Telegrapher's equations

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The telegrapher's equations (or just telegraph equations) are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who developed the transmission line model starting with an August 1876 paper, On the Extra Current.[1]:66–67[dubious ] The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line.

The theory applies to transmission lines of all frequencies including direct current and high-frequency. Originally developed to describe telegraph wires, the theory can also be applied to radio frequency conductors, audio frequency (such as telephone lines), low frequency (such as power lines), and pulses of direct current. It can also be used to electrically model wire radio antennas as truncated single-conductor transmission lines.[2]:7–10 [3]:232

Distributed componentsEdit

 
Schematic representation of the elementary components of a transmission line.

The telegrapher's equations, like all other equations describing electrical phenomena, result from Maxwell's equations. In a more practical approach, one assumes that the conductors are composed of an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

  • The distributed resistance   of the conductors is represented by a series resistor (expressed in ohms per unit length). In practical conductors, at higher frequencies,   increases approximately proportional to the square root of frequency due to the skin effect.
  • The distributed inductance   (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (henries per unit length).
  • The capacitance   between the two conductors is represented by a shunt capacitor C (farads per unit length).
  • The conductance   of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (siemens per unit length). This resistor in the model has a resistance of   ohms.   accounts for both bulk conductivity of the dielectric and dielectric loss. If the dielectric is an ideal vacuum, then  .

The model consists of an infinite series of the infinitesimal elements shown in the figure, and that the values of the components are specified per unit length so the picture of the component can be misleading. An alternative notation is to use  ,  ,  , and   to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the characteristic impedance, the propagation constant, attenuation constant and phase constant. All these constants are constant with respect to time, voltage and current. They may be non-constant functions of frequency.

Role of different componentsEdit

 
Schematic showing a wave flowing rightward down a lossless transmission line. Black dots represent electrons, and the arrows show the electric field.

The role of the different components can be visualized based on the animation at right.

Inductance L
The inductance makes it look like the current has inertia – i.e. with a large inductance, it is difficult to increase or decrease the current flow at any given point. Large inductance L makes the wave move more slowly, just as waves travel more slowly down a heavy rope than a light string. Large inductance also increases the line's surge impedance (more voltage needed to push the same AC current through the line).
Capacitance C
The capacitance controls how much the bunched-up electrons within each conductor repel, attract, or divert the electrons in the other conductor. By deflecting some of these bunched up electrons, the speed of the wave and its strength (voltage) are both reduced. With a larger capacitance, C, there is less repulsion, because the other line (which always has the opposite charge) partly cancels out these repulsive forces within each conductor. Larger capacitance equals weaker restoring forces, making the wave move slightly slower, and also gives the transmission line a lower surge impedance (less voltage needed to push the same AC current through the line).
Resistance R
Resistance corresponds to resistance interior to the two lines, combined. That resistance R dissipates a little of the voltage along the line as heat deposited into the conductor, leaving the current unchanged. Generally, the line resistance is very low, compared to inductive reactance ωL at radio frequencies, and for simplicity is treated as if it were zero, with any voltage dissipation or wire heating accounted for as an afterthought, with slight corrections to the "lossless line" calculation deducted later, or just ignored.
Conductance G
Conductance between the lines represents how well current can "leak" from one line to the other, and higher G dissipates more current as heat, deposited in whatever serves as insulation between the two conductors. Generally, wire insulation (including air) is quite good, and the conductance is almost nothing compared to the capacitive susceptance ωC, and for simplicity is treated as if it were zero; the caveat is that materials that are good insulation at low frequencies are often "leaky" at very high frequencies.

All four parameters L, C, R, and G depend on the material used to build the cable or feedline. All four change with frequency: R, and G tend to increase for higher frequencies, and L and C tend to drop as the frequency goes up. The figure at right shows a lossless transmission line, where both R and G are zero, which is the simplest and by far most common form of the telegrapher's equations used, but slightly unrealistic (especially regarding R).

