro a +1 a −1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Variable Attenuator

Analysis R1 = R0

a

=

R1 R2

+1

Design R1 = R0

R2

=

R0 a −1

R3

=

a −1 R0

COMPUTER FUNDAMENTALS r's complement

n

(r )10 – N (r – 1)’s complement n -m (r – r )10 – N

84-2-1

2421

Biquinary 5043210

0 1 2 3 4 5 6 7 8 9

0000 0111 0110 0101 0100 1011 1010 1001 1000 1111

0000 0001 0010 0011 0100 1011 1100 1101 1110 1111

0100001 0100010 0100100 0101000 0110000 1000001 1000010 1000100 1001000 1010000

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001

Excess-3-code DECIMAL DIGIT Excess-3

0 1 2 3 4 5

0011 0100 0101 0110 0111 1000

0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000

DECIMAL

Types of Binary Coding Binary Coded Decimal Code (BCD) DECIMAL DIGIT BCD Equivalent

0 1 2 3 4 5 6 7 8 9

1001 1010 1011 1100

OPERATIONAL AMPLIFIERS V D = V + – V -

where: VD = differential voltage V+ = voltage at the non-inverting terminal V- = voltage at the inverting terminal CMRR =

Ad Ac

where: Ad = differential gain of the amplifier Ac = common-gain of the amplifier


Slew rate

SR

∆V o = 2π f maxV pk ∆t

=

where: f max = highest undistorted frequency V pk = peak value of output sine wave Differentiator

V o

= − RC

dV in dt

Integrator

V o

=−

1 RC

∫ V dt in

t PHL

=

+ t PLH

2 where: t p = propagation delay tPHL = propagation delay high to low transition tPLH = propagation delay low to high transition Power dissipation is the amount of power that an IC

drains from its power supply. I CCH + I CCL

2

Noise Margin is the maximum noise voltage added

LOGIC GATES

•

Boolean Algebra Postulated and Theorems of Boolean algebra X + 0 = X X • 1 = X

X + X ' = 1

X • X ' = 0

X + X = X

X • X = X

X + 1 = 1

X • 0 = 0 X • Y = Y • X

X • (YZ ) = ( XY ) • Z

(Distributive Law) X (Y + Z ) = XY + YZ X + (YZ ) = ( X + Y )(Y + Z )

(De Morgan’s Theorem) ( X + Y )' = X ' Y '

t p

where: ICCH = current drawn from the power supply at high level ICCL = current drawn from the power supply at low level

R gain = − 2 R1

(Law of Absorption) ( X + Z ) X + XY = X

time for a signal to propagate from input to output.

P D ( AVG ) = I CC ( AVG ) × V CC

Basic inverting amplifier

(Associative Law) X + (Y + Z ) = ( X + Y ) + Z

Logic Family Criterion Propagation delay is the average transition delay

I CC ( AVG ) =

Basic non-inverting amplifier R gain = 1 + 2 R1

(Commutative Law) X + Y = Y + X

•

to the input signal of a digital circuit that does not cause an undesirable change in the circuit output. Low State Noise Margin NM L = V IL

− V OL

where: NM = Noise Margin VIL = low state input voltage VOL = low state output voltage High State Margin NM H

= V OH − V IH

where: NM = Noise Margin VIH = high state input voltage VOH = high state output voltage Logic Swing

V ls

= V OH − V OL

where: Vls = voltage logic swing VOH = high state output voltage VOL = low state output voltage Transition Width

X + ( X + Y ) = X

( XY )' = X '+Y '

V tw

= V IH − V IL

where: Vtw = voltage transition width VIH = high state input voltage VIL = low state input voltage


F = force (Newton) Q = charge (Coulomb) V = voltage across the plates (volt) d = distance between plates (m)

TYPICAL CHARACTERISTICS OF IC LOGIC FAMILIES IC Logic Family

Fan out

Power Dissipation (mW)

Propagation Delay (ns)

Noise Margin (V)

Standard TTL Schottky

10

10

10

0.4

10

22

3

0.4

Low power Schottky TTL ECL CMOS

20

2

10

0.4

Coulomb’s Laws of Electrostatics First Law:

“Unlike charges attract each other while like charges repel.” Second Law:

25 50

25 0.1

2 25

0.2 3

“The force of attraction or repulsion between charges is directly proportional to the product of the two charges but inversely proportional to the square of distance between them.”

