# Q. What term reveals the origin of the line

## Electric lines

Electrical devices and assemblies are connected to one another by electrical lines or cables, sometimes just wires. Colloquially everything leads to the goal, but there is also a differentiated description based on the lexicon of communications technology from SEL - Standard Elektric Lorenz AG. There is also an introduction to general line theory, where the wave resistance is treated theoretically and experimentally. Notes on wavelength and speed of propagation and the skin effect conclude the chapter.

### wire

Wire is a metallic product of small, mostly round cross-section and great length. Diameters up to 5 mm are produced by rolling processes. Smaller diameters of up to 0.01 mm are produced in the repeated drawing process using drawing dies or dies. If platinum wire is coated with silver, platinum wire diameters of a few μm can be produced in the drawing process after etching away the silver (according to Duden Lexicon).

### ladder

In connection technology, it is a metal conductor through which electrical current flows. The conductor diameter is given in millimeters and measured without the insulation.

### management

As a technical term, it is a series connection of supply circuits specified by switching instructions in technical features and switching points. In certain facilities, the line is used to transmit electrical signals.

### Trunk group

The sum of all lines between two end points is called a line bundle.

### electric wire

The cable combines several insulated conductors under a common protective sheath and is used to transmit electrical energy or communication signals.

### Core, core pair, twin core

The conductor with its insulating sleeve is called a wire. In the case of indoor cables, two wires stranded together form a circuit. A symmetrical pair of wires is the arrangement of two conductors with the same diameter and similar insulation, which have a spatially defined position to one another. The twin wire in cables is the transmission element consisting of two wires in a pair.

### Coaxial pair (KxP)

The coaxial pair is a conductor pair with an outer conductor arranged concentrically around the inner conductor. Both conductors are supported against each other by insulating material, the dielectric, at a fixed distance. This arrangement is also called a coaxial tube.

### Coaxial cable (KxK)

These are cables with one or more coaxial pairs for analog transmission of wide frequency bands (TF audio frequency or TV - television video signals) or for the transmission of digital signals at high transmission rates.

### Twisted pair

This is understood to be two-wire lines twisted in pairs.

△### General theory of leadership

The behavior of lines for signal transmission differs from that of supply engineering, where only direct current or very low-frequency alternating current is transported. For the power supply, only the ohmic conductor resistance and, in the case of several conductors, the insulation resistance are of interest.

Signals can contain a wide variety of frequencies, all of which should still be present at the end of a transmission chain. In the simplest case, a grounded signal transmission takes place via a symmetrical line or a coaxial line. The symmetrical line with one pair of wires is usually an ordinary two-wire line. It can also be equipped with special geometric properties. A ribbon cable has two identical single cores embedded in a flat insulation material of a defined width. In the twisted pair cable, two identical single cores are twisted together with a constant sense of rotation.

The main property of a coil is inductance and is created by connecting many individual conductor loops in series. A line can be understood as an unwound coil or at least as a coil with only one turn. It is therefore to be expected that a cable will have low inductive properties.

A capacitor in the simplest design is two parallel and mutually insulated metal plates or foils. Its main characteristic is capacity. In a two-wire line, two narrow metal strips are insulated against each other. A line is expected to have a low capacity.

The line has an ohmic direct current resistance. The value is measured for the entire loop as the sum of the outward and return lines. If you know the wire diameter, the total length and the specific resistance or, as a reciprocal value, the specific conductance, this resistance value can be calculated from the material data.

For analog and digital signals, each line can be written with a passive two-port (four-pole). The representation combines the mentioned properties of resistance, inductance and capacitance. It is agreed that these sizes should be evenly distributed along the entire pipe. If the manufacturing tolerance is good, they are proportional to the cable length. This type of cable is called a homogeneous cable and its properties are called coating values, which means the size in relation to a unit of length.

### The resistance coating R '

In most cases it is sufficient if the DC resistance of a wire is specified with it. In some cases, data sheets for symmetrical cables also specify the loop resistance of a pair of wires. The value of the resistance layer changes with increasing frequency. The ohmic alternating current resistance of the line increases as the current transport to the conductor surface is displaced by the skin effect. The conductor core is de-energized and the conductor cross-section through which the current is still actively flowing becomes smaller. This corresponds to a larger resistance value. The value of the resistance coating is given in Ω / km.

