ELECTRONICS Formulas and Concepts

SEMICONDUCTOR THEORY

The diode current equation kVd

I d = I s (e Tk − 1)

• Atomic Structure Diameter of neutron = 10-13 cm Maximum number of electrons per shell or orbit N e = 2n 2 n = 1,2,3,4 Letter designation K shell – 1 L shell – 2 M shell – 3 N shell – 4

O shell – 5 P shell – 6 Q shell – 7

Mass and Charge of different Particles Particle Mass (kg) Charge (C) − 31 Electron 9.1096 × 10 − 1.6022 × 10−19 Proton 1.6726 × 10−27 + 1.6022 × 10 −19 No charge Neutron 1.6726 × 10−27 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 Ge = .67eV 1 0eV Conductor At room temperature: there are approximately 1.5×1010 of free electrons in a cubic centimeter (cm3) for intrinsic silicon and 2.5×1013 for germanium. •

Diode Theory VthT 1 = VthT 0 + k (T1 − T0 )

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

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 k= 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 and Ge Temperature effects on Is I sT1 = I sT0 e k (T1 −T0 ) Where: IsT1 = saturation current at temperature T1 IsT0 = saturation current at room temperature k = 0.07/°C T1 = new temperature T0 = room temperature (25°C) Reverse Recovery Time (Trr) Trr = t s + tt Where: Trr = time elapsed from forward to reverse bias (ranges from a few ns to few hundreds of ps) Tt = transition time Ts = storage time

DC CIRCUITS 1 1 Coulomb = 6.24×1018 electrons By definition: A wire of 1 mil diameter has a crosssectional area of 1 Circular Mil (CM) 1 mil = 10-3 in 1 in = 1000 mils Asquare = 1 mil2

πD 2 Acircle = mil 2 4 4 1mil 2 = CM π

Quark Up Down Charm Strange Top Bottom

Temperature effects on resistance R2 T + t 2 1 α1 = = R1 T + t1 T + t1

Type/Flavors of Quarks Symbol Charge Baryon no. U +2/3 1/3 D –1/3 1/3 C +2/3 1/3 S –1/3 1/3 T +2/3 1/3 B –1/3 1/3

Proton – 2 Up and 1 Down Neutron – 1 Up and 2 Down

Type D C AA AAA I=

Q t

V =

ρ=

Types of Battery Height (in) Diameter (in) 2 14 1 14 1 34 1 9 178 16 3 3 1 4 8 Ampere( A);

W Q

RA L

Volt (V );

Coulomb(C ) sec ond ( s )

Joule( J ) Coulomb(C )

Ω − m; Ω − cm;

Ω − CM ft

Resistivities of common metals and alloys Material ρ (10-8Ω-m) Aluminum (Al) 2.6 Brass 6 Carbon 350 Constantan (60% Cu and 40% Ni) 50 Copper (Cu) 1.7 Manganin (84% Cu, 12%Mn & 44 4%Ni) Nichrome 100 Silver (Ag) 1.5 Tungsten (W) 5.6 Absolute zero = 0 K = –273°C ρCu = 10.37 Ω–CM/ft

R2 = R1 [1 + α 1 (t 2 − t1 )] where: |T| = inferred absolute temperature, °C R2 = final resistance at final temp. t2 R1 = initial resistance at initial temp. t1 α1 = temp coefficient of resistance at t1 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 -236 Copper, annealed -234.5 Copper, hard-drawn -242 Iron -180 Nickel -147 Silver -243 Steel, soft -218 Tin -218 Tungsten -202 Zinc -250 Temperature-Resistance Coefficients at 20 °C Material α20 Nickel 0.006 Iron, commercial 0.0055 Tungsten 0.0045 Copper, annealed 0.00393 Aluminum 0.0039 Lead 0.0039 Copper, hard-drawn 0.00382 Silver 0.0038 Zinc 0.0037 Gold, pure 0.0034 Platinum 0.003 Bras 0.002 Nichrome 0.00044 German Silver 0.0004 Nichrome II 0.00016 Manganin 0.00003 Advance 0.000018 Constantan 0.000008

1 A A = =σ R ρL L where: σ = specific conductance or conductivity of the material in siemens/m or mho/m. G=

Note: The best is silver with 1.68×1024 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 Q E2 = E = IE = = I 2R t t R where: W = work in Joules (J) t = time in seconds (s) Q = charge in Coulomb (C) P=

Transformations or Conversations: Delta (Δ) to Wye (Y) Pr oduct _ of _ adjacent _ R _ in _ ∆ RY = ∑ of _ all _ R _ in _ ∆ Wye (Y) to Delta (Δ) ∑ of _ cross _ products _ in _ Y R∆ = Opposite _ R _ in _ Y

