The higher the wave resistance, the. Coaxial cable parameters calculator

Transmission line model

The picture shows equivalent circuit an infinitesimal section of coaxial cable. All elements of the circuit are normalized to a unit of length (ohms per meter, farads per meter, siemens per meter, henry per meter in the SI system or ohms per foot, farads per foot, siemens per foot, henry per foot in the British and American units). This equivalent circuit is repeated an infinite number of times along the entire length of the coaxial cable.

Dielectric and magnetic permeability of the dielectric material of the cable

The absolute dielectric constant of the dielectric used in a coaxial cable determines the speed of signal propagation in the cable. This quantity is usually denoted by the Greek letter ε (epsilon) and is a measure of the resistance to an electric field in a given material. In a dielectric, the electric field decreases. In the SI system, dielectric constant is measured in farads per meter (F/m). Vacuum has the lowest dielectric constant. In this regard, the dielectric constant of vacuum was chosen as a constant - the electrical constant ε 0 = 8.854187817...×10 −12 F/m. Previously, it was called the dielectric constant or dielectric constant of vacuum. This constant does not have any physical meaning, it's simple dimensional coefficient and that is why it is now called the electrical constant.

For a given dielectric material, the dielectric constant is usually expressed as the ratio of its dielectric constant to the dielectric constant of a vacuum, that is

Speed ​​of light in vacuum c 0 related to the magnetic constant μ 0 and electrical constant by the following formula:

Magnetic permeability is a measure of a material's ability to maintain a magnetic field within it. It is usually denoted by the Greek letter μ and is measured in SI. Relative magnetic permeability, usually denoted as μ r(from the English relative - relative), represents the ratio of magnetic permeability of this material to the magnetic permeability of vacuum (magnetic constant). Relative magnetic permeability of the absolute majority used in coaxial cables dielectrics is equal to μ r = 1.

Magnetic constant, previously called the magnetic permeability of vacuum, the numerical value of which follows from the definition of the ampere current taking into account the formation magnetic field when current flows through a conductor or when moving electric charge. It is equal

μ 0 = 4π × 10 −7 ≈ 1.256637806 × 10 –6 H/m

Magnetic permeability μ and dielectric constant ε determine the phase velocity of propagation electromagnetic radiation in dielectric

In a vacuum this formula changes to

For non-magnetic materials (that is, dielectrics used in coaxial cables), the formula for phase velocity simplifies:

As we see, the higher the dielectric and magnetic permeability, the lower the phase velocity of propagation of electromagnetic radiation in dielectrics.

Linear capacity of coaxial cable (C")

The linear capacity of a coaxial cable, that is, its capacity per unit length, is one of important characteristics coaxial cables. A coaxial cable can be thought of as a coaxial capacitor, which will necessarily have a non-zero capacitance between the inner and outer conductors. This capacitance is proportional to the length of the cable and depends on its size, shape and the dielectric constant of the dielectric that fills the space between the inner and screen conductors.

Linear capacity C" in farads per meter (F/m) is determined by the formula:

D

d D And d

ε 0 ≈ 8.854187817620...×10 −12 F/m - dielectric constant of vacuum,

ε r is the relative dielectric constant of the insulating material. Relative permittivity of materials commonly used in coaxial cables: polypropylene - 2.2–2.36, polytetrafluoroethylene (PTFE or Teflon) - 2.1, polyethylene - 2.25.

The above formula is used in our calculator.

In English-speaking countries, a linear capacity of 1 foot is used. Considering that 1 ft = 0.3045 m, ln(x) = 2.30259 lg(x), and ε 0 ≈ 8.854187817620... × 10 −12 F/m, this formula for C" in farads per foot (F/ft) can be rewritten as

or in picofarads per foot:

Linear inductance of coaxial cable (L")

For coaxial cable, this is the inductance per unit length L" in henry per meter (H/m), determined by the formula

D- internal diameter of the shielding conductor of the coaxial cable,

d- diameter of the inner conductor of the coaxial cable; quantities D And d must be in the same units,

c

ε 0 = 8.854187817620... × 10 −12 F/m - electrical constant.

The electric constant was previously called the dielectric constant or dielectric constant of vacuum. These names are now considered obsolete, but are still widely used.

