Coaxial Line Impedance Calculator

Enter the dimensions and dielectric properties to calculate the Coaxial Line Impedance.

Coaxial Line Impedance, also known as characteristic impedance (Z0), is a fundamental property of a coaxial cable that determines how radio frequency (RF) signals, video signals, or data pulses propagate along the cable. It is the ratio of the voltage to the current of a wave traveling in one direction along the transmission line when there are no reflections. This impedance is determined by the physical dimensions of the cable – specifically the inner diameter of the outer conductor (D), the outer diameter of the inner conductor (d), and the relative permittivity (εr) of the dielectric material separating them – and is ideally independent of the cable’s length and the frequency of the signal (at least for TEM mode propagation).

Anyone working with RF systems, high-frequency electronics, video distribution, or high-speed data transmission should understand and use the concept of Coaxial Line Impedance. This includes engineers, technicians, and hobbyists involved in radio communication, broadcasting, telecommunications, networking, and electronics design. Matching the impedance of the cable to the source and load impedances is crucial for efficient power transfer and minimizing signal reflections, which can cause signal loss and distortion.

Common misconceptions about Coaxial Line Impedance include thinking it is the same as the DC resistance of the cable (it’s not; DC resistance is related to conductor losses), or that it changes significantly with frequency (for ideal TEM mode, it doesn’t, although losses do increase with frequency). Another is that a 50-ohm cable has 50 ohms of resistance per foot; Z0 is a ratio, not a resistance per unit length.

Coaxial Line Impedance Formula and Mathematical Explanation

The characteristic impedance (Z0) of a coaxial line, assuming Transverse Electromagnetic (TEM) mode propagation, is derived from the inductance per unit length (L) and capacitance per unit length (C) of the cable structure: Z0 = √(L/C).

For a coaxial cable with outer conductor inner diameter D, inner conductor outer diameter d, and a dielectric with relative permittivity εr, the capacitance per unit length (C) and inductance per unit length (L) are given by:

• C = (2 * π * ε0 * εr) / ln(D/d) Farads/meter
• L = (μ0 * μr / (2 * π)) * ln(D/d) Henries/meter

Where ε0 is the permittivity of free space (≈ 8.854 x 10^-12 F/m) and μ0 is the permeability of free space (≈ 4π x 10^-7 H/m), and μr is the relative permeability of the dielectric (usually ≈1 for non-magnetic dielectrics).

Thus, Z0 = √((μ0 * μr * ln(D/d) / (2 * π)) / ((2 * π * ε0 * εr) / ln(D/d))) = (1 / (2 * π)) * √(μ0 * μr / (ε0 * εr)) * ln(D/d).

Since √(μ0 / ε0) is the impedance of free space (≈ 376.7 Ω or 120π Ω), and assuming μr ≈ 1, the formula simplifies to:

Z0 ≈ (1 / (2 * π)) * (376.7 / √εr) * ln(D/d) ≈ (59.958 / √εr) * ln(D/d) Ohms.

Using log base 10 (ln(x) ≈ 2.302585 * log10(x) and 59.958 * 2.302585 ≈ 138), this is often approximated as:

Z0 ≈ (138 / √εr) * log10(D/d) Ohms (More accurately ≈ (60 / √εr) * ln(D/d) Ohms).

Variables Explained

Variable	Meaning	Unit	Typical Range
D	Outer conductor inner diameter	mm (or any unit, as long as d is the same)	1 – 50 mm
d	Inner conductor outer diameter	mm (or any unit, as long as D is the same)	0.1 – 20 mm
εr	Dielectric relative permittivity	Dimensionless	1 (air) – 10 (some ceramics)
Z0	Characteristic Impedance	Ohms (Ω)	30 – 150 Ω

Practical Examples (Real-World Use Cases)

Example 1: Calculating Impedance for RG-58/U Cable

RG-58/U is a common 50-ohm coaxial cable. Let’s assume typical dimensions: inner conductor diameter (d) = 0.9 mm, outer conductor inner diameter (D) = 2.95 mm, and the dielectric is solid polyethylene (εr ≈ 2.25).

• D = 2.95 mm
• d = 0.9 mm
• εr = 2.25
• D/d = 2.95 / 0.9 ≈ 3.278
• √εr = √2.25 = 1.5
• ln(D/d) = ln(3.278) ≈ 1.187
• Z0 ≈ (60 / 1.5) * 1.187 = 40 * 1.187 ≈ 47.5 Ω

Using log10: log10(3.278) ≈ 0.5156, Z0 ≈ (138/1.5) * 0.5156 = 92 * 0.5156 ≈ 47.4 Ω. The actual impedance is designed to be close to 50Ω; manufacturing variations exist.

Example 2: Calculating Impedance for RG-59/U Cable

RG-59/U is often used for video and is typically 75 ohms. Let’s assume inner conductor diameter (d) = 0.58 mm, outer conductor inner diameter (D) = 3.7 mm, and polyethylene dielectric (εr ≈ 2.25).

• D = 3.7 mm
• d = 0.58 mm
• εr = 2.25
• D/d = 3.7 / 0.58 ≈ 6.379
• √εr = 1.5
• ln(D/d) = ln(6.379) ≈ 1.853
• Z0 ≈ (60 / 1.5) * 1.853 = 40 * 1.853 ≈ 74.1 Ω

This is close to the target 75Ω impedance. You can explore how changing dimensions affects the transmission line characteristics.

