# Sinus Wave

When an electric current passes through a wire or conductor, a circular magnetic field is formed around the wire whose power is associated with the current value.

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If this single-wire conductor is moved or rotated within a fixed magnetic field, an "EMF" (Electro-Motivator Force) is induced within the conductor due to the movement of the conductor along the magnetic flux.

From this, as Michael Faraday discovered the effect of "Electromagnetic Induction", we can see that there is a relationship between Electricity and Magnetism, and this is the basic principle that electric machines and generators use to create a **Sinusoidal WaveForm** for our mains supply.

In electromagnetic induction,we said that an EMF is induced within the time when a single wire conductor moves along a permanent magnetic field, thereby cutting the flux lines.

However, if the conductor moves parallel to the magnetic field at points A and B, no flux lines are cut into the conductor and emf is induced, but if the conductor moves at right angles to the magnetic field, as at points C and D, the maximum amount of magnetic flux is cut and produces the maximum amount of induced EMF.

In addition, since the conductor cuts the magnetic field at different angles between^{points} A and C, 0 and 90, the amount of EMF induced will be somewhere between this zero and the maximum value. The amount of EMF then induced in a conductor depends on the angle between the conductor and the magnetic flux and the strength of the magnetic field.

An AC generator uses Faraday's principle of electromagnetic induction to convert mechanical energy such as rotation into electrical energy, a **Sinusoidal WaveForm.**A simple generator consists of a pair of permanent magnets that produce a constant magnetic field between the north and south poles.Inside this magnetic field is a single rectangular wire ring that can be rotated around a fixed axis that allows it to cut magnetic flux at various angles, as shown below.

### Basic Single Coil AC Generator

Because the coil rotates counterclockwise around the central axis perpendicular to the magnetic field, the wire loop cuts the magnetic force lines established between the north and south poles at different angles as it rotates.The amount of EMF induced in the loop at any time is proportional to the angle of rotation of the wire loop.

As this wire loop rotates, the electrons in the wire flow in one direction around the noose.Now when the wire ring passes ^{that} point of 180 and moves in the opposite direction along the magnetic force lines, the electrons in the wire ring change and flow in the opposite direction.Then the direction of electron motion determines the polarity of the induced voltage.

Thus, when the cycle or coil physically rotates a full rotation or turns 360 ^{o,} we can see that a full sinusoidal waveform is produced with a cycle of the waveform produced for each rotation of the coil.When the coil rotates within the magnetic field, electrical connections are made to the coil through carbon brushes and sliding rings used to transfer the induced electric current in the coil.

The amount of EMF induced in a coil that cuts magnetic force lines is determined by the following three factors.

- Speed – the speed at which the coil rotates within the magnetic field.
- Power – the power of the magnetic field.
- Length – the length of the coil or conductor that passes through the magnetic field.

We know that the frequency of a source is the number of appearances of a loop per second, and that frequency is measured in Hertz.Since an induced emk loop is produced at each full speed of the coil along a magnetic field consisting of a north and south pole, as shown above, a fixed number of cycles per second will be produced if the coil rotates at a constant speed.Thus, the rotation speed of the coil will be increased and the frequency will be increased.Therefore, the frequency is proportional to the rotational speed, where ε ∝ Ν ) Ν = rpm

Also, our simple single-coil generator above has only two poles, one north and one south pole, only one double pole.If we add more magnetic poles to the generator above with a total of four poles, two north and two south, two loops will be produced for each rotational speed of the coil.Therefore, the frequency is proportional to the number of magnetic polar pairs of the generator ( ε ∝ P), where P = is the number of "polar pairs".

So based on these two cases, we can say that the frequency output from an AC generator is as follows:

Where: The rotation rate in Ν rpm is the number of P "polar pairs" and converts to 60 seconds.

## Instant Voltage

EmF induced in the coil at any given time depends on the speed at which the coil cuts the magnetic flux lines between the poles, and this depends on the rotation angle Of the manufacturer device Theta (ε).Because an AC waveform constantly changes its value or amplitude, the waveform at any given moment will have a different value than its next moment.

For example, the value in 1ms will differ from the value in 1.2ms, and so on.These values are commonly known as **InstantAneous Values** or V _{i.}Then the instantaneous value and direction of the waveform will vary according to the position of the coil within the magnetic field, as shown below.

### Displacement of the Coil within the Magnetic Field

The instantaneous values of a sinusoidal waveform are given as "Instant value = Maximum value x sin ε" and are generalized with this formula.

Here, the V_{max} coil and the induced maximum voltage ε = ωt , according to the time, is the rotation angle of the coil.

If we know the maximum or peak value of the waveform, instantaneous values can be calculated at various points throughout the waveform using the formula above.By drawing these values on graphics paper, a sinusoidal waveform shape can be created.

To keep things ^{simple,} we will draw instantaneous values for the sinusoidal waveform every 45 o turns and examine a total of 8 points.Again, to keep it simple, we will accept the maximum voltage, V _{MAX} value as 100V.Drawing instantaneous values at shorter intervals, such as every 30 ^{o} (12 points) or 10 ^{o} (36 points), will result in a more accurate sinusoidal waveform structure.

