Transmission line

The transmission line theory is a branch of electrical engineering. It deals with phenomena on electric wires whose length is of the order of the wavelength of the transmitted signal spectrum or above and is mainly used in telecommunications, high-frequency technology, the pulse technique and the electrical power supply at high voltage lines.

The line theory uses the model of the electrical double line and describes this by the equivalent circuit of a " infinitely short " line member whose elements are determined by the governing coverings. Without the electromagnetic field with the help of Maxwell's equations to determine itself, it derives a system of partial differential equations ( the so-called line or telegraph equations) and tries to solve them with different, the respective boundary conditions adapted, mathematical methods.

This makes it possible, the processes and wave phenomena (eg, reflections, standing waves, Negative Overshoot, resistance transformations ), qualitatively understand that occur in connection with lines to take and interpret quantitatively correct for the practical applications. Finally, rules for the use of the line arise as a component in electrical engineering, especially in the telecommunications and electronics.

• 4.1 Special case: Lossless lines
• 4.2 Special case: Sinusoidal signals
• 4.3 Special case: Distortion-free line

• 5.1 Special case: Infinitely long line
• 5.2 Special case: The wave impedance output side terminated line
• 5.3 Special case: The wave impedance input side terminated line

• 6.1 performance of a closed line on both sides
• 6.2 The high-frequency line as quadrupole

• 7.1 Special case short circuit
• 7.2 Special case idle
• 7.3 Special case λ / 4 7.3.1 Application: antenna rod

Background

Only for DC or AC low frequency can describe a line roughly with the ohmic resistance of conductor cross -sectional area, conductivity and length. Once the wavelength of the signals is in the order of the line length or fast gear changes are described on lines, this highly simplified model is not sufficient. Due to the existing on each line capacitance and inductances, the signals propagate maximum speed of light. If the spatial extent of an electrical system is so great that the term of the signals to be processed can not be neglected, occur in addition to the "normal " behavior in a special wave phenomena. The description of such systems requires mathematical procedures that include the location of the signals in the observations.

For example, a AC voltage of 1 GHz in a vacuum has a wavelength of about 30 cm. Therefore wave processes play a major role on the boards of modern computers. Due to the high frequency clocking the data is represented and transmitted by very short pulses with steep edges. Therefore, such systems would not be possible without the application of knowledge of management theory.

However, so that only the transverse electromagnetic fields ( TEM waves ) the technical director spacing must not be no larger than half the wavelength of the generated waves and the resistance per unit length is too large for the methods of classical management theory, play a role. Only then can you describe an "infinitely short" line piece by an equivalent circuit of lumped elements. If this is not the case, Maxwell's equations must be solved directly. Thus, the high and very high frequency technology deal in the theories of cavity waves and antenna systems.

History

The laying of the transatlantic submarine cables and occurring at such long cables severe distortion started in 1850 required a theoretical analysis of the processes to " long lines ". The first dealt about 1855 William Thomson with the description of the operations on lines. 1886 Oliver Heaviside put these findings into its present form as a telegraph equation, thus paving the general management theory. He was named after him Heaviside condition, was to recognize the effect that the problems of distortion were mainly caused by the high transmission capacity. Rudolf Franke looked at the line in 1891 for the first time by the means of the four-pole theory. To solve the distortion problem suggested Mihajlo Pupin in 1900 before the coil-loading of lines to artificially increase its inductance. To ensure that the implementation of long telephone lines, nothing stood in the way. 1903 looked at George Ashley Campbell line as the chain ladder. With the advent of high-frequency technology at the beginning of the 20th century it was necessary to treat even short lines with the means of transmission line theory. They were able to be considered as lossless and used ( for resistance transformation, for example ) except for signal transmission and as a component. All of these applications were based until then to sinusoidal signals and therefore could be solved using the complex AC bill. Middle of the 20th century required the pulse technique and later digital technology the unadulterated transmission of short pulses on lines. For this it was necessary, the analysis using operational calculus to perform graphically with a pulse schedule, or (if memory- free components are connected with non-linear characteristics ) with the Bergeron process. For the analysis of pulse propagation on lossy lines numerical methods have been developed.

