Basic Physical Principles - Broadband Reflectometry for Enhanced Diagnostics and Monitoring Applications - page 13

Fig. 2.2 Schematic of the
cross section of a coaxial
line

dielectric. Fig. 2.2 shows the schematization of the cross section of a typical coaxial
line. The capacitance and the inductance per unit length are

2 πε

ln a

C coax = (2.6)

L coax = 2 ln b

a

(2.7)

π

where is the dielectric

permittivity of the dielectric material. Therefore, the impedance per unit length is

μ is the permeability of the transmission line medium; and ε

L coax
C coax =

μ
ε

1
2

ln b Z coax =

a .

(2.8)

π

μ = μ 0 =

For many materials, the permeability is equal to that of free space (i.e.,

ε 0 =

10 − 7 Hm − 1 ). Additionally, considering that 4 π × ε = ε 0 ε r ,where 8 . 854 ×

10 − 12 Fm − 1 is the dielectric permittivity of free space, for practical purposes, (2.8)

becomes

√ ε r ln b

60

Z coax =

a .

(2.9)

Typically, coaxial line can support transverse electromagnetic (TEM), transverse
electric (TE), and transverse magnetic (TM) modes. In radio-frequency applications
up to a few GHz, the wave propagates in the TEM mode only. However, at frequen-
cies for which the wavelength (in the dielectric) is significantly shorter than the cir-
cumference of the cable, TE and TM modes can also propagate: when more than one
mode can exist, bends and other irregularities in the cable geometry can cause power
to be transferred from one mode to another. For many microwave applications, most
of coaxial lines are designed to work at the TEM mode [4]. Higher-order modes
are prevented from propagating, when the frequency is below the cutoff frequency,
f c − o :

c
π a + 2 √ μ r ε r

f c − o = (2.10)