Considering that 1 ft = 0.3045 m and ln(x) = 2.30259 lg(x), we have:

or in mH/ft

The electric constant ε 0 is by definition related to the speed of light in a vacuum c and magnetic constant μ 0 by the following formula:

where μ 0 = 4π × 10 −7 ≈ 1.256637806 × 10 –6 H/m - magnetic constant, also called magnetic permeability of vacuum (outdated name).

Taking this definition into account, we can rewrite the formula for linear inductance L" in GN/m as

This formula is used in our calculator.

Characteristic impedance of coaxial cable (Z 0)

One of the most important characteristics of a coaxial cable is its characteristic impedance, which can be thought of as the impedance from the signal source connected to an infinitely long piece of cable. The characteristic impedance Z 0 of a coaxial cable is the ratio of voltage to current of a single wave propagating along the cable (without reflections). It is determined by the cable geometry and the dielectric material between the inner conductor and the outer shield and does not depend on the cable length. In SI, characteristic impedance is measured in ohms (Ω). Characteristic impedance can be thought of as the impedance of a transmission line of infinite length, since such a line has no signal reflected from its end. Typically, coaxial cables are available with a characteristic impedance of 50 or 75 ohms, although other values ​​can sometimes be found.

Why 50 and 75 Ohms? There are several versions. According to one of them, 50 Ohms was chosen due to the fact that a coaxial cable with a polyethylene dielectric with a relative dielectric constant ε r = 2.25 ensures minimal signal loss precisely at a characteristic impedance of 50 Ohms; at the same time, it can transmit significant power for the given geometric dimensions of the cable. The 75 Ohm standard is used for inexpensive cables cable television, which do not transmit signals high power and provide best characteristics by losses. Why 75 Ohm? There are several explanations. Some believe that 75 ohms is a compromise between low cable loss and good flexibility. Others believe that these values ​​were chosen quite arbitrarily.

The characteristic impedance Z 0 of a lossy coaxial cable is determined as follows:

R"- linear resistance (per unit length),

L"- linear inductance (per unit length),

G"- linear conductivity of the dielectric material (per unit length),

C"- linear capacity (per unit length),

j - imaginary unit, And

ω - angular frequency.

For a lossless cable, which has zero conductor resistance and no dielectric losses ( R"= 0 and G"= 0), this formula simplifies:

Here the value of Z 0 (in ohms) does not depend on frequency and is truly a quantity, that is, a purely resistive quantity. This lossless transmission line approximation is a useful model for describing low-loss coaxial cables, especially when they are used to carry high-frequency signals.

Replacing L" And C" using their definitions given above, we get:

D- internal diameter of the shielding conductor of the coaxial cable,

d- diameter of the inner conductor of the coaxial cable; quantities D And d must be in the same units,

c- the speed of light in vacuum, equal to 299,792,458 m⋅s −1,

ε 0 = 8.854187817620...×10 −12 F/m - electrical constant.

ε r is the relative dielectric constant of the cable insulator material.

Substituting the values ​​of the electrical constant ε 0 and the speed of light, we obtain:

Considering that ln(x) = 2.30259 lg (x), we obtain practical formula for wave impedance in ohms, which is used in our calculator:

Maximum operating frequency of coaxial cable

The main type of wave in a coaxial cable is a TEM wave (transverse electromagnetic mode). electromagnetic wave). In this propagation mode, the electric and magnetic field lines are perpendicular to each other and to the direction of wave propagation. The electric field lines are located radially, and the magnetic field lines have the form of concentric circles around the central core of the cable. For more high frequencies in coaxial cables, transverse electric TE waves (from the English transverse electric) can be excited, in which only the magnetic field lines are located in the direction of propagation, and transverse magnetic TM waves (from the English transverse magnetic), in which only The electric field lines are located in the direction of wave propagation. However, these two modes are undesirable.

In a coaxial cable the most low frequency, at which waves of type TE 11 are formed, and is the maximum operating frequency f c. This is the upper frequency of coaxial cable use. The signal can propagate as a TE 11 wave if the wavelength in the cable dielectric is shorter than the average circumference of the dielectric; for an air dielectric the formula will look like

λ c- the shortest permissible wavelength in the cable in meters and

D and d- diameters of the outer (screen) and inner cable conductors in meters.

