Lessons In Electric Circuits -- Volume V Chapter 1

USEFUL EQUATIONS AND CONVERSION FACTORS

DC circuit equations and laws

Ohm's and Joule's Laws

NOTE: the symbol "V" ("U" in Europe) is sometimes used to represent voltage instead of "E". In some cases, an author or circuit designer may choose to exclusively use "V" for voltage, never using the symbol "E." Other times the two symbols are used interchangeably, or "E" is used to represent voltage from a power source while "V" is used to represent voltage across a load (voltage "drop").

Kirchhoff's Laws

"The algebraic sum of all voltages in a loop must equal zero."
Kirchhoff's Voltage Law (KVL)

"The algebraic sum of all currents entering and exiting a node must equal zero."
Kirchhoff's Current Law (KCL)

Series circuit rules

• Components in a series circuit share the same current. I_total = I₁ = I₂ = . . . I_n
• Total resistance in a series circuit is equal to the sum of the individual resistances, making it greater than any of the individual resistances. R_total = R₁ + R₂ + . . . R_n
• Total voltage in a series circuit is equal to the sum of the individual voltage drops. E_total = E₁ + E₂ + . . . E_n

Parallel circuit rules

• Components in a parallel circuit share the same voltage. E_total = E₁ = E₂ = . . . E_n
• Total resistance in a parallel circuit is less than any of the individual resistances. R_total = 1 / (1/R₁ + 1/R₂ + . . . 1/R_n)
• Total current in a parallel circuit is equal to the sum of the individual branch currents. I_total = I₁ + I₂ + . . . I_n

Series and parallel component equivalent values

Series and parallel resistances

Series and parallel inductances

Series and Parallel Capacitances

Capacitor sizing equation

A formula for capacitance in picofarads using practical dimensions:

Inductor sizing equation

Wheeler's formulas for inductance of air core coils which follow are useful for radio frequency inductors. The following formula for the inductance of a single layer air core solenoid coil is accurate to approximately 1% for 2r/l < 3. The thick coil formula is 1% accurate when the denominator terms are approximately equal. Wheeler's spiral formula is 1% accurate for c>0.2r. While this is a "round wire" formula, it may still be applicable to printed circuit spiral inductors at reduced accuracy.

The inductance in henries of a square printed circuit inductor is given by two formulas where p=q, and p≠q.

The wire table provides "turns per inch" for enamel magnet wire for use with the inductance formulas for coils.

Time constant equations

Value of time constant in series RC and RL circuits

Time constant in seconds = RC
Time constant in seconds = L/R

Calculating voltage or current at specified time

Calculating time at specified voltage or current

AC circuit equations

Inductive reactance

Capacitive reactance

Impedance in relation to R and X

Z_L = R + jX_L

Z_C = R - jX_C

Ohm's Law for AC

Series and Parallel Impedances

NOTE: All impedances must be calculated in complex number form for these equations to work.

Resonance

NOTE: This equation applies to a non-resistive LC circuit. In circuits containing resistance as well as inductance and capacitance, this equation applies only to series configurations and to parallel configurations where R is very small.

AC power

Decibels

Metric prefixes and unit conversions

• Metric prefixes
• Yotta = 10²⁴ Symbol: Y
• Zetta = 10²¹ Symbol: Z
• Exa = 10¹⁸ Symbol: E
• Peta = 10¹⁵ Symbol: P
• Tera = 10¹² Symbol: T
• Giga = 10⁹ Symbol: G
• Mega = 10⁶ Symbol: M
• Kilo = 10³ Symbol: k
• Hecto = 10² Symbol: h
• Deca = 10¹ Symbol: da
• Deci = 10^-1 Symbol: d
• Centi = 10^-2 Symbol: c
• Milli = 10^-3 Symbol: m
• Micro = 10^-6 Symbol: µ
• Nano = 10^-9 Symbol: n
• Pico = 10^-12 Symbol: p
• Femto = 10^-15 Symbol: f
• Atto = 10^-18 Symbol: a
• Zepto = 10^-21 Symbol: z
• Yocto = 10^-24 Symbol: y

• Conversion factors for temperature
• ^oF = (^oC)(9/5) + 32
• ^oC = (^oF - 32)(5/9)
• ^oR = ^oF + 459.67
• ^oK = ^oC + 273.15

Conversion equivalencies for volume

1 US gallon (gal) = 231.0 cubic inches (in³) = 4 quarts (qt) = 8 pints (pt) = 128 fluid ounces (fl. oz.) = 3.7854 liters (l)

1 Imperial gallon (gal) = 160 fluid ounces (fl. oz.) = 4.546 liters (l)

Conversion equivalencies for distance

1 inch (in) = 2.540000 centimeter (cm)

