Hong Life CalculatorsUnderstanding the coaxial cable impedance calculator — concept and principle
This calculator finds the characteristic impedance of a coaxial line from its geometry and insulation, along with the velocity factor and the capacitance per metre. It is used to check whether a cable or a home-made coaxial section is 50 Ω or 75 Ω, to design matching sections, and to understand cable datasheets.
How it is calculated
- Impedance: Z₀ = (138 / √εᵣ) · log₁₀(D / d), equivalent to (60 / √εᵣ) · ln(D / d).
- Velocity factor: VF = 1 / √εᵣ, the signal speed as a fraction of the speed of light.
- Capacitance: C = 2π ε₀ εᵣ / ln(D / d), with ε₀ = 8.854 × 10⁻¹² F/m, shown in pF/m.
D is the inner diameter of the shield, d the outer diameter of the centre conductor (same units, only the ratio matters), and εᵣ the relative permittivity of the insulation: about 1.0 for air and about 2.25 for solid polyethylene.
The principle
A coaxial line is a distributed chain of tiny series inductances and shunt capacitances. Per metre, the inductance is L = (μ₀ / 2π) ln(D/d) and the capacitance is C as above. A wave travelling along the line sees the impedance Z₀ = √(L / C), which depends only on the ratio D/d and on εᵣ, not on length. The wave speed is 1 / √(LC) = c / √εᵣ, which gives the velocity factor. Because Z₀ grows only logarithmically with D/d, large changes in geometry make modest changes in impedance.
Worked example
- Ratio: D / d = 7.25 / 2.0 = 3.625; log₁₀(3.625) ≈ 0.5593.
- 138 / √2.25 = 138 / 1.5 = 92.
- Z₀ = 92 × 0.5593 ≈ 51.46 Ω.
- Velocity factor: 1 / 1.5 ≈ 0.667.
- Capacitance: 2π × 8.854 × 10⁻¹² × 2.25 / ln(3.625) = 1.2517 × 10⁻¹⁰ / 1.2879 ≈ 97.19 pF/m.
| Centre conductor diameter (mm) | Impedance (Ω) |
|---|---|
| 0.5 | 106.8458562 |
| 1 | 79.1510966 |
| 1.5 | 62.95070077 |
| 2 | 51.456337 |
| 2.5 | 42.54061581 |
| 3 | 35.25594117 |
| 4 | 23.7615774 |
How to read the result
50 Ω is the standard for RF transmitters, test equipment and most antennas; 75 Ω is standard for TV, satellite and video. For an air line, attenuation is lowest near 77 Ω and power handling is highest near 30 Ω; 50 Ω is a practical compromise. With solid polyethylene, 50 Ω needs D/d ≈ 3.49 and 75 Ω needs D/d ≈ 6.5. A velocity factor of 0.66 is typical of solid PE cable; foamed dielectrics reach about 0.8 or more.
Common mistakes
- Using the outer jacket diameter for D instead of the inside of the braid.
- Using the stranded conductor's nominal size; the effective diameter is slightly smaller.
- Assuming εᵣ = 1 for foamed cable; foam is a mix of air and plastic, often 1.4 to 1.6.
- Mixing natural and base-10 logarithms (the constant 60 goes with ln, 138 with log₁₀).
Frequently asked questions
Can I connect 75 Ω cable to a 50 Ω radio?
It works with a mismatch: |Γ| = 0.2, VSWR 1.5, and about 4% of the power reflected. For receiving this is often acceptable; for high-power transmitting, use the correct cable.
Why does capacitance per metre matter?
It sets loading on audio and data lines and lets you identify cable: solid-PE 50 Ω cable is roughly 100 pF/m, 75 Ω cable roughly 67 pF/m.
Sources: Pozar, Microwave Engineering, coaxial line section · The ARRL Handbook for Radio Communications, transmission lines chapter