If the cable uses a non-magnetic material rather than air as a dielectric (magnetic dielectrics such as ferrite are not used in the construction of coaxial cables), it operating frequency can be from 0 to maximum, determined by the formula

D- diameter of the outer conductor in meters,

d- diameter of the inner conductor in meters,

f c- maximum operating frequency in hertz,

ε r is the relative dielectric constant of the dielectric material.

For more practical values ​​in mm and GHz, the formula would be

It is this formula that is used in our calculator. In practice, coaxial cables operate at frequencies less than 90% of this frequency.

Wavelength shortening factor and speed deceleration factor

In a coaxial cable, where the space between the inner conductor and the shield is filled with a dielectric, the signal propagates through this dielectric. The phase speed of the wave that propagates in the dielectric decreases, but its frequency does not change. Spread speed vp(index p from English propagation - distribution), frequency f and wavelength λ in a dielectric are related by the relation

From this relationship it is clear that the wavelength of the signal that propagates in the dielectric also decreases in proportion to the decrease in speed. To compare such a decrease in speed (and the corresponding proportional decrease in wavelength) with the speed of light, in many countries (but not in Russia) the speed deceleration factor VF (from the English Velocity Factor) is used, which is always less than one or less than 100% , if expressed as a percentage.

In Russia and other countries former USSR Traditionally, the inverse value is used - the shortening coefficient, but more on that below. In English literature, if we're talking about about computer networks, not about general physics, the speed of signal propagation in a transmission line vp usually expressed not as a quantity in units of speed, but as a percentage of the speed of light. It would be more correct to call this value the speed deceleration factor VF. For example, in a transmission line with a typical VF = 66%, corresponding to a dielectric constant of 2.25 (solid polyethylene), the signal will travel at 66% of the speed of light. Formula:

VF - speed deceleration factor in percent,

v P- propagation speed in the transmission line (in m/s or ft/s),

c- the speed of light in a vacuum (approximately 3.0 x 10 8 m/s, or 9.8 x 10 8 ft/s).

Note that in the English-language scientific and physical literature not related to computer networks, term propagation speed really means speed, that is, distance per unit time.

Let's assume that we need to measure a short half-wave length of cable with a speed slowdown factor of 66% (corresponding to a wavelength shortening factor of 1.52) for a signal with a frequency of 30 MHz. The wavelength in vacuum corresponding to this frequency will be equal to λ = c/f= 10 m. Therefore, to ensure a delay of half a wave, you need electrical length 5 meters. However, since the signal travels in the cable at 1.52 (66%) less speed, we only need 5 × 0.66 = 3.3 m physical length coaxial cable. That is, we will need a cable that is k = 1/0.66 = 1.52 times shorter than the calculated electrical length. Here k is the same shortening coefficient, which shows how many times the propagation speed less speed light in a vacuum.

If these discussions haven’t given you a headache yet, they definitely will now! Note that in Belarus, Russia, Ukraine and other countries in the post-Soviet space, this length shortening factor, which is always greater than one, is traditionally used instead of the speed deceleration coefficient familiar to English-speaking specialists. By the way, on German this coefficient is called Verkürzungsfaktor, which also means shortening coefficient.

Let's summarize. The speed deceleration factor is the reciprocal of the wavelength shortening factor, which shows how many times the phase or group speed of a wave in a coaxial cable is less than the speed of light in a vacuum. It is this coefficient that is indicated in the characteristics of foreign-made coaxial cables. The deceleration factor shows how many times the speed of light more speed wave propagation in a coaxial cable and is usually (but not always) expressed as a percentage. In the characteristics of coaxial cables Russian production the wavelength shortening factor is indicated, which is always greater than unity. As with waves optical range, when waves pass through a dielectric, their wavelength decreases (compare with refraction!) while maintaining the frequency. Since speed is equal to frequency times wavelength, speed also decreases.

Typically, coaxial cables use non-magnetic dielectrics whose relative permeability is μ r= 1. In such dielectrics, the speed deceleration coefficient VF is equal to the reciprocal square root from the relative dielectric constant of the material through which the signal is transmitted:

IN general case, which includes, for example, dielectrics such as ferrite, the speed deceleration coefficient is determined by the formula

For light propagation in an optical fiber, the speed retardation factor is equal to the reciprocal of the refractive index n material (usually quartz glass) from which the fiber core is made:

One of the parameters of any conductive line is the characteristic impedance. It acquires particular relevance in high-frequency radio transmission technology, where the slightest mismatch in the operation of the circuit leads to significant distortion at the output. On the other hand, each owner of a computer connected to others in local network, encounters the concept of “wave resistance” every day. It is worth noting that the appearance Ethernet networks based on twisted pair allowed end user do not think too much about connectors, grounding, terminators and the quality of connectors, as was the case with coaxial cable lines of 10 megabits (or less). However, even for twisted pair, the term “characteristic impedance” is applicable. In general, we will dwell on the specifics of operating computer networks a little later.

