SINE WAVE: CHARACTERISTICS, PARTS, CALCULATION, EXAMPLES - PHYSICAL - 2025

The sine waves are wave patterns that can be described mathematically by the sine and cosine functions. They accurately describe natural events and time-varying signals, such as the voltages generated by power plants and then used in homes, industries, and streets.

Electrical elements such as resistors, capacitors, and inductors, which are connected to sinusoidal voltage inputs, produce sinusoidal responses. The mathematics used in its description is relatively straightforward and has been thoroughly studied.

Figure 1. A sine wave with some of its main spatial characteristics: amplitude, wavelength and phase. Source: Wikimedia Commons. Wave_new_sine.svg: KraaiennestOriginally created as a cosine wave, by User: Pelegs, as File: Wave_new.svgderivative work: Dave3457

The mathematics of sine or sinusoidal waves, as they are also known, is that of the sine and cosine functions.

These are repetitive functions, which means periodicity. Both have the same shape, except that the cosine is shifted to the left with respect to the sine by a quarter of a cycle. It can be seen in figure 2:

Figure 2. The functions sin x and cos x are displaced with respect to each other. Source: F. Zapata.

Then cos x = sin (x + π / 2). With the help of these functions a sine wave is represented. To do this, the magnitude in question is placed on the vertical axis, while the time is located on the horizontal axis.

The graph above also shows the repetitive quality of these functions: the pattern repeats itself continuously and regularly. Thanks to these functions, it is possible to express sinusoidal voltages and currents varying in time, placing a v or i to represent voltage or current on the vertical axis instead of the y, and on the horizontal axis instead of the x, the t of time is placed.

The most general way to express a sine wave is:

Then we will delve into the meaning of this expression, defining some basic terms in order to characterize the sine wave.

Parts

Period, amplitude, frequency, cycle and phase are concepts applied to periodic or repetitive waves and are important to characterize them properly.

Period

A periodic function like those mentioned, which is repeated at regular intervals, always fulfills the following property:

Where T is a quantity called the period of the wave, and it is the time it takes for a phase of the wave to repeat itself. In SI units, the period is measured in seconds.

Amplitude

According to the general expression of the sine wave v (t) = v _m sin (ωt + φ), v _m is the maximum value of the function, which occurs when sin (ωt + φ) = 1 (remembering that the largest value that admits both the sine function and the cosine function is 1). This maximum value is precisely the amplitude of the wave, also known as peak amplitude.

In the case of a voltage it will be measured in Volts and if it is a current it will be in Amps. In the sine wave shown the amplitude is constant, but in other types of wave the amplitude can vary.

Cycle

It is a part of the wave contained in a period. In the figure above, the period was taken by measuring it from two consecutive peaks or peaks, but it can start to be measured from other points on the wave, as long as they are limited by a period.

Observe in the following figure how a cycle covers from one point to another with the same value (height) and the same slope (inclination).

Figure 3. In a sine wave, a cycle always runs over a period. The important thing is that the starting point and the end are at the same height. Source: Boylestad. Introduction to Circuit Analysis. Pearson.

Frequency

It is the number of cycles that occur in 1 second and is linked to the argument of the sine function: ωt. Frequency is denoted as f and is measured in cycles per second or Hertz (Hz) in the International System.

The frequency is the inverse amount of the period, therefore:

While the frequency f is related to the angular frequency ω (pulsation) as:

Angular frequency is expressed in radians / second in the International System, but radians are dimensionless, so the frequency f and the angular frequency ω have the same dimensions. Note that the product ωt gives radians as a result, and must be taken into account when using the calculator to obtain the value of sin ωt.

Phase

It corresponds to the horizontal displacement experienced by the wave, with respect to a time taken as a reference.

In the following figure, the green wave is ahead of the red wave by time t _d. Two sine waves are in phase when their frequency and phase are the same. If the phase differs, then they are out of phase. The waves in Figure 2 are also out of phase.

Figure 4. Out-of-phase sine waves. Source: Wikimedia commons. No machine-readable author provided. Kanjo ~ commonswiki assumed (based on copyright claims)..

If the frequency of the waves is different, they will be in phase when the phase ωt + φ is the same in both waves at certain times.

Sine wave generator

There are many ways to get a sine wave signal. Home-made electrical outlets provide them.

Faraday's law enforcement

A fairly simple way to obtain a sinusoidal signal is to use Faraday's law. This indicates that in a closed current circuit, for example a loop, placed in the middle of a magnetic field, an induced current is generated when the magnetic field flux through it changes in time. Consequently, an induced voltage or induced emf is also generated.

The magnetic field flux varies if the loop is rotated with constant angular speed in the middle of the field created between the N and S poles of the magnet shown in the figure.

Figure 5. Wave generator based on Faraday's law of induction. Source: Source: Raymond A. Serway, Jonh W. Jewett.

