Free-space impedance¶

The impedance of free space links a plane wave's field strength to its power density, \(s = e^2/\eta_0\). It turns up in EMC field-strength estimates, exposure limits and antenna formulas. Since the 2019 SI it is a measured number:

with \(c\) exact and \(\mu_0\) measured, \(1.25663706127(20) \cdot 10^{-6}\) N/A² in CODATA 2022 (Measured constants). The crate keeps it as an expression over the symbol \(\mu_0\), constants::ETA0, and gives it a value only under a set the caller names. It is not Z0, which in this crate is a line's reference impedance, 50 Ω and the like (Impedance).

A value from a named set¶

use exact_units::{Fixed, FixedScale, Magnitude, Prefix, Rounding, prefix::Micro, units::Ohm};
use exact_units::constants::{Codata2018, Codata2022, ETA0};

/// η₀ as a scale: one of it is the impedance of free space.
#[derive(Copy, Clone, Debug, Default)]
struct Eta0;
impl FixedScale for Eta0 {
    const SCALE: Magnitude = ETA0;
    const SYMBOL: &'static str = "";
    const NAME: &'static str = "Eta0";
    const RUNTIME: Option<Prefix> = None;
}

let eta0 = Fixed::<Ohm, Eta0, u8>::new(1);
let in_2022 = eta0.convert_rounded_with::<Codata2022, Ohm, Micro, i64>(Rounding::NearestEven).unwrap();
let in_2018 = eta0.convert_rounded_with::<Codata2018, Ohm, Micro, i64>(Rounding::NearestEven).unwrap();
assert_eq!((in_2022.get(), in_2018.get()), (376_730_313, 376_730_314)); // µΩ

Without a set the conversion doesn't compile: the factor keeps \(\mu_0\), which has no value of its own (What exact means, rule 4).

use exact_units::{Fixed, FixedScale, Magnitude, Prefix, prefix::Micro, units::Ohm, constants::ETA0};
#[derive(Copy, Clone, Debug, Default)]
struct Eta0;
impl FixedScale for Eta0 {
    const SCALE: Magnitude = ETA0;
    const SYMBOL: &'static str = "";
    const NAME: &'static str = "Eta0";
    const RUNTIME: Option<Prefix> = None;
}
let _ = Fixed::<Ohm, Eta0, u8>::new(1).convert::<Ohm, Micro, i64>();

120π is a convention, and no set's value¶

Many formulas write \(\eta_0 = 120\pi\) Ω, 376.99 Ω. It is not what the old SI fixed either: the 1948 ampere set \(\mu_0 = 4\pi \cdot 10^{-7}\) exactly, so \(\eta_0\) was \(4\pi \cdot 10^{-7} \cdot 299\,792\,458 = 119.9169832\,\pi\) Ω. 120π also rounds \(c\) to \(3 \cdot 10^8\) m/s: two conventions in one number. The crate's Ampere1948 set gives the old value exactly, and none of the sets gives 120π:

use exact_units::Magnitude;
use exact_units::constants::{Ampere1948, C, Codata2018, Codata2022, ETA0};

let old = ETA0.resolve::<Ampere1948>().value;
assert_eq!(old, Magnitude::new(1_199_169_832, 1, -7, 0, 1)); // 119.9169832·π
let pi120 = Magnitude::new(120, 1, 0, 0, 1);
for set in [old, ETA0.resolve::<Codata2018>().value, ETA0.resolve::<Codata2022>().value] {
    assert_ne!(set, pi120);
}
// η₀/(4π) under the 1948 ampere is c·10⁻⁷, 29.9792458 Ω: the "30" of e = √(30p)/d
assert_eq!(ETA0.div(Magnitude::new(4, 1, 0, 0, 1)).resolve::<Ampere1948>().value, C.mul(Magnitude::pow10(-7)));

120π is not folklore: ITU-R P.525-5, in force, uses it. Its eq. 3 is \(s = e^2/(120\pi)\), its eq. 1 is \(e = \sqrt{30 p}/d\) (the 30 is \(120\pi/(4\pi)\)), and its eq. 2 is \(e_{\mathrm{mV/m}} = 173 \sqrt{p_{\mathrm{kW}}}/d_{\mathrm{km}}\) (173 is \(\sqrt{30\,000}\), rounded). In this crate 120π is a named convention, ITU-R P.525's, written by the caller who wants it.

Where the difference matters, and where it doesn't¶

120π is 0.069 % above \(\mu_0 c\). In decibels that is 0.003 dB, both in a power density computed from a field (\(10 \lg\) of the ratio) and in a field computed from a power density (\(20 \lg\) of its square root). The example computes:

No measurement sees 0.003 dB. Even a primary standard field is far coarser: NBS's TEM cells, built to calibrate power-density meters, established field strengths "with uncertainties less than ±1.0 dB" (M. L. Crawford, NBSIR 75-804, 1975), and that report converts field to power density with 377 Ω itself (its eq. 14). In EMC field strength, link budgets and antenna impedances, 120π and \(\mu_0 c\) give the same answer to every digit anyone measures.

Where it bites is consistency. A program that takes \(\varepsilon_0\) from CODATA and \(\eta_0\) as 120π breaks \(\eta_0 = 1/(\varepsilon_0 c)\) by 0.069 %. Then a self-check that holds to \(10^{-6}\) (reciprocity or passivity of computed S-parameters, a golden file, two tools that should agree) fails in the fourth digit, and the cause is a constant, not the code. Naming the convention is the point: 120π is not badly wrong, it is a choice, and two choices in one program disagree.

Going deeper¶

Why μ₀ is measured. The 2019 SI fixed \(h\), \(e\), \(k_{\mathrm{B}}\) and \(N_{\mathrm{A}}\) exactly. \(\mu_0\) followed from the fine-structure constant, \(\mu_0 = 2\alpha h/(e^2 c)\), and \(\alpha\) is measured. Before 2019 the ampere's definition fixed \(\mu_0 = 4\pi \cdot 10^{-7}\) N/A² and left \(h\) and \(e\) measured instead. CODATA 2022 puts \(\mu_0/(4\pi \cdot 10^{-7})\) within about \(10^{-10}\) of 1: the redefinition moved the value by far less than 120π's rounding of \(c\) does.

What the crate checks. eta0_under_each_set takes \(\mu_0\) as each CODATA edition prints it times the exact \(c\) and compares with ETA0 resolved under that set; checks that Ampere1948 gives \(119.9169832\,\pi\) and no set gives \(120\pi\); and that \(\eta_0 \varepsilon_0 c = 1\) exactly, with no set at all.