LEVEL OF INTEGRATION Level of Integration No. of gates per chip

Small Scale Integration (SSI) Medium Scale Integration (MSI) Large Scale Integration (LSI) Very Large Scale Integration (VLSI) Ultra Large Scale Integration (LSI)

F =

Less than 12

k =

12 – 99 100 – 9999 10000 – 99999 100000 or more

CAPACITOR/INDUCTOR TRANSIENT CIRCUITS Capacitors The Gauss Theorem

“The total electric flux extending from a closed surface is equal to the algebraic sum of the charges inside the closed surface.” ψ

≡Q

Electric Flux Density

D =

ψ A

r 2

1

ε

4πε

= ε r ε 0

Permittivity

A measure of how easily the dielectric will “permit” the establishment of flux line within the dielectric. ε

=

D ξ

10 −9 F For vacuum, ε 0 = = 8.854 × 10 −12 36π m Capacitance

C =

•

kQ1Q2

Q V

C = (n − 1)ε

A d

where: Q = charge V = voltage n = number of plates A = plate area d = distance between plates Relative Permittivity (Dielectric Constant) of various dielectrics &r(Average value) Dielectric Material

where: D = flux density, Tesla (T) or Wb/m2 , = electric flux, Weber (Wb) A = plate area, m2 Electric field strength or intensity ( %) F V ξ = = Q d where: - = field strength (N/C, V/m)


Vacuum Air Teflon Paper, paraffined Rubber Transformer oil Mica Porcelain Bakelite

1.0 1.0006 2.0 2.5 3.0 4.0 5.0 6.0 7.0

Glass Distilled water Barium-strontium titanite (ceramic)

7.5 80.0 7500.0

Dielectric strength of some dielectric materials Dielectric Dielectric Strength Material (Average Value) in V/mil

Air Barium-strontium titanite (ceramic) Porcelain Transformer oil Bakelite Rubber Paper, paraffined Teflon Glass Mica

75 75 200 400 400 700 1300 1500 3000 5000

Energy stored

Capacitors in Series

C T

=

1 C 1

QT

+ 1 + C 2

C 3

C =

ε r l

× 10 −9  b  41.4 log   a 

where: a = diameter of single core cable conductor and surrounded by an insulation of inner diameter b .r = relative permittivity of the insulation of the cable l = length of the cylindrical capacitor Capacitance of an isolated sphere C = 4!"r

where: r = radius of the isolated sphere in a medium of relative permittivity .r Capacitance of concentric spheres a.) When outer sphere earthed ab C = 4πε (b − a )

Where: a and b are radii of two concentric spheres . = permittivity of the dielectric between two spheres

1 Q2 2 E = CV = 2 2C 1 1

Cylindrical capacitor

+ ... + 1

C n

= Q1 = Q2 = Q3 = ... = Qn

Capacitors in Parallel C T = C 1 + C 2

+ C 3 + ... + C n QT = Q1 + Q2 + Q3 + ... + Qn

b.) When inner sphere is earthed b2 C = 4πε (b − a )

•

Inductors Inductance (L) is a measure of the ability of a coil

to oppose any change in current through the coil and to store energy in the form of a magnetic field in the region surrounding the coil. In terms of physical dimensions,

Other capacitor configurations Composite medium parallel-plate capacitor ε 0 A C =  d 1 d 2 d 3 

 + +    ε r 1 ε r 2 ε r 3 

where: d1, d2 and d3 = thickness of dielectrics with relative permittivities of .r1, .r2 and .r3 respectively

L

=µ

N 2 A l

Henry

where: + = permeability of the core (H/m) N = number of turns A = area of the core (m2) l = mean length of the core (m) In terms of electrical definition,

Medium partly air parallel-plate capacitor ε 0 A C =   t    d − t −      ε r  


L = N

d φ di

Faraday’s Law

Mutual inductance

“The voltage induced across a coil of wire equals the number of turns in the coil times the rate of change of the magnetic flux.”

It is a measure of the amount of inductive coupling that exists between the two coils.