The limitation of this device is the dependence of the voltage obtained with the frequency of rotation of the loop, as will be seen in greater detail in Example 1 of the Examples section below.

Wien Oscillator

Another way to obtain a sine wave, this time with electronics, is through the Wien oscillator, which requires an operational amplifier in connection with resistors and capacitors. In this way sine waves are obtained whose frequency and amplitude the user can modify according to their convenience, by adjusting with switches.

The figure shows a sinusoidal signal generator, with which other waveforms can also be obtained: triangular and square among others.

Figure 6. A signal generator. Source: Source: Wikimedia Commons. Ocgreg at English Wikipedia.

How to calculate sine waves?

To perform calculations involving sine waves, a scientific calculator is used that has the trigonometric functions sine and cosine, as well as their inverses. These calculators have modes to work the angles either in degrees or in radians, and it is easy to convert from one form to the other. The conversion factor is:

Depending on the calculator model, you must navigate using the MODE key to find the DEGREE option, which allows you to work the trigonometric functions in degrees, or the RAD option, to directly work the angles in radians.

For example sin 25º = 0.4226 with the calculator set to DEG mode. Converting 25º to radians gives 0.4363 radians and sin 0.4363 rad = 0.425889 ≈ 0.4226.

The oscilloscope

The oscilloscope is a device that allows both direct and alternating voltage and current signals to be displayed on a screen. It has knobs to adjust the size of the signal on a grid as shown in the following figure:

Figure 7. A sinusoidal signal measured with an oscilloscope. Source: Boylestad.

Through the image provided by the oscilloscope and knowing the sensitivity adjustment in both axes, it is possible to calculate the wave parameters that were previously described.

The figure shows the sinusoidal voltage signal as a function of time, in which each division on the vertical axis is worth 50 millivolts, while on the horizontal axis, each division is worth 10 microseconds.

The peak-to-peak amplitude is found by counting the divisions that the wave covers vertically, using the red arrow:

5 divisions are counted with the help of the red arrow, so the peak-peak voltage is:

The peak voltage V _p is measured from the horizontal axis, being 125 mV.

To find the period, a cycle is measured, for example the one delimited by the green arrow, which covers 3.2 divisions, then the period is:

Examples

Example 1

For the generator in Figure 3, show from Faraday's law that the induced voltage is sinusoidal. Suppose that the loop consists of N turns instead of just one, all with the same area A and is rotating with constant angular speed ω in the middle of a uniform magnetic field B.

Solution

Faraday's law says that the induced emf ε is:

Where Φ _B is the magnetic field flux, which will be variable, since it depends on how the loop is exposed to the field at each instant. The negative sign simply describes the fact that this emf opposes the cause that produces it (Lenz's law). The flow due to a single turn is:

θ is the angle that the vector normal to the plane of the loop forms with the B field as the rotation proceeds (see figure), this angle naturally varies as:

So that: Φ _B = BAcos θ = BAcos ωt. Now we only have to derive this expression with respect to time and with this we obtain the induced emf:

Since the field B is uniform and the area of ​​the loop does not vary, they leave outside the derivative:

A loop has an area of ​​0.100 m ² and rotates at 60.0 rev / s, with its axis of rotation perpendicular to a uniform magnetic field of 0.200 T. Knowing that the coil has 1000 turns, find: a) The maximum emf that is generated, b) The orientation of the coil in relation to the magnetic field when the maximum induced emf occurs.

Figure 8. A loop of N turns rotates in the middle of a uniform magnetic field and generates a sinusoidal signal. Source: R. Serway, Physics for Science and Engineering. Volume 2. Cengage Learning.

Solution

a) The maximum emf is ε _max = ωNBA

Before proceeding to replace the values, the frequency of 60 rev / s must be passed to the International System units. It is known that 1 revolution is equivalent to one revolution or 2p radians:

60.0 rev / s = 120p radians / s

ε _max = 120p radians x 1000 turns x 0.200 T x 0.100 m ² = 7539.82 V = 7.5 kV

b) When this value occurs sin ωt = 1 therefore:

ωt = θ = 90º, In this case, the plane of the spiral is parallel to B, so that the vector normal to said plane forms 90º with the field. This occurs when the vector in black in figure 8 is perpendicular to the green vector representing the magnetic field.

References

1. Boylestad, R. 2011. Introduction to circuit analysis. 12th. Edition. Pearson. 327-376.
2. Figueroa, D. 2005. Electromagnetism. Physics Series for Science and Engineering. Volume 6. Edited by D. Figueroa. Simon Bolivar University. 115 and 244-245.
3. Figueroa, D. 2006. Physics Laboratory 2. Editorial Equinoccio. 03-1 and 14-1.
4. Sine waves. Recovered from: iessierradeguara.com
5. Serway, R. 2008. Physics for Science and Engineering. Volume 2. Cengage Learning. 881- 884