Black Brown Red Orange Yellow Green Blue Violet Gray White

Multiplier 100 101 102 103 104 105 106 107 108 109

Tolerrance (±%) 20 1 2 3 GMV 5 -

0.1 0.01 -

5 10 20

+100 Bypass -

Fifth band reliability color code Color Failures during 1000 hours of operation Brown 1.0% Red 0.1% Orange 0.01% Yellow 0.001% Ampere − hour _ rating ( Ah) Amperes _ drawn ( A)

Cell Types and Open-Circuit Voltage Cell Name Type Nominal OpenCircuit Voltage Carbon-zinc Primary 1.5 Zinc-chloride Primary 1.5 Manganese Primary or 1.5 dioxide (alkaline) Secondary Mercuric oxide Primary 1.35 Silver oxide Primary 1.5 Lead-acid Secondary 2.1 Nickel-cadmium Secondary 1.25 Nickel-iron Secondary 1.2 (Edison cell) Silver-zinc Secondary 1.2 Silver-cadmium Secondary 1.1 Nickel metal Secondary 1.2 hydride (NiMH)

DIODES

Color Coding Table 2nd significant 0 1 2 3 4 5 6 7 8 9

-

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

Battery life =

Current Division Theorem R2 R1 IT IT I1 = I2 = R1 + R2 R1 + R2

1st significant 0 1 2 3 4 5 6 7 8 9

-

Batteries

Voltage Division Theorem 2 resistors in series with one R1 R2 E E V1 = V2 = R1 + R2 R1 + R2

Color

Gold Silver None

Temp Coef ppm/°C 0 -33 -75 -150 -220 -330 -470 -750 +30 +500

• Diode Applications Half–wave Rectification V VDC = m = 0.318Vm π PIV rating ≥ Vm Full–wave Rectification t

Vrms

1 = V (t ) 2 dt ∫ T 0

VDC = 0.36Vm PIV rating ≥ Vm for bridge-type

PIV rating ≥ 2Vm

for center-tapped

• Other Semiconductor Devices Zener Diode ∆VZ TCC = VZ (T1 − T0 ) where: TCC = temperature coefficient T1–T0 = change in temperature VZ = Zener Voltage at T0

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

III. Fixed Vi, variable RL RVZ RL min = Vi − VZ

Vi max = I R max RS + VZ

RL max =

∆C C O (T1 − T0 ) where: TCC = temperature coefficient T1 – T0 = change in temperature C0 = capacitance at T0 TC C =

Basic Zener Regulator I. Vi and RL 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 RL, variable Vi ( R + RS )VZ Vi min = L RL

In terms of the applied reverse bias voltage: C (0) CT = n  VR   1 +  VT 

VZ I L min

Varactor diode or Varicap diode A CT = ε Wd 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 CT = (VT + VR ) 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 ⅓ for diffused junctions

Photodiode c ; Joules λ where: W = energy associated with incident light waves h = Planck’s constant (6.624×10-34 J-sec) f = frequency W = hf = h

1eV = 1.6×10-9 J 1 Angstrom (Å) = 10-10 m Solar Cell η=

PO = Pi

Pmax  1W  ( Area ) 2   cm 

where: η = 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 0.150 = = 150 width base 0.001

• Basic Operation Relationship between IE, IB and IC: I E = IB + IC

h21 = forward transfer current ratio, hf h22 = output conductance, ho

IC is composed of two components: I C = I majority + I min ority DC Transistor Parameters  ∆I  α =  C   ∆I E  Vcb =cons tan t

 ∆I β =  C  ∆I B I β= C IB

  Vce = cons tan t

IC IE where: IE = emitter current IB = base current IC = collector current α = CB short-circuit amplification factor β = CE forward-current amplification factor α=

Relationship between α and β: β α α= β= β +1 1−α Stability Factor (S): ∆I C ∆I CO ∆I C S ( I CO ) = ∆VBE ∆I S ( I CO ) = C ∆β S ( I CO ) =

Unitless

hi hr hf ho

Comparison between 3 transistor configurations CB CE CC Zi low moderate high Zo high moderate low Ai low high moderate Av high high low Ap moderate high low shift none 180° none B. Re Model Note: Common Base : hib = re ; hfb = –1 Common Emitter: β = hfe ; βre = hie

Siemens Ampere

H-Parameters typical values CE CB CC 1kΩ 20Ω 1kΩ 2.5×10-4 3×10-4 ≈1 50 -0.98 -50 25μS 0.5μS 25μS