Conversion equivalencies for velocity

1 mile per hour (mi/h) = 88 feet per minute (ft/m) = 1.46667 feet per second (ft/s) = 1.60934 kilometer per hour (km/h) = 0.44704 meter per second (m/s) = 0.868976 knot (knot -- international)

Conversion equivalencies for weight

1 pound (lb) = 16 ounces (oz) = 0.45359 kilogram (kg)

Conversion equivalencies for force

1 pound-force (lbf) = 4.44822 newton (N)

Acceleration of gravity (free fall), Earth standard

9.806650 meters per second per second (m/s²) = 32.1740 feet per second per second (ft/s²)

Conversion equivalencies for area

1 acre = 43560 square feet (ft²) = 4840 square yards (yd²) = 4046.86 square meters (m²)

Conversion equivalencies for pressure

1 pound per square inch (psi) = 2.03603 inches of mercury (in. Hg) = 27.6807 inches of water (in. W.C.) = 6894.757 pascals (Pa) = 0.0680460 atmospheres (Atm) = 0.0689476 bar (bar)

Conversion equivalencies for energy or work

1 british thermal unit (BTU -- "International Table") = 251.996 calories (cal -- "International Table") = 1055.06 joules (J) = 1055.06 watt-seconds (W-s) = 0.293071 watt-hour (W-hr) = 1.05506 x 10¹⁰ ergs (erg) = 778.169 foot-pound-force (ft-lbf)

Conversion equivalencies for power

1 horsepower (hp -- 550 ft-lbf/s) = 745.7 watts (W) = 2544.43 british thermal units per hour (BTU/hr) = 0.0760181 boiler horsepower (hp -- boiler)

Conversion equivalencies for motor torque

Locate the row corresponding to known unit of torque along the left of the table. Multiply by the factor under the column for the desired units. For example, to convert 2 oz-in torque to n-m, locate oz-in row at table left. Locate 7.062 x 10^-3 at intersection of desired n-m units column. Multiply 2 oz-in x (7.062 x 10^-3 ) = 14.12 x 10^-3 n-m.


Converting between units is easy if you have a set of equivalencies to work with. Suppose we wanted to convert an energy quantity of 2500 calories into watt-hours. What we would need to do is find a set of equivalent figures for those units. In our reference here, we see that 251.996 calories is physically equal to 0.293071 watt hour. To convert from calories into watt-hours, we must form a "unity fraction" with these physically equal figures (a fraction composed of different figures and different units, the numerator and denominator being physically equal to one another), placing the desired unit in the numerator and the initial unit in the denominator, and then multiply our initial value of calories by that fraction.
Since both terms of the "unity fraction" are physically equal to one another, the fraction as a whole has a physical value of 1, and so does not change the true value of any figure when multiplied by it. When units are canceled, however, there will be a change in units. For example, 2500 calories multiplied by the unity fraction of (0.293071 w-hr / 251.996 cal) = 2.9075 watt-hours.

The "unity fraction" approach to unit conversion may be extended beyond single steps. Suppose we wanted to convert a fluid flow measurement of 175 gallons per hour into liters per day. We have two units to convert here: gallons into liters, and hours into days. Remember that the word "per" in mathematics means "divided by," so our initial figure of 175 gallons per hour means 175 gallons divided by hours. Expressing our original figure as such a fraction, we multiply it by the necessary unity fractions to convert gallons to liters (3.7854 liters = 1 gallon), and hours to days (1 day = 24 hours). The units must be arranged in the unity fraction in such a way that undesired units cancel each other out above and below fraction bars. For this problem it means using a gallons-to-liters unity fraction of (3.7854 liters / 1 gallon) and a hours-to-days unity fraction of (24 hours / 1 day):

Our final (converted) answer is 15898.68 liters per day.

Data

Conversion factors were found in the 78^th edition of the CRC Handbook of Chemistry and Physics, and the 3^rd edition of Bela Liptak's Instrument Engineers' Handbook -- Process Measurement and Analysis.

Contributors

Contributors to this chapter are listed in chronological order of their contributions, from most recent to first. See Appendix 2 (Contributor List) for dates and contact information.

Lessons In Electric Circuits -- Volume V Chapter 2 | COLOR CODES

COLOR CODES

Components and wires are coded are with colors to identify their value and function.

Resistor Color Codes

Components and wires are coded are with colors to identify their value and function.

The colors brown, red, green, blue, and violet are used as tolerance codes on 5-band resistors only. All 5-band resistors use a colored tolerance band. The blank (20%) "band" is only used with the "4-band" code (3 colored bands + a blank "band").

Example #1

A resistor colored Yellow-Violet-Orange-Gold would be 47 kΩ with a tolerance of +/- 5%.

Example #2

A resistor colored Green-Red-Gold-Silver would be 5.2 Ω with a tolerance of +/- 10%.

