Line equation

Under equation or line equations (short for telegraph equation or telegraph line equations , alternatively telegrapher's equations or telegraph equations ) is understood in the electrical engineering a system of coupled partial differential equations of the first order, that the spread of current and voltage on a long, straight, bifilar (two-pole) line describes .

The line theory deals with the analysis of lines by this line equations with different, the respective boundary conditions solves adapted mathematical methods.

General

If current-carrying lines (hereinafter referred to as "wires") are brought into close proximity to one another, if they are even combined in one cable, or if a line has a sufficiently large cross-section, effects become apparent that are negligibly small with thin, spatially separated lines are:

• The cores form capacitances against each other (see the analogy to the plate capacitor ).
• They show inductive properties.
• If the wires are not ideally isolated from one another , cross-conduction losses occur, i.e. a (usually undesired) current flow between the wires.

The telegraph equations can be used to describe two-wire lines, i.e. cables that have two current-carrying "wires". The simplest case of such a line is the so-called Lecher line . These are two parallel wires of finite thickness, which - insulated from each other - are run in a cable and were later used as telegraph lines. More complicated lines than the Lecher line are also used in technology, such as the coaxial line , which play a major role in many high-frequency applications and in modern measurement electronics. One wire of the coaxial line is a hollow cylinder (the so-called outer wire), along the axis of which the second, also cylindrical inner wire (also known as the "core"), separated from the outer wire by an insulator, is guided.

Historically, the line equation was largely developed by Oliver Heaviside for the analysis of problems with long telegraph lines that were laid under water (see article submarine cables ). At first glance, it may seem astonishing that these lines only had one live cable, i.e. in principle only one wire. However, since the seawater served as the return conductor, the ocean-cable system can be understood as a line with two cores and described with the telegraph line equation discussed here.

The system of equations in detail

General shape and characteristic sizes

In the case of a sufficiently straight line that propagates in the direction, the system of telegraph equations is through ${\displaystyle x}$

${\displaystyle {\begin{alignedat}{2}{\frac {\partial U(x,t)}{\partial x}}&=-L'(x){\frac {\partial I(x,t)}{\partial t}}&&\,-R'(x)I(x,t)\\{\frac {\partial I(x,t)}{\partial x}}&=-G'(x)U(x,t)&&\,-C'(x){\frac {\partial U(x,t)}{\partial t}}\end{alignedat}}}$

given. The functions , , and are generally functions of position. For the normal case of homogeneous lines, they are location-independent and thus characteristic constants of the line, the line coverings (also referred to as "primary line constants "): ${\displaystyle R'}$ ${\displaystyle L'}$ ${\displaystyle G'}$ ${\displaystyle C'}$

• R 'means the resistance coating and indicates the ohmic resistance of the cable per unit length.
• C 'means capacitance per unit length and indicates the capacity of the line per unit length.
• L 'means the inductance coating and specifies the inductance of the line per unit length.
• G 'means the discharge layer and indicates the conductance per unit length between the two current-carrying wires.

Motivation of the equations

Figure 1 : The second argument of the functions U and I, the time t, has been suppressed for the sake of clarity

The telegraph equations can be derived from the elementary laws of electrical engineering, namely the knot and mesh rules, if one imagines the entire line to be made up of smaller units of length with an internal circuit structure. However, since the derivation from a mathematical point of view cannot be described as strict, this section is entitled “Motivation of the equation” and not “Derivation”. The "inner" structure of such a line segment is shown in Figure 1: The capacitance of the line section is summarized by a capacitor of the capacitance , its ohmic resistance in a single ohmic component with the resistance value , its inductance accordingly with a coil of self-inductance . Cross-conduction losses are modeled by a cross-resistance between the two wires. This ohmic resistance with (mostly very low) conductance stands for the insulator that separates the wires of the line from one another. If the mesh rule is applied to the mesh that only contains the voltage , the coil, the ohmic resistance and the voltage , we obtain, taking into account the signs : ${\displaystyle \Delta x}$ ${\displaystyle C=C'(x)\Delta x}$ ${\displaystyle R=R'(x)\Delta x}$ ${\displaystyle L=L'(x)\Delta x}$ ${\displaystyle G=G'(x)\Delta x}$ ${\displaystyle U(x,t)}$ ${\displaystyle R}$ ${\displaystyle U(x+\Delta x,t)}$

