674 COMMUNICATION SYSTEMS

Witha=0.445 mm,b=1.473 mm,εr= 2 .26 for polyethylene dielectric, tanδ= 2 × 10 −^4 ,

andρ= 1. 63 × 10 −^8 ·m, calculate the total line attenuation at 100 MHz, and check the value of

Z ̄ 0 (=R 0 )in the specification. If the cable connects an antenna to a receiver 30 m away, determine

the time delay of the cable, the velocity of wave propagation, and the cutoff frequency.

Solution

Attenuation

∣

∣

c=

( 1. 373 × 10 −^3 )

√

1. 63 × 10 −^8 × 108

50

(

1

0. 445 × 10 −^3

+

1

1. 473 × 10 −^3

)

∼= 0 .103 dB/m

Attenuation

∣

∣

d=(^9.^096 ×^10

− (^8) )√ 2. 26 × 108 × 2 × 10 − 4
∼= 0 .00273 dB/m
Losses due to conductors obviously dominate.
Total attenuation= 0. 103 + 0. 00273 ∼= 0 .106 dB/m
Z ̄ 0 =R 0 =√^60
2. 26
ln

1. 473


2. 
   

   445
   ∼= 50 
   Time delay=τ=
   l
   vg

   
   

   30
   √

   
   


3. 26
   3 × 108
   ∼= 0. 15 μs
   The velocity of wave propagationvg=c/
   √
   εr=c/
   √


4. 26 = 0 .665 times the speed of light, or
   vg∼= 2 × 108 m/s
   Cutoff frequencyfc=
   3 × 108
   π
   √


5. 
   

   26 ( 0. 445 + 1. 473 ) 10 −^3

   
   

   3 × 1011
   π
   √

   
   


6. 26 × 1. 918
   ∼=33 GHz
   In practice,f< 0. 95 fcis usually maintained.
   EXAMPLE 15.1.2
   Unlike transmission lines, which operate at any frequency up to a cutoff value, waveguides
   have both upper and lower cutoff frequencies. For rectangular air-filled waveguides [see Figure
   15.1.2(a)], the lower cutoff frequency (for propagation by the dominant mode) is given by
   fc=c/ 2 a, wherecis the speed of light. Since the upper limit cannot be larger than 2fc, practical
   waveguides are designed withb∼=a/2 with a suggested frequency of 1. 25 fc≤f≤ 1. 9 fc. For a
   circular air-filled waveguide [see Figure 15.1.2(b)] with inside radiusa, the lower cutoff frequency
   (for propagation by the dominant mode) isfc= 0. 293 c/a. The operating band is usuallyfc<f
   <1.307fc. The characteristic impedanceZ ̄ 0 (=R 0 )in waveguides is not constant with frequency,
   as it is in transmission lines. For rectangular or circular air-filled waveguides, the expression for
   Z ̄ 0 (=R 0 )is given by
   Z ̄ 0 =R 0 =√^377
   1 −(fc/f )^2