Our modern civilization runs on too many advances to count.

**One in particular is distinguished among all others**—paving the way for all our daily necessities to stand on.

Electricity is the backbone upon which modern civilization relies on, so much so that it has gradually creeped its way into becoming an essential for most. Electrical Engineering requires the thorough understanding of electricity, how it’s generated, what it is used for and much more.

One of the most important concepts to understand when dipping your toes into the world of electricity and electric engineering is **Alternating Current (AC).**

Alternating current, in its most basic definition is just a type of electrical current that changes direction periodically.

Unlike direct Current (DC), which has a unidirectional consistent flow of electricity, AC periodically changes direction which ultimately allows for a more efficient transmission over long distances, in turn, forming the layout that the world’s electrical grids run on.

In this article, we will take a deep dive into what alternating current is, how it is generated, and go into the nature of sinusoidal waveforms—a fundamental concept on the path to understanding AC.

We will break down each concept into its simplest form and give a step-by-step account of how each are related to one another. We will be using example problems to practically clarify complex ideas with the process of getting to a solution, ensuring that a comprehensive understanding of alternating current and its significance to electrical engineering and electricity as a whole is understood.

## What is Alternating Current?

Alternating Current is a type of electrical current that will periodically reverse the direction of the flow of electric charge. The sinusoidal waveform is the most common form that an AC takes. The sinusoidal waveform oscillates symmetrically about a central value— usually a zero, alternating from a peak positive value to a peak negative value.

AC is the form of electricity that is delivered to homes and commercial buildings. AC is used the most often due to its efficiency when it comes to transmitting electricity over large distances with minimal energy loss. This is done via transformers which allow for a controlled increase or decrease in voltage depending on distance, location, and several other factors.

## How is Alternating Current generated?

Typically, AC is generated through **electromagnetic induction**.

This principle is popularly know as Faraday’s Law of Electromagnetic Induction—discovered by scientist, Michael Faraday. Via this principle, we learn that electric current is produced through a conductor when exposed to a changing magnetic field. This can be done in two ways:

**Method 1:**Rotating a conductor (such as a coil of wire) through a magnetic field.**Method 2:**Changing the magnetic field surrounding a stationary conductor .

Let’s take a deeper dive into each one of these methods

**Generating AC by rotating a conductor through a magnetic field**

Whenever a conductor (such as a wire) goes about a magnetic field, the magnetic flux through the conductor changes, in turn producing an electromotive force (EMF) and generating electricity as a result. This is the foundational principle behind many devices like electric generators, whereby mechanical energy is converted into electrical Energy.

Broken down, this is the simplest explanation as to how AC is generated by rotating a conductor through a magnetic field.

**Generating AC by changing the magnetic field surrounding a stationary conductor**

If the magnetic field surrounding a stationary conductor is to change over a period of time, this can also trigger a change in the magnetic flux through the conductor (just as **Method 1**), inducing EMF and as a result, generating electricity . This is the principle behind transformers, whereby AC in one coil induces an EMF in another coil without any physical motion and ultimately generating electricity.

## Why generate AC and not DC?

The reason AC and not DC is generated through these methods is due to the variation in the induced voltage.

Different sections of a conductor (a coil) cuts through the magnetic field lines at varying angles as the coil rotates. As a section of coil cuts through the magnetic field perpendicularly, the induced voltage becomes at its peak. On the other hand, when the conductor is moving parallel to the field lines, no voltage is induced. This alternation of induced current which causes a change in direction periodically is what we know as AC.

## Let’s paint a better picture with an example of this

Imagine that a loop of wire or coil is rotating within a magnetic field. When the loop rotates, the segment of the loop that is moving up cuts through the magnetic field in a single direction, while at the same time, the section moving downward will cut through the magnetic field in the opposite direction. This periodically repeating motion causes current to flow in one direction during the first half of the rotation and in the opposite direction during the second half.

As a result, AC is generated.

## What is a Sinusoidal Waveform in relation to AC?

The most common waveform representation in an AC system is the sinusoidal waveform.

The sinusoidal waveform represents voltage or current, showing how each can vary over time in a smooth, repeating harmonic oscillation. The sinusoidal waveform is particularly crucial in AC systems because it allows for the most stable and efficient transmission of energy.

In relation to AC, the sinusoidal waveform is paramount.

### The sine wave

If you can recall, in your trigonometric classes, you might have come across something referred to as the sine wave—a mathematical curve which describes a smooth periodic oscillation. In the context of AC, the sine wave is by far the most accurate depiction of how voltage or current varies over a period of time. The equation for a sine wave in relation to AC is as follows:

\(V=V_m \sin (\omega t)\) : For Volts

\(Or\)

\(I=I_m \sin (\omega t)\) : For Current

**Where:**

\(V or I\): is the instantaneous voltage or current at time \(t\).