Example #3

A resistor colored White-Violet-Black would be 97 Ω with a tolerance of +/- 20%. When you see only three color bands on a resistor, you know that it is actually a 4-band code with a blank (20%) tolerance band.

Example #4

A resistor colored Orange-Orange-Black-Brown-Violet would be 3.3 kΩ with a tolerance of +/- 0.1%.

Example #5

A resistor colored Brown-Green-Grey-Silver-Red would be 1.58 Ω with a tolerance of +/- 2%.

Example #6

A resistor colored Blue-Brown-Green-Silver-Blue would be 6.15 Ω with a tolerance of +/- 0.25%.

Wiring Color Codes

Wiring for AC and DC power distribution branch circuits are color coded for identification of individual wires. In some jurisdictions all wire colors are specified in legal documents. In other jurisdictions, only a few conductor colors are so codified. In that case, local custom dictates the “optional” wire colors.
IEC, AC: Most of Europe abides by IEC (International Electrotechnical Commission) wiring color codes for AC branch circuits. These are listed in Table below. The older color codes in the table reflect the previous style which did not account for proper phase rotation. The protective ground wire (listed as green-yellow) is green with yellow stripe.
IEC (most of Europe) AC power circuit wiring color codes.

Function	label	Color, IEC	Color, old IEC
Protective earth	PE	green-yellow	green-yellow
Neutral	N	blue	blue
Line, single phase	L	brown	brown or black
Line, 3-phase	L1	brown	brown or black
Line, 3-phase	L2	black	brown or black
Line, 3-phase	L3	grey	brown or black

UK, AC: The United Kingdom now follows the IEC AC wiring color codes. Table below lists these along with the obsolete domestic color codes. For adding new colored wiring to existing old colored wiring see Cook.
UK AC power circuit wiring color codes.

Function	label	Color, IEC	Old UK color
Protective earth	PE	green-yellow	green-yellow
Neutral	N	blue	black
Line, single phase	L	brown	red
Line, 3-phase	L1	brown	red
Line, 3-phase	L2	black	yellow
Line, 3-phase	L3	grey	blue

US, AC:The US National Electrical Code only mandates white (or grey) for the neutral power conductor and bare copper, green, or green with yellow stripe for the protective ground. In principle any other colors except these may be used for the power conductors. The colors adopted as local practice are shown in Table below. Black, red, and blue are used for 208 VAC three-phase; brown, orange and yellow are used for 480 VAC. Conductors larger than #6 AWG are only available in black and are color taped at the ends.
US AC power circuit wiring color codes.

Function	label	Color, common	Color, alternative
Protective ground	PG	bare, green, or green-yellow	green
Neutral	N	white	grey
Line, single phase	L	black or red (2nd hot)
Line, 3-phase	L1	black	brown
Line, 3-phase	L2	red	orange
Line, 3-phase	L3	blue	yellow

Canada: Canadian wiring is governed by the CEC (Canadian Electric Code). See Table below. The protective ground is green or green with yellow stripe. The neutral is white, the hot (live or active) single phase wires are black , and red in the case of a second active. Three-phase lines are red, black, and blue.
Canada AC power circuit wiring color codes.

Function	label	Color, common
Protective ground	PG	green or green-yellow
Neutral	N	white
Line, single phase	L	black or red (2nd hot)
Line, 3-phase	L1	red
Line, 3-phase	L2	black
Line, 3-phase	L3	blue

IEC, DC: DC power installations, for example, solar power and computer data centers, use color coding which follows the AC standards. The IEC color standard for DC power cables is listed in Table below, adapted from Table 2, Cook.
IEC DC power circuit wiring color codes.

Function	label	Color
Protective earth	PE	green-yellow
2-wire unearthed DC Power System
Positive	L+	brown
Negative	L-	grey
2-wire earthed DC Power System
Positive (of a negative earthed) circuit	L+	brown
Negative (of a negative earthed) circuit	M	blue
Positive (of a positive earthed) circuit	M	blue
Negative (of a positive earthed) circuit	L-	grey
3-wire earthed DC Power System
Positive	L+	brown
Mid-wire	M	blue
Negative	L-	grey

US DC power: The US National Electrical Code (for both AC and DC) mandates that the grounded neutral conductor of a power system be white or grey. The protective ground must be bare, green or green-yellow striped. Hot (active) wires may be any other colors except these. However, common practice (per local electrical inspectors) is for the first hot (live or active) wire to be black and the second hot to be red. The recommendations in Table below are by Wiles.He makes no recommendation for ungrounded power system colors. Usage of the ungrounded system is discouraged for safety. However, red (+) and black (-) follows the coloring of the grounded systems in the table.
US recommended DC power circuit wiring color codes.

