# Basic Electrical Engineering – Formulas and Equations

The branch of engineering that deals with the study of design and implementation of various electrical devices and systems used in our everyday life as well as generation, transmission and distribution of electrical power is popularly known as Electrical Engineering.

Electrical engineering primarily convers the study of electric circuits, power systems, electrical machines, power electronics, control system, and many more. Electrical engineering uses formulae and mathematical equations to explain and prove the truth of concepts. These formulae and equations are very useful in understanding the behavior of electrical systems and helps to perform the different calculations in practice.

## Electric Charge

The subatomic property of substances by virtue of which the substances show the electrical behavior is called electric charge. The electric charge is denoted by Q (or q), and is measured in Coulombs (C). The electric charge is carried two elementary atomic particles namely electrons and protons, where an electron carries a negative charge while a proton carries a positive charge.

$$\mathrm{Charge\: on\: an\: electron, \mathit{e} \, =\, -1.6 \times 10^{-19} C}$$

The electric charge is a quantized quantity, which means the electric charge always exists as the integral multiple of elementary charge (e), i.e.,

$$\mathrm{Q = ne;\:\:\: where, n = 0,\, 1,\, 2,\, 3,\, \cdot \cdot \cdot}$$

## Voltage

Voltage, also called potential difference, is defined as the difference of electric potentials of two points in an electrical circuit. It is measured in Volt (V). It may also be defined as the amount of work done required to move a unit charge from one point to another in an electric circuit, i.e.,

$$\mathrm{Voltage, \: V\, =\, \frac{Work\: done\: \left ( W \right )}{Charge\: \left ( Q \right )}}$$

## Electric Current

The directed flow of electric charge, more specifically electrons, through a conductor is known as electric current. Electric current is denoted by I (or i), and is measured in Ampere (A). Electrical current may also be defined as the time rate of change of charge, i.e.,

$$\mathrm{I\, =\, \frac{Q}{t}}$$

In differential form,

$$\mathrm{i\, =\, \frac{dq}{dt}}$$

## Electrical Resistance

Electrical resistance is the measure of opposition offered by a substance in the flow of electric current. It is denoted by R and is measured in Ohms (Ω).

$$\mathrm{R\, =\, \frac{\rho l}{a}}$$

Where, ρ is a constant called resistivity or specific resistance of the material. The resistivity is defined as the property of material by which it opposes the flow of current through it.

## Ohm’s Law

Ohm’s law is a fundamental law related to electrical circuit. It states that the voltage across a conductor is directly proportional to the current flowing through it provided the physical conditions remain constant. It gives the relationship between voltage, current and resistance of a conductor as,

$$\mathrm{V \, =\, IR}$$

## Conductance

The measure of ease that the material offers in the path of current is called conductance, and it is given by as reciprocal of electrical resistance, i.e.

$$\mathrm{Conductance,\, G \, =\, \frac{I}{R}\, =\, \frac{a}{\rho l}\, =\,\frac{\sigma a}{l}}$$

Where, σ is the conductivity of the material, and is given by,

$$\mathrm{Conductivity,\, \sigma \, =\, \frac{1}{\rho }}$$

## Electric Power

The rate of doing work in an electric circuit is called electric power. It is denoted by P and measured in watt (W).

$$\mathrm{P\, \sigma \, =\, \frac{dW}{dt }}$$

In DC circuits,

$$\mathrm{Power,P\, =\,VI\, = \, I^{2}R\, =\,\frac{V^{2}}{R}}$$

In single-phase AC circuits,

$$\mathrm{Active\: power, P = VI\: cos\, \phi }$$

$$\mathrm{Reactive\: power, Q = VI\: sin\, \phi }$$

$$\mathrm{Apparent\: power, S = VI}$$

Where, active power is measured in watts (W), reactive power in volt-ampere reactive (VAr), and apparent power in volt-ampere (VA).

