Illustrated Physics
physics, drawn
Concept

Impedance — where it appears

The property of a medium that decides how much of a wave reflects at a boundary, being the ratio of a driving quantity to a response. Two media with equal impedances pass a wave with no reflection whatever their other properties, which is why matching is a design goal rather than an accident.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

Reflected upside down. A pulse arriving at a join where the impedance rises by a factor of 3, drawn at three moments. The amplitudes are read off the marched wave: the reflected pulse is -0.500 of the incident one and the transmitted pulse is 0.500, against (1−Z₂/Z₁)/(1+Z₂/Z₁) = -0.500 and 2/(1+Z₂/Z₁) = 0.500 from the two matching conditions. The reflection is inverted, which is the same fact as a pulse on a string flipping when it reaches a wall: a wall is a medium of infinite impedance, and the inversion is what keeps the displacement at the join equal to zero. Note that the transmitted amplitude exceeds one where the second medium is lighter, and that this is not a violation of anything: amplitude is not energy.

What happens where the medium changes

Two conditions at a join — the displacement is continuous, and so is the transverse force — fix the reflected and transmitted amplitudes completely. What decides them is one quantity, the impedance, and not the stiffness or the density separately: two quite different media with the same impedance are, to a wave, the same medium.

waves · Impedance What a real coating leaves behind, and over what range. Reflectance against wavelength for a glass surface of index 1.52 in air, uncoated and with quarter-wave layers of 2 different indices, each a quarter of a wave thick at 550 nm. The ideal index is the geometric mean, 1.2329, and the layer made of it takes the reflectance to zero at the design wavelength exactly. No durable solid has that index: magnesium fluoride at 1.38 is the usual compromise and it leaves 1.26% at the design wavelength against 4.26% bare — a reduction of 3.4× rather than a removal. Both curves rise away from the design wavelength, because the thickness is a quarter of a wave only there, and the useful band is wide but not unlimited: the better coating stays under a quarter of the bare reflectance from 415 to 780 nm. The purple cast of a coated lens is that residual — the ends of the visible reflecting while the middle does not.

The layer that makes a reflection vanish

A wave meeting a step in impedance reflects, and nothing can be done about the step. Put a third medium between the two, a quarter of a wavelength thick and of exactly the intermediate impedance, and the reflection stops existing — not reduced, cancelled.

waves · Impedance The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is.

The gap a repeat opens

Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.

waves · Periodic media A graded junction against a quarter-wave layer. Reflectance against wavelength for four ways of joining a medium of index 1 to one of 1.52. The bare interface reflects 4.26 per cent at every wavelength. A single quarter-wave layer of index √(n₀n_s) takes that to zero at 550 nm exactly and rises symmetrically either side, which is the shape of every single-layer coating and every single-section transformer. The remaining curves are graded junctions of 150 nm and 400 nm, built as 120 thin layers whose index climbs geometrically and put through the same matrix product. They do not have a design wavelength at all. Below a cutoff they are flat and negligible; above it they climb steeply toward the bare value, and the cutoff is set by the length: 398 nm for the 150 nm taper, 1062 nm for the 400 nm taper. Roughly, a taper works for every wavelength shorter than about twice its own optical length, which is the statement that a reflection needs a partner a quarter of a wavelength further in to cancel against. The engineering versions of this are everywhere: the horn on a loudspeaker, the moth's eye, the flared transition between two waveguides, and the graded layer that lets an ultrasound probe reach tissue across a hundredfold impedance step.

The taper that matches every note

A quarter-wave layer cancels a reflection at one wavelength and only near it. Spread the same change of impedance over a distance instead, and the reflection vanishes for every wavelength shorter than about twice that distance — not by cancelling one echo against another, but by leaving no step anywhere for an echo to come from.

waves · Impedance The cross term, and the fact that it averages to nothing. The intensity of two waves of amplitude 1 and 0.7 added together, against the phase difference between them, in turns. A detector reads the square of the summed amplitude, which is the sum of the two intensities plus a cross term that swings between plus and minus twice the product. At no phase difference the reading is 2.89 and at half a turn it is 0.09; the flat line is what the two would give with no interference, 1.49, and it is exactly the average of the curve over a whole turn. Interference redistributes and does not create — which answers the question of where the energy goes at a dark fringe by saying that it never left.

What adding does to the energy

Waves add their amplitudes and detectors read squares, so two waves together do not deliver the sum of what each delivers. Where the two get dimmer, the natural question is where the energy went — and the answer depends entirely on whether the sources can feel each other.

waves · Superposition A resonance that goes to zero before it peaks. Fano profiles for asymmetry parameters of 5, 1.5, 0.5, 0, each normalised to its own peak, against detuning in half-widths. A large parameter gives an almost symmetric peak — the resonant path dominates and the shape is nearly Lorentzian. A parameter of zero gives a symmetric dip, a window in which the system transmits nothing on resonance. In between the profile is lopsided, with a zero on one side of the resonance and the maximum on the other, at positions whose product is exactly minus one. The asymmetry is not a defect of the measurement: it is the interference of two ways through the system, and its sign says which side of the resonance the two paths cancel on.

The resonance with a zero in it

Where a resonance is the only way through a system, the response is a symmetric peak. Where there is also a smooth path that does not care about the frequency, the two add before anything is squared — and the result is lopsided, with a frequency at which nothing gets through at all.

waves · Resonance Minima that are not zeros. The amplitude along a line carrying a wave towards a load and its reflection back, for reflection magnitudes of 0, 0.35, 0.7, 1. With everything reflected the pattern touches zero and is a standing wave in the strict sense. With less than everything it does not: the minima sit at one minus the reflection and the maxima at one plus it, so the pattern is a partial standing wave sitting on a travelling one. The spacing is half a wavelength in every case, and the depth is the only thing that changes — which is why one number, the ratio of the maximum to the minimum, is enough to report the whole pattern.

The node that is not standing still

A wave meeting a perfect reflector makes a standing wave with real nodes. A partial reflection makes something that looks the same and is not: the minima are not zeros, energy flows steadily through them, and the depth of the pattern is a measurement of the load that caused it.

waves · Standing waves How close a network gets to a load that stores charge. The fraction of a wave's amplitude reflected from a resistance shunted by a capacitance, with RCωc = 2, against frequency in units of the band edge ωc, for the load alone and for networks of inductors and capacitors whose values, with an ideal transformer at the source, were optimised numerically to keep the reflection low across the band. The bare load holds it to 0.707 across the band; the 1-element network holds it to 0.392 across the band; the 2-element network holds it to 0.320 across the band; the 3-element network holds it to 0.289 across the band. The dashed line is Bode and Fano's floor, exp(−π/RCωc) = 0.208, which no network of any size can go beneath over the whole band; each added element brings the design closer to it, and each buys its flatter band with a reflection that climbs to total just beyond the band edge.

The mismatch no network can remove

A quarter-wave layer or a taper can match a resistance to a resistance as well as anyone likes. Put a capacitance across the load and that stops being true for every network that could ever be built from lossless parts: Bode and Fano proved that the total amount of match available is fixed by the load's resistance and capacitance, so a network can only move it about — and a flat match across a band can never be better than e to the minus π over the load's time constant times the band.

waves · Impedance

Named alongside it

The objects these essays reach for when they reach for this one.

InterferenceImpedance matchingReflection coefficientBandwidthSuperpositionTransfer matrixAntireflectionBoundary conditionsEnergy conservationQuality factorStanding waveTransmission coefficient

All concepts