FIELD EFFECT TRANSISTORS •

JFET  V I D = I DSS 1 − GS VP 

Small Signal Analysis A. Hybrid Model Vi = h11 I in + h12V0

gm =

I o = h21 I in + h22V0 If Vo = 0 h11 =

Vi I in

ohms

If Iin = 0 h12 =

Vi V0

unitless

h21 =

I0 I in

unitless

If Vo = 0

2 I DSS VP

 VGS 1 − VP 

g mo =  ∆I d  =  ∆V gs

2 I DSS VP

Vds = 0

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

I0 siemens V0 where: h11 = input-impedance, hi h12 = reverse transfer voltage ratio, hr

2

0 ≤ VGS ≤ 5

MOSFET I DS = k (VGS − VTH ) 2 k = 0.3mA / V 2

If Iin = 0 h11 =

  

•

FET biasing

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 Vr (rms ) = 0.385Vm Vr (rms ) = 0.308Vm

POWER SUPPLY •

Transformer I p Vs N a= = = s = Is Vp N p

Zs Zp

where: a = turns ratio Vs = secondary induce voltage Vp = primary voltage Ns = no. of turns on the secondary windings Np = no. of turns on the primary windings Ip = current in the primary windings Is = current in the secondary windings Zs = impedance of the load connected to the secondary winding Zp = impedance looking into the primary from source • Rectifier Half–wave signal Vm 2 Vdc = 0.636Vrms PIV = 2Vrms Ripple frequency = AC input frequency Vdc = 0.318Vm

Vrms =

Full–wave rectified signal (bridge type) V Vdc = 0.636Vm Vrms = m 2 Vdc = 0.9Vrms PIV = 2Vrms Ripple frequency = 2×AC input frequency Full–wave with center-tapped transformer Vdc = 0.9Vrms of the half the secondary = 0.45Vrms of the full secondary = 0.637Vpk of half of the secondary = 0.637Vpk of the full secondary PIV = 1.414Vrms of full secondary AC Vr (rms ) r= = DC Vdc Vr (rms ) = Vrms − Vdc where: r = ripple factor Vr(rms) = rms value of the ripple voltage 2

2

•

half–wave rectified signal full–wave rectified signal

Filter Vr (rms ) =

Vr ( p ) 3

=

Vr ( p − p )

2 3 2.4 I dc 2.4Vdc Vr (rms ) = = = C RL C 4 3 fC I 4.17 I dc V ( p − p) Vdc = Vm − r = Vm − dc = Vm − 2 4 fC C 2.4 I dc V (rms ) 2.4 r= r × 100% = × 100% = × 100% Vdc CVdc RL C where: Idc = the load current in mA C = filter capacitor in μF RL = load resistance at the filter stage in kΩ Vm = the peak rectified voltage Idc = the load current in mA C = filter capacitor in μF f = frequency at 60 Hz I dc

• Regulator Voltage Regulation Vnoload − V fload V .R. = × 100% V fload Stability factor (S) ∆Vout S= (constant output current) ∆Vin Improved series regulation R + R2 Vo = 1 (VZ + VBE 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 Ro = equivalent resistance

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

Ro I = I fs Ro + Ru where: Ifs = full scale current Voc = open circuit voltage Ro = internal resistance of ohmmeter D = meter deflection Ru = unknown resistance D=

• Ammeter Shunt I fs Rm Rsh = I t − I fs

Rinsh =

Rm Rsh Rm + Rsh

Vin I fs Rm = I in It where: Rsh = shunt resistance Ifs = full scale current Rm = meter resistance It = total current Rinsh = input resistance of the shunted meter Vin = voltage input Iin = current input

S ac =

Rin sh =

Rin =

V fs I fs

Voltmeter Loading Error V Rin accuracy = wm = Vwom Rin + Ro R V Vwm = in wom Rin + Ro •

Ohmmeter V I fs = oc Ro

0.9 Sensitivity for a full-wave rectifier I fs

• DC Bridges Wheatstone bridge ohmmeter Bridge is balance if R1 R3 = R2 R4 •

Attenuators

Ro = Rins Rino where: Ro = characteristic resistance Rins = input resistance with output terminals shorted Rino = input resistance with output terminals open

• Voltmeter For full scale current Vfs = (Rs + Rm)Ifs V fs Rs = − Rm I fs Rin = Rs + Rm where: Vfs = full scale voltage Rs = series resistor Rin = input resistance Sensitivity of Voltmeter 1 S= I fs

