A worked matching example¶
A tuner measures an antenna's impedance at a few frequencies against a 50 Ω line. A bench tool shows \(\Gamma\), VSWR, return loss and mismatch loss on a host; the firmware on the antenna side only answers "matched or not" per sweep step, from a reflectometer's Q15 \(\Gamma\), on a part without an FPU. cargo run --example 13_impedance_match --features electrical,angle,dimensionless,libm runs the whole thing — Examples has the line and the file.
On the bench, in floats¶
100 Ω on a 50 Ω line (Impedance and admittance, Reflection):
use exact_units::{Admittance, Impedance, Z0, prefix::{Milli, One}, q, units::{Decibel, Loss}};
let z0 = Z0::<One, f64>::bounded(50.0, 0.0);
let z = Impedance::<One, f64>::bounded(100.0, 0.0);
let gamma = z.reflection(z0).finite().unwrap();
assert!(gamma.vswr().is_in(q!(2 V/V +- 0.000001 V/V, Inside)));
assert!(gamma.return_loss().is_in(q!(9.542425... Loss<Decibel>)));
assert!(gamma.mismatch_loss().is_in(q!(0.888... W/W))); // 8/9 of the power, -0.51 dB as its to_db()
// at 440 MHz the antenna looks inductive, 82 + 35j Ω: a shunt element is chosen in admittance
let y: Admittance<Milli, f64> = Impedance::<One, f64>::bounded(82.0, 35.0).to_admittance().finite().unwrap();
assert!(y.im().is_in(q!(-4.403069... mS))); // the susceptance a shunt capacitor cancels
Nothing here can error: a short, an open, or \(Z = -Z_0\) on the way to the match would still give a \(\Gamma\), not a panic (Why no error, ever).
On the firmware, in Q15¶
The sweep step only asks \(|\Gamma| \le -15\ \mathrm{dB}\), from a reflectometer's Q15 \(\Gamma\): a Complex<Per<Volt, Volt>, Pow2<-15>, i16, Saturating>. \(|\Gamma|^2\) is exact in a u32 (Levels, dBm and cable loss), so the test is norm_sqr against a threshold worked out at compile time — no root, no log, no float, and no FPU needed:
use exact_units::prefix::{Pow2, Product};
use exact_units::units::{Per, Times, Volt};
use exact_units::{Complex, Fixed, Saturating, db};
type GammaQ15 = Complex<Per<Volt, Volt>, Pow2<-15>, i16, Saturating>;
type GammaSqr = Fixed<Times<Per<Volt, Volt>, Per<Volt, Volt>>, Product<Pow2<-15>, Pow2<-15>>, u32>;
/// |Γ| ≤ −15 dB as a threshold on |Γ|²: the correct one.
const MATCHED: GammaSqr = db!(norm_sqr - 15, NearestEven);
assert_eq!(MATCHED.get(), 33_954_698);
let retune = GammaQ15::of(7_010, 5_235); // −11.5 dB: should ask for a retune
assert!(retune.norm_sqr() > MATCHED);
let matched = GammaQ15::of(-1_014, 1_356);
assert!(matched.norm_sqr() <= MATCHED);
The classic slip¶
\(\Gamma\) is a ratio of voltage waves, so its level is \(20 \cdot \lg|\Gamma|\). Writing the threshold as db!(-15, NearestEven) — a voltage-ratio level of \(|\Gamma|^2\) itself, not of \(|\Gamma|\) — reads \(10^{-15/20}\) where \(10^{-15/10}\) is meant: it passes antennas down to −7.5 dB (VSWR 2.46) as "matched", not −15 dB (VSWR 1.43):
use exact_units::prefix::{Pow2, Product};
use exact_units::units::{Per, Times, Volt};
use exact_units::{Complex, Fixed, Saturating, db};
type GammaQ15 = Complex<Per<Volt, Volt>, Pow2<-15>, i16, Saturating>;
type GammaSqr = Fixed<Times<Per<Volt, Volt>, Per<Volt, Volt>>, Product<Pow2<-15>, Pow2<-15>>, u32>;
/// The slip: −15 dB read as a voltage-ratio level of |Γ|² itself.
const SLIP: GammaSqr = db!(-15, NearestEven);
assert_eq!(SLIP.get(), 190_941_298);
// −11.5 dB by the spec: a retune by `MATCHED`, "matched" by the slip
let g = GammaQ15::of(7_010, 5_235);
assert!(g.norm_sqr() <= SLIP);
db! and norm_sqr name which is meant at compile time: the two constants above don't get the kind wrong by accident, only by writing the wrong one down. A type that reads Times<Per<Volt, Volt>, Per<Volt, Volt>> where the datasheet says "−15 dB on \(|\Gamma|\)" is the reminder to check which.