Electrical Engineering Formulas (Most Important Equations)

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Key learnings:

Electrical Engineering Defined: Electrical engineering is the branch of science that deals with the study and application of electricity, electronics, and electromagnetism to develop electrical systems and devices.
Voltage Equation: Voltage in an electrical circuit is calculated by dividing the work done by the charge, an essential concept in electrical engineering equations.
Current and Resistance: Understanding electric current as the flow of electrons and resistance as the opposition to this flow is crucial in designing and analyzing circuits.
Power in Electrical Systems: Electric power equations vary by system type, with DC and AC systems using different formulas to determine power usage and efficiency.
Capacitance and Inductance: These properties are vital for storing electric and magnetic energy in circuits, important for the functionality of various electrical devices and systems.

Formulas For Electrical Engineering

Electrical engineering applies electricity, electronics and electromagnetism to circuits, machines, power systems, communications and control.

Topics include power systems, electrical machines, power electronics, computing, signal processing, telecommunications and control system design.

Formulas relate measured quantities, but each equation is valid only under its stated circuit and modelling assumptions.

The sections below collect common introductory relationships and identify several conditions that affect their use.

Voltage

Voltage is electric potential energy transferred per unit charge between two points. Its SI unit is the volt (V), equal to one joule per coulomb.

(1) $\begin{equation*} Voltage (V) = \frac{Work done (W)}{Charge (Q)} \end{equation*}$

The equation gives the unit $\frac{joule}{coulomb}$ .

Current

Electric current is the rate at which charge crosses a surface. Charge carriers may be electrons, ions or other charged particles, depending on the medium.

The SI unit is the ampere (A). Current is commonly denoted by I or by lowercase i for a time-varying value.

(2) $\begin{equation*} I = \frac{dQ}{dt} \end{equation*}$

Resistance

Resistance relates voltage across an element to current through it. Its SI unit is the ohm (Ω).

For a uniform conductor with constant resistivity and cross-sectional area, resistance is proportional to length and inversely proportional to area.

$R \propto \frac{l}{a}$

(3) $\begin{equation*} R = \rho \frac{l}{a} \end{equation*}$

Here, $\rho$ is the material resistivity at the stated temperature.

For an ohmic element at constant physical conditions, ohm’s law is V = IR. The first voltage equation below contains a protected division error; read it as V = IR.

$V \propto I$

(4) $\begin{equation*} Voltage \, V = \frac{I}{R} \, Volt \end{equation*}$

R is resistance in ohms. The following rearrangements, I = V/R and R = V/I, are consistent with Ohm’s law.

(5) $\begin{equation*} Current \, I = \frac{V}{R} \, Ampere \end{equation*}$

(6) $\begin{equation*} Resistance \, R = \frac{V}{I} Ohm \end{equation*}$

Electric Power

Electric power is the rate at which an element absorbs or delivers energy. The sign depends on the chosen voltage and current reference directions.

(7) $\begin{equation*} P = \frac{dW}{dt} \end{equation*}$

For DC System

(8) $\begin{equation*} P = VI \end{equation*}$

(9) $\begin{equation*} P = I^2 R \end{equation*}$

For Single-phase System

(10) $\begin{equation*} P = VI cos \phi \end{equation*}$

(11) $\begin{equation*} P = I^2 R cos \phi \end{equation*}$

(12) $\begin{equation*} P = \frac{V^2}{R} cos \phi \end{equation*}$

The last two single-phase expressions incorrectly attach power factor to resistor-loss formulas. For resistance R, use I²R or V²/R without the cosine factor.

For Three-phase system

(13) $\begin{equation*} P = \sqrt{3} V_L I_L cos \phi \end{equation*}$

(14) $\begin{equation*} P = 3 V_ph I_ph cos \phi \end{equation*}$

(15) $\begin{equation*} P = 3 I^2 R cos \phi \end{equation*}$

(16) $\begin{equation*} P = 3 \frac{V^2}{R} cos \phi \end{equation*}$

The same correction applies here: three equal resistive phase losses total 3I²R, or 3V²/R when V is the phase voltage.

Power Factor

In an AC system, power factor is real power divided by apparent power. For sinusoidal voltage and current it equals the cosine of their phase angle.

(17) $\begin{equation*} Power \, Factor Cos\phi= \frac{Active \, Power}{Apparent \, Power} \end{equation*}$

Power factor is dimensionless. A passive resistive load has a value near 1, while an ideal purely reactive load has zero average real power and a power-factor magnitude of 0. A negative sign indicates reversed real-power direction under a signed convention, not simply a reactive load.