= N

ein

M =

dt

where: N = number of turns of the coil d φ dt

= change in the magnetic flux

Lenz’s Law

“An induced effect is always such as to oppose the cause that produced it.” ein

= − N

LTa

− LTo

4 where: k = coupling coefficient L1 and L2 = self-inductances of coils 1 and 2 LTa and LTo = total inductances with mutual inductance Coupling coefficient (k)

M

k =

d φ dt k =

Induced voltage by Faraday’s Law di e L = L dt

L1 L2

flux _ linkage _ between _ L1 _ and _ L2 flux _ produced _ by _ L1

Formulas for other coil geometries (a) LONG COIL N 2 A L = µ

Energy stored

W L

M = k L1 L2

d φ

= 1 LI 2

l

2

(b) SHORT COIL Inductance without mutual inductance in series LT = L1 + L2 + L3 + ... + Ln

b.) when fields are opposing LTo = L1 + L2 − 2 M Total inductance without mutual inductance (M)

=

1 L1

+

1 L2

+

1 1 L3

+ ... +

With mutual inductance (M) a.) when fields are aiding 2 L1 L2 − M LT ( a ) = L1 + L2 − 2 M b.) when fields are opposing 2 L1 L2 − M LT ( o) = L1 + L2 + 2 M

l

+ 0.45d

where: L = inductance (H) -7 + = permeability (4 /×10 for air) N = number of turns A = cross-sectional area of the coil (m2) l = length of the core (m) d = diameter of core (m)

With mutual inductance (M) a.) when fields are aiding LTa = L1 + L2 + 2 M

LT

L = µ

N 2 A

1 Ln

(c) TOROIDAL COIL with rectangular crosssection N 2 h d 2 L = µ ln 2π d 1

where: h = thickness d1 and d2 = inner and outer diameters (d) CIRCULAR AIR-CORE COIL 0.07( RN ) 2 L = 6 R + 9l + 10b d b R = +

2 2 where: L = inductance (+H) N = number of turns d = core diameter, in


b = coil build-up, in l = length, in

τ

(e) RECTANGULAR AIR-CORE COIL 0.07(CN ) 2 L = 1.908C + 9l + 10b where: L = inductance (+H)

=

L

v R

R

=

τ

0.012 N µ A l

C

v

=

q C

=

l

+

µ

dt C

∫ idt

open short

Current flowing v i = R

i=

1 L

i=

1 − e

R 

=

t

t

E − = eτ R

L

RT

RT

= R1 + R

E R

e

−

−

t RC

with q0 = 0

t

RC

v R τ

−

vC

= Ee

−

= Ee RC = RC

τ

= RC

RLC Transient Circuits

Conditions for series RLC transient circuit: (1) @ t = 0, i = 0 (2) @ t = 0, Ldi/dt = E Current equations Case 1 – Overdamped case 2

 R  1 then when   >  2 L  LC r t r t i = C 1e + C 2 e 1

dv dt

C 1

2

= −C 2

C 2

r 1 = α + β

short open

α

=−

= − R

β

2 L

2

R L

t

 E  −τt    = R 1 − e     


E

2β L

= α − β R  1 =    −  2 L  LC r 2

Case 2 – Critically damped case −

t

t

RC

 R  1 then when   =  2 L  LC i = e αt (C 1 + C 2 t )

−

= Ee L

Discharging Phase:

∫ vdt

i = C

RL Transient Circuit Storage Cycle:

E 

e

L

t  − RC   vC = E 1 − e    

c

di

1

i

Response of L and C to a voltage source Circuit Element @t=0 @t='

L C

R

R

−

 − RC t   q = EC 1 − e    

c

DC Transient Circuits Circuit Voltage Element across R v = iR

v = L

E

q = EC + (q 0 − EC )e

2

where: L = inductance (+H) N = number of turns A = effective cross-sectional area, cm2 l c = magnetic path length, cm l g = gap length, cm 0 = magnetic permeability

L

v L

RC Transient Circuit Charging Cycle:

(g) MAGNETIC CORE COIL (with air gap) 0.012 N 2 A L =

•

=

i

(f) MAGNETIC CORE COIL (no air gap)

l g

R

Decay Phase:

C = d + y + 2b d = core height, in y = core width, in b = coil build-up, in l = length, in

