Coaxial Line Calculator

Coaxial Line Parameters Table

Key parameters for coaxial line calculations.

Parameter	Symbol	Value	Unit	Description
Inner Conductor Diameter	d	—	mm	Diameter of the center conductor.
Outer Conductor Diameter	D	—	mm	Inner diameter of the outer shield.
Dielectric Constant	εr	—	–	Relative permittivity of the insulator.
Frequency	f	—	MHz	Operating frequency.
Conductor Resistivity	ρ	—	Ohm-m	Electrical resistivity of conductor material.
Dielectric Loss Tangent	tan δ	—	–	Dissipation factor of the dielectric.
Characteristic Impedance	Zo	—	Ohms	The ratio of voltage to current on a lossless line.
Attenuation	α	—	dB/m	Signal loss per unit length.
Velocity Factor	VF	—	–	Ratio of signal speed in cable to speed of light.
Skin Depth	δ	—	µm	Depth at which current density drops to 1/e.

Attenuation vs. Frequency Chart

Typical attenuation characteristics across a range of frequencies.

The world of radio frequency (RF) and microwave engineering relies heavily on specialized transmission lines to carry signals efficiently. Among these, coaxial cable stands out as a workhorse, ubiquitous in applications ranging from cable television and internet to professional broadcasting and radar systems. A coaxial line calculator is an indispensable tool for engineers and technicians working with these systems. It allows for the precise determination of critical electrical properties that dictate signal integrity and system performance.

What is a Coaxial Line?

A coaxial line, commonly known as coaxial cable, is a type of electrical cable that features an inner conductor surrounded by an insulating dielectric material, which is in turn enclosed by a braided or solid conductive shield. The outer shield is then covered by an insulating jacket. The defining characteristic of a coaxial line is that its electrical properties are largely independent of external factors, provided the conductors remain perfectly centered. This design is crucial for transmitting high-frequency signals with minimal loss and interference.

Who should use it: RF engineers, microwave engineers, electrical engineers, system designers, ham radio operators, audiovisual technicians, and anyone involved in designing or troubleshooting systems that use RF transmission lines. Understanding the parameters calculated by a coaxial line calculator is key to ensuring optimal signal transmission.

Common misconceptions: One common misconception is that all coaxial cables are the same and offer similar performance. In reality, variations in dielectric material, conductor diameters, and shielding affect impedance, attenuation, and power handling. Another misconception is that coaxial cable is inherently loss-less; all practical coaxial lines exhibit some degree of signal loss (attenuation), especially at higher frequencies.

Coaxial Line Formula and Mathematical Explanation

The performance of a coaxial line is defined by several key parameters, each derived from its physical dimensions and the material properties of its components. A robust coaxial line calculator will compute these values accurately.

Characteristic Impedance (Zo)

Characteristic impedance is perhaps the most critical parameter for a coaxial line. It represents the ratio of voltage to current at any point along a lossless transmission line and is fundamental to matching the cable to the source and load impedances to prevent signal reflections.

The formula for the characteristic impedance of an ideal, lossless coaxial line is:

Zo ≈ (138 / sqrt(εr)) * log10(D/d)

Where:

• Zo is the characteristic impedance in Ohms.
• εr is the relative permittivity (dielectric constant) of the insulating material.
• D is the inner diameter of the outer conductor in the same units as d.
• d is the diameter of the inner conductor in the same units as D.

The constant 138 is derived from fundamental constants (like the speed of light and permeability of free space) and unit conversions. Common coax types are designed for 50 Ohms (e.g., for RF, WiFi) and 75 Ohms (e.g., for video, cable TV).

Attenuation (α)

Attenuation refers to the loss of signal power as it travels down the coaxial line. It's caused by two primary mechanisms: conductor losses (due to the finite conductivity of the inner and outer conductors) and dielectric losses (due to the imperfect insulating material). Attenuation increases with frequency.

A simplified formula for total attenuation (in dB per unit length) is:

α ≈ 4.343 * ( (Rc / Zo) * (1 + 0.5 * (Rc_outer / Rc)) + (ω * ε0 * εr * tan(δ) * Zo / 2) )

Where:

• α is the attenuation in dB per meter.
• Rc is the DC resistance of the inner conductor per unit length.
• Rc_outer is the DC resistance of the outer conductor per unit length.
• Zo is the characteristic impedance.
• ω is the angular frequency (2 * π * f).
• ε0 is the permittivity of free space (≈ 8.854 x 10^-12 F/m).
• εr is the relative permittivity of the dielectric.
• tan(δ) is the loss tangent of the dielectric.

The DC resistance of a conductor is given by ρ / Area. At RF frequencies, we use the AC resistance, which is affected by the skin effect. The term Rc / Zo relates to conductor loss, and the term involving tan(δ) relates to dielectric loss.