So, what is wave impedance? As already indicated, this is one of the characteristics of a conductive line based on metal conductors. The last clause is necessary so as not to confuse modern optical lines data transmission and classic copper wires, where energy carriers are not charged particles, but light - different laws apply there. This value indicates how much resistance the line has to the generator (the source of modulated electrical oscillations). One should not confuse what can be measured with a conventional multimeter and the characteristic impedance of the medium, since these are completely different things. The latter does not depend on the length of the conductor (this is already enough to draw conclusions about the “similarity” of resistances). Physically, it is equal to the ratio of inductance (Henry) to capacitance (Farads). A small note: despite the fact that reactive components of the line are used in the calculations, the characteristic impedance of the circuit is always considered active in the calculations.

It's best to look at everything with an example. Let's imagine a simple circuit consisting of an energy source (generator, R1), conductors with characteristic impedance (R2), and a consumer (load, R3). If all three resistances are equal, all transferred energy reaches the consumer and performs useful work. If in any area this equality is not observed, then an inconsistent operating mode arises. At the point where the correspondence is broken, a reflected wave appears, and part electromagnetic energy returns back to the generator. Accordingly, it is necessary to increase its power to compensate for the amount of reflected energy. In other words, part of the energy is wasted, which means losses and suboptimal operating conditions. In addition, in some cases, mismatch completely disrupts the functioning of the entire line.

Now let's return to computer networks, where characteristic impedance plays important role. For lines based on (50 Ohms), it is important to comply with the condition: the resistance and conductor between them must be equal. Only in this case does the system of terminators and grounding work. If any area cable line physically stretch a little (hang a weight on the conductor), then due to a change in the diameter of the conductors in this place the wave impedance will change, a reflected wave will arise, disrupting the operation of the system. At the same time, the measured active resistance the line may remain virtually unchanged (low-cost devices will not register an increase in resistance at all). Attempts to restore the line by soldering conductors in the damaged area will further aggravate the situation, since not just a transition resistance will appear, but a mixture of different media (tin, copper), in which the waves propagate differently.

WAVE RESISTANCE

PROPAGATION OF SOUND WAVES IN A MEDIUM

The phase speed of sound waves depends only on the elasticity and density of the medium, and therefore on temperature, but does not depend on frequency.

where γ adiabatic index is the ratio of the molar heat capacity of a gas at constant pressure to the molar heat capacity at constant volume, γ = с р / с v. From formula (25) it follows that u does not depend on pressure, but increases with increasing temperature and decreases with increasing molar mass gas For example, in the air at t = 0 o C – , at t = 20 o C – ; in hydrogen at t = 0 o C – u = 1260 m/s, at t = 20 o C – u = 1305 m/s.

In solid and liquid media, the speed of sound is greater than in gases. For water it is equal to 1550 m/s. The average speed of sound in human soft tissues has approximately the same value. IN solids Acoustic waves can be both longitudinal and transverse. The speed of longitudinal sound waves is greater than the speed of transverse ones and amounts to 2 ÷ 6 km/s.

At the interface between two media, sound waves experience reflection and refraction. The laws of reflection and refraction of mechanical waves are similar to the laws of reflection and refraction for light. The transition of a wave from one medium to another leads to a change in the conditions of its propagation, because the density of the medium and the wave speed change. For this reason, the redistribution of energy between the reflected and refracted parts of the wave is determined by the values wave impedances medium ω 1 = ρ 1 u 1 and ω 2 = ρ 2 u 2 . The penetration coefficient of a β wave from medium 1 to medium 2 with normal incidence at the interface is determined by the relation:

. (26)

From this relationship it is clear that sound waves completely, without experiencing reflection, penetrate from medium 1 to medium 2 (β = 1), if ρ 1 u 1 = ρ 2 u 2. If ρ 2 u 2 >> ρ 1 u 1 , then β<< 1. К примеру, волновые сопротивления воздуха и бетона соответственно равны: 400 кг·м -1 ·с -1 и 4 800 000 кг·м -1 ·с -1 . Расчёт коэффициента проникновения звуковой волны из воздуха в бетон даёт – β = 0,037%.

