Skin depth and line input impedance¶

Skin depth¶

At RF the current in a conductor crowds into a skin of depth

(Pozar, Microwave Engineering, 4^th ed., §1.7). For copper at 1 MHz it is about 66 µm; at 100 MHz a tenth of that, 6.6 µm, a fifth of a 35 µm PCB foil. For rational \(\rho\), \(f\) and \(\mu_{\mathrm{r}}\) the square root is still exact: Magnitude holds whole and half exponents of primes, of \(\pi\) and of \(\mu_0\), so \(\delta\) is an exact scale, with \(\mu_0\) left as a symbol until a CODATA set gives it a value.

The resistivity of copper is itself a convention. The International Annealed Copper Standard fixes \(\rho = 1/58\) Ω·mm²/m at 20 °C (IEC 60028), \(1.7241.. \cdot 10^{-8}\) Ω·m; a real board's copper differs by a few percent, and the page names which it uses.

use exact_units::{Fixed, FixedScale, Magnitude, Prefix, Rounding, constants::Codata2022, prefix::Nano, units::Meter};

/// δ² = ρ/(π f μ₀) for ρ = 1/58·10⁻⁶ Ω·m and f = 10⁶ Hz.
const SKIN_SQR: Magnitude = Magnitude::new(1, 58, -12, 0, -1).div(Magnitude::MU0);

/// δ of annealed copper at 1 MHz as a scale.
#[derive(Copy, Clone, Debug, Default)]
struct SkinCu1MHz;
impl FixedScale for SkinCu1MHz {
    const SCALE: Magnitude = SKIN_SQR.sqrt(); // 29^(−1/2), π^(−1/2), μ₀^(−1/2): exact
    const SYMBOL: &'static str = "";
    const NAME: &'static str = "SkinCu1MHz";
    const RUNTIME: Option<Prefix> = None;
}

// exact, no set needed: δ²·π f μ₀/ρ is 1, and 100 times the frequency is a tenth of δ
let delta = SkinCu1MHz::SCALE;
let rho = Magnitude::new(1, 58, -6, 0, 0);
assert_eq!(delta.mul(delta).mul(Magnitude::new(1, 1, 6, 0, 1)).mul(Magnitude::MU0).div(rho), Magnitude::ONE);
let at_100mhz = Magnitude::new(1, 58, -14, 0, -1).div(Magnitude::MU0).sqrt();
assert_eq!(delta.div(at_100mhz), Magnitude::ratio(10, 1));

// a number only under a named set
let nm = Fixed::<Meter, SkinCu1MHz, u8>::new(1)
    .convert_rounded_with::<Codata2022, Meter, Nano, u32>(Rounding::NearestEven)
    .unwrap();
assert_eq!(nm.get() / 10, 6_608); // 66.08 µm

A line's input impedance¶

A lossless line of characteristic impedance \(Z_0\) and electrical length \(\beta l\) turns a load \(Z_{\mathrm{L}}\) into

(Pozar eq. 2.44). \(\tan \beta l\) is rational at a few lengths, and there \(Z_{\mathrm{in}}\) is exact:

• \(\beta l = \pi/4\), an eighth of a wave: \(\tan \beta l = 1\). 100 Ω through 50 Ω is \(40 - 30j\) Ω, exactly.
• \(\beta l = \pi/2\), a quarter wave: \(\tan \beta l\) is the point at infinity, and \(Z_{\mathrm{in}} = Z_0^2/Z_{\mathrm{L}}\). A quarter wave of 50 Ω turns 100 Ω into 25 Ω, the quarter-wave transformer.
• \(\beta l = \pi\), a half wave: \(Z_{\mathrm{in}} = Z_{\mathrm{L}}\).

On a lossless line \(|\Gamma|\) is the same at both ends: the load's \(\Gamma = 1/3\) and the input's \((40 - 30j - 50)/(40 - 30j + 50)\) both have \(|\Gamma|^2 = 1/9\), exactly. The property line checks that for drawn rational loads and \(\tan \beta l\), against the formula on i128 rationals.

A stub's reactance, bracketed¶

A short-circuited stub of length \(l\) is a reactance \(X = Z_0 \tan(2\pi l/\lambda)\). At 0.1 λ that is \(50 \tan(\pi/5)\) Ω, irrational. The example brackets it: \(\pi\) lies between the f64 nearest it and the next one up, \(0.1\) between its f64 and the one below, each product is widened by a step, and tan (increasing below \(\pi/2\)) is widened by two steps each way:

use core::f64::consts::PI;

let widen = |lo: f64, hi: f64, n: u32| (0..n).fold((lo, hi), |(lo, hi), _| (lo.next_down(), hi.next_up()));
let (x_lo, x_hi) = ((2.0 * PI * 0.1f64.next_down()).next_down(), (2.0 * PI.next_up() * 0.1f64).next_up());
let (t_lo, t_hi) = widen(x_lo.tan(), x_hi.tan(), 2);
let (lo, hi) = widen(50.0 * t_lo, 50.0 * t_hi, 1);
assert!(lo < hi && hi - lo < 1e-12);
assert!((lo - 36.327).abs() < 0.001); // X = 36.327.. Ω

These lines stand in for the crate's interval until it is public (NUM-45). They assume the platform's tan is within two steps of the true value, which IEEE 754 doesn't promise for tan; the crate's own interval arithmetic computes in integers and assumes nothing.

Going deeper¶

Why the square root stays exact. \(\rho/(\pi f \mu_0)\) for copper at 1 MHz is \(1/(58 \cdot 10^{12}) \cdot \pi^{-1} \mu_0^{-1}\). Its square root halves every exponent: \(2^{-1/2} \cdot 29^{-1/2} \cdot 10^{-6} \cdot \pi^{-1/2} \cdot \mu_0^{-1/2}\). Magnitude stores twice each exponent, so a half is a whole number there, and \(\delta \cdot \delta\) gives back the rational part, \(\pi\) and \(\mu_0\) exactly. A quarter power (the square root of \(\delta\)) is refused. The property skin_depth checks \(\delta^2 \pi f \mu_0/\rho = 1\) and the factor of 10 for drawn \(\rho\) and \(f\).

The 2019 redefinition, again. \(\mu_0\) in \(\delta\) is the measured one. Under the 1948 ampere (\(\mu_0 = 4\pi \cdot 10^{-7}\)) the same scale gives a \(\delta\) that differs from CODATA 2022's by about \(7 \cdot 10^{-11}\) relative (half of \(\mu_0\)'s \(1.3 \cdot 10^{-10}\)): nothing a conductor's resistivity, known to a few percent, could show.