Load, Amplifier & Enclosure
The triangle that shapes bass — the impedance curve as a dialogue between driver and amplifier, with the enclosure writing the script
The triangle is a coupled loop, not a one-way arrow
The same driver on two different amplifiers gives two different bass characters. The answer isn’t “watts” — it’s the three-way relationship between the driver’s impedance (the load), the amplifier’s output impedance (its character), and the enclosure (which shapes the load). Most texts treat this as a one-way arrow; in reality it is a closed loop.
The amp drives the cone through its own output resistance; that resistance modifies the driver’s Q; the box takes the modified Q and shapes the bass response while reshaping the impedance curve; and that curve loops back to set how much the amp lets the response deviate — back to the amplifier again.
Put an equation on every arrow in the loop, so you can reverse-engineer “how far does Qtc move when I swap amplifiers.” Every chart is computed live from the equivalent circuit of one example driver, so the values in each eq-note cross-check against the plotted curves.
The load is a curve, not a single number
“8 ohms” is a coarse average. The real impedance curve rises and falls with frequency, and its shape is what the amplifier must cope with. This curve is the electrical equivalent circuit into which the driver’s mechanical behaviour is reflected through the force factor Bl.
Fig 1. Impedance magnitude of the example driver in free air, straight from the equivalent circuit — peak 47.5 Ω at 37.5 Hz (fs), down to ~6 Ω in the midband, then climbing slowly with ωLe
So always read two numbers from a spec — the nominal average and the minimum, the point where the amplifier works hardest. An 8 Ω driver dipping to 3 Ω is as demanding as a 4 Ω one.
Peak, minimum and phase — what really squeezes the amp
Phase is a direct by-product of the parallel-RLC: below fs the curve is inductive (positive phase), at fs phase = 0 (a purely resistive peak), above fs it is capacitive (negative phase), then swings positive again in the Le region.
Fig 2. Phase angle of the same driver — positive below fs, crossing zero at the peak, then negative above fs. Where the phase swings hardest is where the amp’s output devices burn hottest
Output-transistor dissipation depends on the instants where voltage and current are both high. A purely resistive load keeps V and I in phase (little heat where V is high, because I is low), but a phase-shifted load overlaps high-V with high-I → the output device burns more than equal rms current would suggest.
IEC 60268-5 requires the rated impedance to be a value such that the minimum |Z| in the working band is no less than 80% of nominal (8 Ω → never below 6.4 Ω). In practice many loudspeakers violate this, and that dip is exactly where the amplifier is loaded hardest.
Where T–S parameters come from: one mechanism, many numbers
T–S parameters aren’t independently measured numbers; they are consequences of a single electro-mechanical mechanism.
Output impedance and damping factor
The single spec that sets how an amp copes with the impedance curve is its own output impedance, usually reported as damping factor.
| Amplifier | R_out | DF (re 8 Ω) | Character |
|---|---|---|---|
| SS | 0.08 Ω | 100 | near voltage source — flat |
| Tube PP | 1.0 Ω | 8 | in between |
| SET | 2.67 Ω | 3 | near current source — follows the curve |
A good solid-state amp has very low Rout, behaving almost like a perfect voltage source — holding voltage constant no matter how the load swings, giving flat response regardless of the impedance shape. A tube amp has much higher Rout because of its output transformer, edging toward a current source; the voltage across the load = I·Z(f), so it rises at the impedance peaks.
The hidden lever: R_out modifies the driver’s Q
Damping factor is only the surface. What actually changes the sound is the effect of Rout on Qes: the total series resistance (amp + speaker cable) adds electrical damping as seen by the driver.
| Amplifier | R_g (≈) | Q_es′ | Q_ts′ | Δ vs SS |
|---|---|---|---|---|
| SS (DF100) | 0.18 Ω | 0.446 | 0.385 | — |
| Tube PP (DF8) | 1.1 Ω | 0.512 | 0.437 | +16% |
| SET (DF3) | 2.77 Ω | 0.633 | 0.523 | +38% |
Fig 3. Same 15 L sealed box (fc=64.9 Hz) — the SET amp pushes Qtc to 0.91, producing a ~+1.5 dB bass hump, while the SS amp lands near a softer Bessel alignment
SOA and peak current — why a reactive load is brutal on the amp
A purely resistive load gives a straight load line on the Vce–Ic plane. A load of Z∠φ shifts voltage and current out of phase, turning the V–I trajectory into an ellipse that reaches into the high-voltage/high-current corner → output-device dissipation overshoots the resistive case, and the ellipse can fall outside the Safe Operating Area.
Otala & Huttunen (1987) measured real commercial loudspeakers by synthesising a Brune equivalent circuit and driving it with a test signal, finding loads drawing peak currents up to 6.6× what an 8 Ω resistor would draw — the combined effect of low Zmin + phase angle + back-EMF + crossover interaction.