A more practical approximation, especially for common 50 and 75 Ohm cables where conductor losses dominate at lower frequencies, can be represented by:

α ≈ K1 * sqrt(f) + K2 * f

Where K1 accounts for conductor losses (related to skin effect, proportional to sqrt(f)) and K2 accounts for dielectric losses (proportional to f). A common approximation for attenuation in dB/m is:

α ≈ (4.343 * R_surface_resistance / Zo) * (1 + 0.5 * (R_outer_surface_resistance / R_inner_surface_resistance)) + (27.3 * εr * tan(δ) * f_GHz)

For simplicity in our calculator, we use a common approximation: Attenuation (dB/m) ≈ 4.343 * ( (SkinEffectResistance / Zo) * (1 + d/D) + (ω * ε0 * εr * tan(δ) * Zo / 2) ) Where SkinEffectResistance is the resistance per unit length at frequency f.

Velocity Factor (VF) or Propagation Velocity

The velocity factor indicates how fast an electromagnetic wave propagates through the dielectric medium of the coaxial cable compared to the speed of light in a vacuum.

VF = 1 / sqrt(εr)

The velocity of propagation (v) is then:

v = VF * c

Where 'c' is the speed of light in a vacuum (approximately 3 x 10^8 m/s). A lower dielectric constant leads to a higher velocity factor and faster signal propagation.

Skin Depth (δ)

Skin depth is a measure of how deeply an electromagnetic wave penetrates a conductor. At higher frequencies, current tends to flow only on the surface of the conductor. The skin depth determines the effective cross-sectional area for current flow, influencing the conductor's resistance.

δ ≈ sqrt(2 / (ω * μ * σ))

Where:

• δ is the skin depth in meters.
• ω is the angular frequency (2 * π * f).
• μ is the magnetic permeability of the conductor material (μ0 for non-magnetic materials like copper).
• σ is the electrical conductivity of the conductor material (1/ρ).

Note: `f` in this formula should be in Hz, so we convert from MHz. `μ` for copper is approximately 4π x 10^-7 H/m. Conductivity `σ` is the reciprocal of resistivity `ρ`.

Variable Explanations Table:

Coaxial Line Calculation Variables

Variable	Meaning	Unit	Typical Range
d	Inner conductor diameter	mm	0.1 – 10+
D	Inner diameter of outer conductor	mm	0.5 – 20+
εr	Dielectric constant	–	1.0 (Vacuum) – 10+ (Some Ceramics)
f	Frequency	MHz	1 – 100,000+
ρ	Conductor resistivity	Ohm-m	~1.68e-8 (Copper) to ~6.4e-8 (Aluminum)
tan(δ)	Dielectric loss tangent	–	0.00001 (PTFE) – 0.1+ (Some lossy dielectrics)
Zo	Characteristic Impedance	Ohms	Typically 50 or 75
α	Attenuation	dB/m	0.001 – 100+ (Highly frequency dependent)
VF	Velocity Factor	–	0.6 – 0.99
δ	Skin Depth	µm	~100 µm (1 MHz) down to <1 µm (GHz range)

Practical Examples (Real-World Use Cases)

Example 1: Designing a 50-Ohm RF Cable Link

An engineer is designing a link for a wireless communication system operating at 2 GHz. They need to choose a coaxial cable with a characteristic impedance of 50 Ohms. They are considering a cable with solid polyethylene dielectric (εr ≈ 2.25) and have measured the relevant diameters.

• Inner Conductor Diameter (d): 1.0 mm
• Outer Conductor Inner Diameter (D): 3.0 mm
• Dielectric Constant (εr): 2.25
• Frequency (f): 2000 MHz
• Conductor Resistivity (ρ): 1.68e-8 Ohm-m (Copper)
• Dielectric Loss Tangent (tan δ): 0.0002 (Typical for PE)

Using the coaxial line calculator:

• Calculated Characteristic Impedance (Zo): Approximately 50.2 Ohms (Close to target).
• Calculated Attenuation (α): Approximately 0.25 dB/m.
• Calculated Velocity Factor (VF): Approximately 0.667.
• Calculated Skin Depth (δ): Approximately 10 µm.

Interpretation: The calculated impedance is very close to the desired 50 Ohms, making this cable suitable for the application. The attenuation of 0.25 dB/m at 2 GHz indicates a moderate loss; for long cable runs, this would need to be considered. The velocity factor of 0.667 means signals travel at about 2/3 the speed of light, impacting propagation delay. The small skin depth highlights why conductor material and surface quality are important at high frequencies.

