Hardware Design¶
cftscal measures whatever hardware you point it at, but a calibration can only be as good as the signal chain underneath it. This page covers the electrical and acoustic limits worth working out before you calibrate — how hard you can safely drive a speaker, how much output to expect from it, how to bring a DAC's voltage down to a level the speaker can take, and the resonances that can put a peak or notch in a measured response that has nothing to do with the device you think you are measuring.
None of this is something cftscal computes for you. It is the arithmetic you do once, on paper, when you build or modify a rig.
Power, voltage, and current¶
Speaker sensitivity is typically reported on a datasheet in \(\frac{dB}{W}\) at a distance of 1 meter — so many dB SPL for one watt of input. To use that number you need to convert between watts and volts, which requires the speaker's impedance.
Starting from \(P = I^2 \times R\) and \(V = I \times R\), and solving for \(I\):
which gives the two forms you actually use:
(\(R\sqrt{P/R}\) and \(\sqrt{PR}\) are the same thing; the second is easier to evaluate in your head.)
Worked example. For an 8 Ω speaker, \(8 \times \sqrt{1/8} = 2.83\) V produces exactly 1 W — which is why datasheet figures for 8 Ω drivers are often quoted at 2.83 V. Checking the other direction: \(2.83^2 / 8 = 1.00\) W.
Maximum safe drive voltage¶
Work backwards from the speaker's power rating to the voltage you must not exceed.
Worked example. An 8 Ω speaker with a 0.5 W handling capacity:
There is no point in exceeding this voltage
Even if your system can generate a larger value, driving the speaker above its rated voltage does not buy you more usable output — it will simply distort, or damage the driver. The calibration will happily record the distorted response as though it were real.
Your system also has to supply the current that voltage implies:
A DAC output or op-amp that cannot source 0.25 A into 8 Ω will clip on current even though its voltage looks fine, which shows up in a calibration as compression at high levels.
These are RMS values; datasheets often quote peaks
Power ratings, and therefore the 2 V and 0.25 A above, are RMS quantities. A converter's full-scale output, and an op-amp's output swing and current limit, are usually peak figures. For a sine wave the two differ by \(\sqrt{2}\):
| RMS | Peak (sine) | |
|---|---|---|
| Voltage | 2.00 V | 2.83 V |
| Current | 0.25 A | 0.35 A |
So the 2 Vrms limit means the amplifier has to swing ±2.83 V cleanly without clipping. Compare like with like before concluding you have headroom — and note that a broadband stimulus such as noise or a chirp has a higher peak-to-RMS ratio than a sine, so it needs more headroom still for the same RMS level.
Many datasheets give both a long-term (continuous) and a short-term (peak) rating. Compute the voltage limit for each, and treat the continuous figure as your working limit:
| Rating | Power | Max. voltage into 8 Ω |
|---|---|---|
| Long-term / continuous | 0.3 W | 1.55 V |
| Short-term / peak | 0.5 W | 2.00 V |
These are nominal specs
Every number in this section comes off a datasheet, and datasheet figures are nominal. Treat them as a ceiling to stay well below, not a target to hit, and verify the actual behavior with a calibration.
Estimating maximum output in dB SPL¶
Once you know the maximum power you can put into the speaker, you can convert the datasheet's rated SPL to the SPL you will actually be able to reach. Power ratios are \(10 \times log_{10}\):
Worked example. A datasheet reports 92 dB at 0.3 W, and you have determined you can safely drive 0.5 W:
So you gain only 2.2 dB, for a maximum of 94.2 dB SPL — doubling the power buys 3 dB, and this is not even a doubling. If instead you can only manage 0.1 W, \(10 \times log_{10}(0.1/0.3) = -4.8\) dB, i.e. 87.2 dB SPL.
Note that this scales the datasheet's SPL figure, which is quoted at 1 metre on axis. If your speaker sits a few centimetres from the ear, or is coupled into a tube, the absolute level will be nothing like the datasheet number — but the \(10 \times log_{10}\) power scaling still tells you how much more you can get out of it than at the rated power.
This calculation is the fastest way to find out early that a speaker cannot physically reach the levels your experiment requires — long before you have wired up a rig and discovered it during calibration.
Sizing a voltage divider¶
If your DAC's full-scale output exceeds the speaker's safe voltage, you need to bring it down. A series resistor forms a voltage divider with the speaker's own impedance:
Solving for the series resistor:
Worked example. An 8 Ω speaker that must not see more than 2 Vrms, driven from a source that delivers 10 Vrms at full scale:
Both voltages must be expressed the same way — both RMS, or both peak. Mixing a peak full-scale figure with an RMS limit gives a divider that is wrong by \(\sqrt{2}\), i.e. 3 dB.