The line equations

The conduction equations of a homogeneous linear two-wire line can be determined dx from the equivalent circuit of such a line segment of infinitesimal length shown in the following figure:

The variables contained in it are the ones with the length dx multiplied line coverings: The inductance L ', the capacitance C', the resistance per unit length R 'and the conductance per unit length G'.

Of voltage u (x, t) and current i ( x, t) in the beginning of voltage u ( x dx, t) and current i ( x dx, t) at the end of this element, the result with the aid of line mesh set and node set of the two partial differential equations of the homogeneous line ( the detailed derivation is carried out in article Telegraph equation):

The main task of leading theory is the solution of this differential equation system for the initial and boundary conditions of various practical engineering applications, and thus the determination of the gradient of voltage u ( x, t) and current i ( x, t) on the line depending on the position x and time t.

Special case: Sinusoidal signals

In many cases, the alternating current technology and classical communication technology, it is sufficient in practice to consider the voltage and current both as a purely sinusoidal (harmonic ) and power-on and transient complete. Then also occur on a line only (stationary ) to sinusoidal signals. In this particular case, the complex AC circuit analysis are applied and because the time dependence is omitted, the line equations reduce to a system of ordinary differential equations for the present on the line from position x -dependent complex amplitude U ( x) and I (x):

They are often referred to in the literature as " complex line equations ". In a linear connection of the two line ends a closed-form solution of the transmission line equations can be specified in this case.

Special case: Lossless lines

For short and / or high-frequency lines can be virtually negligible losses due to the resistance and Leitwertbelag. Thus, the transmission line equations are simplified as follows:

If one converts this ODE system to a single partial differential equation, then we obtain the classical one-dimensional wave equation.

Because then occur on the line neither damping nor distortion ( the Heaviside condition is "automatically" satisfied), the equations in many cases can easily be resolved and the solutions obtained can be interpreted particularly clear. The results of this idealization still provide the essential behavior of a line is correct and also for the "Introduction to the transmission line theory " of educational value.

Special case: Sinusoidal signals on the lossless line

By combining these two special cases we obtain the following particularly simple form of the " complex line equations ":

Since their main importance lies in the radio frequency technology, they are the starting point for the theory of high-frequency lines, the " didactic textbook example " of the transmission line theory.

The general solution of the transmission line equations

The first step of the transmission line theory to the solution of the transmission line equations is the determination of the general solution of the ODE system. This has arbitrary (integration ) constants or functions, which must then be already determined by determining the matched to the specific application, initial and boundary conditions. Since the ODE system of the transmission line equations is linear, the general solution can be determined in the general case, for example, by using the Laplace transform.

In determining the general solution one encounters significant characteristics of the line, such as the speed of propagation, the propagation constant ( propagation constant ) and the characteristic impedance ( characteristic impedance ). In general, the last two quantities are complicated operators in terms of the operational calculus. In particular, the propagation constant is (as the name implies) responsible for the propagation of the waves on the line, their speed, their loss and their deformation.

In the following special cases the general solution is, however, relatively easy to determine.

Special case: Lossless lines

With the definitions of the (real) wave impedance of lossless line

And the (constant ) phase velocity of the lossless line

We obtain the general (so-called d' Alembert's ) solution

Here, uh (t ) and ur (t ) are two time functions that have yet to be determined by the boundary conditions. The proof of correctness can line equations are performed by inserting it into the " lossless ".

This general solution may be interpreted as follows:

• The left term of the solution represents a the end of the line hinlaufende from the beginning ( of any shape ) wave
• The right-hand term represents the beginning of a line running back from the end shaft
• Both waves have the velocity v0.
• You are attenuated on the line nor distorted in shape.
• The characteristic impedance Z0 represents the ratio of voltage and current of each wave dar.

Special case: Sinusoidal signals

With the definitions of the complex impedance ( characteristic impedance )

And the complex propagation constant ( propagation constant )

Obtained according to the solution method (eg Exponentialansatz ) for linear differential equations, the general solution

This UH0 and UR0 are the two yet undetermined complex constants of integration which must be determined by the boundary conditions. The proof of correctness can be provided by inserting it into the " complex line equations ".