Example 2: Analyzing a 75-Ohm Cable TV Drop

A technician is troubleshooting signal issues on a cable television line and needs to understand the performance of the installed RG-6 coaxial cable. The cable uses a foam polyethylene dielectric.

• Inner Conductor Diameter (d): 1.0 mm
• Outer Conductor Inner Diameter (D): 6.9 mm
• Dielectric Constant (εr): 1.5 (Typical for foam PE)
• Frequency (f): 750 MHz (A common TV channel frequency)
• Conductor Resistivity (ρ): 1.68e-8 Ohm-m (Copper-clad steel center conductor)
• Dielectric Loss Tangent (tan δ): 0.0001 (Typical for foam PE)

Using the coaxial line calculator:

• Calculated Characteristic Impedance (Zo): Approximately 75.5 Ohms (Matches RG-6 specification).
• Calculated Attenuation (α): Approximately 0.12 dB/m.
• Calculated Velocity Factor (VF): Approximately 0.816.
• Calculated Skin Depth (δ): Approximately 18 µm.

Interpretation: The calculator confirms the cable's impedance is suitable for 75-Ohm systems like cable TV. The attenuation of 0.12 dB/m is relatively low, meaning a 10-meter cable run would lose about 1.2 dB. The higher velocity factor indicates signals travel faster than in the previous example, reducing delay. Understanding these values helps diagnose signal degradation problems which might be caused by excessive length, damaged cable, or impedance mismatches.

How to Use This Coaxial Line Calculator

Using this coaxial line calculator is straightforward and designed for ease of use by professionals and hobbyists alike.

1. Input Dimensions: Enter the physical dimensions of your coaxial cable: the diameter of the inner conductor (d) and the inner diameter of the outer conductor (D). Ensure you use consistent units (millimeters are recommended).
2. Enter Material Properties: Input the dielectric constant (εr) and the loss tangent (tan δ) of the insulating material. For conductors, enter their resistivity (ρ) in Ohm-meters.
3. Specify Frequency: Enter the operating frequency (f) in Megahertz (MHz) for which you want to calculate performance parameters.
4. Click Calculate: Press the "Calculate" button.
5. Review Results: The calculator will display the primary results: Characteristic Impedance (Zo), Attenuation (dB/m), and Velocity Factor (VF). It also shows intermediate values like skin depth. The main highlighted result typically focuses on Characteristic Impedance or Attenuation, depending on the primary application context.
6. Interpret the Output:
   • Characteristic Impedance (Zo): This value is crucial for matching. Ensure it matches your system's requirements (commonly 50 or 75 Ohms) to minimize reflections and maximize power transfer.
   • Attenuation (α): This tells you the signal loss per meter. For long runs or weak signals, you'll want lower attenuation values. High attenuation at your operating frequency might necessitate using a lower-loss cable type or a signal amplifier.
   • Velocity Factor (VF): This impacts timing and phase. It's important for high-speed digital signals or phase-sensitive RF applications.
   • Skin Depth (δ): Primarily for understanding conductor losses, especially at very high frequencies.
7. Use the Table and Chart: The table provides a detailed breakdown of all input and calculated parameters. The chart visually represents how attenuation changes with frequency, which is critical for systems operating over a range of frequencies.
8. Reset and Copy: Use the "Reset" button to clear fields and start over. Use "Copy Results" to easily transfer the calculated values and assumptions to your documentation or reports.

Decision-making guidance: When selecting or evaluating coaxial cable, prioritize achieving the correct characteristic impedance for your system. Then, consider the attenuation at your specific operating frequency. If the calculated attenuation is too high, look for cables with lower dielectric constants (like foam dielectrics), larger conductor diameters, or better shielding.

Key Factors That Affect Coaxial Line Results

Several factors significantly influence the performance metrics of a coaxial line. Understanding these helps in selecting the right cable and interpreting the calculator's output:

• Dielectric Constant (εr): This is a primary determinant of both characteristic impedance (Zo) and velocity factor (VF). A lower εr leads to a higher Zo and VF. Different dielectric materials (solid polyethylene, foam polyethylene, PTFE/Teflon) have distinct εr values, impacting cable size and performance.
• Conductor Diameters (d and D): The ratio D/d is fundamental to calculating characteristic impedance (Zo). It also influences attenuation. Generally, for a given Zo, increasing d and D proportionally (while maintaining the ratio) can reduce conductor losses by increasing conductor cross-sectional area, though this also increases cable bulk.
• Frequency (f): Frequency has a profound effect, primarily on attenuation. Conductor losses increase with the square root of frequency (due to skin effect), while dielectric losses increase linearly with frequency. This means attenuation is always higher at higher frequencies.
• Conductor Resistivity (ρ) and Conductivity (σ): This material property directly impacts conductor losses. High-conductivity materials like copper result in lower resistance and thus lower attenuation compared to materials like aluminum or even copper-clad steel, especially at lower frequencies. The calculator uses resistivity to derive the AC resistance at the specified frequency via the skin depth calculation.
• Dielectric Loss Tangent (tan δ): This parameter quantifies the energy lost within the insulating material as heat. Materials with a lower loss tangent (e.g., PTFE, low-density foams) are preferred for high-frequency, low-loss applications. Higher tan δ values significantly increase attenuation, particularly at microwave frequencies.
• Shielding Effectiveness: While not directly calculated by basic impedance/attenuation formulas, the quality and type of the outer shield (e.g., single braid, double braid, solid foil) critically affect the cable's shielding effectiveness against external electromagnetic interference (EMI) and its ability to contain its own signal (preventing radiation). Our calculator focuses on intrinsic line parameters.
• VSWR (Voltage Standing Wave Ratio): Although not a direct input, the calculator's results (especially Zo) are fundamental to achieving a low VSWR. Mismatched impedances between the source, cable, and load create standing waves, leading to reflections and effectively increased attenuation.
• Temperature: Conductor resistivity and dielectric properties can change slightly with temperature, which can subtly affect attenuation and impedance, particularly in extreme environments.

Frequently Asked Questions (FAQ)

Common Questions About Coaxial Lines

Q1: What is the difference between 50 Ohm and 75 Ohm coaxial cable?

The characteristic impedance (Zo) is the key difference. 50 Ohm cables (like RG-58, RG-8, LMR series) are optimal for data and RF power transmission, offering a good balance between impedance matching for transmitters/receivers and minimizing attenuation. 75 Ohm cables (like RG-6, RG-11) are commonly used for video and cable television distribution, offering lower attenuation per unit length at common TV frequencies compared to 50 Ohm cables of similar construction. Using the wrong impedance can cause signal reflections and performance degradation.

Q2: Why does attenuation increase with frequency?

Attenuation increases with frequency due to two main factors: the skin effect in conductors and dielectric losses. The skin effect causes current to flow on the surface of conductors, effectively increasing their resistance at higher frequencies. Dielectric losses, related to the polarization and movement of molecules within the insulator as the electric field oscillates, also become more pronounced at higher frequencies.

Q3: What does a Velocity Factor of 0.66 mean?

A Velocity Factor (VF) of 0.66 means that signals travel through the coaxial cable at 66% of the speed of light in a vacuum (c). This impacts the propagation delay (time it takes for a signal to travel a certain length), which is critical in high-speed digital systems and phase-sensitive RF applications. The VF is primarily determined by the dielectric constant (VF = 1 / sqrt(εr)).

Q4: Can I use this calculator for different types of transmission lines?

This calculator is specifically designed for coaxial line calculator parameters based on its geometry (concentric inner and outer conductors). It is not suitable for other transmission line types like microstrip, stripline, twin-lead, or wavegides, which have different physical structures and formulas.

Q5: How does the loss tangent (tan δ) affect performance?

The loss tangent directly quantifies the inefficiency of the dielectric material. A higher loss tangent means more signal energy is absorbed by the dielectric and converted into heat. This significantly increases the overall attenuation of the coaxial cable, especially at higher frequencies. Low-loss dielectrics (low tan δ) are essential for applications requiring minimal signal degradation over long distances or at very high frequencies.

Q6: What are the implications of a large skin depth?

A large skin depth implies that the current is distributed over a larger volume of the conductor. This occurs at lower frequencies. As frequency increases, skin depth decreases, meaning current is confined to a thinner layer near the conductor's surface. This reduces the effective cross-sectional area for current flow, increasing the conductor's AC resistance and thus attenuation.

Q7: How important is the conductor material (e.g., Copper vs. Aluminum)?

The conductor material's resistivity is crucial for calculating conductor losses. Copper has lower resistivity than aluminum, meaning it offers less resistance to current flow and results in lower attenuation, particularly at lower frequencies where the skin effect is less dominant. Many RF cables use copper-clad steel or aluminum for cost or weight reasons, but this typically comes with slightly higher conductor losses.

Q8: What are the limitations of the simplified formulas used in this calculator?

The formulas used provide excellent approximations for ideal or near-ideal coaxial lines. However, real-world cables can have imperfections such as:

• Non-uniform dielectric or conductor spacing.
• Rough conductor surfaces affecting AC resistance.
• Inclusion of multiple dielectric materials (e.g., foam core with solid skins).
• Losses in the outer jacket material.
• Radiation losses from imperfect shielding.

These factors can cause actual measured performance to deviate slightly from calculated values. For highly critical applications, manufacturers' data sheets and more complex electromagnetic simulation software are recommended.