AC Detection 0.45 S ac = Sensitivity for a half-wave rectifier I fs

Voc I= Ro + Ru

L type or the voltage divider V R2 1 attenuation = in = gain = Vout gain R1 + R2 X R1 RC = C1 C1 = 2 2 R2 X C 2 R1 Symmetrical Attenuator R m = 2 ; R2 = mR1 R1 Symmetrical T Analysis R0 = R1 1 + 2m

a=

Vin 1 + m + 1 + 2m = Vout m

Symmetrical Pi Analysis R2 V 1 + m + 1 + 2m R0 = a = in = Vout m 1 + 2m

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

6 7 8 9

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

R1 = R0

R1 +1 R2

Design R1 = R0

R2 =

R0 a −1

R3 =

a −1 R0

COMPUTER FUNDAMENTALS r's complement

(rn)10 – N

(r – 1)’s complement (rn – r-m)10 – N Types of Binary Coding Binary Coded Decimal Code (BCD) DECIMAL DIGIT BCD Equivalent 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 Excess-3-code DECIMAL DIGIT Excess-3 0 0011 1 0100 2 0101 3 0110 4 0111 5 1000

1001 1010 1011 1100

Gray Code (Reflected Code) DECIMAL DIGIT Gray Code 0 0000 1 0001 2 0011 3 0010 4 0110 5 0111 6 0101 7 0100 8 1100 9 1101 10 1111 11 1110 12 1010 13 1011 14 1001 15 1000 DECIMAL

84-2-1

2421

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

Biquinary 5043210 0100001 0100010 0100100 0101000 0110000 1000001 1000010 1000100 1001000 1010000

OPERATIONAL AMPLIFIERS VD = V+ – Vwhere: VD = differential voltage V+ = voltage at the non-inverting terminal V- = voltage at the inverting terminal Ad Ac where: Ad = differential gain of the amplifier Ac = common-gain of the amplifier CMRR =

Slew rate

∆Vo SR = = 2πf max V pk ∆t where: fmax = highest undistorted frequency Vpk = peak value of output sine wave Differentiator Vo = − RC

dVin dt

Integrator Vo = −

1 Vin dt RC ∫

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

R2 R1

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

X •Y = Y • X

X • (YZ ) = ( XY ) • Z

(Distributive Law) X (Y + Z ) = XY + YZ X + (YZ ) = ( X + Y )(Y + Z ) (Law of Absorption) ( X + Z ) X + XY = X (De Morgan’s Theorem) ( X + Y )' = X ' Y '

• Logic Family Criterion Propagation delay is the average transition delay time for a signal to propagate from input to output. t +t t p = PHL PLH 2 where: tp = 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 + I CCL I CC ( AVG ) = CCH 2 PD ( AVG ) = I CC ( AVG ) × VCC where: ICCH = current drawn from the power supply at high level ICCL = current drawn from the power supply at low level Noise Margin is the maximum noise voltage added 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 = VIL − VOL where: NM = Noise Margin VIL = low state input voltage VOL = low state output voltage High State Margin NM H = VOH − VIH where: NM = Noise Margin VIH = high state input voltage VOH = high state output voltage Logic Swing

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

X + (X + Y ) = X ( XY )' = X '+Y '

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

TYPICAL CHARACTERISTICS OF IC LOGIC FAMILIES IC Logic Family Standard TTL Schottky

Fan out

10

22

3

0.4

Low power Schottky TTL ECL CMOS

20

2

10

0.4

10

25 50

Power Propagation Dissipation Delay (ns) (mW) 10 10

25 0.1

2 25

Noise Margin (V) 0.4

0.2 3

LEVEL OF INTEGRATION Level of Integration No. of gates per chip Small Scale Integration Less than 12 (SSI) Medium Scale 12 – 99 Integration (MSI) Large Scale Integration 100 – 9999 (LSI) Very Large Scale 10000 – 99999 Integration (VLSI) Ultra Large Scale 100000 or more Integration (LSI)

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

ψ A where: D = flux density, Tesla (T) or Wb/m2 ψ = electric flux, Weber (Wb) A = plate area, m2 D=

Electric field strength or intensity (ξ) F V ξ= = Q d where: ξ = field strength (N/C, V/m)

F = force (Newton) Q = charge (Coulomb) V = voltage across the plates (volt) d = distance between plates (m) Coulomb’s Laws of Electrostatics First Law: “Unlike charges attract each other while like charges repel.” Second Law: “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.” kQ Q F = 12 2 r 1 k= ε = ε rε 0 4πε Permittivity A measure of how easily the dielectric will “permit” the establishment of flux line within the dielectric. D ε = ξ −9 10 F For vacuum, ε 0 = = 8.854 × 10 −12 36π m Capacitance Q A C = (n − 1)ε V d where: Q = charge V = voltage n = number of plates A = plate area d = distance between plates C=

