Transmission Line Theory: Coaxial Cable Per Unit Length Series Resistance Derivation

When we delve into transmission line theory, particularly with coaxial cables, understanding the per unit length series resistance is crucial for analyzing signal integrity and power loss. Let's break down how we derive this resistance step by step.

Understanding Coaxial Cables

A coaxial cable consists of an inner conductor, an insulating layer, and an outer conductor. This design allows for efficient transmission of electrical signals while minimizing electromagnetic interference. The resistance we are interested in arises primarily from the conductors themselves.

Defining Series Resistance

The series resistance per unit length of a coaxial cable can be expressed as:

• R = Resistance (Ohms per meter)
• L = Length of the cable (meters)

To find the total resistance for a given length, we multiply the resistance per unit length by the length of the cable.

Deriving the Series Resistance

The series resistance of a coaxial cable can be derived using the resistivity of the materials involved. The formula for resistance is given by:

R = ρ * (L / A)

Where:

• ρ = Resistivity of the conductor material (Ohm-meters)
• A = Cross-sectional area of the conductor (square meters)

Calculating the Cross-Sectional Area

For a coaxial cable, we need to consider the inner and outer conductors separately. The cross-sectional area of the inner conductor can be calculated using the formula for the area of a circle:

A_inner = π * (r_inner^2)

For the outer conductor, the area is the annular area between the inner and outer radii:

A_outer = π * (r_outer^2 - r_inner^2)

Combining the Resistances

Since the inner and outer conductors are in parallel, we can find the equivalent resistance. The total resistance per unit length of the coaxial cable can be expressed as:

R_total = 1 / (1/R_inner + 1/R_outer)

Substituting the expressions for R_inner and R_outer into this equation gives us the total series resistance per unit length.

Final Expression

After performing the necessary calculations and simplifications, we arrive at the final expression for the series resistance per unit length of a coaxial cable:

R = (ρ_inner / A_inner) + (ρ_outer / A_outer)

This formula allows engineers to predict how much resistance will affect signal transmission over a given distance, which is essential for designing efficient communication systems.

Practical Implications

Understanding the series resistance of coaxial cables is vital for applications in telecommunications, broadcasting, and networking. High resistance can lead to significant power loss and signal degradation, so selecting materials with low resistivity and optimizing the cable design are key factors in ensuring effective performance.

In summary, the derivation of the per unit length series resistance in coaxial cables involves understanding the geometry of the conductors, calculating their resistances, and combining them appropriately. This knowledge is fundamental for anyone working with transmission lines and communication technologies.