Frequency

Frequency
Frequency
Dimension	T −1
Units
SI	Hz

Frequency is a physical quantity , a characteristic of a periodic process , equal to the number of repetitions or the occurrence of events (processes) per unit of time. It is calculated as the ratio of the number of repetitions or the occurrence of events (processes) to the time period over which they occurred ^[1] . The standard notation in the formulas is ν , f or F.

The unit of frequency measurement in the International System of Units (SI) is Hertz (Russian designation: Hz ; international: Hz ), named after the German physicist Heinrich Hertz .

The frequency is inversely proportional to the oscillation period : ν = 1 / T.

Frequency	1 MHz (10 ⁻³ Hz)	1 Hz (10 ⁰ Hz)	1 kHz (10 ³ Hz)	1 MHz (10 ⁶ Hz)	1 GHz (10 ⁹ Hz)	1 THz (10 ¹² Hz)
Period	1 ks (10 ³ s)	1 s (10 ⁰ s)	1 ms (10 ⁻³ s)	1 µs (10 ⁻⁶ s)	1 ns (10 ⁻⁹ s)	1 ps (10 ⁻¹² s)

Frequency, like time , is one of the most accurately measured physical quantities: up to a relative accuracy of 10 ⁻¹⁷ ^[2] .

In nature, periodic processes are known with frequencies from ~ ^10–16 Hz (the frequency of the orbital revolution around the center of the Galaxy ) to ~ 10 ³⁵ Hz (the frequency of the field oscillations characteristic of the most high-energy cosmic rays ).

In quantum mechanics, the oscillation frequency of the wave function of a quantum mechanical state has the physical meaning of the energy of this state, and therefore the system of units is often chosen in such a way that the frequency and energy are expressed in the same units (in other words, the conversion factor between frequency and energy is constant Planck in the formula E = h ν - is chosen equal to 1).

The human eye is sensitive to electromagnetic waves with frequencies from 4⋅10 ¹⁴ to 8⋅10 ¹⁴ Hz ( visible light ); oscillation frequency determines the color of the observed light. The human hearing analyzer senses acoustic waves with frequencies from 20 Hz to 20 kHz . In different animals, the frequency ranges of sensitivity to optical and acoustic vibrations are different.

Ratios of frequencies of sound vibrations are expressed using musical intervals , such as octave , fifth , third , etc. The interval of one octave between the frequencies of sounds means that these frequencies differ by 2 times , the interval to a pure fifth means the frequency ratio ³ ⁄ ₂ . In addition, to describe the frequency intervals used decade - the interval between frequencies that differ by 10 times . Thus, the range of sound sensitivity of a person is 3 decades ( 20 Hz - 20 000 Hz ). To measure the ratio of very close sound frequencies, units such as a cent (frequency ratio equal to 2 ^1/1200 ) and millioktava (frequency ratio 2 ^1/1000 ) are used.

Instantaneous frequency and spectral frequency

A periodic signal is characterized by an instantaneous frequency, which is (up to a coefficient) the rate of phase change, but the same signal can be represented as a sum of harmonic spectral components having their (constant) frequencies. The properties of the instantaneous frequency and the frequency of the spectral component are different ^[3] .

Sinusoidal waves of different frequencies, the lower waves have higher frequencies than the upper ones. The horizontal axis represents time.

Frequency change

Cyclic frequency

In the theory of electromagnetism , theoretical physics , and also in some applied electrical and radio engineering calculations, it is convenient to use an additional quantity — the cyclic (circular, radial, angular) frequency (usually denoted by ω ). Angular frequency (synonyms: radial frequency, cyclic frequency, circular frequency) is a scalar physical quantity. In the case of rotational motion, the angular frequency is equal to the modulus of the angular velocity vector. In SI and GHS systems, the angular frequency is expressed in radians per second, its dimension is inverse to the dimension of time (radians are dimensionless). The angular frequency in radians per second is expressed in terms of the frequency ν (expressed in revolutions per second or oscillations per second), as ω = 2πν ^[4] .

In the case of using the unit of angular frequency of degrees per second, the connection with the usual frequency will be the following: ω = 360 ° ν .

