Overview

In this article we discuss some useful mathematical formulas in a quick reference format and some helpful heuristics to solve problems.

Contents

  1. Constants
  2. General Formulas
  3. Electrical Engineering
  4. Geometry
  5. Trigonometry
  6. Helpful Heuristics
  7. Additional References
  8. References

Constants

Avogadros Number 1

$N_{A}=6.022$ $140$ $76 \times 10^{23}$ $mol^{-1}$

Avogadros constant is the number of atoms in one mol of a substance.

Vacuum Electric Permittivity 2

$ε_{0}=8.854$ $187$ $8128(13) \times10^-12$ $F$ $m^{-1}$
Vaccum electric permittivity

Mass of an Electron 3

$m_{e}$ $=$ $9.109$ $383$ $7015(28) \times 10^{-31}$ $kg $
Mass of a electron

Mass of a Proton 4

$m_{p}$ $=$ $1.672$ $621$ $923$ $69(51) \times 10^{-27}$ $kg $
Mass of a proton

Magnitude of Charge 5

$e = 1.602$ $176$ $634 \times 10^{-19}$ $C$
Magnitude of charge on a proton or electron

Coulomb Constant

$k_{e}$ $=$ $8.987$ $551$ $7923(14)$ $\times 10^9$


The coulomb constant is often written in the alternative form relating it to the permittivity of free space.

$k_{e}=\frac{1}{4 \pi \epsilon _{0}}$
Alternative Form

General Formulas

Efficiency

Efficiency of a system can be expressed as a unitless percentage of the input work represented by the greek letter eta $\eta$.

$\eta = \frac{W_{OUT}}{W_{IN}}$
Efficiency Formula


Efficiency in electronics is typically calculated in terms of watts.

Electrical Engineering

Current Density

Current density is the charge per unit volume through a cross sectional area of a conductor.

$J=\frac{I}{A}$
Current Density Formula

Where

$I$ is current

$A$ is area in $m^2$

Resistors

Resistors are devices that can be used to limit current in a circuit. They can also be used to form voltage dividers which are typically used in digital multimeter circuits as a range switch with varying ratios.

Resistors in Series

Resistors in series simply add.

$R_{tot}=R_{0}+R_{1}+...R_{n-1}$

Resistors in Parallel

$\frac{1}{R_{tot}}=\frac{1}{R_{0}}+\frac{1}{R_{1}}+...\frac{1}{R_{n-1}}$


$R_{equiv}=\frac{1}{R_{tot}}^{-1}$

Remember that the equivalent resistance is the inverse of the sum.

Resistivity

Resistivity $ρ$ is typically in units of $Ω \cdot m$

Resistivity is the inverse of conductivity. Remember that resistivity is a property of a material and resistance is a property of an object.

Temperature Coefficients

Ohmic materials often have material properties change with temperature. Although not all changes are linear they can be approximated within a small temperature range using the temperature coefficient $α$.


$R_{calc}=R_{ref}(1+α(T-T_{0}))$


Given a reference resistance, typically at $20°C$ a resistance can be calculated at a given temperature according to the given formula.

Sometimes temperature coefficients (TCR) are also given in ppm or ppb which is parts per million or billion. See Calculating PPM Changes for details on how to calculate this.

Calculating Changes from PPM

Electronic components often have parameters listed as a change in parts per million or parts per billion. For this example we will use a thick film resistor and the TCR or temperature coefficient of resistance.

$\Delta R_{Ω}$ = $\frac{R_{ref}}{1E6} \times (T_{amb}-T_{ref})$
Change in TCR Formula

This value is the amount of change that the component undergoes as a function of the environmental change. A common parameter of TCR would be $100ppm/°C$.

Example Parameters

TCR: $100ppm/°C$

$R_{ref}$ : $10kΩ$

$T_{ref}$ : $20°C$

$T_{amb}$ : $80°C$

$\Delta R_{Ω}$ = $\frac{10kΩ}{1E6} \times (80-20) = 0.1 \times 60 = 6Ω$
Change in Resistance Calculation


Given a temperature coefficient of resistance of 100ppm, a reference resistance of $10kΩ$ and a change in temperature of $60°C$ we obtain a change in resistance of $6Ω$. Note that this does not account for the worst case resistance. It is important to also consider the tolerance rating of the device and then use those worst case values to then calculate total parameter deviation. If the TCR was given in parts per billion the denominator would then be 1E9.

Capacitors

For a parallel plate capacitor the formula to determine capacitance is directly proportional to insersected area between the plates and inversely proportional to the distance between them. In other words the larger the surface area of the plates, or the closer they are together will increase the resultant capacitance. The dielectric constant of the plate separating material also becomes a multiplier of the capacitance.

$C = k \frac{ε_{0} \cdot A}{d}$
Parallel Plate Capacitance

Where

$C$ is the capacitance in farads

$k$ is the material permittivity or dielectric constant

$ε_{0}$ is the permittivity of free space

$A$ is the area intersected by the plates in $m^2$

$d$ is the distance between the plates in meters

Energy Stored in a Capacitor

The energy stored in a capacitor can be calculated as follows:

$\frac{1}{2}C \cdot V^2$
Energy Stored

where

$C$ is in Farads

$V$ is in Volts

Dimensional analysis will show that the ouput is in Joules

$F=\frac{C}{V}   V=\frac{J}{C}$


$C \cdot V^2 = \frac{c^2}{J} \times \frac{J^2}{c^2} = J $


Note that big C is capacitance, and little c is coulombs. The formula $\frac{c^2}{J}$ is a rearrangement of $\frac{C}{V}$ where V is $\frac{J}{C}$. Capacitance is $\frac{coulombs}{\frac{joules}{coulombs}}$ or $\frac{coulombs^2}{joules}$.

