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Electromagnetics & Transmission Line Simulator

Coaxial Cable Impedance Calculator — Z_0 and Propagation Speed

Change the inner radius a, outer inner radius b, and dielectric constant ε_r to see how the characteristic impedance Z_0, velocity factor, and per-unit-length capacitance and inductance are determined.

Parameters
Inner radius a
mm
Outer inner radius b
mm
Dielectric constant ε_r
—
Frequency f
MHz

Constants used: c = 2.998×10⁸ m/s, ε_0 = 8.854×10⁻¹² F/m, μ_0 = 4π×10⁻⁷ H/m. Defaults assume PTFE (ε_r ≒ 2.3).

Results (live)
—
Characteristic impedance Z_0
—
Radius ratio b/a
—
Velocity factor VF
—
Dielectric constant ε_r
—
Per-unit-length C
—
Per-unit-length L
E-field animation, propagating pulse and Z_0 design curve

Left = coaxial cross section with the radial E-field (inner→outer) and an expanding signal ring · Center-bottom = a pulse travelling along the cable at the velocity factor VF (dotted = free-space speed c reference) · Right = Z_0 vs b/a with the current point (red) and 50Ω / 75Ω reference lines.

Theory & Key Formulas

A coaxial cable is a transmission line in which an electromagnetic wave propagates in the TEM mode between an inner conductor of radius a and an outer conductor of inner radius b. The per-unit-length capacitance C and inductance L follow analytically from Laplace's equation on the cross section.

Characteristic impedance Z_0 (ε_r is the dielectric constant):

$$Z_0 = \frac{60}{\sqrt{\varepsilon_r}}\,\ln\!\frac{b}{a} = \sqrt{\frac{L}{C}}\ \ [\Omega]$$

Per-unit-length capacitance C and inductance L:

$$C = \frac{2\pi\,\varepsilon_0\,\varepsilon_r}{\ln(b/a)}\ \ [\text{F/m}], \qquad L = \frac{\mu_0}{2\pi}\,\ln\!\frac{b}{a}\ \ [\text{H/m}]$$

Propagation speed v_p, velocity factor VF and wavelength λ:

$$v_p = \frac{1}{\sqrt{LC}} = \frac{c}{\sqrt{\varepsilon_r}}, \quad \text{VF} = \frac{v_p}{c} = \frac{1}{\sqrt{\varepsilon_r}}, \quad \lambda = \frac{v_p}{f}$$

In air (ε_r=1), Z_0=50Ω corresponds to b/a≈2.30 and Z_0=75Ω to b/a≈3.49. Because Z_0 scales with ln(b/a), doubling the ratio b/a does not double Z_0 but only multiplies it by about 1.4. The fact that only the ratio matters — not the absolute size — is a key property of coaxial design.

What is the Coaxial Cable Impedance Calculator

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My radio manual says "50Ω system". Does that mean the cable itself has 50Ω of resistance? I put a multimeter on it and got almost 0Ω.
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Good question. Z_0 is not DC resistance. A coaxial cable is a "transmission line", and electromagnetic waves travel between the inner and outer conductors. The "wave impedance" the wave feels along its path is what we call the characteristic impedance Z_0. The formula is $Z_0 = (60/\sqrt{\varepsilon_r})\,\ln(b/a)$. With the defaults a=0.5 mm, b=3.5 mm, ε_r=2.3 the simulator above gives about 77Ω.
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Wait, only the ratio matters, not the radii themselves?
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Yes — that is the heart of coaxial design. Z_0 depends only on b/a. So a fat 50Ω coax (like RG-8) and a skinny 50Ω coax (RG-58 or 1.5D) all have roughly b/a ≒ 3.4 with PE filling. The thickness only changes power handling and loss: a fatter cable carries more power and has less loss. The reason cellular handsets can use very thin coax is exactly this "only the ratio matters" property.
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The velocity factor reads 0.66 or so. What does that mean?
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It is the speed of the EM wave inside the cable divided by the speed of light in vacuum. For coaxial cable it is set entirely by the dielectric: $\text{VF}=1/\sqrt{\varepsilon_r}$. PTFE and solid PE give 0.66 to 0.70; foamed PE goes above 0.80. When ham radio operators cut a quarter-wave stub, cutting at the free-space wavelength is too long — they need to multiply by VF. The "wavelength λ (at f)" card in the simulator already includes VF for you.
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What is the practical difference between 50Ω and 75Ω? I heard TV uses 75Ω.
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For the same dielectric and same outer diameter, the optimum b/a is at different points for different goals. Maximum power transfer is near 30Ω, minimum loss near 77Ω. 50Ω is the compromise — and so radio and instrumentation standardize on it. TV and CATV care about minimizing loss over long runs, so they chose 75Ω, very close to the minimum-loss point. Mixing 50Ω and 75Ω gear creates a mismatch and reflections, which can show as ghosts or weak signals.

