Final answer:
The characteristic impedance of a coaxial cable is derived using the relationship Z0 = √(L/C), with L as inductance and C as capacitance per unit length. Substituting in the formulas for L and C, Z0 simplifies to Z0 = √(μ0ε) / 2π, where μ0 is the permeability of free space and ε is the permittivity of the insulating material.
Step-by-step explanation:
To derive the characteristic impedance of a coaxial cable, we consider both its inductance and capacitance. The characteristic impedance (Z0) can be determined using the primary relationship Z0 = √(L/C), where L is the inductance and C is the capacitance per unit length of the cable.
The inductance per unit length (L) of a coaxial cable can be found using the equation L = (μ0/2π) ln(ro/ri), where μ0 is the permeability of free space, ro is the outer radius, and ri is the inner radius of the cable. The capacitance per unit length (C) is given by C = (2πε) / ln(ro/ri), where ε is the permittivity of the insulating material between the conductors.
Substituting the expressions for L and C into the characteristic impedance formula, we obtain Z0 = √((μ0/2πε) ln(ro/ri)/(1/ln(ro/ri))). Simplifying this, Z0 simplifies to Z0 = √(μ0ε) / 2π. This characteristic impedance of a coaxial cable is an important parameter as it determines how signals will propagate along the cable and is critical for ensuring maximum power transfer and minimizing reflection.