Calculating Voltage from Charge

Voltage across a capacitor can be determined if the capacitance of the system is known along with the charge.

$C = \frac{Q}{V}$
Capacitance Formula


$V = \frac{Q}{C}$
Rearranging for Voltage

Where

$Q$ is charge, in coulombs

$V$ is Volts

$C$ is capacitance, in farads

Voltage Divider

$V_{out}=V_{in} \frac{R2}{R1+R2}$

Geometry

Area of a circle = $π \cdot r^2$

Circumference = $2 \cdot π \cdot r$

Area of a triangle = $\frac{1}{2} b \cdot h$

Trigonometry

Law of Sines 6

$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$

Law of Cosines 7

Standard Form: $a^2=b^2+c^2-2bc \cos(A)$


Alternative Form: $\cos(A) = \frac{b^2+c^2-a^2}{2bc}$

Pythagorea Theorem

$A^2+B^2=C^2$

Given two sides of a right triangle we can determine the unknown side. Note that this formula applied to right triangles only.

SOH CAH TOA

SOH CAH TOA is a useful mnemonic for remembering the trigonometric rules to get lengths or angles from right triangles.

$\sin(\theta)=\frac{opposite}{hypotenuse}$ $\cos(\theta)=\frac{adjacent}{hypotenuse}$ $\tan(\theta)=\frac{opposite}{adjacent}$


Helpful Heuristics for Problem Solving

Dimensional Analysis

When calculating mathematical formulas do not forget units. Often it is helpful to perform a dimensional analysis to verify the correct use of a formula or order of operations.

Example Analysis

$R = \frac{kg \cdot m^2}{A^2 \cdot s^3}=\frac {V}{A}$   $C=\frac{s^4 \cdot A^2}{m^2 \cdot kg}=\frac {A \cdot s}{V}$


$R_{Ω} \cdot C_{F} = \frac{kg \cdot m^2}{A^2 \cdot s^3} \times \frac{s^4 \cdot A^2}{m^2 \cdot kg} = s$


In this example we multiply capacitance by resistance. Notice that the units of $R \cdot C$ is seconds and therefore the units of $τ$ in RC and RL calculations is seconds. Performing a dimensional analysis is one additional level of sanity check to verify that your answer was calculated correctly.

Factoring

When a problem presents itself that appears to be challenging look for patterns to attempt to factor the formula. Can the equation be rewritten in a different format such as fractions instead of roots and so forth.

Graphical Methods

If you have trouble solving a problem try to graph the problem, draw a picture, or write down the known information or X-Y values. Try to think about what the formula is doing and how it is manipulating the terms. Viewing the formula in terms of what it is doing as opposed to simply numbers and words will often give additional insight.

Greek Alphabet

Many mathematical formulas include greek letters. This table can be used to aid in understanding.

Symbol English Example Use
$Α$ $α$ Alpha Coefficients
$Β$ $β$ Beta Transistor Gain
$Γ$ $γ$ Gamma  
$Δ$ $δ$ Delta Relative or Change
$Ε$ $ε$ Epsilon Permittivity
$Ζ$ $ζ$ Zeta  
$Η$ $η$ Eta Efficiency
$Θ$ $θ$ Theta Angle
$Ι$ $ι$ Iota  
$Κ$ $κ$ Kappa  
$Λ$ $λ$ Lambda Wavelength
$Μ$ $μ$ Mu  
$Ν$ $ν$ Nu  
$Ξ$ $ξ$ Xi  
$Ο$ $ο$ Omicron  
$Π$ $π$ Pi  
$Ρ$ $ρ$ Rho  
$Σ$ $σ$ Sigma Sum, Statistical Deviation
$Τ$ $τ$ Tau Time constant, $2π$
$Υ$ $υ$ Upsilon  
$Φ$ $φ$ Phi  
$Χ$ $χ$ Chi  
$Ψ$ $ψ$ Psi  
$Ω$ $ω$ Omega Reistances, Angular Frequency

Alt Codes

Alt codes are useful to reproduce symbols that are needed in engineering, science, and mathematics. To enter these symbols ensure the num-lock is on, press and hold Alt while typing the code on the num-pad then release the alt key. The table below includes select symbols.

Alt Code Symbol Description
234 Ω Omega (Ohms)
0176 ° Degrees
230 µ Mu (Micro)
224 α Alpha
238 ε Epsilon
231 τ Tau
227 π Pi

References

  1. NIST: Avogadro Constant ↩

  2. NIST: Vaccum Electric Permittivity ↩

  3. NIST: Electron Mass ↩

  4. NIST: Proton Mass ↩

  5. Magnitude of Elementary Charge ↩

  6. Larson, Ron, Algebra and Trigonometry. 7th ed. pp.544-550. ↩

  7. Larson, Ron, Algebra and Trigonometry. 7th ed. pp.553-559. ↩