Phase Velocity Calculator
Find the phase velocity of a signal on a transmission line from its per-unit-length inductance and capacitance, or solve backwards for L or C — with step-by-step math, a live curve, and line-delay metrics.
L ↑ or C ↑ → slower wave
The wave slows as the line stores more magnetic energy (L) or electric energy (C) per metre. vₚ falls with the square root of the product, so quadrupling L alone halves vₚ.
Dielectric loading
A dielectric raises C but leaves L almost unchanged. Velocity factor ≈ 1/√εᵣ, which is why coax with a solid polyethylene core runs near 0.66c.
Impedance is the ratio, speed the product
Z₀ = √(L/C) depends on the ratio of L to C, while vₚ depends on their product. Two lines can share one speed but have very different impedances.
The vacuum limit
For a line in vacuum, L·C = μ₀ε₀ and vₚ = c. A passive line with L·C below that has no physical meaning, and the tool warns you.
How to Use This Calculator
Phase velocity links three per-length quantities: inductance L, capacitance C and speed vₚ. Pick the one to solve, then fill in the other two.
1. Choose what to solve for
Select vₚ, L or C. The formula is rearranged automatically and that field becomes read-only.
2. Enter values and units
Type a value, drag the log-scale slider, or load a preset cable. Each field has its own unit menu (nH/m to H/m, pF/m to F/m, m/s to % of c).
3. Read result and steps
The result card shows the answer in your chosen unit, engineering notation, velocity factor and the full working.
4. Go Advanced
Advanced mode adds Z₀, ε_eff, delay per metre, total delay for a line length, and a 1% sensitivity check.
Transmission Line Model
A line is modelled as many tiny sections. Each section holds a series inductance L·Δx and a shunt capacitance C·Δx. A signal passes along the chain at vₚ = 1/√(L·C), and the labels below update with your inputs.
Quick Output: Real-World Examples
Typical per-metre values for common lines, calculated live with vₚ = 1/√(L·C). Press Use to load a row into the calculator and see the full result. Always confirm with the manufacturer datasheet.
| Line | L (nH/m) | C (pF/m) | vₚ (×10⁸ m/s) | Velocity factor | Z₀ (Ω) | Delay (ns/m) |
|---|
Frequently Asked Questions
They are per-unit-length values: inductance in henries per metre and capacitance in farads per metre. If you use total L and C of a lumped section, the result is not a velocity.
On a lossless line, yes. On dispersive media phase velocity can differ from group velocity, which is the speed at which information travels.
The product L·C has a floor of μ₀ε₀ for ordinary materials, so vₚ is capped at c. Results above c indicate inputs that do not describe a physical passive line.
Divide vₚ by c (299,792,458 m/s). The calculator shows it for you under every result.
Calculating...
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| v_p | Phase velocity (speed of propagation) | m/s |
| L | Inductance per unit length (distributed) | H/m |
| C | Capacitance per unit length (distributed) | F/m |
| VF | Velocity factor = v_p / c (ratio to speed of light, optional) | dimensionless |
What it means
The phase velocity v_p is the speed at which the phase of a sinusoidal wave propagates along a transmission line. It is given by v_p = 1/√(LC), where L and C are the distributed inductance and capacitance per unit length. For a lossless line, the phase velocity is constant and independent of frequency. The phase velocity determines the wavelength on the line: λ = v_p / f. In a coaxial cable, the phase velocity is typically about 2/3 of the speed of light due to the dielectric constant. Example: For a line with L=0.5µH/m and C=0.2nF/m, v_p = 1/√(0.5e-6 * 0.2e-9) = 1/√(1e-16) = 1×10⁸ m/s. The wavelength at 100MHz is λ = 1e8 / 1e8 = 1m.
Worked example
Phase Velocity – Practical Example
Real‑World| Parameter | Value |
|---|---|
| L | 0.25 µH/m |
| C | 100 pF/m |
| Formula | vp = 1 / √(LC) |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
Phase velocity v_p = 1/√(LC) is the speed of a sinusoidal wave along a transmission line. It determines the time delay of signals. Engineers use it to compute propagation delays, to design phase shifters, and to analyse dispersion. This velocity is less than or equal to the speed of light, depending on the dielectric. It is essential for high‑speed digital and RF design.
- Propagation delay calculation in interconnects
- Phase shifter and delay line design
- Transmission line length determination for matching
- Signal integrity analysis
- Educational understanding of wave speed
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