Physics and Engineering of Radiation Detection

478 Chapter 8. Signal Processing

All coaxial cables have characteristic resistance, capacitance and inductance which
can be modeled as shown in Fig.8.2.1(b). It can be shown that the capacitance and
inductance are given by

C =

2 π

ln(b/a)Lc

(8.2.3)

L =

μ

2 π

ln

(

b

a

)

Lc (8.2.4)

whereandμare the effective permittivity and permeability of the dielectric (i.e.
insulators between central wire and outer shield) respectively.aandbare the radii
of the central wire and the shield respectively. Lc, as before, represents the total
length of the cable. These relations can be used to determine the characteristic
impedance of the cable as

Zc =

(

L

C

) 1 / 2

=

√

μ

2 π

√



ln

(

b

a

)

. (8.2.5)

The characteristic impedance of most of the commercially available standard coaxial
cables lies between 50 Ω and 200 Ω. For exampleZcfor RG58 is 50 Ω.
Another quantity of interest is the velocity of signal propagation, which becomes
important in timing applications. It can be approximated from the relation

v =

Lc

√

LC

=

1

√

μ

. (8.2.6)

Typically signal velocities are on the order of 80% to 99% of the velocity of light
in vacuum. This implies that the signal in a cable actually propagates as an elec-
tromagnetic wave. If this wave encounters an imperfection on its way, it may get
distorted or reflect back. These imperfections could be due to cable joints or termi-
nations. If the end of the cable is not connected to anything, then the wave will see
an infinite load impedance and will completely reflect back. This can also be seen
from the relation of reflection coefficient Γ defined as

Γ=

Zl−Zc

Zl+Zc

, (8.2.7)

whereZis the characteristic impedance and the subscriptslandcstand for the load
and cable respectively. If the end of the conductor is open the reflection coefficient
will be 1 (Zl=∞⇒Γ = 2). In this situation the signal will be reflected back with
the samephase. The other extreme is to short the other end of the cable. In this
case theZl=0andΓ=−1. Hence the reflected pulse will beinverted in phase.
Based on this discussion it be concluded that any impedance mismatch between two
cables connected together or the cable and the load may produce reflected pulses
and deteriorate the signal transmission. It is therefore imperative that care is taken
to ensure impedance matching at any point of cable discontinuity.