Page 349 - The ROV Manual - A User Guide for Remotely Operated Vehicles 2nd edition

Secondary parameters describe two main characteristics: characteristic impedance and propaga- tion constant. Characteristic impedance is the input impedance of an infinitely long transmission line and is given by the formula:
Z0 5 OðR 1 jωLÞ=ðG 1 jωCÞ
where R is resistance of conductors in series per unit length, L is inductance per unit length, G is dielectrical conductance per unit length, C is capacitance per unit length, j is an imaginary unit, and ω is angular frequency. Or simply,
Z0 5 OL=C
for a lossless line since R and G go to zero.
If a transmission line is terminated in its characteristic impedance, its input impedance is Z0
(note: some texts reference the characteristic impedance function with the notation Zc) irrespective of its length. As an example, the characteristic impedance of a coaxial cable is typically 75Ω— hence the standard convention “75Ω Coax.”
The propagation constant (γ) determines the attenuation and the phase change of the time vary- ing sinusoidal wave traveling along the transmission line given by:
γ5α1jβ5OðR1jωLÞðG1jωCÞ
where α is attenuation of the signal having frequency ω over a unit distance, and β is the phase change in the signal when it travels a unit distance. Both α and β are called the “attenuation con- stant” and the “phase constant” of the transmission line, respectively.
Theoretically, a linear transmission line should not distort an electrical signal (Figure 13.13). However, it can cause the received signal to differ by having been multiplied by some constant and be delayed for a period pending reception. Therefore:
1. The line will introduce attenuation distortion if its attenuation constant α varies with frequency (in a linear fashion) and
2. phase distortion if the phase constant β is nonlinear with the frequency (i.e., all frequency components are delayed by the same amount).
13.2 Transmission 341
β
α
ω
FIGURE 13.13
Attenuation and phase characteristics of distortion-less transmission.