L =

 − R L t  = E 1 − e    

t

C 1 = 0

=

C 2 α

=−

Total impedance, Z Z = R − jX C

E L

R

Z = R

2 L

Case 2 – Underdamped case

 R  < 1 then   2 L  LC i = e αt (C 1 cos+ C 2 sin β t ) when 

α

=−

β

2 L

R  1 =    −  2 L  LC

C 1 = 0

C 2

=

E β L

V T

=

2

V R

AC CIRCUITS 1

•

Form factor =

Peak factor =

X L

V avg

V m V rms

=

=

= 1.11

0.637V m V m

0.707V m

= 2π fL

X C

=

1 2π fC

V T

=

2

V R

+ V L2

Total impedance, Z Z = R + jX L

Z = R

2

+ X L2

Series RC Circuit Total voltage, VT V T = V R

V T

=

2

V R

θ

=

= tan

θ

−1 X L

= tan −1

Y

V R

=

2

V T Z

+ I L2

I R

=

G

I T

=

2

I R

V R


Y = G

2

I T ∠θ

= − tan −1

+ jI C =

+ I C 2

+ BC 2

I L I R

= Y ∠θ = − tan −1

θ

Total Admittance, Y Y = G + jBC

V C

=

θ

+ B L2

2

Parallel RC Circuit Total Current, I T I T = I R

R

− jV C = V T ∠θ

+ V C 2

=

Parallel AC circuits Parallel RL Circuit Total Current, I T I T = I R − jI L

V L

Z ∠θ θ

R

Total admittance, Y Y = G − jB L

= tan −1

Z ∠θ

•

•

= V T ∠θ

V R

+ ( X L − X C ) 2 −1 ( X L − X C )

I T

= 1.4142

− jX C =

(V L − V C )

Total Current, I T

I T

Series AC circuits Series RL Circuit Total voltage, VT V T = V R + jV L

= ± tan

θ

Introduction to AC: Formulas V rms 0.707V m

2

R

= ± tan −1

θ

Total impedance, Z Z = R + jX L

X C

− jV C = V T ∠θ

+ (V L − V C ) 2

Z = R

= − tan −1

θ

Series RLC Circuit Total voltage, VT V T = V R + jV L

2

R

+ X C 2

2

= Z ∠θ

B L G

I T ∠θ θ

= tan −1

I C I R

= Y ∠θ θ

= tan −1

BC G

Parallel RLC Circuit Total Current, I T I T = I R + jI C − jI L 2

=

I T

I R

= ± tan

θ

Y = G

= ± tan

θ

=

V rms

V avg

= 0.5V p

V rms

=

2

V DC

+

V p

2

+ ( BC − BL ) −1 ( BC − B L )

V rms

V T

2

2

= V p

V avg

= V p

White Noise

= I T Z

V rms

Power of AC Circuits True/Real/Average/Active Power

≈ 1 V p 4

ENERGY CONVERSION

2

= I RV R = V T I T cos θ

R

Types of three-phase alternators A. Wye or Star-connected

Reactive Power 2

Q = I X X eq

=

V x

2

X eq

= I X V X = V T I T sin θ

Apparent Power 2

Q = I T Z =

cosθ = sin θ =

P

S Q

V T

2

Z

= V T I T

= Reactive Factor (RF) S S = P ± jQ = S ∠θ S = P θ

= 3V phase I Line = I phase P 3φ = 3V L I L cos θ P 3φ = 3V P I P cosθ V Line

= Power Factor (PF)

2

b

Square wave

G

Y

P = I R R =

= V p

b

= 0.577V p

= Y ∠θ

Total voltage, V T

V R

a

V avg

Sine wave on dc level

1

2

a

Triangular or Sawtooth

I R

Total impedance, Z

Z =

= V p

V rms

I T ∠θ

+ ( I C − I L ) 2 −1 ( I C − I L )

Total admittance, Y Y = G + jBC − jB L 2

DC Pulse

+Q

2

B. Delta or Mesh-connected

= ± tan −1 Q

P

•

Values of other alternating waveforms Symmetrical Trapezoid a + 0.577(b − a) a+b V rms = V p V avg = V b 2b p


V Line I Line

= V phase

= 3I phase

= 3V L I L cos θ P 3φ = 3V P I P cos θ

P 3φ

Mathematically,

≥1

β A φ

Frequency of the AC Voltage Generated in an Alternator PN f =

= n × 360°

n = 1, 2,3...