${\displaystyle U(x+\Delta x,t)-U(x,t)+RI(x+\Delta x,t)+L{\frac {\partial I(x+\Delta x,t)}{\partial t}}=0.}$

If we now insert as well into the equation, we see: ${\displaystyle L=L'(x)\Delta x}$ ${\displaystyle R=R'(x)\Delta x}$

${\displaystyle U(x+\Delta x,t)-U(x,t)=-R'\Delta xI(x+\Delta x,t)-L'\Delta x{\frac {\partial I(x+\Delta x,t)}{\partial t}},}$

If it is sufficiently small, then: ${\displaystyle \Delta x}$

${\displaystyle U(x+\Delta x,t)=U(x,t)+{\frac {\partial U(x,t)}{\partial x}}\Delta x}$

and it results:

${\displaystyle {\frac {\partial U(x,t)}{\partial x}}\Delta x=-R'\Delta xI(x+\Delta x,t)-L'\Delta x{\frac {\partial I(x+\Delta x,t)}{\partial t}}.}$

We divide by and get: ${\displaystyle \Delta x}$

${\displaystyle {\frac {\partial U(x,t)}{\partial x}}=-R'I(x+\Delta x,t)-L'{\frac {\partial I(x+\Delta x,t)}{\partial t}}.}$

Now also applies:

${\displaystyle I(x+\Delta x,t)=I(x,t)+{\frac {\partial I(x,t)}{\partial x}}\Delta x}$

such as

${\displaystyle {\frac {\partial I(x+\Delta x,t)}{\partial t}}={\frac {\partial I}{\partial t}}(x,t)+{\frac {\partial ^{2}I(x,t)}{\partial x\partial t}}\Delta x}$

For small , that is, for , these expressions go into or . Insertion gives the equation: ${\displaystyle \Delta x}$ ${\displaystyle \Delta x\rightarrow 0}$ ${\displaystyle I(x,t)}$ ${\displaystyle \partial I(x,t)/\partial t}$

${\displaystyle {\frac {\partial U(x,t)}{\partial x}}=-R'I(x,t)-L'{\frac {\partial I(x,t)}{\partial t}},}$

which is the first of the two telegraph equations. From the knot rule we get the equation

${\displaystyle I(x+\Delta x,t)=I(x,t)-GU(x,t)-C{\frac {\partial U(x,t)}{\partial t}}.}$

Insertion of

${\displaystyle I(x+\Delta x,t)=I(x,t)+{\frac {\partial I(x,t)}{\partial x}}\Delta x}$

such as

${\displaystyle G=G'(x)\Delta x}$and after subsequent division by the second telegraph equation yields : ${\displaystyle C=C'(x)\Delta x}$ ${\displaystyle \Delta x}$

${\displaystyle {\frac {\partial I(x,t)}{\partial x}}=-G'U(x,t)-C'{\frac {\partial U(x,t)}{\partial t}}.}$

The consistency of the model could be seen as jeopardized by the two cross-connection connections, since it is purely arbitrary at which point in the circuit they are to be incorporated into the model of a “line segment”. Ultimately, these connections represent summaries of processes that are inherently continuous in nature. So we could switch the cross-conductor connection, which contains the capacitor, between the ohmic resistor and the coil . However, since the derivation does not depend on the specific position of the cross-connections (for the knot rule only the existence of the connections is necessary, the mesh rule was applied to a mesh that does not contain the two cross-connections), the model is self-consistent in this sense . ${\displaystyle R}$ ${\displaystyle L}$