Function	label	Color
Protective ground	PG	bare, green, or green-yellow
2-wire ungrounded DC Power System
Positive	L+	no recommendation (red)
Negative	L-	no recommendation (black)
2-wire grounded DC Power System
Positive (of a negative grounded) circuit	L+	red
Negative (of a negative grounded) circuit	N	white
Positive (of a positive grounded) circuit	N	white
Negative (of a positive grounded) circuit	L-	black
3-wire grounded DC Power System
Positive	L+	red
Mid-wire (center tap)	N	white
Negative	L-	black

Lessons In Electric Circuits -- Volume V Chapter 3

CONDUCTOR AND INSULATOR TABLES

• Copper wire gage table
• Copper wire ampacity table
• Coefficients of specific resistance
• Temperature coefficients of resistance
• Critical temperatures for superconductors
• Dielectric strengths for insulators
• Data

Copper wire gage table

Soild copper wire table:

Size        Diameter         Cross-sectional area       Weight
AWG          inches        cir. mils     sq. inches   lb/1000 ft
================================================================
4/0 -------- 0.4600 ------- 211,600 ------ 0.1662 ------ 640.5
3/0 -------- 0.4096 ------- 167,800 ------ 0.1318 ------ 507.9
2/0 -------- 0.3648 ------- 133,100 ------ 0.1045 ------ 402.8 
1/0 -------- 0.3249 ------- 105,500 ----- 0.08289 ------ 319.5 
1 ---------- 0.2893 ------- 83,690 ------ 0.06573 ------ 253.5 
2 ---------- 0.2576 ------- 66,370 ------ 0.05213 ------ 200.9 
3 ---------- 0.2294 ------- 52,630 ------ 0.04134 ------ 159.3 
4 ---------- 0.2043 ------- 41,740 ------ 0.03278 ------ 126.4 
5 ---------- 0.1819 ------- 33,100 ------ 0.02600 ------ 100.2 
6 ---------- 0.1620 ------- 26,250 ------ 0.02062 ------ 79.46 
7 ---------- 0.1443 ------- 20,820 ------ 0.01635 ------ 63.02 
8 ---------- 0.1285 ------- 16,510 ------ 0.01297 ------ 49.97 
9 ---------- 0.1144 ------- 13,090 ------ 0.01028 ------ 39.63 
10 --------- 0.1019 ------- 10,380 ------ 0.008155 ----- 31.43 
11 --------- 0.09074 ------- 8,234 ------ 0.006467 ----- 24.92 
12 --------- 0.08081 ------- 6,530 ------ 0.005129 ----- 19.77 
13 --------- 0.07196 ------- 5,178 ------ 0.004067 ----- 15.68 
14 --------- 0.06408 ------- 4,107 ------ 0.003225 ----- 12.43 
15 --------- 0.05707 ------- 3,257 ------ 0.002558 ----- 9.858 
16 --------- 0.05082 ------- 2,583 ------ 0.002028 ----- 7.818 
17 --------- 0.04526 ------- 2,048 ------ 0.001609 ----- 6.200
18 --------- 0.04030 ------- 1,624 ------ 0.001276 ----- 4.917 
19 --------- 0.03589 ------- 1,288 ------ 0.001012 ----- 3.899 
20 --------- 0.03196 ------- 1,022 ----- 0.0008023 ----- 3.092 
21 --------- 0.02846 ------- 810.1 ----- 0.0006363 ----- 2.452 
22 --------- 0.02535 ------- 642.5 ----- 0.0005046 ----- 1.945 
23 --------- 0.02257 ------- 509.5 ----- 0.0004001 ----- 1.542 
24 --------- 0.02010 ------- 404.0 ----- 0.0003173 ----- 1.233 
25 --------- 0.01790 ------- 320.4 ----- 0.0002517 ----- 0.9699
26 --------- 0.01594 ------- 254.1 ----- 0.0001996 ----- 0.7692
27 --------- 0.01420 ------- 201.5 ----- 0.0001583 ----- 0.6100 
28 --------- 0.01264 ------- 159.8 ----- 0.0001255 ----- 0.4837 
29 --------- 0.01126 ------- 126.7 ----- 0.00009954 ---- 0.3836 
30 --------- 0.01003 ------- 100.5 ----- 0.00007894 ---- 0.3042 
31 -------- 0.008928 ------- 79.70 ----- 0.00006260 ---- 0.2413 
32 -------- 0.007950 ------- 63.21 ----- 0.00004964 ---- 0.1913 
33 -------- 0.007080 ------- 50.13 ----- 0.00003937 ---- 0.1517 
34 -------- 0.006305 ------- 39.75 ----- 0.00003122 ---- 0.1203 
35 -------- 0.005615 ------- 31.52 ----- 0.00002476 — 0.09542 
36 -------- 0.005000 ------- 25.00 ----- 0.00001963 — 0.07567 
37 -------- 0.004453 ------- 19.83 ----- 0.00001557 — 0.06001 
38 -------- 0.003965 ------- 15.72 ----- 0.00001235 — 0.04759 
39 -------- 0.003531 ------- 12.47 ---- 0.000009793 — 0.03774 
40 -------- 0.003145 ------- 9.888 ---- 0.000007766 — 0.02993 
41 -------- 0.002800 ------- 7.842 ---- 0.000006159 — 0.02374 
42 -------- 0.002494 ------- 6.219 ---- 0.000004884 — 0.01882 
43 -------- 0.002221 ------- 4.932 ---- 0.000003873 — 0.01493 
44 -------- 0.001978 ------- 3.911 ---- 0.000003072 — 0.01184 