In three-phase AC circuits,

$$\mathrm{Active\: power,P\, =\, 3V_{p}I_{p}\, cos\, \phi \, = \, \, =\, \sqrt{3}V_{L}I_{L}\, cos\, \phi }$$

$$\mathrm{Reactive\: power,Q\, =\, 3V_{p}I_{p}\, sin\, \phi \, = \, \, =\, \sqrt{3}V_{L}I_{L}\, sin\, \phi }$$

$$\mathrm{Apparent\: power,S\, =\, 3V_{p}I_{p}\, = \, \, =\, \sqrt{3}V_{L}I_{L}}$$

## Power Factor

Power factor is measure of utilization of electric power in an ac electric circuit. It provides information about the part of total power utilized by load in an electrical system. It is given by the ratio of active power to the apparent power, i.e.,

$$\mathrm{Power\: factor,cos\, \phi \, =\, \frac{Active\: power (P)}{Apparent\: power (S)}}$$

The power factor of an electrical load varies from -1 to 1. For resistive load it is unity (1), for inductive load it is lagging, and for capacitive load it is leading.

## Frequency and Time Period

The number of cycle that an alternating quantity completes in one second is called frequency of the quantity. It is denoted by f and is measured in Hertz (Hz).

$$\mathrm{f \, =\, \frac{No.\: of\: cycles}{time}}$$

The time required by an alternating quantity to complete one cycle is called its time period. It is denoted by T, and is measured in seconds (s).

The frequency of an alternating quantity is inversely proportional to its time period, i.e.,

$$\mathrm{f \propto \frac{1}{T}}$$

## Wavelength

For an alternating quantity, the distance between two consecutive crests in the adjacent cycles of a wave is called wavelength of the signal. It is denoted by Greek letter Lambda (λ).

$$\mathrm{\lambda \, =\, \frac{ u }{f}}$$

Where, v is the velocity of wave and f is the frequency

## Capacitance

The property of a substance of storing electric charge in the form of electrostatic field is called capacitance of the substance. The circuit element used to introduce the capacitance effect in an electric circuit is called capacitor. The capacitance of capacitor is denoted by C, and measured in Farad (F).

The capacitance of a capacitor is given by,

$$\mathrm{C \, =\, \frac{Q }{V}}$$

Where, Q is the electric charge accumulated on each plate of the capacitor, and V is the voltage across the plates of the capacitor.

We may also express the capacitance of a capacitor in terms of its physical dimensions as,

$$\mathrm{C \, =\, \frac{\epsilon A }{d}}$$

Where, ε is the permittivity of medium between plates, A is the area of cross-section of capacitor plate, and d is the distance between plates of the capacitor.

## Inductance

Inductance is a property of materials by which they store electrical energy in the form of magnetic field. The element is known as inductor. The inductance is denoted by L and measured in Henry (H). It is given by the ratio of magnetic flux linkage and current as,

$$\mathrm{L \, =\, \frac{N \phi }{I}}$$

Where, N is the number of turn in inductor coil, and ϕ is the magnetic flux.

## Electric Field Intensity

The space around an electrically charge body in which a test charge experiences a force of attraction or repulsion is known as electric field. The strength of force acting on the charge placed in the field is called electric field intensity. It is denoted by E and is measured in Newton per coulomb (N/C).

$$\mathrm{E \, =\, \frac{F}{Q}}$$

## Coulomb’s Law

The Coulomb’s law is a fundamental law of electrostatics, which states that the electrostatic force acting between two charges is directly proportional to the product of magnitude of charges and inversely proportional to square of the distance between charges, i.e.,

$$\mathrm{F \propto \frac{Q_{1}Q_{2}}{d^{2}}}$$

$$\mathrm{\Rightarrow F \, =\, k \frac{Q_{1}Q_{2}}{d^{2}} \, =\,\frac{1}{4\pi \epsilon }\frac{Q_{1}Q_{2}}{d^{2}} }$$

Where, Q1 and Q2 are static charges, and d is the distances between these charges

## Gauss’s Law for Electric Flux

The Gauss’s law of electrostatics gives the value of electric flux, which is,

$$\mathrm{\phi _{e}\, =\, \frac{Q}{\epsilon } }$$

## EMF Equation of DC Generator

The emf generated by a DC generator is given by,

$$\mathrm{E_{g}\, =\, \frac{NP\phi Z}{60A } }$$

Where, N is the speed of rotation of armature, P is the number of field poles, ϕ is the magnetic flux per poles, Z is the number of armature conductors and A is the no. of parallel paths.