This is why “watts” alone says nothing — an amp needs current capability and an SOA that handles a real reactive load, not just power into a paper 8 Ω resistor. Driver/crossover design that raises the minimum and controls phase directly eases the SOA burden.
Sealed box — second-order high-pass, single peak
This is where the designer takes control: the box type sets both the impedance shape and the bass behaviour at once. A sealed box adds air-spring stiffness, pushing fs→fc and raising Q by the same factor.
Fig 4. The sealed box pushes fs 37.5 → fc 64.9 Hz; the peak stays a single peak but shifts right — the curve stays smooth and predictable
Vented box — Helmholtz, fourth-order high-pass, twin peaks
The port is a Helmholtz resonator adding a second resonance; the air mass in the port works against the air spring in the box.
Fig 5. The vented box splits the single peak into two peaks straddling a dip at fb=40 Hz (peaks ~20.2 and 73.8 Hz). Approximate relation: fL·fH ≈ fs·fb
The upside is more depth and output than a sealed box of equal size, but below fb the rolloff steepens to 24 dB/oct with more group delay.
Aperiodic box — self-damped, amp-independent
The aperiodic box sits in the middle — a quasi-sealed box with a vent stuffed with an acoustic resistance (packed fibre / variovent). This resistance acts like the cone’s shock absorber, dissipating energy at resonance; reflected into the mechanical domain it adds directly to Rms.
Fig 6. The acoustic resistance presses the peak from 47.5 down to ~21.4 Ω into a single broad hump — the smoothest curve of the three box types
| Box | Zmax | Curve shape | Qtc amp-dependent? |
|---|---|---|---|
| Sealed | 47.5 Ω | single high peak | yes |
| Vented | ~55 Ω | twin peaks + dip | partly |
| Aperiodic | 21.4 Ω | low broad hump | barely |
The cone is damped by the box’s acoustic losses, not the amp’s electrical damping → Qtc stops depending on Qes′ (the amp). This is the root of “works with any amp” — you move the damping burden from the amp into the box.
Ripple from R_out — “tube follows the curve,” quantified
When Rout is non-zero, amp and load form a voltage divider. At impedance peaks more voltage reaches the driver (louder); at the dips more voltage drops across Rout (quieter).
Fig 7. Response deviation (peak-normalised) on the same sealed box — the SET lets ~2.7 dB of ripple track the impedance peak at fc, while the SS amp is nearly flat
Matching it up + conjugate networks
Put the three together and the impedance curve is the dialogue, the box is the scriptwriter. With a high-damping SS amp you have near-total freedom — pick a vented box and chase depth. With a tube amp the game changes: you want a flat curve + self-damped bass.
Pick a smooth-curve box (sealed/aperiodic) plus a conjugate network if needed, so the term Z/(Z+Rout) ≈ constant. A high-Rout amp then produces no ripple along the curve.
Use acoustic resistance (aperiodic) to damp the cone with acoustic losses, cutting Qtc loose from Qes′ (the amp).
Minimum and flatness are different problems — Zmin is a resistive floor set by the driver + crossover. A box or L/C network can’t raise it without burning power. The box fixes flatness and damping, not the current floor the amp must supply. A tube amp must still check Zmin to pick its tap, even after the curve is flattened.
A loudspeaker isn’t “8 ohms” — it’s a curve, and that curve is a dialogue with your amp. A solid-state amp is indifferent to the script; a tube amp answers every line. The box designer’s job is to write that script well — shape it to a favourite amp, or make it flat and self-damped so it plays with anything. Either is valid; just know what you’re shaping.
References
- aesThiele, A. N. “Loudspeakers in Vented Boxes, Part I & II,” JAES 19(5):382–392 (1971); 19(6):471–483 (1971).
- aesSmall, R. H. “Closed-Box Loudspeaker Systems — Part I & II,” JAES 20(10):798–808 (1972); 21(1):11–18 (1973).
- aesSmall, R. H. “Vented-Box Loudspeaker Systems — Parts I–IV,” JAES 21 (1973).
- aesOtala, M. & Huttunen, P. “Peak Current Requirement of Commercial Loudspeaker Systems,” JAES 35(6):455–462 (1987).
- stdIEC 60268-5, Sound system equipment — Part 5: Loudspeakers.
- bookBeranek, L. L. & Mellow, T. J. Acoustics: Sound Fields and Transducers, Academic Press 2012.
- bookSelf, D. Audio Power Amplifier Design, 6th ed., Focal Press 2013.
- bookDickason, V. The Loudspeaker Design Cookbook, 7th ed., Audio Amateur Press 2006.