0195136047.pdf

15.1 WAVES, TRANSMISSION LINES, WAVEGUIDES, AND ANTENNA FUNDAMENTALS 673

and current are given by

v 2 (t)=v 1 (t−tl) (15.1.5)

i 2 (t)=i 1 (t−tl) (15.1.6)

where the delay timetlis

tl=l/vg (15.1.7)

On substituting Equations (15.1.3) and (15.1.4), we obtain

v 2 (t)=

R 0

RS+R 0

vS

(

t−

l

vg

)

(15.1.8)

i 2 (t)=

1

RS+R 0

vS

(

t−

l

vg

)

(15.1.9)

Because of the line’s behavior, the impedance at terminalscandd, which isV ̄ 2 /I ̄ 2 , is stillZ ̄ 0 (=R 0 ).

As a consequence, the line can be terminated by an impedanceZ ̄ 0 (=R 0 )at the output terminals

with no effect on voltages and currents anywhere else on the line. This result is very significant,

because if a wave is launched on a real, finite-length transmission line, the wave will dissipate

itself in the terminating load impedance if that impedance is equal to the line’s characteristic

impedance. The load is then said to bematchedto the line. In Figure 15.1.3, the source and

load resistances are shown matched so thatRS=R 0 =RL. Otherwise, a mismatch at the load

(RL=R 0 ) reflects some of the signal energy back toward the source where any mismatch (RS=

R 0 ) further reflects energy in the forward direction. Impedance matching on both ends eliminates

these undesired multiple reflections. Note that the value ofR 0 will be different for different types

of transmission lines.

EXAMPLE 15.1.1

An RG-213/U (radio guide 213/universal coaxial cable) is a small-sized, flexible, double-braided

cable with silvered-copper conductors, and a characteristic impedance of 50. The characteristic

impedanceZ ̄ 0 (=R 0 )is related to the cable’s geometrical parameters by

Z ̄ 0 =R 0 =√^60

εr

ln

b

a

whereεris the relative permittivity (dielectric constant) of the dielectric, andbandaare the radii

of the outer and inner conductors, respectively. The velocity of wave propagation in a coaxial line

isvg=c/

√

εr, wherec= 3 × 108 m/s is the velocity of light. The cutoff frequency, given by

fc=

c

π

√

εr(a+b)

Hz

puts an upper bound for wave propagation. The attenuation due to conductor losses is approxi-

mately given by

Attenuation

∣

∣

c=

( 1. 373 × 10 −^3 )

√

ρf

Z 0

(

1

a

+

1

b

)

dB/m

whereρis the resistivity of the conductors. Attenuation due to dielectric losses is given by

Attenuation

∣

∣

d=(^9.^096 ×^10

− (^8) )√ε
rftanδdB/m
where tanδis known as the loss tangent of the dielectric.