Relative Permittivity (Dielectric Constant) of various dielectrics Dielectric Material εr(Average value) Vacuum 1.0 Air 1.0006 Teflon 2.0 Paper, paraffined 2.5 Rubber 3.0 Transformer oil 4.0 Mica 5.0 Porcelain 6.0 Bakelite 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 75 Barium-strontium 75 titanite (ceramic) Porcelain 200 Transformer oil 400 Bakelite 400 Rubber 700 Paper, paraffined 1300 Teflon 1500 Glass 3000 Mica 5000 Energy stored E=

1 Q2 CV 2 = 2 2C

Capacitors in Series 1 CT = 1 1 1 1 + + + ... + C1 C 2 C 3 Cn QT = Q1 = Q2 = Q3 = ... = Qn Capacitors in Parallel CT = C1 + C 2 + C 3 + ... + C n QT = Q1 + Q2 + Q3 + ... + Qn Other capacitor configurations Composite medium parallel-plate capacitor ε0A C=  d1 d 2 d 3    + +  ε r1 ε r 2 ε r 3  where: d1, d2 and d3 = thickness of dielectrics with relative permittivities of εr1, εr2 and εr3 respectively Medium partly air parallel-plate capacitor ε0 A C=     d − t − t      εr  

Cylindrical capacitor ε rl × 10 −9 b  41.4 log  a  where: a = diameter of single core cable conductor and surrounded by an insulation of inner diameter b εr = relative permittivity of the insulation of the cable l = length of the cylindrical capacitor C=

Capacitance of an isolated sphere C = 4πεr where: r = radius of the isolated sphere in a medium of relative permittivity εr Capacitance of concentric spheres a.) When outer sphere earthed ab C = 4πε (b − a ) Where: a and b are radii of two concentric spheres ε = permittivity of the dielectric between two spheres 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, N2A L=µ Henry l where: μ = 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, dφ L=N di

Faraday’s Law “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.” dφ ein = N dt where: N = number of turns of the coil dφ = change in the magnetic flux dt Lenz’s Law “An induced effect is always such as to oppose the cause that produced it.” dφ ein = − N dt Induced voltage by Faraday’s Law di eL = L dt Energy stored WL =

1 2 LI 2

Mutual inductance It is a measure of the amount of inductive coupling that exists between the two coils. M = k L1 L2 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 M =

Coupling coefficient (k) k= k=

M L1 L2

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

Formulas for other coil geometries (a) LONG COIL N2A L=µ l (b) SHORT COIL

Inductance without mutual inductance in series LT = L1 + L2 + L3 + ... + Ln With mutual inductance (M) a.) when fields are aiding LTa = L1 + L2 + 2 M b.) when fields are opposing LTo = L1 + L2 − 2M Total inductance without mutual inductance (M) 1 LT = 1 1 1 1 + + + ... + L1 L2 L3 Ln With mutual inductance (M) a.) when fields are aiding L1 L2 − M 2 LT ( a ) = L1 + L2 − 2 M b.) when fields are opposing L1 L2 − M 2 LT ( o) = L1 + L2 + 2 M

N2A l + 0.45d where: L = inductance (H) μ = permeability (4π×10-7 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) L=µ

(c) TOROIDAL COIL with rectangular crosssection N 2h d2 L=µ ln 2π d1 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 (μH) 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 (μH) C = d + y + 2b d = core height, in y = core width, in b = coil build-up, in l = length, in (f) MAGNETIC CORE COIL (no air gap) 0.012 N 2 µA L= lc (g) MAGNETIC CORE COIL (with air gap) 0.012 N 2 A L= l lg + c µ where: L = inductance (μH) N = number of turns A = effective cross-sectional area, cm2 l c = magnetic path length, cm l g = gap length, cm Μ = magnetic permeability •

DC Transient Circuits Circuit Voltage Element across R v = iR L C

di v=L dt q 1 v = = ∫ idt C C

Current flowing v i= R 1 i = ∫ vdt L dv i=C dt

Response of L and C to a voltage source Circuit Element @ t = 0 @ t = ∞ L open short C short open RL Transient Circuit Storage Cycle: t R − t  − E E  L  τ  i = 1 − e  = 1 − e R  R

R − t   v R = E 1 − e L   

L R

v L = Ee

Decay Phase: t

R

E − t E − i= e L = e τ R R τ =

L RT

RT = R1 + R

RC Transient Circuit Charging Cycle: q = EC + (q 0 − EC )e t  −  q = EC 1 − e RC    t E − i = e RC R t  −  RC  vC = E 1 − e   

