Characteristic impedance

The characteristic impedance of a Transmission line is the ratio of voltage to current for a single propagating wave:

$Z_0=\frac{V_0^{+}}{I_0^{+}}=-\frac{V_0^{-}}{I_0^{-}}.$

Forward-propagating wave: $Z_0=V/I$. Backward-propagating: $Z_0=-V/I$ (the minus sign reflects that current flows in $-\hat{\mathbf{z}}$ for the backward wave but is conventionally measured in $+\hat{\mathbf{z}}$).

In terms of line parameters:

$Z_0=\sqrt{\frac{R^{\prime }+j\omega L^{\prime }}{G^{\prime }+j\omega C^{\prime }}}(\Omega ).$

For a lossless line ($R^{\prime }=G^{\prime }=0$):

$Z_0=\sqrt{\frac{L^{\prime }}{C^{\prime }}}\text{(real, frequency-independent)}.$

Common values:

• 50 Ω — RF and microwave coax. The historical standard, a compromise between minimum loss (~77 Ω) and maximum power handling (~30 Ω) for air-dielectric coax.
• 75 Ω — TV cable, video. Optimized for minimum loss.
• 300 Ω — twin-lead TV antenna cable.
• 100 Ω — twisted pair, Cat-5/6 Ethernet differential. (USB is 90 Ω differential; RS-485/CAN industrial pairs use 120 Ω.)

What Z₀ is not

Three common misunderstandings:

1. $Z_0$ is not a resistance. It’s not an actual resistor anywhere in the line. A 50 Ω cable doesn’t dissipate energy at the rate $V^2/50$; it carries the wave to the load (or to a reflection), without loss in the ideal case. The “Ω” units come from the $V/I$ ratio, not from dissipation.

2. $Z_0$ doesn’t depend on line length. It depends only on the line’s per-unit-length parameters, which depend only on geometry and material. A 50 Ω cable is 50 Ω whether it’s 1 cm or 1 km long.

3. $Z_0$ is not the input impedance. $Z_{\text{in}}$, what a source actually sees looking into the line, depends on the load and the line length. $Z_0$ is the intrinsic property of the line itself.

Why it’s the V/I ratio of a single wave

The telegrapher’s equations in phasor form imply, for the forward-propagating component $\tilde{V}\propto e^{-\gamma z}$:

$\tilde{I}=-\frac{1}{R^{\prime }+j\omega L^{\prime }}\frac{d\tilde{V}}{dz}=\frac{\gamma }{R^{\prime }+j\omega L^{\prime }}\tilde{V}.$

Substituting $\gamma =\sqrt{(R^{\prime }+j\omega L^{\prime })(G^{\prime }+j\omega C^{\prime })}$:

$\frac{\tilde{V}}{\tilde{I}}=\frac{R^{\prime }+j\omega L^{\prime }}{\gamma }=\sqrt{\frac{R^{\prime }+j\omega L^{\prime }}{G^{\prime }+j\omega C^{\prime }}}.$

So for a single propagating wave on the line, voltage and current are locked in this constant ratio. The wave “lives” with this $Z_0$ as its impedance, separate from any external load.

Why is the matched load special?

If the load impedance equals $Z_0$, the wave reaching the load can “see” what looks like infinite more line. No reflection occurs: $V_0^{-}=0$. Energy is delivered cleanly to the load.

If $Z_L\ne Z_0$, the load can’t absorb the wave at its arrival ratio of $V/I$. Some energy reflects back as a backward wave, and a standing wave forms on the line.

This is why impedance matching matters so much in RF and high-speed design. A mismatch wastes power (reflected back to source), distorts pulses (standing waves), and at high frequencies can damage the source (reflected energy overheating amplifiers).

Z₀ for common geometries

Coaxial line (inner radius $a$, outer $b$, dielectric $ϵ,\mu ={\mu }_0$):

$Z_0=\frac{1}{2\pi }\sqrt{\frac{{\mu }_0}{ϵ}}\ln \frac{b}{a}=\frac{{\eta }_0}{2\pi \sqrt{{ϵ}_r}}\ln \frac{b}{a},$

with ${\eta }_0=\sqrt{{\mu }_0/{ϵ}_0}\approx 377$ Ω.

For ${ϵ}_r=2.3$ (polyethylene) and $b/a=3.5$: $Z_0\approx 50$ Ω. This is the standard RG-58 design.

Two-wire line (wire radius $a$, separation $D\gg a$, dielectric $ϵ$):

$Z_0\approx \frac{1}{\pi }\sqrt{\frac{{\mu }_0}{ϵ}}\ln \frac{D}{a}=\frac{{\eta }_0}{\pi \sqrt{{ϵ}_r}}\ln \frac{D}{a}.$

Parallel plate line (width $w$, separation $d$, dielectric $ϵ$):

$Z_0=\sqrt{\frac{{\mu }_0}{ϵ}}\cdot \frac{d}{w}=\frac{{\eta }_0}{\sqrt{{ϵ}_r}}\cdot \frac{d}{w}.$

(Assumes $w\gg d$ so fringing is negligible.)

Measurement trick: open + short

For a lossless line of length $l$, the input impedance with the far end open is $Z_{\text{in}}^{\text{oc}}=-j{Z}_0\cot (\beta l)$, and with the far end shorted is $Z_{\text{in}}^{\text{sc}}=j{Z}_0\tan (\beta l)$. Multiplying:

$Z_{\text{in}}^{\text{oc}}\cdot Z_{\text{in}}^{\text{sc}}=-j^2{Z}_0^2\cdot \cot (\beta l)\tan (\beta l)=Z_0^2.$

So $Z_0=\sqrt{Z_{\text{in}}^{\text{oc}}{Z}_{\text{in}}^{\text{sc}}}$. By measuring input impedances under both terminations with a network analyzer, you can extract $Z_0$ of an unknown line. This is the standard technique used by RF labs.