Copper wire ampacity table

Ampacities of copper wire, in free air at 30^o C:

======================================================== 
|                      INSULATION TYPE:                |
|         RUW, T           THW, THWN        FEP, FEPB  | 
|           TW               RUH            THHN, XHHW | 
======================================================== 
Size    Current Rating   Current Rating   Current Rating 
AWG     @ 60 degrees C   @ 75 degrees C   @ 90 degrees C 
======================================================== 
20 -------- *9 ----------------------------- *12.5 
18 -------- *13 ------------------------------ 18  
16 -------- *18 ------------------------------ 24  
14 --------- 25 ------------- 30 ------------- 35  
12 --------- 30 ------------- 35 ------------- 40  
10 --------- 40 ------------- 50 ------------- 55  
8 ---------- 60 ------------- 70 ------------- 80  
6 ---------- 80 ------------- 95 ------------ 105  
4 --------- 105 ------------ 125 ------------ 140  
2 --------- 140 ------------ 170 ------------ 190  
1 --------- 165 ------------ 195 ------------ 220  
1/0 ------- 195 ------------ 230 ------------ 260   
2/0 ------- 225 ------------ 265 ------------ 300  
3/0 ------- 260 ------------ 310 ------------ 350  
4/0 ------- 300 ------------ 360 ------------ 405  

* = estimated values; normally, wire gages this small are not manufactured with these insulation types.

Coefficients of specific resistance

Specific resistance at 20^o C:

Material      Element/Alloy        (ohm-cmil/ft)        (ohm-cm·10-6) 
====================================================================
Nichrome ------- Alloy ---------------- 675 ------------- 112.2
Nichrome V ----- Alloy ---------------- 650 ------------- 108.1
Manganin ------- Alloy ---------------- 290 ------------- 48.21
Constantan ----- Alloy ---------------- 272.97 ---------- 45.38
Steel* --------- Alloy ---------------- 100 ------------- 16.62
Platinum ------ Element --------------- 63.16 ----------- 10.5
Iron ---------- Element --------------- 57.81 ----------- 9.61
Nickel -------- Element --------------- 41.69 ----------- 6.93
Zinc ---------- Element --------------- 35.49 ----------- 5.90
Molybdenum ---- Element --------------- 32.12 ----------- 5.34
Tungsten ------ Element --------------- 31.76 ----------- 5.28
Aluminum ------ Element --------------- 15.94 ----------- 2.650
Gold ---------- Element --------------- 13.32 ----------- 2.214
Copper -------- Element --------------- 10.09 ----------- 1.678
Silver -------- Element --------------- 9.546 ----------- 1.587 

* = Steel alloy at 99.5 percent iron, 0.5 percent carbon.

Temperature coefficients of resistance

Temperature coefficient (α) per degree C:

Material     Element/Alloy         Temp. coefficient 
===================================================== 
Nickel -------- Element --------------- 0.005866    
Iron ---------- Element --------------- 0.005671    
Molybdenum ---- Element --------------- 0.004579    
Tungsten ------ Element --------------- 0.004403    
Aluminum ------ Element --------------- 0.004308    
Copper -------- Element --------------- 0.004041    
Silver -------- Element --------------- 0.003819    
Platinum ------ Element --------------- 0.003729    
Gold ---------- Element --------------- 0.003715    
Zinc ---------- Element --------------- 0.003847    
Steel* --------- Alloy ---------------- 0.003  
Nichrome ------- Alloy ---------------- 0.00017
Nichrome V ----- Alloy ---------------- 0.00013
Manganin ------- Alloy ------------ +/- 0.000015    
Constantan ----- Alloy --------------- -0.000074    