Any real medium has viscosity, and therefore, as sound propagates, attenuation is observed, ᴛ.ᴇ. reduction in the amplitude of sound vibrations. Attenuation is due to: absorption of the energy of sound waves by the medium, ᴛ.ᴇ. irreversible conversion of mechanical energy into other forms (mainly thermal); reflection of waves from the interfaces between layers of matter with different acoustic resistance; as well as scattering on elements of the microstructure of the medium. These factors play a particularly important role in the propagation of mechanical waves in biological objects.

The decrease in sound intensity upon penetration into the medium occurs according to an exponential law:

where I and I 0 are the wave intensities on the surface of the substance and at depth l from her. Attenuation coefficient for a homogeneous medium –

where λ is the sound wavelength; u is its speed in a given environment; ρ – density of matter; η – viscosity coefficient.

The phenomenon of gradual attenuation of sound in enclosed spaces (in the process of numerous reflections from walls and other obstacles) is usually called reverberation of sound. The time during which the sound intensity decreases by a million times (amplitude 1000) is usually called reverberation time. The room has good acoustics if the reverberation time is 0.5 - 1.5 s.

9. CHARACTERISTICS OF AUDITORY SENSATION

THEIR RELATIONSHIP WITH THE PHYSICAL CHARACTERISTICS OF SOUND WAVES

WEBER-FECHNER LAW

Sound, as an object of auditory perception, is assessed by a person subjectively. Those. sound has physiological characteristics that are a reflection of its physical parameters. One of the tasks of acoustics is to establish a correspondence between the objective parameters of sound waves and the subjective assessment of the auditory sensation that these waves cause in the human ear. Solving this problem makes it possible to objectively judge the condition of a particular person’s hearing aid based on the results of physical measurements.

The auditory sensation has three basic characteristics: pitch, timbre and volume.

The vibration frequency of a sound wave is estimated by the ear as pitch(pitch) . The higher the vibration frequency, the higher ("subtle") the sound is perceived.

Timbre– physiological characteristics of complex tones. Having the same fundamental frequencies, complex vibrations can differ in their sets of overtones. This difference in the spectra is perceived as timbre (sound color). For example, by the timbre of a sound it is easy to distinguish the same tone reproduced on different musical instruments.

Volume characterizes the level of auditory sensation (strength of auditory sensation). This subjective value associated with the sensitivity of the ear depends, first of all, on the intensity, as well as on the frequency of the sound wave. The relationship between volume and frequency is complex.
Posted on ref.rf
At constant sound strength (intensity), sensitivity first increases as the frequency increases, reaching a maximum in the frequency range 2000 ÷ 3000 Hz, then decreases again, going to zero at 20 kHz. With age, the ability to perceive high-frequency vibrations deteriorates. Already in middle age, a person, as a rule, is not able to perceive sounds with a frequency above 12-14 kHz. The dependence of ear sensitivity on frequency means that the range of intensities that can cause an auditory sensation will also be different for different frequencies (Fig. 6). The upper curve on the graph corresponds to the pain threshold. The bottom graph is called the threshold volume curve, ᴛ.ᴇ. I 0 = f(ν) at a volume level equal to zero.

A person with normal hearing perceives a change in volume only if the intensity of the wave changes by approximately 26%. At the same time, he quite accurately captures the difference when comparing two sensations of different intensities. This feature lies

in the basis of the comparative method of loudness measurement. Loudness is quantified by comparing the auditory sensation of two sound sources. In this case, it is not the absolute value of loudness that is determined, but its relationship with the loudness, the value of which is taken as the initial (or zero) value. Those. determine the volume level E: how much louder a given sound is in comparison with the sound whose volume is taken as the initial volume. Loudness, like intensity level, is measured in bels (B). At the same time, 0.1B of volume is usually called background (background), and not a decibel.

When comparing the volumes of sounds, we agreed to proceed from a tone with a frequency of 1000 Hz. Those. The volume of a tone with a frequency of 1000 Hz is taken as the reference for the volume scale. In this case, the energy costs, expressed by the intensity level, at a frequency of 1000 Hz are numerically equal to the volume: the intensity level L = 1B (10 dB) corresponds to the volume E = 1 B (10 background), the intensity level L = 2B (20 dB) corresponds to the volume E = 2 B (20 background), etc.