Values of primary parameters for telephone cableEdit

Representative parameter data for 24 gauge telephone polyethylene insulated cable (PIC) at 70 °F (294 K)

Frequency R L G C
Hz Ωkm Ω1000 ft µHkm µH1000 ft µSkm µS1000 ft nFkm nF1000 ft
1 Hz 172.24 52.50 612.9 186.8 0.000 0.000 51.57 15.72
1 kHz 172.28 52.51 612.5 186.7 0.072 0.022 51.57 15.72
10 kHz 172.70 52.64 609.9 185.9 0.531 0.162 51.57 15.72
100 kHz 191.63 58.41 580.7 177.0 3.327 1.197 51.57 15.72
1 MHz 463.59 141.30 506.2 154.3 29.111 8.873 51.57 15.72
2 MHz 643.14 196.03 486.2 148.2 53.205 16.217 51.57 15.72
5 MHz 999.41 304.62 467.5 142.5 118.074 35.989 51.57 15.72

More extensive tables and tables for other gauges, temperatures and types are available in Reeve.[4] Chen[5] gives the same data in a parameterized form that he states is usable up to 50 MHz.

The variation of   and   is mainly due to skin effect and proximity effect.

The constancy of the capacitance is a consequence of intentional, careful design.

The variation of G can be inferred from Terman: “The power factor ... tends to be independent of frequency, since the fraction of energy lost during each cycle ... is substantially independent of the number of cycles per second over wide frequency ranges.”[6] A function of the form   with   close to 1.0 would fit Terman’s statement. Chen [5] gives an equation of similar form. Whereas G(·) is conductivity as a function of frequency, G1, f1, and ge are all real constants.

G(·) in this table can be modeled well with

 
 
 

Usually the resistive losses grow proportionately to   and dielectric losses grow proportionately to   with   so at a high enough frequency, dielectric losses will exceed resistive losses. In practice, before that point is reached, a transmission line with a better dielectric is used. In long distance rigid coaxial cable, to get very low dielectric losses, the solid dielectric may be replaced by air with plastic spacers at intervals to keep the center conductor on axis.

The equationsEdit

The telegrapher's equations are:

 

They can be combined to get two partial differential equations, each with only one dependent variable, either   or  :

 

Except for the dependent variable (  or  ) the formulas are identical.

General solution for terminated lines of finite lengthEdit

Let

 

be the Fourier transform of the input voltage   then the general solutions for voltage and current are[citation needed]

 

and

 

with   being the Fourier transform of the input current   similar to   and

 

being the transfer function of the line,

 

the series impedance per unit length, and

 

the shunt impedance (reciprocal of the shunt admittance per unit length). The parameter   represents the total length of the line, and   locates an arbitrary intermediate position along the line.   is the impedance of the electrical termination.

With no termination (broken line),   is infinite, and the   terms vanish from the numerator and denominator of the transfer function,   If the end is perfectly grounded (shorted line),   is zero and the   terms vanish.

Remarks on notation

As in other sections of this article, the formulas can be made somewhat more compact by using the secondary parameters

 

replacing the product and ratio square root factors written out explicitly in the definition of  

The Fourier transforms used above are the symmetric versions, with the same factor of   multiplying the integrals of both the forward and inverse transforms. This is not essential; other versions of the Fourier transform can be used, with appropriate juggling of the coefficients that ensures their product remains  

Lossless transmissionEdit

When ωL >> R and ωC >> G, resistance can be neglected, and the transmission line is considered as an ideal lossless structure. In this case, the model depends only on the L and C elements. The telegrapher's equations then describe the relationship between the voltage V and the current I along the transmission line, each of which is a function of position x and time t:

 
 

The equations for lossless transmission linesEdit

The equations themselves consist of a pair of coupled, first-order, partial differential equations. The first equation shows that the induced voltage is related to the time rate-of-change of the current through the cable inductance, while the second shows, similarly, that the current drawn by the cable capacitance is related to the time rate-of-change of the voltage.

 
 

The telegrapher's equations are developed in similar forms in the following references: Kraus,[7] Hayt,[8] Marshall,[9] Sadiku,[10] Harrington,[11] Karakash,[12] and Metzger.[13]

These equations may be combined to form two exact wave equations, one for voltage V, the other for current I:

 
 

where

 

is the propagation speed of waves traveling through the transmission line. For transmission lines made of parallel perfect conductors with vacuum between them, this speed is equal to the speed of light.