The numerically cyclic frequency is equal to the number of cycles (oscillations, revolutions) in 2π seconds. The introduction of a cyclic frequency (in its main dimension - radians per second) allows us to simplify many formulas in theoretical physics and electronics. So, the resonant cyclic frequency of an oscillatory LC-circuit is equal $\omega _{LC}=1/{\sqrt {LC}},$ ${\ displaystyle \ omega _ {LC} = 1 / {\ sqrt {LC}},}$ whereas the usual resonant frequency $\nu _{LC}=1/(2\pi {\sqrt {LC}}).$ ${\ displaystyle \ nu _ {LC} = 1 / (2 \ pi {\ sqrt {LC}}).}$ At the same time, a number of other formulas are complicated. The crucial consideration in favor of the cyclic frequency was that the multipliers $2\pi$ ${\ displaystyle 2 \ pi}$ and $1/2\pi$ ${\ displaystyle 1/2 \ pi}$ , appearing in many formulas when using radians to measure angles and phases, disappear with the introduction of a cyclic frequency.

In mechanics, when considering rotational motion, the analogue of the cyclic frequency is angular velocity .

Discrete Event Frequency

The frequency of discrete events (pulse frequency) is a physical quantity equal to the number of discrete events occurring per unit of time. The unit of the frequency of discrete events is a second to a minus of the first degree (Russian designation: s ⁻¹ ; international: s ⁻¹ ). The frequency 1 s ⁻¹ is equal to the frequency of discrete events at which one event occurs during a time of 1 s ^[5] ^[6] .

Speed

Rotation frequency is a physical quantity equal to the number of full revolutions per unit of time. The unit of rotational speed is second to minus first degree ( s ⁻¹ , s ⁻¹ ), revolution per second. Often used units such as revolutions per minute, revolution per hour, etc.

Other values related to frequency

Bandwidth - $\nu _{max}-\nu _{min}$ ${\ displaystyle \ nu _ {max} - \ nu _ {min}}$
Frequency interval - $\log(\nu _{max}/\nu _{min})$ ${\ displaystyle \ log (\ nu _ {max} / \ nu _ {min})}$
Frequency deviation - $\Delta \nu /2$ ${\ displaystyle \ Delta \ nu / 2}$
Period - $1/\nu$ ${\ displaystyle 1 / \ nu}$
Wavelength - ${v}/\nu$ ${\ displaystyle {v} / \ nu}$
Angular speed (rotational speed) - $d\phi /dt;2\pi F_{BP.}$ ${\ displaystyle d \ phi / dt; 2 \ pi F_ {BP.}}$

Units of measure

In the SI system, the unit of measurement is Hertz. The unit was originally introduced in 1930 by the International Electrotechnical Commission ^[7] , and in 1960 was adopted for general use by the 11th General Conference on Weights and Measures as the SI unit. Prior to this, the unit of frequency was used cycle per second ( 1 cycle per second = 1 Hz ) and derivatives (kilocycle per second, megacycle per second, kilomegacycle per second, equal respectively to kilohertz, megahertz and gigahertz).

Metrological aspects

Frequency meters of different types are used to measure the frequency, including: electron-counting and capacitor frequency meters for measuring pulse frequencies, resonant and heterodyne frequency meters, and spectrum analyzers for determining the frequencies of spectral components. To reproduce the frequency with a given accuracy , different measures are used - frequency standards (high accuracy), frequency synthesizers , signal generators , etc. They compare frequencies with a frequency comparator or with an oscilloscope using Lissajous figures .

Standards

For checking the means of measuring frequency, national frequency standards are used. In Russia, the national frequency standards are:

The state primary standard for units of time, frequency, and national time scale GET 1-98 is located at VNIIFTRI .
The secondary standard of the unit of time and frequency VET 1-10-82 - is in SNIIM (Novosibirsk).

Calculations

The frequency of a recurring event is calculated by taking into account the number of occurrences of this event during a specified period of time . The resulting amount is divided by the duration of the corresponding time period. For example, if 71 uniform events occurred during 15 seconds , the frequency will be

\nu ={\frac {71}{15\,{\mbox{s}}}}\approx 4.7\,{\mbox{Hz}}

{\ displaystyle \ nu = {\ frac {71} {15 \, {\ mbox {s}}}} \ approx 4.7 \, {\ mbox {Hz}}}

If the received number of samples is small, then a more accurate technique is to measure the time interval for a given number of occurrences of the event in question, rather than finding the number of events within a given period of time ^[8] . The use of the latter method introduces a random error between the zero and first readings, averaging half the reference; This may lead to the appearance of the average error in the calculated frequency Δν = 1 / (2 T _m ) , or the relative error Δ ν / ν = 1 / (2 v T _m ) , where T _m is the time interval, and ν is the measured frequency. The error decreases with increasing frequency, so this problem is most significant for low frequencies, where the number of samples N little.

