Coaxial Cable Characteristics and Calculations

Overview
Coaxial cables are widely used in RF systems due to their ability to minimize signal loss and electromagnetic interference. This write-up provides calculations for determining key characteristics such as impedance, capacitance, and inductance of coaxial cables.

Key Parameters
- Inner conductor radius: r1 (in mm or meters)
- Outer conductor inner radius: r2 (in mm or meters)
- Relative permittivity of the dielectric: εr

Impedance of Coaxial Cable
The characteristic impedance (Z0) of a coaxial cable is calculated using the following formula:

Z0 = (138 / √εr) * log10(r2 / r1)

Where:
- Z0: Characteristic impedance (Ohms)
- r2: Inner radius of the outer conductor
- r1: Radius of the inner conductor
- εr: Relative permittivity of the dielectric

Capacitance Per Unit Length
Capacitance per unit length (C) is given by:

C = (2 * π * ε0 * εr) / loge(r2 / r1)

Where:
- C: Capacitance per unit length (F/m)
- ε0: Permittivity of free space (8.854 × 10^-12 F/m)
- εr: Relative permittivity of the dielectric
- loge: Natural logarithm

Inductance Per Unit Length
The inductance per unit length (L) of the coaxial cable is:

L = (μ0 / 2 * π) * loge(r2 / r1)

Where:
- L: Inductance per unit length (H/m)
- μ0: Permeability of free space (4π × 10^-7 H/m)

Attenuation in Coaxial Cable
The attenuation constant (α) is determined by:

α = (R / 2 * Z0) + (G * Z0 / 2)

Where:
- α: Attenuation constant (Np/m)
- R: Resistance of the conductors (Ohms)
- G: Conductance of the dielectric (S/m)

Velocity Factor
The velocity factor (VF) is a measure of the speed of signal propagation relative to the speed of light:

VF = 1 / √εr

Where:
- VF: Velocity factor (unitless)
- εr: Relative permittivity of the dielectric

Skin Effect Resistance
The resistance due to the skin effect is calculated as:

Rs = √(π * f * μ / σ)

Where:
- Rs: Skin effect resistance (Ohms/m)
- f: Frequency (Hz)
- μ: Magnetic permeability (H/m)
- σ: Conductivity of the material (S/m)

Practical Applications
These equations are critical for designing and evaluating RF systems. By inputting parameters specific to a coaxial cable, engineers can determine the cable's behavior under various conditions.

For instance:
- Impedance mismatches can be avoided by designing cables with appropriate Z0.
- Capacitance and inductance values directly affect the transmission line's frequency response.
- The velocity factor is essential for determining signal delay in communication systems.

Use these formulas to design efficient coaxial cable systems and ensure proper RF signal transmission with minimal loss.