Electrical Length vs. Physical Length: A Clear Explanation

This article explains the difference between electrical length and physical length, concepts crucial in RF and microwave circuit design. Understanding these lengths is essential when calculating the physical length needed for a specific electrical length, particularly when working with dielectric substrates. While RF calculators can simplify the process, it’s important to grasp the underlying equations.

Electrical Length

Electrical length represents the phase shift a signal experiences as it travels along a cable or transmission line. It’s often expressed in degrees, which is particularly useful when working with Smith charts for impedance matching. Determining electrical length requires considering the properties of the dielectric material used in the cable.

The core principle involves calculating the full wavelength (λ) corresponding to a 360-degree phase shift. This is achieved using the following equations:

λ = (1 / frequency) *(Vf * 3 x 10⁸) …(Equation 1)

Where:

• Vf = Velocity of propagation = 1 / √(εr)
• εr = Dielectric constant of the material

The relationship between electrical length and physical length is then:

(Phase shift in degrees / 360) * λ = cable physical length …(Equation 2)

Here, “Phase shift in degrees” represents the desired electrical length. A simplified version for calculating wavelength:

λ = c / frequency, where c = 3 x 10⁸ m/sec (speed of light).

Physical Length

Physical length, as the name suggests, is the actual measured length of the cable or transmission line. Let’s illustrate how to calculate physical length with an example:

Example:

• Electrical length = 45 degrees
• Frequency = 500 MHz
• PTFE dielectric with Vf = 0.69

Using the formulas above, the physical length of the cable would be approximately 5.137 cm.