Because The energy range of sound waves is divided into 13 levels in bels (or 130 levels in dB), then, accordingly, the volume scale will have 13 levels in bels (or 130 levels in backgrounds).

At the root of the creation of the loudness scale is the psychophysical law of Weber-Fechner. According to this law, for all types of sensations it is true: if you successively increase the strength of the stimulus in a geometric progression (ᴛ.ᴇ. by the same number of times), then the sensation of this irritation increases in an arithmetic progression (ᴛ.ᴇ. by the same amount). Mathematically, this means that the loudness of a sound is directly proportional to the logarithm of the intensity.

If there is a sound stimulus with intensity I, then, based on the Weber-Fechner law, the volume level E is related to the intensity level as follows:

E = kL = k log, (27)

where I / I 0 is the relative strength of the stimulus, k is a certain proportionality coefficient depending on frequency and intensity (k = 1 for a frequency of 1000 Hz). Dependence of volume on intensity and the vibration frequency in the sound measurement system is determined on the basis of experimental data using graphs (Fig. 7), which are called equal loudness curves, ᴛ.ᴇ. I = f(ν) at

E= const. When studying hearing acuity, a zero loudness level curve is usually constructed, ᴛ.ᴇ. dependence of the hearing threshold on frequency – I 0 = f (ν). This curve is the main one in a system of similar curves constructed for different volume levels, for example, in steps through 10 backgrounds (Fig. 7). This system of graphs reflects the relationship between frequency, intensity level and volume, and also allows you to determine any of these three values, if the other two are known.

WAVE RESISTANCE - concept and types. Classification and features of the category "WAVE RESISTANCE" 2017, 2018.

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There is a persistent prejudice and, one might even say, misconception among many people regarding high-frequency cables. As a developer of antennas, who is also the head of a company that produces them, I am constantly plagued by this question. I will try to put an end to this issue once and for all and close the topic of using 75 Ohm cables instead of 50 Ohms for the purpose of transmitting low power signals. I will try not to bore the reader with complex terms and formulas, although a certain minimum of mathematics is still necessary to understand the issue.

In low-frequency radio engineering, to transmit a signal with given current-voltage parameters, we need a conductor that has certain insulation properties from the environment and linear resistance, so that at the point of receiving the low-frequency signal we receive a signal sufficient for subsequent processing. In other words, any conductor has resistance, and it is desirable that this resistance be as small as possible. This is a simple condition for low frequency technique. For signals with low transmitted power, a thin wire is enough for us; for signals with high power, we must choose a thicker wire.

Unlike low-frequency radio technology, in high-frequency technology many other parameters have to be taken into account. Undoubtedly, as in LF technology, we are interested in the power and resistance transmitted through the transmission medium. What we usually call transmission line resistance at low frequencies is called loss at high frequencies. At low frequencies, losses are primarily determined by the transmission line's own linear resistance, while at HF ​​the so-called Skin effect appears. Skin effect - leads to the fact that the current displaced by a high-frequency magnetic field flows only along the surface of the conductor, or rather in its thin surface layer. Because of this, the effective cross-section of the conductor can be said to decrease. Those. under equal conditions, pumping the same power at low and high frequencies requires wires of different sections. The thickness of the skin layer depends on the frequency; with increasing frequency, the thickness of the skin layer decreases, which leads to losses greater than at lower frequencies. The skin effect is present with alternating current of any frequency. For clarity, I will give some examples.

So for a current with a frequency of 60 hertz, the thickness of the skin layer is 8.5 mm. And for a current of 10 MHz, the thickness of the skin layer will be only 0.02 mm. Isn't it a striking difference? And for frequencies of 100, 1000 or 2000 MHz, the thickness of the conductive layer will be even less! Without going into mathematics, I will say that the thickness of the skin layer depends, first of all, on the specific conductivity of the conductor and frequency. Therefore, to transmit the maximum possible power to HF, we need to take a cable with the largest surface area of ​​the central core. Moreover, given that at microwave frequencies the thickness of the skin layer is small, we do not necessarily need to use a solid copper cable. You probably won't even notice the difference from using a cable with a steel center conductor coated with a thin layer of copper. Unless it will be more rigid in bending. Of course, it is desirable to have a thicker layer of copper on the steel conductor. There are, of course, advantages to using solid copper cable; it is more flexible and can carry more power at lower frequencies. Also, the DC power supply voltage of preamplifiers is often transmitted via coaxial cable, and here, too, copper cable has no competition. But for transmitting small power of no more than 10-200 mW to a microwave, from an economic point of view, the use of copper-plated cable would be more justified. We will assume that the question of choosing between copper-plated and copper cables has been closed.