This general solution may be interpreted as follows:

• The left term of the solution is a the end of the line hinlaufende from the beginning sinusoidal wave ( traveling wave ) dar.
• The right-hand term represents the beginning of a line running back from the end of sinusoidal wave
• These waves are damped due to the α (generally frequency dependent ) damping constant and according to the ( always) the frequency-dependent phase constant β in the phase rotated. Because of the generally non-linear frequency dependence of the phase constant, one must distinguish between phase velocity and group velocity.
• The complex impedance ZLtg provides means that the ratio of the complex amplitudes of the voltage and current of each wave dar. contrast, by the interference of incident and reflected current wave is not constant, the ratio of total voltage U ( x ) and the total current I ( x), caused it standing waves, which are characterized by their standing wave ratio.

Special case: Distortion-free line

Oliver Heaviside already has shown that on a line, the line coverings named after him Heaviside condition

Meet, which are (despite attenuation ) is not distorted " through the line current waves " in shape. This also means that the phase velocity does not depend on the frequency and therefore no dispersion occurs, not " diverge " So adjacent frequency groups. This desirable property but usually not fulfilled by a real line because of the primarily capacitance layer. To realize nevertheless as long as possible distortion-free telephone lines, are practically the inductance artificially increased ( coil-loaded line Krarupkabel ).

The operational behavior of a line

In the second step, the transmission line theory determined from the general solution of a concrete ( particulate ) solution by the remaining degrees of freedom are eliminated by defining boundary and initial conditions.

• On the one hand a conduit is used for signal transmission, then an active (generator ) the polarized is supported at its top and is connected at its end, a passive Lastzweipol. When a line running through the shaft strikes the end or the beginning of the line, then it can be reflected there. She is generally in size and in shape ( if it is not sinusoidal) changed. The impedance of the respective financial statements and the characteristic impedance thereby determine how that happens. For a quantitative description of the reflections, the reflection factors are at the beginning and end of the line.
• On the other hand, it is possible to consider the conduit as a quadrupole and to determine from the general solution of the quadripole parameters.

Under the following specific conditions the transmission line theory has developed methods to determine closed particular solutions:

• Include both the generator and the Lastzweipol a lossless line only linear ohmic resistances, then there is a closed form solution as an infinite ( by the occurrence of multiple reflections ) series. The propagation of a single pulse edge can graphically as pulse Timetable: to be represented and calculated (English Lattice Diagram).
• Include the generator and / or the Lastzweipol a lossless line nonlinear memory free resistors, then the propagation of a single pulse edge can be determined graphically using the Bergeron process.
• Include both the generator and the Lastzweipol any linear components, then the solution is to be determined with the help of the Laplace transform (or other operational calculus ). If the line is lossless, then that is doable manually, it is lossy, however, then procedures are generally computational / numerical necessary.
• Include both the generator and the Lastzweipol a lossy line any linear components and is the generator voltage (in) sinusoidal, then the solution is to determine closed using the complex AC bill.

Special case: Infinitely long line

In this case, there are only a forward wave ( the left-hand term of the general solution). The ratio of voltage to current for each location of the line corresponding to the characteristic impedance. It follows an important "interpretation " of the characteristic impedance: The input impedance of an infinitely long line is equal to its characteristic impedance. The Generatorzweipol thus acts as a voltage divider from its internal resistance and the wave resistance.

This theoretical case is then approximately reached in practice, when the line by its attenuation is very long and so large that the reflected wave at the end of practice at the line beginning " not measurable " is.

Special case: The wave impedance output side terminated line

If a finite-length line is terminated with a passive two-terminal network whose impedance equal to the characteristic impedance of the line, then the effect is just as if the line would continue indefinitely. Therefore, a completed with the characteristic impedance (adjusted ) line behaves as an infinitely long line, in particular, there is only a forward wave and its input resistance is also equal to the characteristic impedance.

Special case: The wave impedance input side terminated line

In this case, a wave from input to output is running, there is partially reflected ( and these are generally deformed ) and running back to the entrance, where their energy in the generator internal resistance is "consumed".

Example: Sinusoidal signals on the lossless line

For the " perfect example " of the transmission line theory, the practically important high-frequency line, should be included here the complete solutions. The general solution is simplified in this case to

Wherein the propagation constant due to the lack of damping is represented only by the linear frequency dependent phase constant:

Here, λ is the wavelength on the line, which is smaller by the reduction factor when the wavelength of an electromagnetic wave of the same frequency in vacuo.