Frequently Asked Questions

Connecting lines of different characteristic impedance causes a mismatch and part of the incident wave is reflected. Plugging a 75Ω cable into a 50Ω system gives voltage reflection coefficient |Γ| = |75−50|/|75+50| = 0.2 and VSWR = 1.5, which sends reflected power back to the transmitter and can foldback the output or trigger protection. Matching is done with a λ/4 transformer (a quarter-wave section of √(50×75) ≒ 61Ω coax) or a resistive matching pad.
The skin effect concentrates current near the conductor surface, so the effective resistance grows as the square root of frequency (conductor loss ∝ √f). On top of that, the dielectric's loss tangent tan δ causes dielectric loss roughly proportional to f. Above 1 GHz the dielectric loss starts to dominate, which is why low-tan-δ materials like PTFE and foamed PE are used. Above ~10 GHz designs typically switch to semi-rigid cable or waveguide.
A VSWR of 1.5 reflects about 4% of incident power, 2.0 about 11%, and 3.0 about 25%. Consumer transceivers and ham radios usually aim for VSWR ≤ 2.0, while commercial and broadcast transmitters target VSWR ≤ 1.5. Above VSWR ≈ 3.0 the transmitter's protection circuit foldback typically kicks in, and solid-state finals risk thermal damage. The standard practice is to tune the antenna match while watching a real-time VSWR / antenna analyzer.
Pure air gives ε_r ≒ 1.0006 ≒ 1 and VF ≒ 1, which minimizes dielectric loss. In practice, periodic PTFE spacers (rings or helices) are needed to hold the inner conductor, so the effective ε_r is typically 1.05 to 1.20. Designers compensate by making b/a slightly smaller than the theoretical value. Air-dielectric cables are used in high-end calibration cables and base-station transmit feeders, with loss less than half that of solid-dielectric cables.

Real-World Applications

Radio communication input/output: Almost all RF equipment — radios, transceivers, SDR dongles — standardizes the input and output on the 50Ω system. Feeders between antenna and transmitter, internal section-to-section wiring, and SMA / N / BNC connectors are all built around Z_0 = 50Ω, so that matching keeps reflection loss low and protects the transmitter.

Television, CATV and video signals: Terrestrial TV, satellite, CATV and professional video gear use the 75Ω system. Because long cable runs are common, the near-minimum-loss point of 75Ω is preferred, with F-type and BNC (75Ω) connectors. 50Ω and 75Ω BNCs look similar but have different internal dimensions; mixing them causes small mismatches and physical damage.

Measurement and calibration: Vector network analyzers (VNA), signal generators, and high-speed scope probes all calibrate to 50Ω. Semi-rigid coax and precision SMA hold Z_0 to within ±0.5Ω. PCB microstrip lines are also designed to 50Ω so that the impedance is continuous from the connector to the chip.

High-speed digital signaling: USB 3.x differential is 90Ω, HDMI is 100Ω differential, PCIe is 85Ω differential — modern high-speed digital interfaces are all treated as transmission lines with characteristic impedance controlled at PCB design time. They are not coax, but the TEM approximation and $v_p = c/\sqrt{\varepsilon_r}$ are used directly for propagation delay and skew calculations.

Common Misconceptions and Cautions

The most common misconception is that "characteristic impedance is a resistance you can measure with a multimeter". Z_0 is the input impedance of an infinitely long line, the ratio of voltage and current of a wave on the line. Under DC, the inner and outer conductors are insulated (infinite resistance), and the inner conductor itself has a few ohms or less. Neither becomes 50Ω. Remember that 50Ω is the impedance "felt by waves" on the line. Moving the b/a slider changes Z_0, but not because the conductors' DC resistance changed — only the wave impedance shape did.

The next most common error is to assume that "a thicker cable has a lower Z_0". In reality Z_0 depends only on the ratio b/a, not on absolute size. A thick 50Ω coax of 10 mm OD and a thin 50Ω coax of 1 mm OD both have b/a ≒ 3.4 (with PE filling). Only power handling and per-length loss change with size. Try scaling a and b by the same factor in the simulator: Z_0 stays exactly the same. This is the physical meaning of the "ratio-only" formula.

Finally, note that this simulator covers the ideal frequency-independent Z_0 of the TEM mode. In real coaxial cables, well above the cable's maximum recommended frequency higher-order modes (such as TE_11) appear and the single-TEM assumption breaks down. A small frequency-dependent imaginary part of Z_0 due to conductor loss is also ignored. The formulas here are sufficient up to typical GHz use of standard coaxial cable, but for millimeter-wave or extreme designs you should turn to a 3D EM solver (HFSS, CST, OpenEMS) to model the real geometry.

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How to Use

  1. Set inner conductor radius a (mm) using slAVal slider: typical range 0.5–2.0 mm for 50 Ω cables.
  2. Set outer conductor inner radius b (mm) using slBVal slider: maintain b/a ratio between 2.4–4.0 for standard impedances.
  3. Enter dielectric constant εr using slErVal slider: polyethylene ≈ 2.25, PTFE ≈ 2.1, foam ≈ 1.5.
  4. Set frequency f (GHz) using slFVal slider to compute wavelength λ = c·VF/f.
  5. Read Z₀ (ohms), velocity factor VF, λ (mm), and capacitance per meter from output labels.

Worked Example

RG-58 coaxial cable: inner radius a = 0.58 mm, outer radius b = 2.35 mm, εr = 2.25 (polyethylene). Formula: Z₀ = (138/√εr) × log₁₀(b/a) = (138/1.5) × log₁₀(4.05) ≈ 50.2 Ω. Velocity factor VF = 1/√εr ≈ 0.667. At f = 1 GHz: λ = 300 × 0.667 / 1 = 200.1 mm. Capacitance C = 2πε₀εr / ln(b/a) ≈ 100 pF/m.

Practical Notes

  1. Minimize Z₀ ripple in transmission systems by maintaining consistent b/a ratio; deviation ±5% causes impedance mismatch loss exceeding 0.5 dB at 10 GHz.
  2. Foam dielectrics (εr = 1.5) reduce velocity factor to 0.82, extending electrical length by 22% versus solid polyethylene—critical for phased-array antenna feed networks.
  3. Skin effect dominates conductor loss above 100 MHz; use silver-plated copper outer conductors for <0.1 dB/m attenuation at microwave frequencies.
  4. RG-6 (75 Ω video): b/a ≈ 4.4; RG-214 (50 Ω, shielded): stranded center supports flexing in lab harnesses without kinking.