To understand the differences between cables in the characteristic impedance, I will not tell you what the characteristic impedance of a cable is. Oddly enough, this is not necessary to understand the difference. First, let's figure out why there are cables with different characteristic impedances. First of all, this is connected with the history of the formation of radio engineering. At the dawn of radio engineering, the choice of insulating materials for coaxial cables was very limited. Now we normally perceive the presence of a huge range of plastics, foamed dielectrics, rubber with conductor properties or ceramics. 80 years ago none of this existed. There was rubber, polyethylene, paraffin, bakelite, and fluoroplastic (also known as Teflon) was invented in the 30s. The characteristic impedance of cables is determined by the ratio of the diameters of the central inner conductor and the outer diameter of the cable.

Below is the nomogram.

The thickness of the center conductor is determined by its ability to transmit the greatest power. The outer diameter is selected depending on the dielectric used - the filler located between the two conductors. Using the nomogram, it becomes clear that the range of cable wave impedances convenient for industrial production lies in the range of 25 - 100 Ohms.

So, one of the criteria is manufacturability. The next criterion is the maximum transmitted power. Omitting the mathematics, I will say that to transmit maximum power using the most widely used dielectrics, the optimal wave impedance is in the range of 20-30 Ohms. At the same time, wave impedances of 50-75 Ohms correspond to the minimum attenuation. Moreover, cables with a characteristic impedance of 75 Ohms have less attenuation than cables with a characteristic impedance of 50 Ohms. It becomes more or less clear that it is more profitable to use a 75 Ohm cable for transmitting low powers, and 50 Ohms for transmitting high powers.

Now I consider it necessary to consider the less important issue of coordinating the transmission line. I will simply try to answer questions about whether it is possible to connect a 75 Ohm cable instead of a 50 Ohm one.

Understanding coordination issues requires special knowledge in radio engineering. Therefore, we will limit ourselves to just stating the facts. But the facts are that in order to transmit a signal with minimal losses, the internal resistance of the signal source must be equal to the characteristic impedance of the cable. At the same time, the characteristic impedance of the cable must be equal to the characteristic impedance of the load. In other words, the signal source is the transmitter, the load is the antenna. Let's look at several situations in which, for simplicity, we will consider the cable to be ideal without losses, and the power transmitted through the cable is small - up to 100-200 milliwatts (20 dBm).

Let's consider a situation where the output impedance of the transmitter is 50 Ohms, we connect a 50 Ohm cable and a 75 Ohm antenna to it. In this case, the losses will be 4% of the output power. Is this too much? The answer is ambiguous. The fact is that in HF radio engineering they operate mainly with logarithmic quantities reduced to decibels. And if 4% is converted into decibels, then the loss in the line will be only 0.18 dB.

If we connect a transmitter with a 50 Ohm output to a 75 Ohm cable and then to a 50 Ohm antenna. In this case, 8% of power is lost. But bringing this value to decibels, it turns out that the loss will be only 0.36 dB.

Now let's look at typical cable attenuation for a frequency of 2000 MHz. And let’s compare what is better to use: 20 meters of 75 Ohm cable or 20 meters of 50 Ohm cable.

The attenuation at 20 meters for the well-known expensive Radiolab 5D-FB cable is 0.3 * 20 = 6 dB.

The attenuation at 20 meters for a high-quality Cavel SAT703 cable is 0.29 * 20 = 5.8 dB.

Taking into account the mismatch loss - 0.36 dB, we find that the gain from using a 50 Ohm cable is only 0.16 dB. This roughly corresponds to 2 extra meters of cable.

Now let's compare the price. 20 meters of Radiolab 5D-FB cable cost, at best, about 80*20=1600 rubles. At the same time, 20 meters of Cavel SAT703 cable costs 25*20=500 rubles. The difference in price is 1100 rubles. very noticeable. The advantages of 75 Ohm cables also include ease of cutting and accessibility of connectors. Therefore, if someone once again starts being smart and tells you that there is no way to use a 75 Ohm cable for a 3G modem, then with a clear conscience send it to ... or to me for our wonderful antennas. Thank you for your attention.

