Spectrum map who covers what, and what falls in the gaps
Solid bars are fundamentals (the note itself); thin bars are harmonics and attack (character). Shaded red is a band no driver in the chosen system covers.
The teal band, 1–4 kHz, carries speech clarity. Wavelengths are under the axis: λ = 343 ÷ f.
Wavelength
One full cycle is λ: peak to the next peak on the same side. Zero to the first peak is a quarter of that, λ/4.
An object much smaller than λ is invisible to that frequency, which is why bass wraps around cabinets and heads while treble beams and reflects.
Response spec · f3
f3 is where bass has fallen 3 dB (half power) below the speaker's normal level. "40 Hz–20 kHz ±3 dB" means the curve stays inside the shaded window over that range.
A range quoted at −6 or −10 dB looks deeper for the same speaker. Below f3 a sealed box falls ~12 dB/oct, a ported box ~24 dB/oct.
Trace an input ears · translator · brain · translator · muscle · mouth
The six stages of a DSP-based active speaker. Only analog sources need an ADC; digital sources need a receiver. Everything lands on one I²S bus at one sample rate.
The DSP does volume, EQ, the crossover split, delay and limiting. Each output then gets a DAC channel; each driver gets an amp channel. Digital-input amps fold the DAC into the amp chip; a line-level sub out still needs a DAC.
Where the crossover lives
Passive, powered and active differ in one thing: where the crossover sits relative to the amplifier. Dashed outline is the cabinet. Marketing uses "powered" and "active" interchangeably; engineers mean crossover before the amps.
Air displacement
Peak volume of air needed for a target level at one frequency: Vd = p·4πr / (ρ·ω²). Each boundary halves the requirement. Excursion is Vd ÷ cone area; pass means travel × margin fits inside Xmax.
Every octave lower needs four times the excursion, so excursion-limited output falls 12 dB per octave.
Sensitivity and level
2.83 V equals 1 W only into 8 Ω. Into 4 Ω it is 2 W, so the 1 W figure is 3 dB lower: sens₁W = sens₂.₈₃V − 10·log(8/Z). Level at 1 m: SPL = sens₁W + 10·log(W).
The line shows level against power: every doubling of watts adds only 3 dB.
Beaming
When the wavelength nears the driver's size, sound narrows into a forward beam. It starts near f = 343 ÷ (π·d). Ideal piston in a baffle.
This is why the woofer hands over before it beams and why a waveguide narrows the tweeter to match at the crossover.
Stereo geometry
Angle at the listener: 2·atan(spacing/2 ÷ distance). Reference is 60°: spacing equal to listening distance. Below ~40° the image narrows; above ~70° the centre goes hollow.
Height offset gives the vertical angle to the tweeter; past ~15° most domes lose treble.
Sealed box
Closed box from Thiele-Small values. α = Vas/Vb, Fc = Fs·√(1+α), Qtc = Qts·√(1+α). Qtc 0.707 is maximally flat; higher peaks before rolling off.
Stuffing makes the box act larger. Lift is the DSP boost needed at the target frequency; it becomes amp power and excursion.
Vented box EBP · f3 · port speed
EBP = Fs ÷ Qes: under ~50 suits sealed, over ~100 suits ported. Response is Small's vented-box model with box losses QL = 7. f3 is where the curve crosses −3 dB.
Port speed at the tuning frequency for the target level at 1 m (half space). Keep it under ~17 m/s (5% of the speed of sound) to avoid chuffing.
Port tuning
Round port length for a tuning: L = 23562.5·d²·n / (Fb²·Vb) − 0.732·d (d cm, Vb L, one flanged end). Port area at 10–15% of cone area is a common start; smaller risks chuffing.
Baffle step
Bass wraps around the cabinet into full space; treble reflects forward off the front panel. In open space the cabinet loses up to 6 dB of bass below roughly 115 ÷ width(m) Hz. A desk or wall returns part of it, so desktop speakers usually compensate 2–4 dB.