### The inductance coating L '

The value is precisely composed of an outer and an inner inductance coating. The external inductance coating depends on the geometric line structure and the magnetic properties of the conductor. Since no ferromagnetic conductors are normally used, this inductance coating is independent of the current flow. A much smaller inner inductance coating is based on the magnetic alternating fields in the conductor. The value decreases with increasing frequency, since the inside of the conductor becomes field-free as the skin effect increases. To a good approximation, the inductance coating can therefore be viewed as frequency-independent. Its value is given in mH / km or μH / m.

### The capacitance per unit length C '

The value is determined by the geometric structure of the line and the dielectric constant of the insulation. As with any capacitor, the smaller the distance between the wires, the larger the conductor surface and the higher the dielectric constant, the greater the value. High quality lines have smaller values for C '. The capacitance per unit length is given in nF / km or pF / m.

### The dissipation coating G '

The insulation losses and the dielectric losses of the insulation are recorded in this value. The dissipation layer depends on the frequency. Its value is given in mS / km or μS / m. The loss factor tan δ is sometimes used instead of G '. Its value is calculated as tan δ = G '/ (ω · C').

### The damping layer α

The value of alpha depends on the frequency and the insulation material. In the case of a cable that is adapted on both sides, it can be determined by measurement from the attenuation factor a in relation to the conductor length l. The damping coverage is given in Neper / km or dB / 100 m.

The equivalent circuit diagram of the line shows that each line has the character of a low-pass filter.

△### The wave resistance

The cable properties mentioned above are specified by the manufacturer in data sheets. For simple lines, the characteristic impedance Z can be derived from the geometric and physical properties. In practice, Z can be determined with an automatic L-C measuring bridge by taking two measurements at a higher frequency. The wavelength of the measuring frequency must be much larger than the length of the conductor section. One measurement takes place in idle mode with the cable end open. It behaves capacitively and delivers Zleer. The second measurement is carried out at the same frequency with a short-circuited line and inductive behavior and delivers Zshort. The wave resistance is calculated from both.

### Experimental determination of the wave resistance of a line

The measurements were carried out on a 19 m long flat double cable made of copper braid with a conductor cross-section of 1.5 mm^{2}. A type of cable that is widely used in home electronics as a connection cable for loudspeaker boxes. An adjustable sine frequency generator served as the signal source. The current and voltage were measured with electronic multimeters, the display accuracy of which is guaranteed by the manufacturer within a bandwidth of 30 kHz. A previous test series of measurements also yielded correct values for sinusoidal frequencies up to 50 kHz. The shortest wavelength of 6 km at the highest measuring frequency of 50 kHz fulfilled the measuring conditions with regard to the line length.

The ohmic loop resistance of the 19 m long line was determined to be 0.45 Ω with a DC voltage and current measurement with the end of the line short-circuited. This value could also be calculated from the conductor length, the conductor cross-section and the conductivity of copper. The capacitive line property was measured with the line end open. The table shows the values of the simultaneous current and voltage measurement at the line input. Ohm's law is used to calculate the impedance Z for each frequency, which can be equated with the capacitive reactance, since the ohmic line resistance is negligibly small.

The impedance was determined with a short-circuited line end by measuring the current and voltage at the same time. When evaluating the inductive line properties, the ohmic line resistance must be taken into account, as it is in the order of magnitude of the impedances. The table shows the measured and calculated values.

An average capacitance value of 1.2 nF and an average inductance value of 16 μH are calculated for this line. These values can be used to specify the coating resistance of the line. The insulation resistance of the line could not be measured. It is very likely that the value is greater than 10 MΩ.

With these values and the equivalent circuit for a homogeneous line, Bode diagrams were created in a simulation circuit for a 10 m long line for different output loads. For load resistance values that are smaller than the characteristic impedance of the line determined with Z = 110 Ω, the course of the attenuation corresponds to a 1st order low pass. If the load is low due to high resistance values, a resonance point appears, since L 'and C' form a series resonant circuit and the damping corresponds to a 2nd order low-pass filter. As the line length increases, the corner frequencies shift in the direction of lower frequencies.

### Reflections along a line

If alternating voltage is applied to a homogeneous line, a voltage and current wave moves with the propagation speed v = 1 / √ (LC) towards the end of the line. In the case of a homogeneous line, the voltage / current ratio is the same for each measuring point on the line and is called the characteristic impedance Z.

If the end of the line is terminated or loaded with the wave impedance, the energy supplied is completely converted at the terminating resistor. If the line termination is different from Z, a new voltage / current ratio is created at the end of the line. It can be interpreted as a new signal source that emits a wave returning to the input. Depending on the size of the terminating resistor, the reflection wave is the same or opposite to the incoming wave.

