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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" 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

Inductor sizing equation

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)

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.

Gerald Gardner (January 2003): Addition of Imperial gallons conversion.