Measurement Methods

Stroboscopic method

The use of a special instrument - a stroboscope - is one of the historically early methods for measuring the frequency of rotation or vibration of various objects. In the process of measurement, a stroboscopic light source is used (as a rule, a bright lamp, which periodically gives short light flashes), the frequency of which is adjusted using a pre-calibrated timing circuit. The light source is directed to a rotating object, and then the frequency of flashes gradually changes. When the frequency of flashes is equalized with the frequency of rotation or vibration of an object, the latter manages to complete a full oscillatory cycle and return to the initial position in the interval between two flashes, so that when illuminated with a stroboscopic lamp this object will appear stationary. This method, however, has a drawback: if the object's rotational speed ( x ) is not equal to the strobe frequency ( y ), but is proportional to it with an integer coefficient (2 x , 3 x , etc.), then the object will still be illuminated look fixed.

The stroboscopic method is also used to fine tune the rotational speed (oscillations). In this case, the frequency of the flashes is fixed, and the frequency of the periodic movement of the object changes until it begins to seem fixed.

Beat Method

Close to the stroboscopic method is the beat method . It is based on the fact that when mixing oscillations of two frequencies (reference ν and measured ν ' ₁ ) in a nonlinear circuit, the difference frequency also appears in the vibration spectrum Δν = | ν - ν ' ₁ |, called the beat frequency (with linear addition of oscillations, this frequency is the envelope frequency of the total oscillation). The method is applicable when it is more preferable to measure low-frequency oscillations with a frequency Δ f . In radio engineering, this method is also known as the heterodyne frequency measurement method. In particular, the beat method is used to fine tune musical instruments. In this case, the sound vibrations of a fixed frequency (for example, from a tuning fork ), heard simultaneously with the sound of the tuned instrument, create periodic amplification and attenuation of the overall sound. When fine tuning the instrument, the frequency of these beats tends to zero.

Frequency meter application

High frequencies are usually measured using a frequency counter . It is an electronic device that evaluates the frequency of a particular repetitive signal and displays the result on a digital display or analog display. Discrete logic elements of a digital frequency meter allow you to take into account the number of periods of oscillation of a signal within a predetermined period of time, measured by reference quartz watches . Periodic processes that are not electrical in nature (such as, for example, rotation of an axis , mechanical vibrations or sound waves) can be converted into a periodic electrical signal using a measuring transducer and are fed to the input of the frequency meter in this form. Currently, devices of this type are capable of covering a range up to 100 Hz; this indicator is a practical ceiling for direct counting methods. Higher frequencies are already measured by indirect methods.

Indirect measurement methods

Outside of the range available to the frequency meters, the frequencies of electromagnetic signals are often indirectly estimated using local oscillators (i.e. frequency converters). The reference signal of a previously known frequency is combined in a non-linear mixer (such as a diode , for example) with a signal whose frequency must be set; as a result, a heterodyne signal is formed, or - alternatively - beats , generated by the frequency differences of the two original signals. If the latter are close enough to each other in their frequency characteristics, then the heterodyne signal is small enough to be measured with the same frequency meter. Accordingly, as a result of this process, only the difference between the unknown frequency and the reference frequency is estimated, which should be determined by other methods. To cover even higher frequencies, several stages of mixing can be involved. Currently, studies are being conducted aimed at expanding this method in the direction of infrared and visible light frequencies (the so-called optical heterodyne detection).

Examples

Electromagnetic Radiation

Full spectrum of electromagnetic radiation with a selected visible part

Visible light is an electromagnetic wave consisting of oscillating electric and magnetic fields moving in space. The frequency of the wave determines its color: 4 × 10 ¹⁴ Hz - red , 8 × 10 ¹⁴ Hz - purple ; between them in the range (4 ... 8) × 10 ¹⁴ Hz are all the other colors of the rainbow. Electromagnetic waves having a frequency of less than 4 × 10 ¹⁴ Hz are invisible to the human eye, such waves are called infrared (IR) radiation . Below the spectrum is microwave radiation and radio waves . Light with a frequency higher than 8 × 10 ¹⁴ Hz is also invisible to the human eye; such electromagnetic waves are called ultraviolet (UV) radiation . With increasing frequency, the electromagnetic wave passes into the region of the spectrum where the X-rays are located, and at even higher frequencies, into the region of gamma rays .

