Transmission-Line Characteristic Impedance

Equivalent forms

Z_0 = \sqrt{\frac{R + j\omega L}{G + j\omega C}}

v_p = \frac{1}{\sqrt{LC}}

An impedance with the units of resistance that dissipates nothing — it's the wave's own ratio of volts to amps, pure geometry of L and C.

Symbol	Name	SI unit	Dimension	Range
$Z_0$	Characteristic impedanceoutput Voltage-to-current ratio of a traveling wave	Ω	M·L²·T⁻³·I⁻²	1 – 1000
$L$	Inductance per length Series inductance per metre	H/m	M·L·T⁻²·I⁻²	1e-8 – 0.00001
$C$	Capacitance per length Shunt capacitance per metre	F/m	M⁻¹·L⁻³·T⁴·I²	1e-12 – 1e-9

Unit systems

Symbol	SI
$Z_0$	Ω
$L$	H/m
$C$	F/m

Where it holds

The simple

\sqrt{L/C}

holds for a lossless line (R,

G \approx 0)

, an excellent approximation at RF. With significant loss,

Z_{0} = \surd ((R+j\omega L)/(G+j\omega C))

becomes complex and frequency-dependent (the distortionless and low-frequency regimes differ).

Dimensional analysis

$\sqrt{[L]/[C]} = \surd ((H/m)/(F/m)) = \sqrt{H/F} = \sqrt{\Omega ^{2}} = \Omega \checkmark$

Discovery

Oliver Heaviside · 1885

Heaviside reduced Maxwell's equations to the 'telegrapher's equations' for cables and introduced characteristic impedance, explaining signal reflections that had plagued long telegraph and telephone lines.

Try this

A cable can be 'matched' or it can echo your signal back as a ghost. Why does a wire have a 50- or 75-ohm 'impedance' even though it's just metal?

A coaxial cable has 250 nH/m of inductance and 100 pF/m of capacitance. Find its characteristic impedance.

Research status: stable

Real-world applications

$50 \Omega RF / 75 \Omega$ video cabling
PCB trace impedance control
Antenna feed matching
High-speed digital signal integrity

Common misconceptions

$Z_{0} is$ a resistance that heats up — it's a wave ratio; a lossless line stores no net energy in $Z_{0}$
A cable's impedance depends on its length — $Z_{0} is$ per-unit-length geometry, independent of length

Experimental verification

Time-domain reflectometry: send a step into a cable and an impedance mismatch reflects part of it; the reflection coefficient

\Gamma = (Z_L - Z_{0})/(Z_L + Z_{0})

gives

Z_{0}

directly. Matched

50 \Omega

loads produce no echo.

Derivation

From the telegrapher's equations

\partial V/\partial x = -L \partial I/\partial t

,

\partial I/\partial x = -C \partial V/\partial t

, a forward wave

V = f(t - x/v)

gives

V/I =

constant.

Substituting yields

Z_{0} = \sqrt{L/C}

and propagation speed

v = 1/\sqrt{LC}

.

Losses generalize

L\to R+j\omega L

,

C\to G+j\omega C

.

Limiting cases

C \to

large (closely spaced conductors)⟶

Z_{0}

→ smallMore capacitance per metre lowers the impedance —

why 75 \Omega

coax has a thinner dielectric gap than

50 \Omega

.

load

= Z_{0}

(matched)⟶ no reflectionA matched termination absorbs the wave completely — the goal of impedance matching.

What if…

What if the load doesn't match

Z_{0}

?

Part of the wave reflects $(\Gamma = (Z_L-Z_{0})/(Z_L+Z_{0}))$ , creating standing waves and a VSWR > 1 — wasted power and possible damage.

What if you squeeze the conductors closer together?

Capacitance per metre rises, so $Z_{0}$ drops — the knob cable makers turn to hit 50 or $75 \Omega$ .

1

Coax with 250 nH/m and 100 pF/m

Given ·

L:: 2.5e-7
C:: 1e-10

Find · Z_0

Steps

$L/C = 2.5\times 10^{-7} / 10^{-10} = 2500 \Omega ^{2}$
$Z_0 = \sqrt{2500}$
$= 50 \Omega$ — the standard RF impedance

Result ·

50 \Omega

— exactly standard RF coax.

telegraphers equationsvswrInductive Reactance→speed of light em