082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

450 Chapter 13 Integration
More generally, capacitance is defined asC = Q/V, whereV is the voltage difference.
Therefore
C =
Q = 2πε r ε 0
(
V a
−V b b
ln
a)
Note that this is the capacitance per unit length of cable. Using ε r
= 1.55, a =
5.1 ×10 −4 m,b= 1.5 ×10 −3 m, weget
2π ×1.55 ×8.85 ×10−12
C = ( ) = 7.99 ×10 −11 ≈ 80 pF m −1
1.5 ×10
−3
ln
5.1 ×10 −4
Engineeringapplication13.5
Characteristicimpedanceofacoaxialcable
Acommonlyquotedparameterofacoaxialcableisitscharacteristicimpedance,Z 0
.
Thecharacteristicimpedanceistheratioofthevoltagetothecurrentforapropagating
wave travelling on an electrical transmission line in the absence of reflections.
ThevalueofZ 0
iseasytoselectatthedesignstagebycarefullychoosingthedimensionsaandbtogether
with the type of insulating material within the cable. The two
mostcommoncharacteristicimpedancesforflexiblecablesareapproximately50
and 75 . The main reason for selecting 50 is that it represents a good compromise
between the ability to handle high power and the minimization of losses that
occur in thermoplastic dielectrics. The value of 75 is mainly considered optimal
for situations of low power transmission and where losses are the most important
consideration. Often these 75 cables are of the air-dielectric type where the inner
conductor is supported by a spacer rather than a solid plastic dielectric. An example
of an application for a 75 cable is the connection from a rooftop TV antenna to a
TV set.
Itcanbeshownfromfundamentaltransmissionlinetheorythatthecharacteristic
impedanceofaloss-freecableis
√
L
Z 0
=
C
Itcan be shown thatthe expression for the inductance of a coaxial cable isgiven by
L = µ ( )
0 b
2π ln a
As shown inEngineering application 13.4, the expression for the capacitance is
C = 2πε r ε 0
( b
ln
a)