All these waves, from the lowest frequencies of radio waves to the high frequencies of gamma rays, are fundamentally the same, and they are all called electromagnetic radiation. They all propagate in vacuum at the speed of light .

Another characteristic of electromagnetic waves is the wavelength . The wavelength is inversely proportional to frequency, so that electromagnetic waves with a higher frequency have a shorter wavelength, and vice versa. In vacuum wavelength

\lambda =c/\nu ,

{\ displaystyle \ lambda = c / \ nu,}

where c is the speed of light in a vacuum. In an environment in which the phase velocity of propagation of an electromagnetic wave c ′ differs from the speed of light in a vacuum ( c ′ = c / n , where n is the refractive index ), the relationship between the wavelength and frequency is as follows:

\lambda ={\frac {c}{n\nu }}.

{\ displaystyle \ lambda = {\ frac {c} {n \ nu}}.}

Ещё одна часто использующаяся характеристика волны — волновое число (пространственная частота), равное количеству волн, укладывающихся на единицу длины: k = 1/λ . Иногда эта величина используется с коэффициентом 2 π , по аналогии с обычной и круговой частотой k _s = 2π/λ . В случае электромагнитной волны в среде

k=1/\lambda ={\frac {n\nu }{c}}.

k=1/\lambda ={\frac {n\nu }{c}}.

k_{s}=2\pi /\lambda ={\frac {2\pi n\nu }{c}}={\frac {n\omega }{c}}.

k_{s}=2\pi /\lambda ={\frac {2\pi n\nu }{c}}={\frac {n\omega }{c}}.

Звук

The properties of sound (mechanical elastic oscillations of the medium) depend on frequency. A person can hear vibrations with a frequency of 20 Hz to 20 kHz (with age, the upper limit of the frequency of the audible sound decreases). A sound with a frequency lower than 20 Hz (corresponds to the notes of the subcontractase ) is called infrasound ^[9] . Infrasonic vibrations, although not audible, can be felt palpably. A sound with a frequency above 20 kHz is called ultrasound , and with a frequency above 1 GHz, it is called hypersound .

In music, sounds are usually used whose pitch (main frequency) lies from the subcontractava to the 5th octave. So, the sounds of a standard 88-key piano keyboard are stacked in the range from the note A of a subcontractava ( 27.5 Hz ) to the note to the 5th octave ( 4186.0 Hz ). However, a musical sound usually consists not only of the pure sound of the fundamental frequency, but also of the overtones or harmonics mixed in with it (sounds with frequencies that are multiples of the fundamental frequency); The relative amplitude of the harmonics determines the timbre of the sound. Overtones of musical sounds lie in the entire range of frequencies accessible to hearing.

AC Frequency

Voltage and frequency: 220-240 V / 60 Hz 220-240 V / 50 Hz 100-127 V / 60 Hz 100-127 V / 50 Hz

The workplace of the flight attendant aircraft An-26 . A 400 Hz frequency meter is visible in the upper right corner.

In Europe (including Russia and all countries of the former USSR), most of Asia, Oceania (except Micronesia), Africa, and part of South America, the industrial frequency of alternating current in the power network is 50 Hz . In North America (USA, Canada, Mexico), Central and in some countries of northern South America (Brazil, Venezuela, Colombia, Peru), as well as in some Asian countries (in the southwestern part of Japan, in South Korea, Saudi Arabia in the Philippines and Taiwan) uses a frequency of 60 Hz . See Standards of connectors, voltages and frequencies of the power grid in different countries . Almost all household appliances work equally well in networks with a frequency of 50 and 60 Hz, provided that the network voltage is the same. At the end of the 19th - first half of the 20th century, before standardization, frequencies from 16 ² ₃ to 133 ¹ ⁄ ₃ Hz were used in various isolated networks. The first is still used on some 15 kV railway lines of the world, where it was adopted for use by electric locomotives without rectifiers - DC traction motors were fed directly from a transformer .