Performance of a closed line on both sides

Possess on a line of length L of the Generatorzweipol an internal (complex) impedance ZG and an open circuit voltage UG and Lastzweipol an internal (complex) impedance Z2, then is obtained from the general solution after the determination of the two constants UH0 and UR0 the final solution as a superposition of the departing and returning wave

During the derivation of this solution of the complex reflection coefficient at the output are defined as

And the complex reflection coefficient at the input

Thus, the signals on the line at any point x can be determined for each specific linear connection of the training and input. It turns out that the ratio of back and reflected power depends solely on the wiring at the output and that it is advantageous to the generalized (complex) reflection coefficient for an arbitrary point x to define the line as follows:

In practice often interested in the easy measurable variation of the amplitude and the RMS value of most existing standing waves. It is calculated relative to the departing wave at the output to

If you set the maximum and minimum of the standing wave in relationship, one obtains as an important measure of the mismatch the standing wave ratio (SWR ):

The high-frequency line as quadrupole

If you specify an alternative as boundary conditions for determining a particular solution of voltage U2 and current I2 at the end of the line before, then you I (x) obtained for the voltage U ( x) and the current on the line:

For x = 0 is used to calculate the voltage U1 and the current I1 at the beginning of the line and thus obtains the four-pole chain equations of management:

They are the basis for the use of a line piece as high-frequency technical component.

Line transformation

As line transformation is defined as the effect that can appear as complex impedance of an entirely different nature and size, in special cases, therefore, as a capacitor or coil a (complex) resistance at the end of a high-frequency generator -powered line at the beginning of the line.

Using the above-mentioned Four-pole chain equations can be ( by simple division and introduction of the wavelength λ, the line characteristic impedance ZLtg and line length LLtg ) Z1 immediately determine the input impedance of the line depending on the impedance of Lastzweipols Z2:

Which are more frequently used notations.

Based on this fundamental for the management transformation relation analyzes the transmission line theory, the transformation behavior of the line at certain cable lengths ( λ/4-Leitung, λ/2-Leitung ) and certain statements ( matched, short circuit, open circuit, real conclusion, reactance as a conclusion, a general complex statements).

Alternatively, line transformations are relatively easy to perform using the Smith chart: One turns to the normalized termination resistance in the Smith chart only in the angle

Around the point ( line length, generator frequency, relative permittivity, vacuum speed of light). The normalized input impedance can then be read directly from the Smith chart.

Special case of short circuit

For a short-circuited at the end of line ( Z2 = 0), the equation simplifies to

The ratio LLtg / λ = m determined on the basis of the sign rules of the tangent function, whether this U-shaped line as a capacitance, an inductance or a resonant circuit behaves:

• For 1/4 > m> 0, it is an inductance
• 1/4 = m, there is a parallel resonance circuit with the resonance wavelength λ; λ / 3; λ / 5; ...
• For 1/2 > m > 1/ 4, it is a capacitor.
• For 1/2 = m, there is a series resonant circuit with the resonant wavelength λ; λ / 2; λ / 4; ...
• For 3/4 > m> 1/2 there is an inductance
• ...

Note: In all equations should be used as LLtg the actual geometric line length. The influence of the velocity factor is already contained in the wavelength " on the line ".

In radio devices for very high frequencies, a movable shorting bar is used to set the desired property can with stub cables or Lecher lines. As a rule, no symmetric pair cable is used, the energy radiates and therefore only has a low quality factor, but a closed, cylindrical symmetric resonant cavity.

Special case idle

For an open end line ( Z2 → ∞ ), the equation simplifies to

The ratio LLtg / λ = m determines whether this idle line as a capacitance, an inductance or a resonant circuit behaves:

• For 1/4 > m > 0, there is a capacitance
• For 1 /4 = m, there is a series resonant circuit with the resonance wavelength λ; λ / 2; λ / 4; ...
• For 1/2 > m > 1/ 4, it is an inductor.
• For 1/2 = m, there is a parallel resonance circuit with the resonance wavelength λ; λ / 3; λ / 5; ...
• For 3/4 > m> 1/2 there is a capacity
• ...