* = Steel alloy at 99.5 percent iron, 0.5 percent carbon

Critical temperatures for superconductors

Critical temperatures given in Kelvins

Material     Element/Alloy      Critical temperature(K)
=======================================================       
Aluminum -------- Element --------------- 1.20
Cadmium --------- Element --------------- 0.56
Lead ------------ Element --------------- 7.2
Mercury --------- Element --------------- 4.16
Niobium --------- Element --------------- 8.70
Thorium --------- Element --------------- 1.37
Tin ------------- Element --------------- 3.72 
Titanium -------- Element --------------- 0.39
Uranium --------- ELement --------------- 1.0
Zinc ------------ Element --------------- 0.91
Niobium/Tin ------ Alloy ---------------- 18.1
Cupric sulphide - Compound -------------- 1.6

Critical temperatures, high temperature superconuctors in Kelvins
Material                        Critical temperature(K)
=======================================================       
HgBa2Ca2Cu3O8+d ---------------- 150 (23.5 GPa pressure)
HgBa2Ca2Cu3O8+d ---------------- 133
Tl2Ba2Ca2Cu3O10 ---------------- 125
YBa2Cu3O7 ----------------------  90
La1.85Sr0.15CuO4 ----------------  40
Cs3C60 -------------------------  40 (15 Kbar pressure)
Ba0.6K0.4BiO3 -------------------  30
Nd1.85Ce0.15CuO4 ----------------  22
K3C60 --------------------------  19
PbMo6S8 ------------------------  12.6 

Note: all critical temperatures given at zero magnetic field strength.

Dielectric strengths for insulators

Dielectric strength in kilovolts per inch (kV/in):

Material*         Dielectric strength 
=========================================    
Vacuum --------------------- 20            
Air ------------------------ 20 to 75      
Porcelain ------------------ 40 to 200     
Paraffin Wax --------------- 200 to 300    
Transformer Oil ------------ 400           
Bakelite ------------------- 300 to 550    
Rubber --------------------- 450 to 700    
Shellac -------------------- 900           
Paper ---------------------- 1250          
Teflon --------------------- 1500          
Glass ---------------------- 2000 to 3000  
Mica ----------------------- 5000          

* = Materials listed are specially prepared for electrical use

Data

Tables of specific resistance and temperature coefficient of resistance for elemental materials (not alloys) were derived from figures found in the 78th edition of the CRC Handbook of Chemistry and Physics. Superconductivity data from Collier's Encyclopedia (volume 21, 1968, page 640).

Lessons In Electric Circuits -- Volume V Chapter 4

ALGEBRA REFERENCE

• Basic identities
• Arithmetic properties
  • The associative property
  • The commutative property
  • The distributive property
• Properties of exponents
• Radicals
  • Definition of a radical
  • Properties of radicals
• Important constants
  • Euler's number
  • Pi
• Logarithms
  • Definition of a logarithm
  • Properties of logarithms
• Factoring equivalencies
• The quadratic formula
• Sequences
  • Arithmetic sequences
  • Geometric sequences
• Factorials
  • Definition of a factorial
  • Strange factorials
• Solving simultaneous equations
  • Substitution method
  • Addition method
• Contributors

click Below of this Post for Remaining Content..

Basic identities

Note: while division by zero is popularly thought to be equal to infinity, this is not technically true. In some practical applications it may be helpful to think the result of such a fraction approaching positive infinity as a positive denominator approaches zero (imagine calculating current I=E/R in a circuit with resistance approaching zero -- current would approach infinity), but the actual fraction of anything divided by zero is undefined in the scope of either real or complex numbers.

Arithmetic properties

The associative property

In addition and multiplication, terms may be arbitrarily associated with each other through the use of parentheses:

The commutative property

In addition and multiplication, terms may be arbitrarily interchanged, or commutated:

The distributive property

Properties of exponents

Radicals

Definition of a radical

When people talk of a "square root," they're referring to a radical with a root of 2. This is mathematically equivalent to a number raised to the power of 1/2. This equivalence is useful to know when using a calculator to determine a strange root. Suppose for example you needed to find the fourth root of a number, but your calculator lacks a "4th root" button or function. If it has a y^x function (which any scientific calculator should have), you can find the fourth root by raising that number to the 1/4 power, or x^0.25.

It is important to remember that when solving for an even root (square root, fourth root, etc.) of any number, there are two valid answers. For example, most people know that the square root of nine is three, but negative three is also a valid answer, since (-3)² = 9 just as 3² = 9.

Properties of radicals

Important constants

Euler's number

Euler's constant is an important value for exponential functions, especially scientific applications involving decay (such as the decay of a radioactive substance). It is especially important in calculus due to its uniquely self-similar properties of integration and differentiation.

e approximately equals:
2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996

Pi

Pi (π) is defined as the ratio of a circle's circumference to its diameter.

Pi approximately equals:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511

Note: For both Euler's constant (e) and pi (π), the spaces shown between each set of five digits have no mathematical significance. They are placed there just to make it easier for your eyes to "piece" the number into five-digit groups when manually copying.