Frequency

Frequency is the number of cycles per unit time, denoted by f. Its SI unit is the hertz (Hz), equal to one cycle per second.

Public AC power systems commonly use 50 Hz or 60 Hz, but signals and engineered systems use many other frequencies.

The time period is defined as the time required to produce one complete waveform cycle, denoted as T.

The frequency is inversely proportional to time period (T).

(18) $\begin{equation*} F \propto \frac{1}{T} \end{equation*}$

Wavelength

Wavelength is the distance between consecutive points at the same phase, such as adjacent crests.

It is defined as a ratio of velocity and frequency for sinusoidal waves.

(19) $\begin{equation*} \lambda = \frac{v}{f} \end{equation*}$

Capacitance

A capacitor stores electrical energy in an electric field when voltage is supplied. The effect of capacitors in electrical circuits is known as capacitance.

The electric charge Q accumulated in capacitor is directly proportional to the voltage developed across the capacitor.

$Q \propto V$

$Q = CV$

(20) $\begin{equation*} C = \frac{Q}{V} \end{equation*}$

For ideal parallel plates with negligible edge effects, capacitance depends on plate spacing d, overlapping area A and dielectric permittivity.

(21) $\begin{equation*} C = \frac{\epsilon A}{d} \end{equation*}$

Inductor

An inductor stores energy in a magnetic field while current flows. Depending on its function, it may be called a coil, choke or reactor.

The unit of inductance is henry (H).

For a linear inductor, inductance is flux linkage divided by current. In the protected equation below, ф_B must represent total flux linkage; if it represents flux per turn, multiply it by the number of turns.

(22) $\begin{equation*} L = \frac{\phi_B}{I} \end{equation*}$

Electric Charge

Electric charge is a property of particles and bodies that determines their electromagnetic interaction. A charged object can experience force in an electromagnetic field.

Charge may be positive or negative. It is denoted by Q and measured in coulombs (C); protons and electrons carry equal-magnitude charges with opposite signs.

One coulomb is the charge transferred by a current of one ampere in one second.

(23) $\begin{equation*} Q = IT \end{equation*}$

Electric Field

An electric field assigns a force per unit positive test charge to each point in space.

Electric field strength is denoted by E and is a vector quantity.

Its magnitude for a sufficiently small positive test charge is:

(24) $\begin{equation*} E = \frac{F}{Q} \end{equation*}$

For an ideal parallel-plate capacitor with uniform field magnitude, potential difference equals field magnitude times plate spacing.

$V = \frac{Work done}{charge} = \frac{Fd}{Q} = Ed$

(25) $\begin{equation*} E = \frac{V}{d} \end{equation*}$

Electric Force

Two stationary point charges exert forces described by Coulomb’s law.

Charges with the same sign repel, while charges with opposite signs attract.

For point charges in vacuum separated by distance d, Coulomb’s law gives:

(26) $\begin{equation*} F = \frac{Q_1 Q_2}{4 \pi \epsilon_0 d^2 } \end{equation*}$

The corresponding field magnitude from one point charge is:

$E = \frac{F}{Q} = \frac{kQq}{Qd^2}$

(27) $\begin{equation*} E = \frac{kq}{d^2} \end{equation*}$

Electric Flux

The gauss’s law relationship states that the net electric flux through a closed surface equals enclosed charge divided by vacuum permittivity:

(28) $\begin{equation*} \phi = \frac{Q}{\epsilon_0} \end{equation*}$

DC Machine

Back EMF

(29) $\begin{equation*} E_b = \frac{P \phi NZ}{60A} \end{equation*}$

Losses in DC Machine

Copper loss

The copper losses occur due to current flowing through the windings. The copper loss is directly proportional to the square of current flowing through the winding, and is also known as I2R loss or ohmic loss.

Armature copper loss: $I_a^2 R_a$

Shunt field copper loss: $I_{sh}^2 R_{sh}$

Series field copper loss: $I_{se}^2 R_{se}$

Copper loss in interpole: $I_a^2 R_i$

The displayed resistance model gives brush contact loss as $I_a^2 R_b$ . Many DC-machine calculations instead model an approximately fixed brush voltage drop and use that drop multiplied by armature current.

Hysteresis Loss

Hysteresis loss results from repeated magnetisation of the armature core. The displayed Steinmetz-type expression is empirical, so its coefficient and exponent depend on material and operating range.

(30) $\begin{equation*} P_h = \eta B_{max}^1.6 f V \end{equation*}$