Crossover
Ideal electrical filters. Linkwitz-Riley pairs are −6 dB each at the crossover and sum flat with both drivers in phase; Butterworth pairs are −3 dB each and bump or notch. Invert the tweeter to see the polarity case.
Band plan
Each crossover point is a low-pass on the driver below and a high-pass on the driver above. The middle driver gets both, which makes a band-pass.
Delay and phase
A path difference is a delay: t = d ÷ 343. At a given frequency that delay is a phase shift: φ = 360·f·t. Two equal sources sum to +6 dB in phase, +3 dB at 90°, and cancel at 180°.
The same delay is a small phase shift in the bass and a large one in the treble. Fix with DSP delay or polarity.
Room modes
Axial modes between parallel walls: f = n·c / 2L. Marks from different dimensions within 5 Hz pile up into a boomy note; wide gaps become thin spots. A sub crossover of 80–100 Hz sits among the first few.
Distance and boundaries
Open-space level falls 6 dB per doubling of distance: SPL₂ = SPL₁ − 20·log(r₂/r₁). Each large nearby boundary adds up to 6 dB at low frequencies. Rooms reflect sound back, so real drop-off is smaller.
Comb filtering
A delayed copy of the same sound (desk, wall, or two drivers overlapping off-axis) cancels where the delay equals half a wavelength. First notch at 343 ÷ (2·path difference), then a notch every 343 ÷ path difference.
Schroeder & critical distance
Uses the room size from the Room modes card. Below the Schroeder frequency 2000·√(RT60/V) individual modes rule, so placement and EQ matter; above it the room behaves statistically and treatment matters.
Critical distance 0.057·√(Q·V/RT60) is where direct and reflected sound are equal. Closer than that (a desk), you mostly hear the speaker.
Watts needed
SPL = sens + 10·log(W) − 20·log(r) + pair. A stereo pair adds about 3 dB. Open-space figures; a room adds a few dB, so these are conservative. Red mark: your amp.
Clipping
Dashed lines are the supply rails. When the wave asks for more voltage than the rails hold, the tops flatten and odd harmonics appear. 6 dB more level needs four times the watts.
Power claim normaliser
Reduces a marketing wattage to continuous power per channel at about 1% THD. Peak→continuous and total→per channel are exact halvings for a sine. The 10%→1% THD (×0.8) and 4→8 Ω (÷1.6) factors are rules of thumb.
dB and levels
Power ratio 10^(dB/10), voltage ratio 10^(dB/20); +10 dB sounds about twice as loud. dBV is referenced to 1 V, dBu to 0.775 V; consumer line ≈ −10 dBV, pro line +4 dBu.
Driver wiring
Strings in series, strings in parallel: Z = Z₁ × series ÷ parallel. Halving the load doubles current, power and amp heat. Below the amp's minimum load the amp is at risk.
Series adds, parallel divides. A+B speaker switches put pairs in parallel. Bridged amps see half the load per channel.
Amp to driver match
Rail voltage implied by the rating: V = √(2·W·Z) peak. Under 50% of the driver's rating wastes headroom; over 200% risks the driver on sustained signals. A DSP limiter can cap the difference.
Class is the idle current
Push-pull pair. Class A idles hot so neither device ever switches off. Class AB idles cool: at low volume it behaves like Class A; on loud peaks one device switches off (red marks). Class D switches the output fully on and off and isn't shown.
Impedance curve nominal vs minimum
A speaker's impedance moves with frequency: a peak at the bass resonance (two around the tuning for a ported box), a minimum just above it near the voice coil's DC resistance Re, and a rise in the treble from coil inductance.
The amp has to survive the minimum, not the nominal. IEC 60268-5 lets the minimum fall to 80% of the nominal rating, so nominal ≈ minimum ÷ 0.8, and an amp rated for 4 Ω is expected to cope with a 4 Ω speaker's 3.2 Ω dip.
Volts · amps · ohms · watts
Ohm's law V = I·R and power P = V·I = V²/R = I²R. dB from power is 10·log; dB from voltage is 20·log.