Sinusoidal steady-stateEdit

In the case of sinusoidal steady-state (i.e., when a pure sinusoidal voltage is applied and transients have ceased), the voltage and current take the form of single-tone sine waves:

 
 

where   is the angular frequency of the steady-state wave. In this case, the telegrapher's equations reduce to

 
 

Likewise, the wave equations reduce to

 
 

where k is the wave number:

 

Each of these two equations is in the form of the one-dimensional Helmholtz equation.

In the lossless case, it is possible to show that

 

and

 

where   is a real quantity that may depend on frequency and   is the characteristic impedance of the transmission line, which, for a lossless line is given by

 

and   and   are arbitrary constants of integration, which are determined by the two boundary conditions (one for each end of the transmission line).

This impedance does not change along the length of the line since L and C are constant at any point on the line, provided that the cross-sectional geometry of the line remains constant.

The lossless line and distortionless line are discussed in Sadiku,[14] and Marshall,[15]

Loss-free case, general solutionEdit

In the loss-free case ( ), the most general solution of the wave equation for the voltage is the sum of a forward traveling wave and a backward traveling wave:

 

where

f1 represents the amplitude profile of a wave traveling from left to right in a positive x direction whilst f2 represents the amplitude profile of a wave traveling from right to left. It can be seen that the instantaneous voltage at any point x on the line is the sum of the voltages due to both waves.

Using the current I and voltage V relations given by the telegrapher's equations, we can write

 

Lossy transmission lineEdit

 
In the presence of losses the solution of the telegrapher's equation has both damping and dispersion, as visible when compared with the solution of a lossless wave equation.

When the loss elements  R  and  G  are not negligible, the differential equations describing the elementary segment of line are

 

By differentiating both equations with respect to x , and some algebra, we obtain a pair of hyperbolic partial differential equations each involving only one unknown:

 

These equations resemble the homogeneous wave equation with extra terms in V and I and their first derivatives. These extra terms cause the signal to decay and spread out with time and distance. If the transmission line is only slightly lossy (Rω L and Gω C), signal strength will decay over distance as eα x, where  [16](p 130)

Signal pattern examplesEdit

 
Changes of the signal level distribution along the single dimensional transmission medium. Depending on the parameters of the telegraph equation, this equation can reproduce all four patterns.

Depending on the parameters of the telegraph equation, the changes of the signal level distribution along the length of the single-dimensional transmission medium may take the shape of the simple wave, wave with decrement, or the diffusion-like pattern of the telegraph equation. The shape of the diffusion-like pattern is caused by the effect of the shunt capacitance.

AntennasEdit

Because the equations governing the flow of current in wire antennas are identical to the telegrapher's equations,[2]:7–10 [3]:232 antenna segments can be modeled as two-way, single-conductor transmission lines. The antenna is broken into multiple line segments, each segment having approximately constant primary line parameters, R, L, C, and G.[a]

At the tip of the antenna, the transmission-line impedance is essentially infinite (equivalently, the admittance is almost zero) and after a brief "pile-up" of electrons (or electron holes) at the tip, with nowhere to go, the high accumulated voltage pushes a (nearly) equal-amplitude wave away from the tip, and the reversed wave with equal voltage but sign-reversed current flows back towards the feedpoint.

The consequence of the reflected wave is that the antenna wire carries waves from the feedpoint to the tip, and then from the tip, back to the feedpoint, with a net zero current at the tip. The combination of the overlapping, oppositely-directed waves with a guaranteed current node at the tip, forms the familiar standing waves most often considered for practical antenna-building. Further, partial reflections occur within the antenna where ever there is a mismatched impedance at the junction of two or more elements, and these reflected waves also contribute to standing waves along the length of the wire(s).[2][3]

Solutions of the telegrapher's equations as circuit componentsEdit

 
Equivalent circuit of an unbalanced transmission line (such as coaxial cable) where: 2/Zₒ = trans-admittance of VCCS (Voltage Controlled Current Source), x ≡ length of transmission line, Z(s) ≡ Zₒ(s) ≡ characteristic impedance, T(s) ≡ propagation function, γ(s) ≡ propagation "constant", s ≡ j ω , j² ≡ −1 .
 