Basic Configuration of a Resonant Circuit Oscillator

120

where: f = frequency (Hz) P = number of poles (even number) N = speed of prime mover (rpm) Speed Characteristics of DC Motors E H = k s c φ

where: Ec = counter emf k s = speed constant φ = flux

•

Torque Characteristics of DC Motors T = k t φI a

where: Ia = armature current k t = torque constant φ = flux

LC Oscillators Resonant-Frequency Feedback Oscillators Oscillator Type X1 X2 X3

Hartley Colpitts Clapp Pierce Crystal

L C C C

Amplifier gain without feedback, AV

120 f P

where: N = synchronous speed (rpm) f = frequency (Hz) P = number of poles

= − R r e

for a common-emitter configuration The feedback factor, β

=−

OSCILLATORS

•

Introduction Oscillator Requirements

a. Amplifier b. Tank circuit c. Feedback Overall gain with feedback A A f = 1 + β A

C L Series LC (net L) Crystal (net L)

A. Hartley Oscillator

Speed of an AC Motor

N =

L C C C

L2 L1

To maintain the oscillation, AV

= R = r e

L1 L2

The frequency of oscillation is 1 f 0 = 2π Leq C where

Barkhausen Criterion for Oscillation

a. The net gain around the feedback loop must be no less than one; and b. The net phase-shift around the loop must be a positive integer multiple of 2/ radians or 360°. PDF created with pdfFactory trial version www.pdffactory.com

Leq

= L1 + L2 + 2 M M = L1 L2

B. Colpitts Oscillator

Parallel

Amplifier gain without feedback,

f rp

= − R

AV

r e

C 2

RC Oscillators RC Phase-Shift Oscillator

The gain of the basic inverting amplifier is,

where

R f

=−

AV

=

C 1C 2 C 1

+ C 2

AV

=

r e

β

=

AV

C 1

+

1 1 C 2

+

=−

R f R s

1 C 3

Wien Bridge Oscillator

The open-loop gain is AV

•

Crystal Oscillators Frequency drift

LC: 0.8% Crystal: 0.0001% (1 ppm)

= 1+

R f R s

β

thicknessα

1

=1

3

To maintain the oscillation,

f

R f

The thicker the crystal, the lower its frequency of vibration Series and Parallel Resonant Frequencies Series

=

=3

The feedback factor is

Natural frequency of vibration

f rs

= −29

The frequency of oscillation is, 1 f 0 = 2π RC 6

where 1

29

To maintain the oscillation,

C 1

The frequency of oscillation is 1 f 0 = 2π LC eq

=

=− 1

C 2

C. Clapp Oscillator

C eq

R s

The feedback factor is,

To maintain the oscillation, R

+ C m

•

The frequency of oscillation is 1 f 0 = 2π LC eq C eq

C s

Note: Series resonant frequency, f rs is slightly lower than parallel resonant frequency, f rp.

C 1

=−

C s C m

2π L

The feedback factor, β

1

=

1 2π LC s

R s

=2

The frequency of oscillation is, 1 f 0 = 2π R1C 1 R2 C 2 Neglecting loading effects of the op-amp input and output impedances, the analysis of the bridge results in


R f R s

=

R1 R 2

+

C 2 C 1

Equations of closed-loop gain for different types of feedback connections Feedback Gain with Type of Type Feedback Amplifier Voltage Voltage Av A = vf Series Amplifier 1 + β Av

(bridge-balance condition)

Therefore, for the bridge to be balanced, R 1 = R 2 = R and C1 = C2 = C The frequency of oscillation f 0

=

1 2π RC

FEEDBACK AMPLIFIERS

•

Types of Feedback Connections Equations of open-loop gain, feedback factor and closed-loop gain for different types of feedback $ Feedback Source Output A Af Connection Signal Signal Voltage Voltage Voltage vo v f vo

Series

Current Series

Voltage Current

Voltage Shunt

Current Voltage

Current Shunt

Current Current

vi

vo

v s

io

v f

io

vi

io

v s

vo

i f

vo

ii

vo

i s

io

i f

io

ii

io

i s

Note: Some references try to designate the following terms to describe the four main types of feedback equations. 2. Series-shunt = Voltage series 3. Series-series = Current series 4. Shunt-shunt = Voltage shunt 5. Shunt-series = Current-shunt