Logarithms

Definition of a logarithm

"log" denotes a common logarithm (base = 10), while "ln" denotes a natural logarithm (base = e).

Properties of logarithms

These properties of logarithms come in handy for performing complex multiplication and division operations. They are an example of something called a transform function, whereby one type of mathematical operation is transformed into another type of mathematical operation that is simpler to solve. Using a table of logarithm figures, one can multiply or divide numbers by adding or subtracting their logarithms, respectively. then looking up that logarithm figure in the table and seeing what the final product or quotient is.
Slide rules work on this principle of logarithms by performing multiplication and division through addition and subtraction of distances on the slide.

Marks on a slide rule's scales are spaced in a logarithmic fashion, so that a linear positioning of the scale or cursor results in a nonlinear indication as read on the scale(s). Adding or subtracting lengths on these logarithmic scales results in an indication equivalent to the product or quotient, respectively, of those lengths.
Most slide rules were also equipped with special scales for trigonometric functions, powers, roots, and other useful arithmetic functions.

Factoring equivalencies

The quadratic formula

Sequences

Arithmetic sequences

An arithmetic sequence is a series of numbers obtained by adding (or subtracting) the same value with each step. A child's counting sequence (1, 2, 3, 4, . . .) is a simple arithmetic sequence, where the common difference is 1: that is, each adjacent number in the sequence differs by a value of one. An arithmetic sequence counting only even numbers (2, 4, 6, 8, . . .) or only odd numbers (1, 3, 5, 7, 9, . . .) would have a common difference of 2.
In the standard notation of sequences, a lower-case letter "a" represents an element (a single number) in the sequence. The term "a_n" refers to the element at the n^th step in the sequence. For example, "a₃" in an even-counting (common difference = 2) arithmetic sequence starting at 2 would be the number 6, "a" representing 4 and "a₁" representing the starting point of the sequence (given in this example as 2).
A capital letter "A" represents the sum of an arithmetic sequence. For instance, in the same even-counting sequence starting at 2, A₄ is equal to the sum of all elements from a₁ through a₄, which of course would be 2 + 4 + 6 + 8, or 20.

Geometric sequences

A geometric sequence, on the other hand, is a series of numbers obtained by multiplying (or dividing) by the same value with each step. A binary place-weight sequence (1, 2, 4, 8, 16, 32, 64, . . .) is a simple geometric sequence, where the common ratio is 2: that is, each adjacent number in the sequence differs by a factor of two.

Lessons In Electric Circuits -- Volume V Chapter 4 (Part 2)

Lessons In Electric Circuits -- Volume V Chapter 4 (Part 2)

Factorials

Definition of a factorial

Denoted by the symbol "!" after an integer; the product of that integer and all integers in descent to 1.
Example of a factorial:

Strange factorials

Solving simultaneous equations

The terms simultaneous equations and systems of equations refer to conditions where two or more unknown variables are related to each other through an equal number of equations. Consider the following example:

For this set of equations, there is but a single combination of values for x and y that will satisfy both. Either equation, considered separately, has an infinitude of valid (x,y) solutions, but together there is only one. Plotted on a graph, this condition becomes obvious:

Each line is actually a continuum of points representing possible x and y solution pairs for each equation. Each equation, separately, has an infinite number of ordered pair (x,y) solutions. There is only one point where the two linear functions x + y = 24 and 2x - y = -6 intersect (where one of their many independent solutions happen to work for both equations), and that is where x is equal to a value of 6 and y is equal to a value of 18.
Usually, though, graphing is not a very efficient way to determine the simultaneous solution set for two or more equations. It is especially impractical for systems of three or more variables. In a three-variable system, for example, the solution would be found by the point intersection of three planes in a three-dimensional coordinate space -- not an easy scenario to visualize.

Substitution method

Several algebraic techniques exist to solve simultaneous equations. Perhaps the easiest to comprehend is the substitution method. Take, for instance, our two-variable example problem:

In the substitution method, we manipulate one of the equations such that one variable is defined in terms of the other:

Then, we take this new definition of one variable and substitute it for the same variable in the other equation. In this case, we take the definition of y, which is 24 - x and substitute this for the y term found in the other equation:

Now that we have an equation with just a single variable (x), we can solve it using "normal" algebraic techniques:

Now that x is known, we can plug this value into any of the original equations and obtain a value for y. Or, to save us some work, we can plug this value (6) into the equation we just generated to define y in terms of x, being that it is already in a form to solve for y:

Applying the substitution method to systems of three or more variables involves a similar pattern, only with more work involved. This is generally true for any method of solution: the number of steps required for obtaining solutions increases rapidly with each additional variable in the system.
To solve for three unknown variables, we need at least three equations. Consider this example:

