Example data table
| Case | r | D | εr | f | Z0 | C′ | L′ | Delay |
|---|---|---|---|---|---|---|---|---|
| Sample | 5 mm | 50 mm | 1.0 | 10 MHz | 274.90 Ω | 12.13 pF/m | 0.917 µH/m | 3.336 ns/m |
Formula used
This tool models a TEM two‑wire transmission line made from two parallel circular conductors with radius r and center spacing D, embedded in a homogeneous dielectric.
- A = arccosh(D / (2r))
- C′ = (π ε) / A (capacitance per unit length)
- L′ = (μ A) / π (inductance per unit length)
- Z0 = √(L′ / C′) (characteristic impedance)
- v = 1 / √(μ ε) (phase velocity)
- λ = v / f and β = 2π / λ when f > 0
Here ε = ε0 εr and μ = μ0 μr. Results assume negligible conductor and dielectric losses.
How to use this calculator
- Enter the conductor radius r and choose its unit.
- Enter the center spacing D; ensure D > 2r.
- Select a dielectric preset or type your own εr.
- Keep μr = 1 unless you have a special material.
- Add frequency to get wavelength and phase constant.
- Add line length to get total L, total C, and delay.
- Click Calculate, then download CSV or PDF if needed.
Technical article
Geometry inputs that matter
Parallel two‑wire lines are defined by conductor radius r and center spacing D. Larger spacing increases impedance and lowers capacitance. Larger radius lowers impedance and increases capacitance. The calculator converts your units to meters and checks that D is greater than 2r so the cylinders do not overlap. For the sample r = 5 mm and D = 50 mm in air, the ratio D/(2r) equals 5.
Why arccosh(D/2r) appears
The electric field of two parallel cylinders leads to a logarithmic potential relationship. The key term becomes A = arccosh(D/2r). When conductors are far apart, A grows slowly, so parameter changes become less dramatic at large D. In the sample above, A is about 2.293, which drives both C′ and L′.
Capacitance per meter C′
Capacitance is C′ = (π ε)/A, where ε = ε0 εr. Changing the dielectric preset updates εr, and C′ scales almost linearly. Doubling εr nearly doubles C′, which is why plastic insulation produces higher capacitance than air. With εr = 1, the sample gives roughly 12.13 pF/m.
Inductance per meter L′
Inductance is L′ = (μ A)/π, with μ = μ0 μr. For most non‑magnetic materials μr is about 1, so geometry dominates. Increasing spacing raises A and therefore increases L′, which generally pushes Z0 upward. The sample geometry produces about 0.917 µH/m.
Characteristic impedance Z0
Impedance follows Z0 = √(L′/C′). Since L′ rises with A and C′ falls with A, spacing has a strong effect. Increasing εr lowers Z0 because C′ increases while L′ stays nearly the same. In air, the sample calculates about 274.9 Ω.
Velocity, delay, and wavelength
Phase velocity is v = 1/√(με). In air, v is near the speed of light; in dielectrics it drops by about 1/√εr. The calculator reports delay per meter and one‑way delay for the chosen length. If frequency is entered, it also gives λ = v/f and β = 2π/λ. At 10 MHz in air, λ is about 30 m.
Practical design notes and limits
The model assumes a lossless TEM line. Real conductors have resistance and skin effect, and nearby objects change the field distribution. Use these results for fast sizing and comparison, then apply full-loss or field‑solver methods when you need tight tolerances in practice.
FAQs
1) Why must spacing D be greater than 2r?
D is measured center‑to‑center. If D ≤ 2r, the conductors overlap or touch, and the two‑cylinder field model breaks. The calculator blocks those cases to keep results physical.
2) What does εr change in the results?
εr scales permittivity ε, which increases capacitance C′ and lowers velocity v. Because Z0 depends on √(L′/C′), higher εr usually reduces characteristic impedance.
3) When should I change μr?
Most cables in air or plastics use non‑magnetic conductors and dielectrics, so μr ≈ 1. Only change μr if you know the surrounding medium is magnetic, otherwise you may distort L′ and v.
4) Why is frequency optional?
Geometry and dielectric set C′, L′, and Z0 for a TEM line. Frequency is only needed to compute wavelength λ and phase constant β. Losses and dispersion are not modeled here.
5) What do the total L and total C mean?
They are per‑unit‑length values multiplied by your line length. Total L and C help estimate delay, resonance, and matching networks. They are not the same as lumped values when the line is electrically long.
6) How close should results match real measurements?
Good agreement is possible when the environment is uniform and far from other conductors. Nearby metal, insulation thickness variations, and conductor surface effects can shift Z0 and delay, especially at high frequency.