## Back EMF of DC Motor

The emf induced in a dc motor due to electromagnetic induction is called back emf or counter emf. It is given by,

$$\mathrm{E_{b}\, =\, \frac{NP\phi Z}{60A } }$$

## EMF Equation of Transformer

The expression that gives the value of emf induced in windings of a transformer is called emf equation. It is given by,

$$\mathrm{E\, =\,4.44\, f\phi _{m}N }$$

Where, f is the frequency of ac supply, ϕm is the maximum flux and N is the number of turns in winding.

## Hysteresis Loss

The power loss in iron core of electrical machines (motor, generator, transformer, etc.) due to magnetic reversal is called hysteresis loss, and it is calculated by,

$$\mathrm{P_{h}\, =\,\eta\, B_{m}^{1.6}\, fV }$$

Where, η is the hysteresis coefficient, Bm is maximum flux density in core, V is volume of the core, and f is the frequency of magnetic reversal.

## Eddy-Current Loss

The power loss occurs due to eddy currents induced in iron core of machines is called eddy current loss, and is given by,

$$\mathrm{P_{e}\, =\,k_{e}\, B_{m}^{2}\, f^{2}t^{2}\, V }$$

Where, ke is a constant, and t is the thickness of each lamination of core.

## Transformer Turns Ratio and Transformation Ratio

Turns ratio of a transformer is defined as the ratio of number of turns in primary winding to the number of turns in secondary winding, i.e.

$$\mathrm{Turns \: ratio, a\, =\,\frac{N_{1}}{N_{2}}\,=\,\frac{E_{1}}{E_{2}}\,=\,\frac{V_{1}}{V_{2}}\,=\,\frac{I_{2}}{I_{1}}}$$

Transformation ratio of a transformer is defined as the ratio of output voltage to the input voltage, i.e.,

$$\mathrm{Transformation \: ratio\, =\,\frac{V_{2}}{V_{1}}\,=\,\frac{E_{2}}{E_{1}}\,=\,\frac{N_{2}}{N_{1}}\,=\,\frac{I_{1}}{I_{2}}\,=\,\frac{1}{a}}$$

## Synchronous Speed

In rotating electric machines such as motors and generators, the magnetic field rotates at a constant speed, which is called synchronous speed.

$$\mathrm{N_{s}\, =\, \frac{120\, f}{P}}$$

Where, f is the supply frequency, and P is the field poles in the machine.

## Efficiency

For an electrical machine, the ratio of output power to the input power is known as efficiency of the machine, i.e.,

$$\mathrm{Efficiency,\eta \, =\, \frac{Output\: power\left ( P_{o} \right )}{Input\: power\left ( P_{i} \right )}}$$

## EMF Equation of 3-Phase Alternator

The EMF equation of a 3-phase alternator gives the magnitude of generated EMF. Generated EMF per phase is,

$$\mathrm{E_{ph} \, =\,2.22k_{p}k_{d}f\phi Z\, =\, 4.44k_{p}k_{d}f\phi T }$$

Where, kp and kd are pitch factor and distribution factor of armature winding respectively, Z is the number of conductors per phase, and T is the number of turns per phase.

## Electrical Impedance

In AC circuits, the combined opposition offered by the resistance, inductance and (or) capacitance in the flow of current is called impedance. It is denoted by Z and measured in Ohms (Ω).

$$\mathrm{Z \, =\,R+jX}$$

Where, X is the reactance, which is opposition offered by inductor or capacitor.

For inductor,

$$\mathrm{Inductive\: reactance,\, X_{L} \, =\,\omega L\, =\, 2\pi fL}$$

For capacitor,

$$\mathrm{Capacitive\: reactance,\, X_{C} \, =\,\frac{1}{\omega C}\, =\, \frac{1}{2\pi fC}}$$

## Conclusion

In this article, we listed all the important formulae and equations of basic electrical engineering. Also, we defined each quantity for your reference. Every electrical engineer must know all these formulae because they are frequently used in different calculations.