Decoupling the system

In the case of constant deposits , the system of telegraph equations can be decoupled. To do this, the first equation has to be derived partially according to location , the second equation according to time . The resulting second equation can be inserted into the first because of the interchangeability of partial derivatives, and one obtains a separate differential equation for each of the current and voltage: ${\displaystyle x}$ ${\displaystyle t}$

${\displaystyle {\frac {\partial ^{2}U(x,t)}{\partial x^{2}}}-L'C'{\frac {\partial ^{2}U(x,t)}{\partial t^{2}}}=(L'G'+R'C'){\frac {\partial U(x,t)}{\partial t}}+R'G'U(x,t)}$

and

${\displaystyle {\frac {\partial ^{2}I(x,t)}{\partial x^{2}}}-L'C'{\frac {\partial ^{2}I(x,t)}{\partial t^{2}}}=(L'G'+R'C'){\frac {\partial I(x,t)}{\partial t}}+R'G'I(x,t)}$

Note, however, that these two equations are no longer equivalent to the initial system, since the formation of the partial derivative does not represent an equivalent conversion. Although every solution to the telegraph equation is also a solution to the decoupled equations, not every solution to the decoupled equations has to be a solution to the telegraph equation. Nevertheless, information can be obtained from the decoupled equations: The decoupled equations are wave equations . Since the solutions to the telegraph equation are below those of the decoupled equations, we expect wave-like voltage and current curves as solutions to the telegraph equation.

Solving the telegraph equation

In the case of constant surface areas and the operation of the line with stationary sinusoidal signals, the telegraph equation can be solved by applying plane waves :

${\displaystyle {\begin{aligned}U(x,t)&=u_{0}e^{j\omega t-\gamma x}\\I(x,t)&=i_{0}e^{j\Omega t-\Gamma x}\end{aligned}}}$

We admit that , and complex numbers. If you insert these plane waves into the telegraph equation, you can see that the solutions can only hold for arbitrary times if holds. Likewise, the solution can only apply to any if is. ${\displaystyle \gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle t}$ ${\displaystyle \omega =\Omega }$ ${\displaystyle x}$ ${\displaystyle \gamma =\Gamma }$

Furthermore, by inserting the approach into the telegraph equation , one finds that the amplitudes and solutions of the linear homogeneous system of equations ${\displaystyle u_{0}}$ ${\displaystyle i_{0}}$

${\displaystyle {\begin{aligned}-\gamma u_{0}+(L'j\omega +R')i_{0}&=0\\(C'j\omega +G')u_{0}-\gamma i_{0}&=0\end{aligned}}}$

are. However, this only has non-trivial solutions if the determinant of the coefficient matrix vanishes:

${\displaystyle \det {\begin{pmatrix}-\gamma &L'j\omega +R'\\C'j\omega +G'&-\gamma \end{pmatrix}}=0,}$

which exactly is the case when the equation ${\displaystyle \gamma }$

${\displaystyle \gamma ^{2}=(R'+j\omega L')(G'+j\omega C')\Leftrightarrow \gamma =\pm {\sqrt {(R'+j\omega L')(G'+j\omega C')}}}$

enough. Since the telegraph equation is linear , the sum of two solutions is again a solution ( superposition principle ), so that the most general solution that we can obtain from the approach of plane waves is:

${\displaystyle {\begin{aligned}U(x,t)&=u_{1}e^{j\omega t-\gamma x}+u_{2}e^{j\omega t+\gamma x}\\I(x,t)&=i_{1}e^{j\omega t-\gamma x}+i_{2}e^{j\omega t+\gamma x}\end{aligned}}}$

with . also means (complex) transfer constant or propagation constant . ${\displaystyle \gamma ={\sqrt {(R'+j\omega L')(G'+j\omega C')}}}$ ${\displaystyle \gamma }$