•

Negative Feedback Equations A A f = 1 + β A

•

Current Series

G mf

=

Voltage Shunt

Rmf

=

Current Shunt

Aif

=

Gm

1 + βGm

Transconductance Amplifier Transresistance Amplifier

Rm

1 + β Rm

Currrent Amplifier

Ai

1 + β Ai

Performance Characteristics of Negative Feedback Networks Equations of amplifier impedance levels when using negative feedback connection Feedback Input Output Type Resistance Resistance Voltage Ri (1 + βA) Ro

Series

increased

Current Series Voltage Shunt

Ri (1 + βA)

Current Shunt

Ri

A f

where: A = gain without feedback (open-loop gain) Af = gain with feedback (closed-loop gain) 1 + )A = desensitivity or sacrifice factor )A = loop gain

dA f A f dA A

Ri

Ro

1 + β A decreased

1 + β A decreased Ro (1 + βA) increased

1 + β A decreased dA f

where:

increased

1 + β A decreased Ro (1 + βA) increased

=

1

dA

1 + β A A

= change in gain with feedback

= change in gain without feedback magnitude, |)A| = 1 phase-shift, 2 = 180°

The limiting condition is for the negative feedback amplifiers.


•

AC CIRCUITS 2

•

Parallel Resonance A. Theoretical Parallel Resonant Circuit

Series Resonance

Characteristics of parallel resonance

f r

1. 2. 3. 4.

At resonance, BL = BC, XL = XC, IL = IC. At resonance, Z is maximum. Z = R P. At resonance, IT is minimum. IT = IRP. At resonance, Z is resistive. 2 = 0° (I in phase with E). 5. At f < f r, Z is inductive. 2 = – (I Lags E). 6. At f > f r, Z is capacitive. 2 = + (I Leads E).

1

=

2π LC where: f r = resonant frequency L = Inductance C = Capacitance

Q of a Theoretical circuit:

Characteristics of series resonance

1. 2. 3. 4.

At resonance, XL = XC, VL = VC. At resonance, Z is minimum. Z = R. At resonance, I is maximum. I = E/R. At resonance, Z is resistive. 2 = 0° (I in phase with E). 5. At f < f r , Z is capacitive. 2 = + (I Leads E). 6. At f > f r , Z is inductive. 2 = – (I Lags E).

Q=

X L R

=

X C R

Resonant Rise in Voltage V L = V C

=

R P X C

= R P

BW = f 2

− f 1 =

C

= QE

which the operation is satisfactory and is taken between two half-power (3dB down) points.

− f 1 =

f r

Equivalent Theoretical Circuit

Q

If Q 3 10; then f r bisects BW f 1

= f r −

BW

2

f 2

= f r +

L

Bandwidth (BW)

Bandwidth (BW) is the range of frequencies over

BW = f 2

C

f r Q

B. Practical Parallel Resonant Circuit

1 L R

X L

=

Resonant Rise in tank current I tan k = QI T = I L = I C

Quality Factor (Q) of a resonant circuit: Re active _ power _ of _ either _ L _ or _ C Q= Active _ power _ of _ R

Q=

R P

BW

2


Impedance transformation:

Alloys commonly magnetized Alloy Percentage Content

Permalloy Hipernik Perminvar Alnico

22% Fe, 78% Ni 40% Fe, 60% Ni 30% Fe, 45% Ni, 25% Co 24% Co, 51% Fe

Coulomb’s Laws First Law

“The force of attraction or repulsion between two magnetic poles is directly proportional to their strengths.”

Q of Equivalent Theoretical Circuit R Q = P X Leq

Second First Law

“The force of attraction or repulsion between two poles is inversely proportional to the square of the distance between them.”

Q of Practical Circuit

Q =

X L RS

F = k

Resonant frequency (practical circuit)

f r

=

1

2

1−

RS C

; if R S = 0; f r =

where: k = 1

L 2π LC 2π LC 2 1 1 Q = f f r = ; if Q 3 10; r 2π LC 1 + Q 2 2π LC

Total Impedance Z Z = RS (1 + Q 2 ) ≈ Q 2 RS

if Q 3 10

MAGNETISM AND MAGNETIC CIRCUITS

•

Magnetism Curie temperature (Pierre Curie) – the critical

temperature such that when ferromagnets are heated above that temperature their ability to possess permanent magnetism disappears. Curie temperatures of ferromagnets Ferromagnet Temperature (°C)

