A.The traveling wave and its characteristicsequations (2.33) and (2.34) show that, on a line terminated in its characteristic impedance, the voltage and current decrease exponentially in amplitude by the factor e as the distance from the generator increases. This is caused by the factor line loses, which absorb energy from the wave as it travels. A second important effect is the progressively lag in phase as Z increases, as shown by the factor e= -Bx. This lag in caused by the finite time required for the wave to the travel the distance z.

The traveling wave on the line can be expressed most neatly by using the method of Sec.2.1 for obtaining instantaneous values. First express eq.(2.33) in terms of the maximum value, then multiply by the unit rotating vector e*, and finally take the real part of the resulting expression. If we take E, tobe real, this process gives use=pppppNow, we can follow the same reasoning regarding traveling waves that we used in Chap 1. Imagine that an observer is following the wave cos(wt-bx) so that he stays with a point of constant phase, for which the angle wt-bx=a constant. To find the velocity of the point, we take the time derivative of this expression and obtain.w-**but dx/dt is the velocity we desire (the phase velocity) and this is seen to bev = w/BIt should be observed that this is the velocity at which the steady state A-C wave and its accompanying electric and magnetic fields are propagated, but it is not the velocity of the electrons in the wire. The electrons may be visualized as executing an oscillatory motion as shown in Fig 2.5 (this motion is superimposed on their usual random velocity).The phase of the oscillation lags in the direction of the motion of the wave, and this gives rise to a sinusoidal distribution of charge which apparently travels along the wire, as suggested in the illustration. The portions marked A are regions of deficiency of charge , and the one marked B is a region of excess charge. The charged regions travel to the right with the phase velocity . An analogous situation is the propagation of sound , which travels at about 1,100t/sec in air at sea level . The actual motion of the air molecules , however , s only a sinusoidal one , an the velocity of an individual molecule is not comparable with the velocity of propagation of the wave. For a line with negligible losses , Eq. (2.31 ) becomes merely v=jwXXXXXXXXXXXXXXXXXXXXXXXXXXX. The velocity X/X in this case is simple XXXXX

TRANSMISSION LINES As mentioned in Sec. 1.5, the product LC for perfect conductors immersed in a losuless dielectric is a constant which depends only in the dielectric constant and the permeability of the dielectric . For air the velocity turns X X X

Fig.2.5. generations of a traveling wave by individual sinusoidal oscillations.

XXXFig.2.6. A traveling wave on a lossy line terminated in its characteristic impedance Drawn for a line of XXX wavelengths long and a total attenuation of 0.9 neper .

out to be approximately XXXX meters/sec . A solid dielectric will increase C and thus reduce the velocity . This effect is frequently expressed in terms of a velocity constant :

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX (2.39 )A low loss coaxial line with a solid state dielectric may have a velocity constant about 0.6 or 0.7. as shown by Eq (1.1), the wavelength for a given frequency is reduced by the same factor as the phase velocity.Figure 2.6 shows a traveling wave a voltage on a lossy line at three successive instant of time, as plotted from Eq (2.36). The amplitude of the diminishes exponentially by the factor xxx as it travels. The accompanying current wave will be similar in form but will be out of phase with the voltage by an amount equal to the angle of Zx, since I= E/Zx. The transmitted power will decrease down the line by the factor.XxxxxxxxxxxxxxxxxxxxxxxxFigure 2.7 the rms voltage and current along a lossy line terminated in its characteristic impedance. Drawn for a total attenuation of 0.9 neper.The instantaneous voltage at any point on the line will vary sinusoidally as the traveling wave slides past, and will have an amplitude xxxx and an rms value xxx. The phase lags progressively down the line; fpr instantance, a quarter wavelength from the generator the voltage lags by a quarter of a cycle, or 90 degree. The phase lag at any point with respect to the input is Bx radians.The wavelength lamda is equal to the distance between successive crests of the wave at any moment ;xxx ,it is equal to the change in x which makes the angle Bx increase by 2phe radians. Therefore, B lamda andXxxxxxxxxxxxxxxxxComparison with Eq.(2.37) shows that the phase velocity and wavelength are related byXxxxxxxxxxxxxxxxxxxxThis relation was first stated in chapter 1 eq (1.1). As an exercise, the student should return to the example given in the preceding section and shows that the phase velocity was 176.200 miles/sec and the wavelength 176.2 miles.The variation of rms voltage and current along a line is sketched in fig.2.7 . when the overall losses are small, the factor xx is nearly unity, and if the line is terminated in Zo, the current and voltage will be practically uniformin magnitude over the whole length. Such a line is said to be flat.XxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxFigure 2.8. polar plot of E for a line terminated in Ze. Drawn for I= 7x/8 and an attenuation of 0.8 neper per wavelength.a logarithmic spiral, for the magnitude decreases exponentially with increasing angle. At this point the student should review in his mind the process by which any one of these vectors can be translated into the corresponding instantaneous line-to-line voltage.

B. THE CONSTANT OF TWO-CONDUCTOR LINES3.1 A qualitative picture of skin effectWhen an alternating current flows in a conductor, the alternating magnetic flux within the conductor induces an emf. This emf causes the current density to decrease in the interior of the wire and to increase at the outer surface. The result, which is known as the skin effect, increases in prominence as the frequency is raised. When ferromagnetic conductors are used, the effect may be appreciable even at commercial power frequencies. At radio frequencies, the current in a wire of moderate size is concentrated in a thin skin at the surface. Analysis shows that when the cross-sectional dimensions of the conductor are much larger than the effective thickness of the skin of current, the current density varies exponentially inward from the surface. The distance in which current density decreases to 1/e of its surface. Value is called the nominal depth of penetration (e=2.718) . the name may be somewhat misleading, for there is, of course, an appreciable amount of current below this depth. The nominal depth of penetration is analysis ogous to the time constant of an exponential transient. In sec 3.2 it is shown that the nominal depth of penetration is given by relation.