Characteristic impedance and reflectivity

In the case of constant coverings and lossless conduction (i.e. , ) it can be shown by insertion that a stress of the form ${\displaystyle R'=0}$ ${\displaystyle G'=0}$

${\displaystyle U(x,t)=u_{1}e^{j\omega t-\gamma x}+u_{2}e^{j\omega t+\gamma x}}$

always a stream of form

${\displaystyle I(x,t)={\frac {u_{1}}{Z_{0}}}e^{j\omega t-\gamma x}-{\frac {u_{2}}{Z_{0}}}e^{j\omega t+\gamma x}}$

With

${\displaystyle Z_{0}={\sqrt {\frac {L'}{C'}}}.}$

has the consequence. As the peak value of the current and that of the voltage , given by ${\displaystyle i_{0}}$ ${\displaystyle u_{0}}$

${\displaystyle {\begin{aligned}u_{0}&={\sqrt {u_{1}^{2}+u_{2}^{2}}}\\i_{0}&={\sqrt {\left({\frac {u_{1}}{Z_{0}}}\right)^{2}+\left({\frac {u_{2}}{Z_{0}}}\right)^{2}}},\end{aligned}}}$

about the relationship

${\displaystyle {\frac {u_{0}}{i_{0}}}=Z_{0}}$

which is very reminiscent of Ohm's law is called the wave resistance of the line. In order to understand the concept of reflectivity, let's look again at the above illustration of the voltage curve. Since the line is lossless, the complex transmission constant is simplified to the ${\displaystyle Z_{0}}$ ${\displaystyle \gamma =j\omega {\sqrt {L'C'}}}$

Figure 2 : Line, terminated with a complex resistor.

The stress curve is therefore a superposition of two plane waves, namely a wave with a wave vector (hereinafter referred to as “incoming wave”) and a wave with a wave vector (“returning wave”). The first of these two waves runs in the direction, the second wave against the direction. If the proportion of the returning wave in the total wave is attributed to the reflection of a part of the incoming wave at the end of the line, the ratio is given ${\displaystyle k_{1}=+\omega {\sqrt {L'C'}}}$ ${\displaystyle k_{2}=-\omega {\sqrt {L'C'}}}$ ${\displaystyle x}$ ${\displaystyle x}$

${\displaystyle r={\frac {u_{2}}{u_{1}}}}$

just the fraction of the incoming wave that was reflected at the end of the line. It is therefore called the reflectivity of the line. It should be noted that the amplitude of the wave with wave vector , i.e. the amplitude of the incoming wave, represents the amplitude of the wave with wave vector , i.e. the returning wave. It can now be shown that the reflectivity of a line terminated with a complex resistor (Figure 2) is about ${\displaystyle r}$ ${\displaystyle u_{1}}$ ${\displaystyle +\omega {\sqrt {L'C'}}}$ ${\displaystyle u_{2}}$ ${\displaystyle -\omega {\sqrt {L'C'}}}$ ${\displaystyle Z}$

${\displaystyle r={\frac {Z-Z_{0}}{Z+Z_{0}}}}$

can be calculated. We obtain the limit case of an open line from the evaluation of the limit value for : It results that the entire wave is reflected (without a phase jump). The short-circuited line corresponds to the case . This results in a reflectivity of −1, so the entire wave is reflected, but there is a phase jump of 180 °. ${\displaystyle r}$ ${\displaystyle \left|Z\right|\rightarrow \infty }$ ${\displaystyle r=1}$ ${\displaystyle Z=0}$

Web links

• Klaus Wille: Telegraph equation ( Memento from January 24, 2014 in the Internet Archive ) (PDF; 1.6 MB), script for the lecture "ELECTRONICS 2013", Technical University of Dortmund, April 15, 2013, pp. 58–60.

Individual evidence

1. ↑ Ernst Weber and Frederik Nebeker, The Evolution of Electrical Engineering , IEEE Press, Piscataway, New Jersey USA, 1994 ISBN 0-7803-1066-7