Using a divider rather than just turning the software level down is worth it when you want to use the full range of your DAC: attenuating in software throws away bits of resolution and leaves you closer to the converter's own noise floor, whereas attenuating in hardware lets the DAC run near full scale where its signal-to-noise ratio is best.
Size the resistor for power, not just resistance
Most of the power now lands in the resistor, not the speaker. In the example above the current is \(10\,V / 40\,\Omega = 0.25\) A, so the series resistor dissipates \(0.25^2 \times 32 = 2\) W while the speaker gets 0.5 W. A common 1/4 W resistor would cook. Pick one rated well above the dissipation you calculate, and remember the source has to supply the full 2.5 W.
A series resistor also changes the speaker's damping
Adding resistance in series raises the source impedance the driver sees, which reduces its electrical damping and alters the response around its resonance. This is not a reason to avoid the divider — you calibrate the system as built, and the calibration captures whatever the response turns out to be — but it does mean the divider is part of the system, so changing or removing it invalidates the calibration.
Account for gain elsewhere in the chain
\(V_{out}\) is the voltage that actually arrives at the divider, which is not necessarily the DAC's output. Do not forget to compensate for any gain built into an op-amp or buffer circuit between the DAC and the speaker.
Cable resonance and grounding¶
Signal cables resonate when their physical length is a quarter wavelength of the signal they carry. The wavelength of an electrical signal is set by the speed of light:
Running the numbers for the audio range makes the conclusion obvious:
| Frequency | Quarter wavelength |
|---|---|
| 100 Hz | 749,481 m (749 km) |
| 1 kHz | 74,948 m |
| 100 kHz | 749 m |
A 3 m cable resonates at about 25 MHz — three orders of magnitude above anything in an acoustic measurement.
Cable resonance is a non-issue for acoustic work
Since nobody is running 750 meters of cable inside a sound booth, cable resonance can be ruled out as a source of artifacts across the entire 100 Hz – 100 kHz range. If you are chasing noise or an odd response, look at grounding, shielding, and ground loops instead — not cable length.
(Strictly, a signal travels through real cable at a velocity factor of roughly 0.6–0.85 of \(c\), depending on the dielectric, so the lengths above are overestimates by that factor — 100 kHz needs more like 450–640 m rather than 749 m. That does not change the conclusion in the slightest.)
Acoustic tube resonance¶
Acoustic resonance is a completely different story, because sound travels at roughly 340 m/s instead of \(3 \times 10^{8}\) m/s. The relevant lengths therefore come out in millimeters, which is exactly the scale of a probe tube, a coupler, or an ear canal.
The wavelength of a 14 kHz tone in air is only \(340 / 14000 = 24.3\) mm; its quarter wavelength is 6.1 mm.
For a tube of length \(L\), the resonances depend on its boundary conditions — and so does the pattern of higher modes, which is easy to get wrong:
| Tube | Lowest resonance | Higher modes | 20 mm tube |
|---|---|---|---|
| Closed at one end (quarter-wave) | \(f_1 = \frac{c}{4L}\) | odd multiples: \(3f_1, 5f_1, \ldots\) | 4.25, 12.75, 21.25 kHz |
| Open at both ends, or closed at both (half-wave) | \(f_1 = \frac{c}{2L}\) | all integer multiples: \(2f_1, 3f_1, \ldots\) | 8.5, 17.0, 25.5 kHz |
A quarter-wave tube (one open end, one closed — the closest simple model for a probe tube sealed against an eardrum) skips the even multiples entirely: its modes are at \(c/4L\), \(3c/4L\), \(5c/4L\). Only the half-wave case has modes at every integer multiple. Either way, a single tube puts a whole series of peaks and notches into the response, not just one.
These are idealizations
Real probe tubes are neither ideally open nor ideally closed at their ends, have finite wall losses, and couple into a cavity rather than free space, so measured resonances land near these frequencies rather than exactly on them, with finite Q. Use the formulas to know roughly where to expect structure, not to predict a measured curve. The speed of sound itself is also only roughly 340 m/s — it rises about 0.6 m/s per °C — so across a realistic range of room temperatures every frequency in the table moves by a percent or two.
This is what most odd-looking notches actually are
A 20 mm probe tube has resonances squarely inside the frequency range of a typical CFTS measurement. This is why the Starship and Starship Check workspaces care so much about the probe tube being fully seated and unobstructed, and why the coupler used for a calibration has to match how the starship is actually used: change the length of the acoustic cavity and you move every one of these resonances.
A calibration correctly measures these resonances and compensates for them, so they are not inherently a problem — but they only stay compensated if the geometry does not change between calibration and use.