Equivalent circuit of a balanced transmission line (such as twin-lead) where: 2/Zₒ = trans-admittance of VCCS (Voltage Controlled Current Source), x ≡ length of transmission line, Z(s) ≡ Zₒ(s) ≡ characteristic impedance, T(s) ≡ propagation function, γ(s) ≡ propagation "constant", s ≡ j ω , j² ≡ −1 .

The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the top figure implements the solutions of the telegrapher's equations.[17]

The bottom circuit is derived from the top circuit by source transformations.[18] It also implements the solutions of the telegrapher's equations.

The solution of the telegrapher's equations can be expressed as an ABCD type two-port network with the following defining equations[19]

 

where

 

and

 

just as in the preceding sections. The line parameters Rω, Lω, Gω, and Cω are subscripted by ω to emphasize that they could be functions of frequency.

The ABCD type two-port gives   and   as functions of   and   The voltage and current relations are symmetrical: Both of the equations shown above, when solved for   and   as functions of   and   yield exactly the same relations, merely with subscripts "1" and "2" reversed, and the   terms' signs made negative ("1"→"2" direction is reversed "1"←"2", hence the sign change).

In the bottom circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have connections to ground not shown. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedance Zₒ(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the blocks marked "T(s)" carries the forward wave and the other carries the backward wave. The depicted circuit is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from   to   in the sense that  ,  ,   and   would be same whether this circuit or an actual transmission line was connected between   and   There is no implication that there are actually amplifiers inside the transmission line.

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which is called the shield, sheath, common, earth, or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential mode and common mode. The circuit shown in the bottom diagram only can model the differential mode.

In the top circuit, the voltage doublers, the difference amplifiers, and impedances Zₒ(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable or a microstrip line.

These are not unique: Other equivalent circuits are possible.

See alsoEdit

NotesEdit

  1. ^ Since voltage lost due to radiation is typically small compared to the voltages required due to the antenna's surge impedance, and since dry air is a very good insulator, the antenna is often modeled as lossless: R = G = 0 . The essential loss or gain of voltage due to transmission or reception is usually inserted post-hoc, after the transmission line solutions, although it can be modeled as a small value of R at the expense of working with complex numbers.

CitationsEdit

  1. ^ Hunt 1961
  2. ^ a b c Raines, Jeremy Keith (2007). Folded Unipole Antennas: Theory and applications. Electronic Engineering (1st ed.). McGraw Hill. ISBN 978-0-07-147485-6.ISBN 0-07-147485-4
  3. ^ a b c Schelkunoff, Sergei A.; Friis, Harald T. (July 1966) [1952]. Antennas: Theory and practice. John Wiley & Sons. LCCN 52-5083.
  4. ^ Reeve 1995, p. 558
  5. ^ a b Chen 2004, p. 26
  6. ^ Terman 1943, p. 112
  7. ^ Kraus 1989, pp. 380–419
  8. ^ Hayt 1989, pp. 382–392
  9. ^ Marshall 1987, pp. 359–378
  10. ^ Sadiku 1989, pp. 497–505
  11. ^ Harrington 1961, pp. 61–65
  12. ^ Karakash 1950, pp. 5–14
  13. ^ Metzger 1969, pp. 1–10
  14. ^ Sadiku 1989, pp. 501–503
  15. ^ Marshall 1987, pp. 369–372
  16. ^ Miano, Giovanni; Maffucci, Antonio (2001). Transmission Lines and Lumped Circuits. Academic Press. ISBN 0-12-189710-9. The book uses the symbol  μ  instead of  α .
  17. ^ McCammon 2010
  18. ^ Hayt 1971, pp. 73–77
  19. ^ Karakash 1950, p. 44

ReferencesEdit