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Line Characteristics | Propagation of Electric Waves Along Transmission Lines

Line Characteristics: 

Line transmission is the theory of the propagation of electric waves along transmission lines. These transmission lines are assumed to consist of a pair of wires that are uniform throughout their whole length. When this uniformity holds good, it is immaterial, for the general theory, whether the two wires are air-spaced on telegraph poles, are two conductors in our underground cable, or form a pair in a field quad cable.

The infinite line

The propagation of elastic waves along any uniform and symmetrical transmission line may be deduced in terms of the results for a hypothetical line of infinite length having electrical constants per unit length identical to those of the line under consideration. For this reason, the propagation of electric waves along an infinite line will be considered first.

When an alternating voltage is applied to the sending end of an infinite length of line, a finite current will flow due to the capacitance and the leakage conductance between the two wires constituting the line.

The ratio of the voltage applied, to the current flowing, will give the input impedance. This input impedance is known as the “ characteristic impedance” of the line, and is denoted by Z_O.

 The characteristic impedance of any line is defined as the impedance looking into an infinite length of the line.

 Short line terminated in Zo

Consider an infinite line having input terminals 1 and 2 as in fig 2(a). The impedance looking in at terminals 1 and 2 will, by definition, be Zo.

 Suppose that a short section AB at the near end of the line is now removed [fig 2(b)], so that the line now starts at terminals 3 and 4. The impedance looking in at terminals 3 and 4 will still be Z_O, since the removal of the short section does not affect the infinite nature of the line. This means that the short section AB, from the electrical point of view, was originally terminated in impedance Z_O at B. If the short section AB is now terminated in actual impedance Z_O, the current and voltage at all points along its length will be exactly the same as if it were terminated in an infinite length of line.

Therefore, it follows that any short line terminated in Z_O behaves electrically, at all points along its length, as if it were an infinite line.

Determination of Z_O for a short line

 A short line may be considered as a complex network and can be represented as by a T section. If the short line is terminated in Z_O, it will behave as an infinite line, and have input impedance Z_O. Since the T section represents the line, it also must have input impedance Z_O.

Let the equivalent T section have series arms Z₁/2, Z₁/2 and shunt arm Z₂ as in fig.3.

Hence for a short line, Z_O can be determined if Z₁ and Z₂ can be found. This will require two equations, which may be obtained by measuring the impedance using two different terminating impedances. For convenience these termination will be taken as zero and infinity.

Let the input impedance with an infinite- impedance termination i.e. open- circuit, be Zoc. As in fig. 4(a)

Let the input impedance with a zero impedance termination i.e., short- circuit, be Zsc. As in fig 4(b).

_.Currents and Voltages along an infinite line

 Consider a current I_S applied to the sending end A of an infinite line as in fig 5(a). At the point B, at a distance of one mile down the line, let the current be I_1.

Due to the loss introduced by the line, the current I₁ will be less than I_S and also a phase-shift will be introduced. Therefore the ratio I_S / I₁ will be a vector quantity.

A convenient way of representing a vector quantity is in the form e^g, where g is a complex quantity. {g=Gamma}

Hence, let I_S / I₁= e^g.

Where g is known as the “propagation constant” per mile of the line.

Attenuation and phase constants

Attenuation and phase constants

Line constants

A practical line has a characteristic impedance, a propagation constant g an attenuation constant a , and a phase constant b . These are known as “ secondary line constants.” Although they are referred to as constants, in general, all will vary if the frequency is changed.

The “ primary line constants’ (which, for the purpose of transmission theory, are assumed to be independent of frequency) are R, G, L and C where

R is the resistance per mile of the line, measured in ohms.

G is the leakance per mile of the line, measured in mhos.

L is the inductance per mile of the line, measured in henries

C is the capacitance per mile of the line, measured in farads.

They are measured considering both conductors, i.e. per mile loop. These primary constants may be obtained by measurements on a sample of the line.

Relationship between primary and secondary line constants

 Consider a short length of line, l mile long. This short section will have a resistance Rl, a leakance Gl, an inductance Ll, and a capacitance Cl. Its characteristic impedance will be Z, the same as that of the complete line. Its propagation constant will be gl, where g is the propagation constant per mile of the complete line. This short section of line may be represented as

If the length of the section is very small, Z₁ will be approximately equal to the series impedance of the section. i.e. Rl + jwLl; and Z₂ will be approximately equal to the shunt impedance of the section, i.e. 1/Gl+jwCl.

The accuracy of the statement increases as l decreases, and in order to obtain an accurate answer it will be assumed that the section is so small that l tends to zero.

 Determination Z in terms of primary constants

It has been shown that for a T section

Determination of g in terms of primary constants