If the load resistance at the end of the line is greater than the line impedance, the voltages are added and the corresponding currents are subtracted. If the load resistance is smaller than that of Z, the situation is reversed. The reflected wave travels to the line entrance with the unchanged propagation speed v. On the line, its voltage / current ratio is equal to the line impedance. If the signal source connected to the beginning of the line has an internal resistance different from Z, there will also be a reflection at the input.

We speak of total reflection when the end of the line is open and therefore unloaded or short-circuited. In both cases the reflection wave has a maximum. At the open end of the line it is a voltage maximum and at a short circuit it is a current maximum. For all other mismatches, the amplitude of the reflection is always smaller. All reflections that occur are superimposed along the line and a steady state with standing waves is established on the line.

The magnitude of the characteristic impedance Z is calculated from the ratio of the signal voltage to the signal current. The phase angle of Z is formed from the phase difference between the two quantities. At any point on the line, the voltage / current ratio is only equal to Z if there are no reflections on the line. The signal reflection is described in more detail elsewhere in this web project using an interactive simulation.

### Reflection measurement

The impedance of this line could be determined in the laboratory by measuring the signal reflection. A 1 MHz square pulse and a duty cycle of 10% was applied to the input of the line via an input resistor. The end of the line was loaded with an adjustable terminating resistor. The reflection (blue signal) can be displayed with the oscilloscope connected to the line input. If the reflected signal is adjusted to the minimum, then the set value of the terminating resistor corresponds to the line impedance. The source signal (green) in front of the input resistor is shown as a reference. The following figure shows the measurement results for R

For the line examined above, this measurement method yielded the value 115 Ω. The experimental value corresponds to the line impedance ZL = 115.4 Ω, which follows from the coating values in the measurement tables. Here, ZL = √ (L '/ C'). Comparable values are also provided by calculations with the table values at the same frequency and the formula ZL = √ (Zkap · Zind).

### The dependence of the wave resistance on the line geometry

The following sketches show that there is a connection between the line impedance, the geometric line values and the insulating material. In the case of twisted cables, in addition to the dielectric constant of the insulating material, the number of loops per centimeter with which the twist took place also has an effect. Therefore, instead of εr, an effective value εr eff has been used in the equation.

### The characteristic impedance of a coaxial line

For a coaxial line defined as ideal, the line impedance is derived in more detail. The wave resistance is an effective resistance, but not a loss resistance. The line absorbs energy at the feed point and transports it to the end of the connected load. If the load resistance is matched to the wave resistance of the line, the power has been matched and the entire energy is completely absorbed and converted by the load resistance.

### The wave resistance of free space

If you remove all values associated with the wire line in the mathematical derivation of the equation of the wave resistance, only the natural constants remain. The wave resistance of the free space is calculated from them. It is effective in the propagation of an electromagnetic wave in space.

△### Wavelength and speed of propagation

In a vacuum, both the permittivity (dielectric constant) εr and the permeability number μr have the value 1. If the insulating material of a line does not consist of air, the permittivity is greater than 1. The permeability values of non-ferromagnetic conductors do not differ until the fourth digit after Comma. For them, the value 1 still applies to a very good approximation. If the inductance and capacitance per unit length are multiplied, the propagation speed of an electromagnetic wave in a vacuum or along a conductor is calculated from this.

Only in a vacuum are μr and εr equal to 1 and the electromagnetic wave propagates at the speed of light. The signal moves more slowly along a line with εr> 1 than in free space. The ratio of the speeds is called the shortening factor. Its value is always less than 1. The shortening factors of coaxial lines have values between 0.65 and 0.85.

For electromagnetic waves in a vacuum, the speed of light can also be used to calculate the associated wavelength for each frequency. In any medium, the speed of propagation is lower, which also changes the wavelength. In any medium, the shortening factor must be taken into account, which affects both the speed of propagation and the wavelength. The frequency remains unchanged.

△### The skin effect

If there is electrical voltage on a conductor, current flows, which generates a magnetic field in and around the conductor. In the case of direct voltage, a static state occurs shortly after switching on, in which the direct current fills the entire cross-section of the conductor.The current density J is a vector quantity and is calculated from the quotient of the current and the conductor cross-section through which it passes. In general, the current is calculated using Ohm's law. In theoretical electrical engineering, the vector quantities of current density and electric field strength are used.

