Basic Physical Principles - Broadband Reflectometry for Enhanced Diagnostics and Monitoring Applications - page 16

where Γ is the reflection coefficient and V I is the incident voltage. Γ is given by

Z L − Z 0

Γ = (2.17)

Z L + Z 0

with | Γ |≤ 1. By rearranging (2.17)

+ Γ

Z 0 1 Z L = (2.18)

1 − Γ

Generally, Γ is a complex quantity and it is often defined through magnitude ( ρ )

and angle phase ( φ ):

Γ = ρ φ . (2.19)

It is worth noting that usually both are referred to as reflection coefficient;

however, the former is a complex quantity, whereas the latter is scalar. As long
as there is no change in the impedance of the TL, the magnitudes of the incident
and reflected voltages do not change with position; therefore,

Γ and ρ

does not change
with position. On the other hand, the phase of the reflection coefficient changes as
position changes [29].
When a sinusoidal signal propagates down a transmission line and the termina-
tion of the line is not matched, the incident voltage and the reflected voltage create
a typical interference pattern. The envelope of the sinusoidal voltage will maintain
a constant shape ( standing wave ). The magnitude varies with distance, but the volt-
age at each point on the line varies sinusoidally. The voltage standing wave ratio
(VSWR) is the ratio of the maximum V max and minimum V min of the envelope:

ρ

V max
V min .

VSWR = (2.20)

The VSWR is related to the magnitude of the complex reflection coefficient by the
following equation:

1 + | Γ |

VSWR =

−| Γ | .

(2.21)

1

VSWR is greater than or equal to one: VSWR = 1whennomismatchoccurs;

VSWR → ∞ for an open or short circuit.

2.3 Characteristic Parameters of Electrical Networks

An electrical network is typically used to indicate a circuit (either single or multi-
component) or a generic TL. Generally, an electrical network is represented as a
black box with input and output terminals, and the analysis of the network is done
at the terminals. A network can be characterized through different parameters that
relate current to voltage; the most common parameters are summarized in Table 2.3.
When dealing with multi-port networks, it is particularly useful to represent network
parameters through matrices. The choice of the most suitable representation depends
not only on which the characteristics of interest are, but also on the frequency range
of analysis and, obviously, on the available measurement instrumentation.