Iron (Fe) Nickel (Ni) Cobalt (Co) Gadolinium

770 358 1130 16

m1m 2 r 2

1 4πµ

(Newtons, N) µ

= µ r µ 0

Magnitude of the Force F = BIl sin θ

(Newtons, N) where: B = flux density (Wb/m2) I = current (A) l = length of conductor (m) 2 = angle between the conductor and field Magnitude of the flux surrounding a straight conductor Φ = 14 Il log R (Maxwells, Mx) r

where: I = current (A) l = length of conductor (ft) R = radius to the desired limiting cylinder r = radius of the conductor The force between two parallel conductors 2 I I l F = 1 2 × 10 −7 (Newtons, N) d

where: l = length of each conductor (m) d = distance between conductors (m) I1 = current carried by conductor A I2 = current carried by conductor B


Magnitude of the flux between two parallel conductors − Φ = 28 Il log (d r ) (Maxwells, Mx) r

Ohm’s Law for Magnetic Circuits Cause Effect = Opposition

where: I = current (A) l = length of conductor (ft) r = radius of each conductor (m) d = distance of the conductors from center to center (m)

Then,

•

Magnetic Circuits

B

=Φ

A

where: B = Flux density in Tesla (T) 4 = Flux lines in Webers (Wb) A = Area in square meters (m2) Note: 1 Tesla = 1 Wb/m2

Φ=

where: ℜ = reluctance ℑ = magnetomotive force, mmf (Gb or At) Φ = flux (Weber or Maxwells) Comparison bet. Magnetic and Electric Circuits Electric Circuits Magnetic Circuits Resistance, R (!) Reluctance, ℜ (Gb/Mx) Current, I (A) Flux, 4 (Wb or Mx)

Total reluctance in series

ℜ T = ℜ1 + ℜ 2 + ... + ℜ n

= 4π × 10 −7

Weber

H or Ampere − meter m

Note: + = +0; +r = 1 5 non–magnetic + < +0; +r < 1 5 diamagnetic + > +0; +r > 1 5 paramagnetic + >> +0; +r >> 1 5 ferromagnetic (+r 3 100)

ℜ=

L

c.)

Maxwell

= 1 + 1 + ... + 1 ℜ T ℜ1 ℜ 2 ℜn Total flux in series

Φ T = Φ1 = Φ 2 = ... = Φ n Φ T = Φ 1 + Φ 2 + ... + Φ n

Different units of Reluctance ( ℜ ) Ampere − turn Ampere − turn

1

µ A

Note: The t in the unit A-t/Wb is the number of turns of the applied winding.

Weber Gilbert

Total reluctance in parallel

Total flux in parallel

where: ℜ = reluctance L = the length of the magnetic path A = the cross-sectional area

a.)

mmf, ℑ (Gb or At)

emf, V (V)

Permeability

µ0

ℑ ℜ

b.)

d.)

Maxwell Gilbert Weber

Note: 1 Weber = 1×108 maxwells 1 Gilbert = 0.7958 ampere-turns 1 Gauss = 1 maxwell/cm2

Energy stored

W m

1 2

= ℜΦ 2

Joules

Magnetomotive force (mmf, ℑ ) Ampere – turns, At ℑ = NI Gilberts, Gb ℑ = 0.4π NI mmf of an air gap dB mmf = µ0

Ampere-turns

Tractive force or lifting force of a magnet


1  AB 2   F =  2  µ 0  

Newtons

Magnetizing Force (H)

H =

ℑ

H =

l

NI l

Note: The unit of H is At/m Permeability – the ratio of flux density to the

magnetizing force. µ

= B

H

B and H of an infinitely long straight wire µ I I B = H = 2πr 2πr Steinmetz’s Formula of Hysteresis Loss J W h = ηfBm1.6 m3 where: ( = hysteresis coefficient

f = frequency Bm = maximum flux density Ampere’s Circuital Law

“The algebraic sum of the rises and drops of the mmf a closed loop of a magnetic circuit is equal to zero; that is, the sum of the mmf rises equals the sum of the mmf drops around a closed loop.” ∑ ∩ℑ = 0 (for magnetic circuits) Source of mmf is expressed by the equation (At) ℑ = NI For mmf drop,

ℑ = Φℜ

(At)

A more practical equation of mmf drop (At) ℑ = H l


ELECTRONICS Formulas and Concepts.pdf

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