Understanding Characteristic Impedance: Definition, Formulas, and Transmission Line Types

This page explains the basics of characteristic impedance and provides equations for calculating it for various transmission lines. These include parallel lines, coaxial lines, shielded parallel lines, striplines, and more.

When the reflected wave in a transmission line is zero, the ratio of V(z)/I(z) in the direction of propagation (z) is called the characteristic impedance. It’s denoted by the symbol Z₀.

Characteristic impedance (Z₀) is defined as:

$Z_0 = \sqrt{\frac{Z}{Y}} = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$

Where:

• Z is the impedance per unit length
• Y is the admittance per unit length
• R is the resistance per unit length
• L is the inductance per unit length
• G is the conductance per unit length
• C is the capacitance per unit length
• $\omega$ is the angular frequency ($2\pi f$)
• $j$ is the imaginary unit

Characteristic Impedance of Different Transmission Lines

Characteristic Impedance of a Lossy Transmission Line

$Z_o = \sqrt{\frac{R + j\omega L}{G + j\omega C}}$

Characteristic Impedance of a Lossless Transmission Line

$Z_o = \sqrt{\frac{L}{C}}$

Characteristic Impedance Equations for Specific Transmission Line Types

Parallel Line

$Z_o = \frac{276}{\sqrt{\epsilon_r}} * log(\frac{2S}{d})$

Where:

• S is the spacing between the two conductors.
• d is the diameter of the conductor.
• $\epsilon_r$ is the relative dielectric constant (dielectric constant) between the conductors.

Coaxial Line

$Z_o = \frac{138}{\sqrt{\epsilon_r}} * log(\frac{D}{d})$

Where:

• D is the diameter of the outer conductor.
• d is the diameter of the inner conductor.
• $\epsilon_r$ is the relative dielectric constant (dielectric constant) of the insulator between the conductors.

Shielded Parallel Line

$Z_o = \frac{276}{\sqrt{\epsilon_r}} * log (\frac{2A *(1-B^2)}{1+B^2})$

Where:

• A = s/d
• B = s/D
• s is the center-to-center spacing of the parallel conductors
• d is the diameter of the conductors
• D is the inner diameter of the shield
• $\epsilon_r$ is the relative dielectric constant (dielectric constant) of the medium between the conductors

Stripline

$Z_o = \frac{377}{\sqrt{\epsilon_r}} * (\frac{T}{W})$

Where:

• T is the thickness of the dielectric material (e.g., PCB substrate).
• W is the width of the etched line (the signal trace).
• $\epsilon_r$ is the relative dielectric constant (dielectric constant) of the dielectric material.