In the on-board networks of aircraft, submarines, etc., a frequency of 400 Hz is used . Higher frequency of the power network allows reducing the weight and dimensions of transformers and obtaining high rotational speeds of induction motors , although it increases transmission losses over long distances due to capacitive losses , an increase in inductive line resistance and radiation losses.

Notes

↑ Frequency Article in the Scientific and Technical Encyclopedic Dictionary.
↑ A new record of atomic clocks accuracy is set (Uncased) . Membrana (February 5, 2010). The appeal date is March 4, 2011. Archived February 9, 2012.
↑ L. Fink. Signals, interference, errors ... Notes about some surprises, paradoxes and delusions in communication theory. - M .: Radio and communication, 1978, 1984.
↑ Angular Frequency (Unsolved) . Big encyclopedic polytechnic dictionary . The appeal date is October 27, 2016.
↑ Devil A. G. Units of physical quantities. - M .: " High School ", 1977. - p. 33. - 287 p.
↑ Dengub V. M. , Smirnov V. G. Units of magnitude. Dictionary reference. - M .: Standards Publishing House, 1990. - P. 104. - 240 p. - ISBN 5-7050-0118-5 .
↑ IEC History (Unc.) . Iec.ch. Circulation date June 2, 2013. Archived June 2, 2013.
↑ Bakshi, KA Electronic Measurement Systems . - US: Technical Publications, 2008. - p. 4–14. - ISBN 978-81-8431-206-5 .
↑ Sometimes abroad between the infrasound and the audible sound take the frequency of 16 Hz.

Literature

Fink L. M. Signals, interference, errors…. - M .: Radio and communication, 1984.
Burdun GD, Bazakutsa V. A. Units of physical quantities. - Kharkov: Vishcha school, 1984.
Yavorsky B. M., Detlaf A. A. Physics Handbook. - M .: Science, 1981.

Links

Radio circuits and signals
Essay by A. B. Sergienko “Analog Modulation” (inaccessible link from 05/22/2013 [2265 days] - history , copy )
Signals and linear systems
THEORETICAL BASES OF RADIO ENGINEERING

[1] Frequency Article in the Scientific and Technical Encyclopedic Dictionary.

[record-2] A new record of atomic clocks accuracy is set (Uncased) . Membrana (February 5, 2010). The appeal date is March 4, 2011. Archived February 9, 2012.

[3] L. Fink. Signals, interference, errors ... Notes about some surprises, paradoxes and delusions in communication theory. - M .: Radio and communication, 1978, 1984.

[4] Angular Frequency (Unsolved) . Big encyclopedic polytechnic dictionary . The appeal date is October 27, 2016.

[5] Devil A. G. Units of physical quantities. - M .: " High School ", 1977. - p. 33. - 287 p.

[6] Dengub V. M. , Smirnov V. G. Units of magnitude. Dictionary reference. - M .: Standards Publishing House, 1990. - P. 104. - 240 p. - ISBN 5-7050-0118-5 .

[7] IEC History (Unc.) . Iec.ch. Circulation date June 2, 2013. Archived June 2, 2013.

[8] Bakshi, KA Electronic Measurement Systems . - US: Technical Publications, 2008. - p. 4–14. - ISBN 978-81-8431-206-5 .

[9] Sometimes abroad between the infrasound and the audible sound take the frequency of 16 Hz.

Frequency

Instantaneous frequency and spectral frequency

Cyclic frequency

Discrete Event Frequency

Speed

Other values related to frequency

Units of measure

Metrological aspects

Standards

Calculations

Measurement Methods

Stroboscopic method

Beat Method

Frequency meter application

Indirect measurement methods

Examples

Electromagnetic Radiation

Звук

AC Frequency

See also

Notes

Literature

Links

More articles:

Frequency
$\nu ={\frac {n}{t}}$ ${\ displaystyle \ nu = {\ frac {n} {t}}}$ $\ nu = {\ frac {n} {t}}$
Dimension	T ⁻¹
Units
SI	Hz

Frequency

Instantaneous frequency and spectral frequency

Cyclic frequency

Discrete Event Frequency

Speed

Other values ​​related to frequency

Units of measure

Metrological aspects

Standards

Calculations

Measurement Methods

Stroboscopic method

Beat Method

Frequency meter application

Indirect measurement methods

Examples

Electromagnetic Radiation

Звук

AC Frequency

See also

Notes

Literature

Links

More articles:

Other values related to frequency