### Sinusoidal WaveForm Structure

Coil Angle ( ε ) | 0 | 45 | 90 | 135 | 180 | 225 | 270 | 315 | 360 |

e = Vmax.sinε | 0 | 70.71 | 100 | 70.71 | 0 | -70.71 | -100 | -70.71 | -0 |

Points on the sinusoidal waveform are obtained from various rotation positions between 0 ^{o} and 360 ^{o} , to the ordinate of the waveform corresponding to the angle, by reflecting a full rotation of the ε and wire loop or coil, or 360^{o,} and a full waveform is produced.

From the graph of the sinusoidal waveform, when ε 0^{o} equals 180 o or 360^{o,} we can see^{that} the EMF produced is zero, since the coil cuts the flux line in the minimum amount.However, when ε 90^{equals o} and 270^{o,} emf flux is the highest value since it has cut the maximum amount.

Therefore, a sinusoidal waveform has a positive peak in 90^{o}and a negative peak of 270 ^{o.} These positions correspond to the following in the chart: B, D, K and H .

The waveform shape, which is then produced by our simple single loop generator, is generally called **Sinus Wave** because its shape is said to be sinusoidal.This type of waveform is called sinus wave because it is based on the trigonometric sinus function used in mathematics, ( x(t) = Amax.sinε).

When dealing with sine waves in the time area and especially current-related sine waves, the unit of measurement used along the horizontal axis of the waveform can be time, degree or radian.In electrical engineering, it is more common to use **Radiant** as an angular measurement of the angle along the horizontal axis instead of the degree.For example, ω = 100 rad/s or 500 rad/s.

## Radian

**Radians** (radi) are the distance radius on which the circle extends around it.Since the circumference of a circle is equal to a radius of 2π x, the radial around 360 ^{o} of a circle should be 2π.

In other words, the radians are an angular unit of measurement, and the length of a radia (r) will fit 6,284 (2*π times around the entire circumference of a circle.Thus, a radians 360 ^{o} /2π = **57.3 ^{o equals.}**The use of radian in electrical engineering is very common, so it is important to remember the following formula.

### Description of Radia

Using radians as a unit of measure for a sinusoidal waveform will give a full cycle of 360^{o} to 2π radians. Then the half sinusoidal waveform should be equal to 1π radians or only π (pi). Then, knowing that pi (π) equals 3,142, the relationship between degrees and radiians for a sinusoidal waveform is given as follows:

### Relationship Between Degrees and Radians

If we apply these two equations to various points throughout the waveform:

For the more common equivalents used in sinusoidal analysis, the conversion between degree and radians is given in the table below.

### Relationship Between Degrees and Radians

Degrees | Radian | Degrees | Radian | Degrees | Radian |

0 ^{o} | 0 | 135 ^{O} | 3π 4 | 270 ^{O'O} | 3π 2 |

30 ^{o} | π 6 | 150 ^{O} | 5π 6 | 300 ^{O} | 5π 3 |

45 ^{O} | π 4 | 180 ^{O} | π | 315 ^{O} | 7π 4 |

^{60 O} | π 3 | 210 ^{O} | 7π 6 | 330 ^{O} | 11π 6 |

^{90 O} | π 2 | 225 ^{O'O} | 5π 4 | 360 ^{O} | 2π |

120 ^{O} | 2π 3 | 240 ^{O} | 4π 3 |

The speed at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform.Since the frequency of the waveform is given in ε Hz or revolution per second, the waveform also has angular frequency in radians/second, ω , (Greek letter omega).It is then given as the angular speed of the sinusoidal waveform.

### Angular Velocity of a Sinusoidal WaveForm

and 50Hz, the angular speed or frequency of the mains supply is given as follows:

Since the mains supply frequency in Turkey is 60Hz, it can be given as follows: 377 rad/s

Now the high speed determines the frequency of the sinusoidal waveform while rotating the generator around the middle axis, and we also know that it can be named so **the angular velocity**, ω .However, we must now know that the time required to complete a full cycle is equal to the periodic duration of the sinusoidal waveform (T).

Since the frequency is inversely proportional to the time period, ε = 1/T , so we can replace the amount of frequency in the equation above with the equivalent periodic time amount, and this equation gives us this:

The equation above states that for a smaller periodic duration of the sinusoidal waveform, the angular velocity of the waveform must be greater.Similarly, in the equation above for the frequency amount, the higher the frequency, the higher the angular speed.

### Sinusoidal WaveForm Questionnaire 1

A sinusoidal waveform is defined as: V _{m} = 169.8 sin(377t) volts.After six milliseconds (6ms), calculate the RMS voltage, frequency, and instantaneous value (V_{i)} of the waveform.

We know from above that the general expression for a sinusoidal waveform is as follows:

Then, the expression given above for the sinusoidal waveform is in this way V _{m} = 169.8 sin (377t) of the peak voltage value.

Waveforms RMS voltage is calculated as follows:

Angular velocity ( ω ) is given as 377 rad/s.Then 2πε = 377 .Thus, the frequency of the waveform is calculated as follows:

After a period of 6mS, the instant voltage V _{i} value is given as follows:

Note that the angular speed at t = 6mS is given in radians (radi).If desired, we can convert it to an equivalent angle in degrees and instead use this value to calculate the instantaneous voltage value.The angle of the instantaneous voltage value in degrees is therefore given as follows:

## Sinusoidal WaveForm

The generalized format used to analyze and calculate the various values of a **Sinusoidal WaveForm** is as follows:

In the next content, we will examine the phase difference. The phase difference is the relationship between two sinusoidal waveforms that are at the same frequency but pass through the horizontal zero axis at different time intervals.