Being that the first equation has the simplest coefficients (1, -1, and 1, for x, y, and z, respectively), it seems logical to use it to develop a definition of one variable in terms of the other two. In this example, I'll solve for x in terms of y and z:

Now, we can substitute this definition of x where x appears in the other two equations:

Reducing these two equations to their simplest forms:

So far, our efforts have reduced the system from three variables in three equations to two variables in two equations. Now, we can apply the substitution technique again to the two equations 4y - z = 4 and -3y + 4z = 36 to solve for either y or z. First, I'll manipulate the first equation to define z in terms of y:

Next, we'll substitute this definition of z in terms of y where we see z in the other equation:

Now that y is a known value, we can plug it into the equation defining z in terms of y and obtain a figure for z:

Now, with values for y and z known, we can plug these into the equation where we defined x in terms of y and z, to obtain a value for x:

In closing, we've found values for x, y, and z of 2, 4, and 12, respectively, that satisfy all three equations.

Addition method

While the substitution method may be the easiest to grasp on a conceptual level, there are other methods of solution available to us. One such method is the so-called addition method, whereby equations are added to one another for the purpose of canceling variable terms.
Let's take our two-variable system used to demonstrate the substitution method:

One of the most-used rules of algebra is that you may perform any arithmetic operation you wish to an equation so long as you do it equally to both sides. With reference to addition, this means we may add any quantity we wish to both sides of an equation -- so long as its the same quantity -- without altering the truth of the equation.
An option we have, then, is to add the corresponding sides of the equations together to form a new equation. Since each equation is an expression of equality (the same quantity on either side of the = sign), adding the left-hand side of one equation to the left-hand side of the other equation is valid so long as we add the two equations' right-hand sides together as well. In our example equation set, for instance, we may add x + y to 2x - y, and add 24 and -6 together as well to form a new equation. What benefit does this hold for us? Examine what happens when we do this to our example equation set:

Because the top equation happened to contain a positive y term while the bottom equation happened to contain a negative y term, these two terms canceled each other in the process of addition, leaving no y term in the sum. What we have left is a new equation, but one with only a single unknown variable, x! This allows us to easily solve for the value of x:

Once we have a known value for x, of course, determining y's value is a simply matter of substitution (replacing x with the number 6) into one of the original equations. In this example, the technique of adding the equations together worked well to produce an equation with a single unknown variable. What about an example where things aren't so simple? Consider the following equation set:

We could add these two equations together -- this being a completely valid algebraic operation -- but it would not profit us in the goal of obtaining values for x and y:

The resulting equation still contains two unknown variables, just like the original equations do, and so we're no further along in obtaining a solution. However, what if we could manipulate one of the equations so as to have a negative term that would cancel the respective term in the other equation when added? Then, the system would reduce to a single equation with a single unknown variable just as with the last (fortuitous) example.
If we could only turn the y term in the lower equation into a - 2y term, so that when the two equations were added together, both y terms in the equations would cancel, leaving us with only an x term, this would bring us closer to a solution. Fortunately, this is not difficult to do. If we multiply each and every term of the lower equation by a -2, it will produce the result we seek:

Now, we may add this new equation to the original, upper equation:

Solving for x, we obtain a value of 3:

Substituting this new-found value for x into one of the original equations, the value of y is easily determined:

Using this solution technique on a three-variable system is a bit more complex. As with substitution, you must use this technique to reduce the three-equation system of three variables down to two equations with two variables, then apply it again to obtain a single equation with one unknown variable. To demonstrate, I'll use the three-variable equation system from the substitution section:

Being that the top equation has coefficient values of 1 for each variable, it will be an easy equation to manipulate and use as a cancellation tool. For instance, if we wish to cancel the 3x term from the middle equation, all we need to do is take the top equation, multiply each of its terms by -3, then add it to the middle equation like this:

We can rid the bottom equation of its -5x term in the same manner: take the original top equation, multiply each of its terms by 5, then add that modified equation to the bottom equation, leaving a new equation with only y and z terms:

At this point, we have two equations with the same two unknown variables, y and z:

By inspection, it should be evident that the -z term of the upper equation could be leveraged to cancel the 4z term in the lower equation if only we multiply each term of the upper equation by 4 and add the two equations together:

Taking the new equation 13y = 52 and solving for y (by dividing both sides by 13), we get a value of 4 for y. Substituting this value of 4 for y in either of the two-variable equations allows us to solve for z. Substituting both values of y and z into any one of the original, three-variable equations allows us to solve for x. The final result (I'll spare you the algebraic steps, since you should be familiar with them by now!) is that x = 2, y = 4, and z = 12.

Contributors

Contributors to this chapter are listed in chronological order of their contributions, from most recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Chirvasuta Constantin (April 2, 2003): Pointed out error in quadratic equation formula.