−

t RC

with q0 = 0 v R = Ee

t RC

τ = RC

Discharging Phase: vC = Ee

−

t 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

1  R  then when   > LC  2L  i = C1 e r1t + C 2 e r2 t E 2βL r2 = α − β

C2 = −

C1 = −C 2 r1 = α + β α =−

 R  1 β =  −  2 L  LC

R 2L

Case 2 – Critically damped case 2

   

−

1  R  when   = then LC  2L  i = e αt (C1 + C 2 t )

R − t L

C1 = 0

C2 = α =−

Total impedance, Z Z = R − jX C = Z ∠θ

E L

R 2L

Case 2 – Underdamped case

Series RLC Circuit Total voltage, VT VT = VR + jVL − jVC = VT ∠θ

2

1  R  when   < then LC  2L  i = e αt (C1 cos+ C 2 sin β t ) α =−

R 2L

β=

C1 = 0

 R  1  −  2 L  LC E C2 = βL

AC CIRCUITS 1 •

Vm Vm = = 1.4142 Vrms 0.707Vm

X L = 2πfL

XC =

1 2πfC

VT = VR2 + VL2

θ = tan −1

VL VR

θ = tan −1

XL R

Series RC Circuit Total voltage, VT VT = VR − jVC = VT ∠θ VT = VR2 + VC2

Z = R2 + (X L − X C )2

θ = tan −1

(X L − XC ) R

Total Current, IT IT =

VT Z

• Parallel AC circuits Parallel RL Circuit Total Current, IT I T = I R − jI L = I T ∠θ θ = − tan −1

IL IR

Total admittance, Y Y = G − jBL = Y ∠θ

Total impedance, Z Z = R + jX L = Z ∠θ Z = R 2 + X L2

(VL − VC ) VR

Total impedance, Z Z = R + jX L − jX C = Z ∠θ

I T = I R2 + I L2

• Series AC circuits Series RL Circuit Total voltage, VT VT = VR + jVL = VT ∠θ

θ = ± tan −1

VT = VR2 + (VL − VC ) 2

θ = ± tan −1

Introduction to AC: Formulas 0.707Vm V Form factor = rms = = 1.11 Vavg 0.637Vm Peak factor =

XC R

θ = − tan −1

Z = R 2 + X C2

Y = G 2 + BL2

θ = − tan −1

BL G

Parallel RC Circuit Total Current, IT I T = I R + jI C = I T ∠θ I T = I R2 + I C2

θ = tan −1

IC IR

Total Admittance, Y Y = G + jBC = Y ∠θ VC VR

Y = G 2 + BC2

θ = tan −1

BC G

Parallel RLC Circuit Total Current, IT I T = I R + jI C − jI L = I T ∠θ I T = I R2 + ( I C − I L ) 2

DC Pulse Vrms = V p

a b

Vavg = V p

Triangular or Sawtooth Vrms = 0.577V p

(I C − I L ) θ = ± tan IR −1

Total admittance, Y Y = G + jBC − jBL = Y ∠θ

Vavg = 0.5V p

Sine wave on dc level Vrms = VDC + 2

Y = G 2 + ( BC − BL ) 2 θ = ± tan −1

( BC − BL ) G

Total impedance, Z 1 Z= Y

Vp

2

2

Square wave Vrms = V p

Total voltage, VT VT = I T Z

Power of AC Circuits True/Real/Average/Active Power 2 V 2 P = I R R = R = I RVR = VT I T cos θ R

Vavg = V p

White Noise 1 Vrms ≈ V p 4

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

Reactive Power 2

Q = I X X eq = 2

Vx = I X V X = VT I T sin θ X eq

Apparent Power 2

V Q = I T Z = T = VT I T Z P cos θ = = Power Factor (PF) S Q sin θ = = Reactive Factor (RF) S S = P ± jQ = S ∠θ 2

S = P2 + Q2 θ = ± tan −1

a b

VLine = 3V phase I Line = I phase P3φ = 3VL I L cos θ P3φ = 3VP I P cosθ B. Delta or Mesh-connected

Q P

• Values of other alternating waveforms Symmetrical Trapezoid a + 0.577(b − a) a+b Vrms = Vp Vavg = Vp b 2b

VLine = V phase I Line = 3I phase

P3φ = 3VL I L cos θ

Mathematically, βA ≥ 1

P3φ = 3VP I P cosθ

φ = n × 360°

Frequency of the AC Voltage Generated in an Alternator PN f = 120 where: f = frequency (Hz) P = number of poles (even number) N = speed of prime mover (rpm) Speed Characteristics of DC Motors E H = ks c φ where: Ec = counter emf ks = speed constant φ = flux Torque Characteristics of DC Motors T = k t φI a where: Ia = armature current kt = torque constant φ = flux Speed of an AC Motor 120 f P where: N = synchronous speed (rpm) f = frequency (Hz) P = number of poles N=