\(V_m\): is the peak voltage or current reached during the cycle.

\(\omega \): is the angular frequency\((f)\), given by \( \omega = 2\pi f \).

\(t\): is the time at a specific point.

To fully understand the principles behind AC sine waves, there are **some key terminologies that need to be understood first. **They are the following:

**Cycle:** A cycle is a complete set of positive and negative values of an alternating quantity. A cycle can also be defined in terms of angular measure—one cycle being equivalent to 360 degrees or \(2\pi\) radians.

**Period (\(T\)):** A period is the time required for a sine wave to complete one full cycle. A cycle is a set of complete positive and complete negative alternations.

\(T=\frac{1}{f} \)

**Frequency (\(f\)):** Frequency is the number of cycles that occur in one second. Meaning the full set of positive and negative alternations that occur in a single second. Frequency is measured in cycles/ sec \((c/s) \) or Hertz \((Hz)\).

\(f=\frac{1}{T} \)

**Alternation:** An alternation is half a cycle of an alternating quantity whether positive or negative. Alternations span 180 degrees or about \(\pi\) radians.

**Angular Velocity:** Angular velocity is the rate at which the phase of the wave changes with respect to time. In other words, it is the angle turned in one revolution per cycle all over time taken. This is given by:

\(\omega = 2\pi f\)

**Amplitude (Peak Value):** Amplitude simply put, is the highest value (positive or negative) that an alternating quantity is able to achieve—also known as peak value. Amplitude/Peak value is represented by either \(V_p\) and \(I_p\) or as \(V_m\) and \(I_m\).

**Average Value:** The average value of a sine wave is the total area under one cycle over the time period it takes to complete a full cycle. The average value of a sine wave over a full cycle is zero because the negative and positive alternations cancel each other out, But typically, average value refers to the average of the absolute value (also called the rectified average). The rectified average is always equal to \(\frac{2}{\pi}\), approximately \(0.637 \cdot V_p\).

**Root Mean Square (RMS):** This is the measure of the effective voltage or current. The RMS is the same as the DC voltage or current that would produce the same power in a resistive load. When it comes to the sinusoidal waveform, the RMS value is related to the amplitude \(V_m\) given by the formula:

\(\text{RMS} =\frac{V_m}{\sqrt{2}}\)

**Instantaneous Value: **The Instantaneous value is the value of an alternating quantity at a point in its alternation. The Instantaneous value is denoted by either a \(v\) or \(I\) depending on if it is voltage or current being calculated.

**Form Factor: **This is the ratio of RMS value of a waveform to its average value. This is given as:

\(Form\:Factor = \frac{RMS\:Value}{Average\:Value}\)

**Peak Factor: **The peak factor is the ratio of peak value to RMS value given by the following:

\(Peak\:Factor = \frac{Peak\:Value}{RMS\:Value}\)

## AC Sine Waves example problems

As you can see, the concepts surrounding the sinusoidal waveform are many. **In other to paint a better picture, let’s go through some example problems to solve.**

**Example Problem 1:** Write the mathematical expression for a sinusoidal voltage having a frequency of 60(Hz) and a peak voltage of \(80_v\).

Solution:

Let’s first determine our givens.

Data:

f = 60(Hz)

\(V_m\) = \(80_v\)

From the angular velocity formula, we can determine that:

\(\omega = 2\pi \cdot 60 \)

\(\omega \approx 376 rads\)

From the data above, we can now express this problem mathematical as:

\(V=V_m\sin \omega t \)

**\(V=80\sin 376 t\)**

**Example Problem 2: **An alternating current is given by the expression below:

\(I = 141.4_A\sin 314 t \)

- Determine the maximum value.
- What is the frequency?
- Determine the time period (T).
- What is the Instantaneous current at \(3_ms\)

**Solution:**

- \(Maximum\:value = 141.4_A\)
- To find the frequency, we can use the formula:

\(f=\frac{\omega}{2\pi}\)

\(f=\frac{314}{2\pi}\)

\(f=50_Hz\)

- To determine the Time period, we can use the formula below:

\(T=\frac{1}{f}\)

\(T=\frac{1}{50}\)

\(T=0.02_s\)

- To determine the Instantaneous value at \(3_ms\), we can use the equation given by the problem:

\(I=141.4\sin 314 t \)

When \(t=3_ms)

\(I=141.4\sin (314)\cdot(3\times 10^-3)\)

\(I=114.35_A)

## In summary…

Whether you’re an engineering student or simply just a curious person wanting to learn how Alternating Current powers your world, understanding AC, how it’s generated, and the significance of the sinusoidal waveform to it, is essential to grasping the principles of electrical engineering and the functioning of our modern day electrical systems.