Strokov Andrey.

So, the second article in the series, which I have already mentioned several times. Today I will try to cram into the readers’ heads a few key points without which one cannot live in the world. So far I have talked about coordination, coordinated load. I mentioned something about the width of the line, which seems to have to be strictly defined. It's time to set the record straight. You will need a plastic bottle and scissors, an endless pair of wires and a little patience, welcome to the cat!

Let's come from afar.
Let's take a generator with internal resistance R. And connect load R1 to it. This is a common scheme.

The question is how effective is this scheme? At what load resistance can you get maximum power?

A few calculations:

To get maximum power, remember the derivative and equate it to zero.

and now we already get that maximum power is released when R = R1. In this case, the generator-load system is said to be coordinated.

Well, now the tricks begin. We feed our circuit a higher frequency. Last time we saw that the voltage can be completely different in different parts of the line. Let our diagram look like this:

yes, forget about antinodes for now, there are no standing waves, we consider only the incident one. In any case, Ohm’s law cannot be applied “head-on” to this picture. That's when such trouble begins, which means we are dealing with long line. At the same time, you can remember our solder snot and 1206 capacitors, which begin to behave randomly at some frequencies, again due to the fact that the dimensions are comparable to the wavelength and all sorts of loops, standing waves and resonances appear there. Everyone calls it devices with distributed parameters. Usually we talk about distributed parameters when the dimensions of the elements are at least 10 times larger than the wavelength.
So what should we do with our circuit? Last time we talked about the length of lines without touching on other parameters. It's time to correct this misunderstanding.
Imagine that a generator (or output stage, for example) pumps power into the line. There is no reflected wave (yet), our generator does not know at all what is on the other side of the line, it pumps to nowhere. It’s as if we take a speaker, bring it to a pipe, and sound waves go into the pipe.

The parameters of such a system can be defined in different ways. It is possible to determine (though it is not yet clear how) current and voltage. And you can determine the power (the product of current and voltage) and the ratio of current to voltage in the line. The last value has the meaning of resistance. That's what they call it - wave resistance. And this value for a specific line (and at a specific frequency, to be precise) is always the same and does not depend on the generator.
If you take an infinite line with some given Z (this is how characteristic impedance is usually denoted) and connect your multimeter to it, it will show this resistance. Although, it would seem, just a couple of wires. But if the pair is finite, as is usually the case in our lives, a reflection will appear at the end of the line, a standing wave. Therefore, your multimeter will show infinite resistance (this will, in principle, be an antinode).

So, a wave runs along the line. The characteristic impedance of the line does not change (they say that the line is regular), the voltage to current ratio is the same. And now - bang! — the line resistance makes a jump.

Since further on the relationship between current and voltage will be different, the “extra” or missing current at the jump point forms a reflected wave. For a more detailed understanding of the process, it would be nice to write down telegraph equations for the point, but first, it is enough to remember that
When reflected from XX, the phase does not change
When reflected from a short circuit, the phase is reversed by 180°

Well, it remains to say about connecting the line to the load. In principle, the load can be considered as an infinite line with a characteristic impedance equal to the load resistance. The previous example with a multimeter, I think, shows this very clearly for those who stocked up on endless wire at the beginning of the post. So if the load resistance is equal to the line resistance, the system is matched, nothing is reflected, the SWR is one. Well, if the resistances are different, all the above arguments about reflection are valid.
Actually, last time we looked at short circuit and short circuit, these things can be looked at as loads with zero or infinite resistance.

Using re-reflections on impedance jumps and lines with different impedances, you can get many different things in the microwave. We need to talk about the Smith chart and complex wave impedance, this is not today. I'll give just a couple of examples:
1. If a line segment is half a wavelength long, its characteristic impedance is not important. The characteristic impedance at the input is equal to the characteristic impedance at the output.

2. For a quarter-wave segment with characteristic impedance of line Z, the characteristic impedance at the input is calculated by the formula

This way you can match lines with different characteristic impedances in a narrow range (in which one-three-five-... quarters of the wavelength corresponds to the length of the loop)

Now let's take a closer look at the transmission line.