Basic Configuration of a Resonant Circuit Oscillator

•

LC Oscillators Resonant-Frequency Feedback Oscillators Oscillator Type X1 X2 X3 Hartley L L C Colpitts C C L Clapp C C Series LC (net L) Pierce Crystal C C Crystal (net L)

A. Hartley Oscillator Amplifier gain without feedback, R AV = − re for a common-emitter configuration The feedback factor, β =−

OSCILLATORS • Introduction Oscillator Requirements a. Amplifier b. Tank circuit c. Feedback Overall gain with feedback Af =

A 1 + βA

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 2π radians or 360°.

n = 1, 2,3...

L2 L1

To maintain the oscillation, R L AV = = 1 re L2 The frequency of oscillation is 1 f0 = 2π Leq C where Leq = L1 + L2 + 2M M = L1 L2

B. Colpitts Oscillator Amplifier gain without feedback, R AV = − re

Parallel

The feedback factor,

Note: Series resonant frequency, frs is slightly lower than parallel resonant frequency, frp.

C β =− 1 C2 The frequency of oscillation is 1 f0 = 2π LC eq where C eq =

C1C 2 C1 + C 2

To maintain the oscillation, R C AV = = 2 re C1 C. Clapp Oscillator The frequency of oscillation is 1 f0 = 2π LC eq where C eq =

1 1 1 1 + + C1 C 2 C 3

• Crystal Oscillators Frequency drift LC: 0.8% Crystal: 0.0001% (1 ppm) Natural frequency of vibration 1 thicknessα f The thicker the crystal, the lower its frequency of vibration Series and Parallel Resonant Frequencies Series 1 f rs = 2π LC s

f rp =

1 CC 2π L s m C s + Cm

• RC Oscillators RC Phase-Shift Oscillator The gain of the basic inverting amplifier is, Rf AV = − Rs The feedback factor is, β =−

1 29

To maintain the oscillation, Rf = −29 AV = − Rs The frequency of oscillation is, 1 f0 = 2πRC 6 Wien Bridge Oscillator The open-loop gain is AV = 1 +

Rf Rs

=3

The feedback factor is β=

1 3

To maintain the oscillation, Rf =2 Rs The frequency of oscillation is, 1 f0 = 2π R1C1 R2 C 2 Neglecting loading effects of the op-amp input and output impedances, the analysis of the bridge results in

Rf Rs

=

R1 C 2 + R2 C1

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 Current Transconductance Gm G = mf Series Amplifier 1 + βG

(bridge-balance condition)

Therefore, for the bridge to be balanced, R1 = R2 = R and C1 = C2 = C The frequency of oscillation f0 =

1 2πRC

m

Voltage Shunt

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 β Af Connection Signal Signal Voltage Voltage Voltage v o vf vo Series vi vs vo Current Voltage Current vf io io Series v v i Voltage Shunt

Current Voltage

Current Shunt

Current

Current

i

o

s

vo ii

if

vo is

io ii

vo if io

io is

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

Current Shunt •

Transresistance Amplifier Currrent Amplifier

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 + β A) Ro Series increased 1 + βA decreased Current Ri (1 + β A) Ro (1 + β A) Series increased increased Voltage Ri Ro Shunt 1 + βA 1 + βA decreased decreased Current Ro (1 + β A) Ri Shunt increased 1 + βA decreased dA f Af

•

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

Rm Rmf = 1 + β Rm Ai Aif = 1 + β Ai

where:

dA f Af

=

1 dA 1 + βA A

= change in gain with feedback

dA = change in gain without feedback A magnitude, |βA| = 1 phase-shift, θ = 180° The limiting condition is for the negative feedback amplifiers.

AC CIRCUITS 2 •

• Parallel Resonance A. Theoretical Parallel Resonant Circuit

Series Resonance

fr =

1

2π LC where: fr = resonant frequency L = Inductance C = Capacitance Characteristics of series resonance 1. At resonance, XL = XC, VL = VC. 2. At resonance, Z is minimum. Z = R. 3. At resonance, I is maximum. I = E/R. 4. At resonance, Z is resistive. θ = 0° (I in phase with E). 5. At f < fr, Z is capacitive. θ = + (I Leads E). 6. At f > fr, Z is inductive. θ = – (I Lags E). Quality Factor (Q) of a resonant circuit: Re active _ power _ of _ either _ L _ or _ C Q= Active _ power _ of _ R Q=