If alternating current flows through a conductor, this creates alternating magnetic fields. According to the law of induction, they generate eddy currents in the conductor, which weaken the excitation current towards the center of the conductor. In the edge areas, the excitation and eddy current paths tend to run in the same direction. Starting from the surface of the conductor, the current density decreases from the maximum value of 100% or 1 towards the interior of the conductor according to an exponential function. This reduces the effective conductor cross-section and the resistance increases with increasing signal frequency. For a better overview, only one magnetic field line and one eddy current loop are shown in the sketch.

The exponent k contains the angular frequency of the signal and, along with the conductivity and permeability, the characteristic material properties of the conductor. The higher the signal frequency, the lower the current density inside the conductor. The current then only flows in a narrow area near the conductor surface, the conductor skin, while the interior remains almost current-free. The influence of the skin effect is negligibly small in the LF range and only gains importance in the HF range.

### Penetration depth / skin depth

Comparable with the time constant τ during the recharging processes on the capacitor, an equivalent conductive layer thickness or skin depth δä, English skin depth, is defined for good conductors. The non-indexed δ is usually written as a shape symbol. In relation to the conductor surface, the current density is reduced by a factor of 1 / e to 37% at a distance of the penetration depth x = δä = 1 / k.

The damping factor k can be determined with the aid of theoretical electrophysics. The equivalent conductive layer thickness or skin depth can thus be determined as shown opposite. The following table shows the skin depths for three conductors and increasing frequency. The permeability number of these conductors is μr = 1 and the field constant μo = 4 · π · 10^{−7} Vs / Am.

The DC resistance depends on the electrical conductivity, the cross-section and the conductor length. The skin effect acts at higher frequencies and leads to a greater AC resistance. If the conductor diameter is large compared to the skin depth with D> 10 · δ, the current flows evenly distributed in an annular surface of thickness δ and the interior of the conductor remains current-free. The ohmic alternating current resistance Rω is proportional to the √f dependent.

If the skin depth is large enough with δ ≥ r, the skin effect can be neglected in this frequency range and the ohmic direct current resistance RDC is the determining variable.

The following table shows the values of the frequency-dependent AC resistance Rω for two copper wires of length l = 1 m with the radius r1 = 0.5 mm and r2 = 0.05 mm. The two direct current resistances are calculated as RDC1 = 22 mΩ and RDC2 = 2.195 Ω.

The values show that RDC is the determining factor at low frequencies. By reducing the wire diameter, the skin effect only becomes effective from higher frequencies. For this reason, coils in the high frequency range of transmission and reception technology are not wound with solid wire, but with HF litz wire. It consists of many thin wires, individually insulated with varnish, which are stranded according to a certain system and have a silk wrapping as outer insulation. At the connection point, all the individual wires are recorded as being conductive, with the coil being given a low ohmic resistance by connecting the individual wires in parallel. The coil quality is therefore higher in a much larger frequency range than with a solid wire with the same RDC and is only negatively influenced by the skin effect from very high frequencies.

A so-called starting frequency can be defined for the effective start of the skin effect. It is proportional to the conductor cross-section. If the cross-section is halved, the skin effect only takes effect from twice the frequency. The frequency of use for the wire examples used above is calculated to be 17.5 kHz for the cross section 0.785 mm² and 1.7 MHz for the cross section 7.85 x 10^{−3}mm². At these frequencies, the ohmic alternating current resistance is around half as large as the associated direct current resistance.

### Proximity effect

The English word proximity means proximity. The proximity effect describes the current displacement that occurs in closely spaced conductors through which alternating current flows. The effect is particularly pronounced at high frequencies. The current flowing through one conductor A generates a magnetic field around it, which penetrates parallel conductor B and generates induction eddy currents there. Unlike the skin effect, they are not rotationally symmetrical to the center of the conductor. Illustrated in the illustration, the acting magnetic field lines generate eddy current areas perpendicular to the direction of entry, which lead to an eddy current ring in the longitudinal direction of the conductor. In the closely spaced insides of the conductors, the excitation current in conductor A and the eddy current in conductor B flow in the same direction. In the outer side of the B-conductor that is further away from conductor A, the eddy current flows in the opposite direction to the excitation current.

A corresponding process can be made for an excitation current in conductor B, which acts with its external magnetic field on conductor A and there generates an entire eddy current ring running in the direction of the conductor. If both excitation currents run in opposite directions, then the current density is greater in the area of the inside of the conductor and less in the outside area. If the direction of the excitation current is the same in both conductors, the current density is reduced at the directly adjacent edges and increases towards the outside. In both cases, the effective conductor cross-section for the current flow is reduced and the impedance increases.

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