Same voltage into half the ohms: double the current and power, +3 dB. Double the voltage into the same ohms: four times the power, +6 dB.
Speaker cable loss
Cable resistance forms a divider with the speaker. Loop resistance is both conductors: 2 × length × Ω/m. Keep it under ~5% of the speaker's impedance. Damping factor at the speaker includes the cable.
Headphone drive
From dB/mW: P = 10^((SPL − S)/10) mW. From dB/V: V = 10^((SPL − S)/20). Then V = √(P·Z). Aim for amp output impedance at or below 1/8 of the headphone's impedance so the response doesn't shift.
Distortion & noise units
THD as % and as dB are the same thing: dB = 20·log(% / 100). 1% = −40 dB, 0.1% = −60 dB, 0.01% = −80 dB. A noise floor below the signal is the SNR in dB. Speakers distort far more than electronics; electronics are usually not the weak link.
Sampler
A tone sampled at the chosen rate and rounded to the bit depth. Above half the sample rate the samples describe a lower tone (aliasing). Each bit adds about 6 dB: DR ≈ 6.02·bits + 1.76.
Bitrate
Uncompressed PCM: rate × bits × channels. Bluetooth figures are maximums (cyan); links step down as the signal weakens. Log scale.
dBFS to watts
0 dBFS is the digital ceiling. A volume setting is attenuation: it subtracts from the dBFS level. The DAC turns 0 dBFS into its full-scale voltage; the amp's gain multiplies that; P = V²/R.
Heat & power budget
Music averages far below its peaks (crest factor). Output power is small; the rest of the input becomes heat inside the cabinet. Class D at ~88%; Class AB efficiency approximated from the class B law η ≈ (π/4)·Vpk/Vrail, so AB wastes most at normal levels.
Supply sizing uses the ⅛-power rule for music, and full-power sine for test.
Supply rail sizing
Peak voltage Vpk = √(2·P·R). A bridged (BTL) output on one supply swings about ±the supply; a single-ended half-bridge swings about half of it. 0.9 allows for output-stage losses, giving power near 1% THD; 10% THD ratings run about 25% higher.
Hiss at the seat
The amp's output noise voltage drives the speaker like a tiny signal: hiss = sens + 20·log(Vn / 2.83 V) − 20·log(d). Below 0 dB SPL it's inaudible; well below the room's own noise it's hidden. Desktop speakers are judged at 60 cm and closer.
Latency budget
DSP buffering ≈ 2 blocks (in and out) ÷ sample rate. A linear-phase FIR adds half its length. Bluetooth figures are typical and vary by device. ITU-R BT.1359: late audio is noticeable past ~125 ms and unacceptable past ~185 ms; instruments and games want under ~20 ms.
Production limits end-of-line frequency response
Each unit's response is compared with the golden sample inside a test window, against ± limit lines. Lines often normalise level at 1 kHz first so overall sensitivity spread doesn't fail units. Yield is a simulated batch of 300 units with the spread you set.
Tighter limits catch more faults but fail more good units. That trade is the end-of-line limit decision.
Gated measurement
Without an anechoic chamber, the measurement is cut off (gated) before the first reflection arrives. The reflection-free window sets the lowest reliable frequency, roughly 1 ÷ window. Below that, measure close to the cone (near-field) and splice.
Pair matching
A level difference between left and right pulls the phantom centre toward the louder speaker (stereophonic law of sines). Pairs are commonly matched within about ±0.5–1 dB across the midrange; driver sensitivity alone can vary ±1–2 dB, which is why matching or DSP trim happens on the line.
Sampling plan · OC curve
Inspect n units from a lot; accept the lot if c or fewer are defective. The OC curve is the chance of accepting a lot at each true defect rate (binomial). Producer's risk: good lots rejected. LTPD: the defect rate accepted only 10% of the time.
Presets follow ANSI/ASQ Z1.4 single sampling, general level II, AQL 1.0; check the standard for your lot.