How to Use This Coaxial Line Impedance Calculator

1. Enter Outer Diameter (D): Input the inner diameter of the outer conductor in millimeters.
2. Enter Inner Diameter (d): Input the outer diameter of the inner conductor in millimeters. Ensure D is greater than d.
3. Enter Dielectric Constant (εr): Input the relative permittivity of the insulating material between the conductors. For air, it’s approximately 1. For materials like PTFE or Polyethylene, it’s typically around 2.1 to 2.25.
4. Calculate: The calculator will automatically update the Coaxial Line Impedance (Z0) in Ohms and other intermediate values as you type or when you click “Calculate”.
5. Read Results: The primary result is the calculated Z0. Intermediate values like D/d ratio, √εr, and log10(D/d) are also shown.
6. Use the Chart and Table: The chart visually represents how impedance changes with the D/d ratio for different dielectrics. The table provides quick lookups for common values.
7. Decision-Making: Use the calculated impedance to ensure it matches your system requirements (e.g., 50Ω for most RF systems, 75Ω for video). Mismatched impedances lead to reflections and power loss. Our guide on understanding dielectric constant can be helpful.

Key Factors That Affect Coaxial Line Impedance Results

1. Ratio of Diameters (D/d): The Coaxial Line Impedance is directly proportional to the logarithm of the ratio D/d. A larger ratio results in higher impedance.
2. Dielectric Relative Permittivity (εr): The impedance is inversely proportional to the square root of εr. A higher dielectric constant lowers the impedance.
3. Manufacturing Tolerances: Small variations in the diameters D and d, and inconsistencies in the dielectric material during manufacturing can cause the actual impedance to deviate slightly from the calculated value along the cable’s length.
4. Dielectric Material Consistency: Uniformity of the dielectric material is crucial. Voids or changes in density can alter εr and thus Z0 at different points along the cable.
5. Frequency (at very high frequencies): While the basic formula suggests impedance is frequency-independent for TEM mode, at very high frequencies (microwaves and above), the skin effect and dielectric losses become more significant, and other propagation modes can exist, slightly affecting the effective impedance and increasing losses. It’s also vital to consider the VSWR at your operating frequency.
6. Physical Deformities: Bending the cable too sharply, crushing it, or other physical damage can change the D/d ratio locally, altering the Coaxial Line Impedance at that point and causing reflections.
7. Connector Impedance Matching: The impedance of connectors used with the coaxial cable must match the cable’s impedance to avoid reflections at the connection points.

Frequently Asked Questions (FAQ)

1. What is the difference between 50 Ohm and 75 Ohm coaxial cables?

The primary difference is their characteristic impedance, determined by their physical construction (D/d ratio and dielectric). 50 Ohm cables (like RG-58, RG-213) are standard for most RF applications, radio communications, and data networks like Ethernet (older types). 75 Ohm cables (like RG-59, RG-6) are mainly used for video signals (cable TV, CCTV) and some antenna installations due to historical reasons and impedance matching with certain antenna types like dipoles.

2. Why are 50 and 75 Ohms common values for Coaxial Line Impedance?

These values emerged as compromises. 50 Ohms is near the point of maximum power handling for air dielectric cables, and 75 Ohms is close to the point of minimum attenuation for air dielectric cables and matches the impedance of a half-wave dipole antenna in free space.

3. How does the dielectric material affect Coaxial Line Impedance?

The dielectric material’s relative permittivity (εr) directly affects the impedance. Higher εr reduces impedance and also slows down the signal propagation speed in the cable. Foamed dielectrics have lower εr than solid ones, leading to higher impedance for the same D/d ratio or allowing different D/d for the same impedance, often with lower loss.

4. Can I use a 75 Ohm cable in a 50 Ohm system?

It’s generally not recommended. Mismatching impedances (connecting a 75 Ohm cable to a 50 Ohm device) causes signal reflections, leading to power loss (Return Loss) and potential signal distortion, especially at higher frequencies. Check your RF power levels and system sensitivity.

5. Does the length of the cable affect its characteristic impedance?

The characteristic impedance (Z0) itself is a property determined by the cable’s cross-sectional geometry and materials, and is ideally independent of length. However, the total attenuation (signal loss) increases with length.

6. What happens if the impedance is not uniform along the cable?

Variations in impedance along the cable length act like multiple small mismatches, causing reflections and increasing the overall signal loss and distortion. This is often due to manufacturing inconsistencies or physical damage.

7. How accurate is this Coaxial Line Impedance calculator?

This calculator uses the standard formula for TEM mode propagation, which is very accurate for most practical purposes, especially at frequencies where coaxial cables are typically used and behave ideally. For very high frequencies or non-TEM modes, more complex models might be needed.

8. What is the velocity factor of a coaxial cable?

The velocity factor (VF) is the speed at which a signal travels through the cable relative to the speed of light in a vacuum. It is given by VF = 1/√εr. A higher dielectric constant means a lower velocity factor (slower signal speed).

Related Tools and Internal Resources

• VSWR Calculator: Calculate Voltage Standing Wave Ratio based on forward and reflected power or impedance mismatch.
• Understanding Dielectric Constant: A guide explaining the importance of dielectric properties in cables and capacitors.
• RF Power Calculator: Convert between different RF power units and calculate power density.
• Transmission Line Basics: An introduction to the fundamental principles of transmission lines, including impedance.
• Coaxial Cables Selection: Browse our range of coaxial cables for various applications.
• Troubleshooting RF Systems: Tips and guides for diagnosing issues in RF setups, often related to impedance matching.