Air line

An air line is a Transmission line whose dielectric is air (or vacuum), with perfectly conducting conductors. Equivalently: a lossless TEM line with $R^{\prime }=0$, $G^{\prime }=0$, $ϵ={ϵ}_0$, $\mu ={\mu }_0$.

This is the simplest possible TEM line — and the canonical starting case for transmission-line problems before adding dielectric materials, conductor losses, or leakage.

Parameter values

The four distributed parameters reduce to two:

$R^{\prime }=0,G^{\prime }=0,L^{\prime }=\text{geometry only},C^{\prime }=\text{geometry only},$

with the universal TEM identity $L^{\prime }{C}^{\prime }={\mu }_0{ϵ}_0=1/c^2$ pinning down their product. For a specific geometry (coax, parallel-wire, etc.), $L^{\prime }$ and $C^{\prime }$ follow from the cross-section dimensions.

For a coaxial line with inner radius $a$, outer radius $b$:

$L^{\prime }=\frac{{\mu }_0}{2\pi }\ln \frac{b}{a},C^{\prime }=\frac{2\pi {ϵ}_0}{\ln (b/a)}.$

Check: $L^{\prime }{C}^{\prime }={\mu }_0{ϵ}_0$. ✓

Why it’s useful as a baseline

Phase velocity equals $c$:

$u_p=\frac{1}{\sqrt{L^{\prime }{C}^{\prime }}}=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}=c\approx 3\times 10^8\text{ m/s}.$

Signals on an air line travel at the speed of light — the fastest any TEM line can carry them. Any dielectric (${ϵ}_r>1$) slows the signal by a factor of $\sqrt{{ϵ}_r}$, see Phase velocity.

Propagation constant is purely imaginary:

$\gamma =j\omega \sqrt{L^{\prime }{C}^{\prime }}=j\omega /c=j\beta ,\alpha =0.$

No attenuation. The wave amplitude stays constant along the line (for as long as the model holds — real conductors and air have nonzero losses, just tiny).

Characteristic impedance is real:

$Z_0=\sqrt{\frac{L^{\prime }}{C^{\prime }}}.$

For air-dielectric coax, $Z_0=(60\Omega )\ln (b/a)$. Inverting: 50 Ω requires $b/a=e^{50/60}\approx 2.3$; 75 Ω requires $b/a=e^{75/60}\approx 3.5$ (the canonical air-dielectric values). Filling the line with a dielectric of relative permittivity ${ϵ}_r$ scales these by $1/\sqrt{{ϵ}_r}$ — see Versus dielectric-filled line below.

When the air-line approximation is valid

The model holds when:

• The dielectric losses ($G^{\prime }$) are negligible — true for air, vacuum, dry gases.
• The conductor losses ($R^{\prime }$) are negligible — approximately true at low frequency in good conductors, or in superconducting lines.
• No radiation losses — TEM lines are mostly self-contained, but at very high frequencies and with imperfect geometries, some power can leak as radiation.

Real-world “air-dielectric” coaxial cables (like rigid copper coax, used for low-loss RF and microwave) have actual finite $R^{\prime }$ from conductor resistance — especially at high frequency due to skin effect — but their $G^{\prime }$ is essentially zero. They behave like air lines for many purposes, with small corrections for conductor loss.

When it’s used in problems

The air-line idealization shows up as:

• The reference case in textbook transmission-line problems. “Assume a lossless air line” simplifies $\gamma$ to $j\beta$ and $Z_0$ to a real number, and makes Smith chart analysis exact.
• High-precision measurements. Air-dielectric precision coaxial standards (Type N, 3.5 mm, 2.4 mm connectors with rigid bead-supported inner conductors) are calibrated to air-line behavior at the metrology level.
• RF and microwave instrumentation. Network analyzers use air-dielectric reference standards because their behavior is fully predicted by geometry alone — no dielectric properties to characterize.

Beyond about 10 GHz, even “air lines” need corrections for skin-effect conductor loss; but in the GHz range and below, the air-line model is highly accurate for rigid coax.

Versus dielectric-filled line

	Air line	Dielectric-filled
$ϵ$	${ϵ}_0$	${ϵ}_r{ϵ}_0$
$u_p$	$c$	$c/\sqrt{{ϵ}_r}$
$\lambda$ at 1 GHz	30 cm	30/$\sqrt{{ϵ}_r}$ cm
$L^{\prime }$	same (depends on $\mu$, not $ϵ$)	same
$C^{\prime }$	$C_0^{\prime }$	${ϵ}_r{C}_0^{\prime }$
$Z_0$	$\sqrt{L^{\prime }/C_0^{\prime }}$	$\sqrt{L^{\prime }/({ϵ}_r{C}_0^{\prime })}$

Filling with a dielectric (${ϵ}_r>1$) lowers $Z_0$ by $\sqrt{{ϵ}_r}$ and slows the wave by the same factor. Hence the practical fact that 50 Ω coax with PE dielectric (${ϵ}_r\approx 2.3$) requires $b/a\approx 3.5$ (standard RG-58: $60/\sqrt{2.3}\cdot \ln 3.5\approx 49.6\Omega$), while 50 Ω air-dielectric coax needs only $b/a\approx 2.3$.

In context

Air line is the cleanest base case for TEM transmission lines: zero loss, light-speed propagation, real $Z_0$. From there, adding dielectric filling (${ϵ}_r>1$) gives the typical practical coax; adding conductor loss ($R^{\prime }>0$) and dielectric leakage ($G^{\prime }>0$) gives the Lossy transmission line used to model real cables at high frequency or with imperfect materials.