XL XC 1 L = = R R R C

Characteristics of parallel resonance 1. At resonance, BL = BC, XL = XC, IL = IC. 2. At resonance, Z is maximum. Z = RP. 3. At resonance, IT is minimum. IT = IRP. 4. At resonance, Z is resistive. θ = 0° (I in phase with E). 5. At f < fr, Z is inductive. θ = – (I Lags E). 6. At f > fr, Z is capacitive. θ = + (I Leads E). Q of a Theoretical circuit: R R C Q = P = P = RP X L XC L Resonant Rise in tank current I tan k = QI T = I L = I C Bandwidth (BW) BW = f 2 − f1 =

B. Practical Parallel Resonant Circuit

Resonant Rise in Voltage VL = VC = QE Bandwidth (BW) is the range of frequencies over which the operation is satisfactory and is taken between two half-power (3dB down) points. f BW = f 2 − f1 = r Q If Q ≥ 10; then fr bisects BW BW BW f1 = f r − f2 = fr + 2 2

fr Q

Equivalent Theoretical Circuit

Alloys commonly magnetized Alloy Percentage Content Permalloy 22% Fe, 78% Ni Hipernik 40% Fe, 60% Ni Perminvar 30% Fe, 45% Ni, 25% Co Alnico 24% Co, 51% Fe

Impedance transformation:

Q of Equivalent Theoretical Circuit R Q= P X Leq Q of Practical Circuit Q=

XL RS

1 2π LC

1 fr = 2π LC

1−

2 S

R C 1 ; if RS = 0; f r = L 2π LC

1 Q2 ; if Q ≥ 10; f r = 2 1+ Q 2π LC

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

Second First Law “The force of attraction or repulsion between two poles is inversely proportional to the square of the distance between them.” F =k

Resonant frequency (practical circuit) fr =

Coulomb’s Laws First Law “The force of attraction or repulsion between two magnetic poles is directly proportional to their strengths.”

if Q ≥ 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) 770 Nickel (Ni) 358 Cobalt (Co) 1130 Gadolinium 16

where: k =

m1m 2 r2

1 4πµ

(Newtons, N) µ = µ r µ0

Magnitude of the Force (Newtons, N) F = BIl sin θ where: B = flux density (Wb/m2) I = current (A) l = length of conductor (m) θ = angle between the conductor and field Magnitude of the flux surrounding a straight conductor R Φ = 14 Il log (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 2I 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 (d − r ) Φ = 28 Il log (Maxwells, Mx) r 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) •

Magnetic Circuits

Φ A where: B = Flux density in Tesla (T) Φ = Flux lines in Webers (Wb) A = Area in square meters (m2) B=

Note: 1 Tesla = 1 Wb/m2 Permeability µ 0 = 4π × 10 −7

Weber H or Ampere − meter m

Note: μ = μ0; μr = 1 → non–magnetic μ < μ0; μr < 1 → diamagnetic μ > μ0; μr > 1 → paramagnetic μ >> μ0; μr >> 1 → ferromagnetic (μr ≥ 100) ℜ=

L µA

where: ℜ = reluctance L = the length of the magnetic path A = the cross-sectional area Note: The t in the unit A-t/Wb is the number of turns of the applied winding. Different units of Reluctance ( ℜ ) Ampere − turn Ampere − turn a.) b.) Weber Maxwell Gilbert Gilbert c.) d.) Maxwell Weber Note: 1 Weber = 1×108 maxwells 1 Gilbert = 0.7958 ampere-turns 1 Gauss = 1 maxwell/cm2

Ohm’s Law for Magnetic Circuits Cause Effect = Opposition Then, Φ=

ℑ ℜ

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 (Ω) Reluctance, ℜ (Gb/Mx) Current, I (A) Flux, Φ (Wb or Mx) emf, V (V) mmf, ℑ (Gb or At) Total reluctance in series ℜ T = ℜ1 + ℜ 2 + ... + ℜ n Total reluctance in parallel 1 1 1 1 = + + ... + ℜ T ℜ1 ℜ 2 ℜn Total flux in series Φ T = Φ 1 = Φ 2 = ... = Φ n Total flux in parallel Φ T = Φ 1 + Φ 2 + ... + Φ n Energy stored Wm =

1 ℜΦ 2 2

Joules

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

Ampere-turns

Tractive force or lifting force of a magnet 1  AB 2   Newtons F =  2  µ 0 

Magnetizing Force (H) ℑ NI H= H= l l Note: The unit of H is At/m Permeability – the ratio of flux density to the magnetizing force. B µ= H B and H of an infinitely long straight wire I µI B= H= 2πr 2πr Steinmetz’s Formula of Hysteresis Loss J Wh = ηfBm1.6 m3 where: η = 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 = Hl