Special case of λ / 4

A λ/4-lange line with the wave impedance Zk transformed between input and output according to the formula

If a λ/4-lange line ( coax or ribbon cable ) at the end shorted ( Ze = 0), it acts on the input as an idle (ie, high impedance, Za → ∞ ). Conversely, acts an open end λ/4-lange line at the entrance like a short circuit. The Zk is irrelevant in both cases. It is worth mentioning that the λ/4-Transformation is periodic, so also occur. This represents an extension by non-transforming λ/2-Elemente.

In the picture, a method is shown, as can be a high-frequency double line supported and grounded, although they isolated λ at the target wavelength. The inner conductor of an air-filled coaxial cable for high transmission performance can be based on comparable manner by a λ/4-Topfkreis.

If you remove the lower lever, the λ/4-lange stub acts as a selective short-circuit for very specific frequencies, while isolated at DC. Therefore the undesired propagation of RF energy defined frequency can be suppressed.

Many components of radar technology as Branch duplexer and ring couplers are based on the impedance transformation from λ/4-Leitungen.

In a waveguide, the electric energy with a probe ( rod antenna ) is coupled, which extends λ / 4 away from the closed end in the waveguide. The detaching from the probe of the coupler electromagnetic wave "sees" on three sides of the infinite resistance of the shorted λ/4-Leitung, so it can only propagate in the remaining direction.

In the microwave oven is the " door seal " a revolving belt of a λ/4-Kanal which is exactly 3 cm wide and its metal surfaces do not touch. This width matches the wavelength produced by the magnetron 12 cm. This makes it possible even without using failure-prone contacts, " imprison " the electromagnetic radiation field in the interior.

The Wilkinson divider is most easily explained as a power adder: Two transmitters or antennas, each with the source impedance Z0 provide in-phase signals to the gates P2 and P3. The following λ/4-Leitungen each with the impedance transform to 2 × Z0 at port P1, resulting by the parallel connection again gives the total impedance Z0. The resistance of 2 × Z0 between P2 and P3 has no effect on the right, as long as there incoming signals are in phase. The power -of-phase signals, it converts into heat.

Application: antenna rod

In the radio art dipole antennas are often used, the object is the impedance of the transmitter output to the wave impedance of vacuum ( 377 Ω ) transform, so that the energy can be radiated efficiently.

• If one feeds a (non- interrupted ) λ/2-Dipol at one end, to measure where an impedance Ze ≈ 2200 Ω. Because this value too much from the wave impedance of a coaxial cable (Z ≈ 50 Ω ) is different, a direct connection would lead to an unacceptable mismatch.
• In the method just described were - in fact - two λ/4-Stäbe operated electrically in parallel, so each of the two impedance Ze ≈ 4400 Ω has.

• To reduce the Dipolimpedanz, the dipole is usually separated in the middle, because there you measure the much lower value Z ≈ 70 Ω, which fits better to the cable impedance. This result can also be interpreted as a series connection of two 35 Ω resistors - each λ/4-Stab has the impedance Za = 35 Ω. Such groundplane antennas are used at medium-wave broadcasting stations, at the lower end actually this impedance Za = 35 Ω is measured when the upper end remains free.

Substituting these values ​​4400 Ω and 35 Ω on both sides of the transformation law in λ/4-Stab

A, we obtain the result Zk = 392 Ω, which is the value of the free space characteristic impedance Z0 close.

Special case λ / 2

A λ/2-lange line with the wave impedance Zk does not transform between input and output; One can not speak of a "transformation" so actually.

If a λ/2-lange line ( coax or ribbon cable ) at the end shorted ( Ze = 0), it also acts like a short circuit at the input. Conversely, acts an open end λ/2-lange line also at the entrance like an idle. If you connect as many λ/2-lange lines, you get the same result. The Zk is irrelevant in all cases.

Other branches of management theory

To address some important practical problems, the classical transmission line theory is supplemented on the basis of homogeneous double-circuit line with constant line coverings by the following sub-areas:

• Influence of the skin effect on the unit length and the wave propagation
• Refraction through rough spots on the line or to coupling of pipes
• Equivalent circuits for short lines
• Line circuits and use the line as a component
• Wave propagation on multi-core systems
• Coupled lines and directional couplers
• Crosstalk between different lines