Playground · research instrument

DSP · Audio

Speaker Power Compression

A lumped thermal RC model of voice-coil heating under continuous drive. Dial in power, cooling architecture, and copper TCR — watch temperature, resistance rise, SPL sag, and the steady-state destination in real time.

Independent research instrument — not claimed as MakerPortal shipped product code. Methods, equations, assumptions, and limitations are disclosed so you can inspect what the page does and does not establish.

Thermal Controls

Input power Continuous RMS (W)150Re at 20°C Voice-coil DC resistance (Ω)6.2Tamb Ambient temperature (°C)20Time acceleration Simulation speed8×

Cooling architecture presets

Rth_v Coil-to-magnet Rth (K/W)1.5Cth_v Coil thermal cap (J/K)15α Copper TCR (1/K) — default 0.003930.00393

Hardware hook:Get the MEMS Microphone Breakout ↗

Coil temp

20.0°C

°C

6.20

Current

4.92

A RMS

Elect. power

150

SPL compression

0.00

Coil tau

22.5

τ (s)

Steady-state T

395

°C

Time history — Temperature & SPL compression0.0 s simulated

Motor cross-section & thermal gradient

Power sweep — Steady-state temperature vs. input power

Essence

Your speaker gets quieter the longer you punish it. That's copper doing math.

Set $P_{in} = 150$ W, walk away, come back five minutes later. The amp still reads 150 W on the dial, but the cone is moving less. You lost 2 dB. Not to the amp, not to the air — to heat.

Every $I^2 R$ watt that doesn't become sound becomes temperature inside a few grams of copper wire. Copper's resistance climbs about $0.39\%$ per degree — $\alpha = 0.00393$ $K^{-1}$ — so as it heats, it draws less current from the same voltage. Less current means less $Bl \cdot I$ force, so less SPL. The system quietly negotiates itself down. Engineers call it power compression, and in live sound it eats headroom for breakfast.

Past 200°C the glue that holds the coil to the former softens. Past 300°C you smell it. This simulation lets you watch that negotiation in real time: temperature, resistance, current, electrical power, SPL sag, all coupled. No hidden knobs.

The intuition — heat has mass and insulation

If you've seen an electrical RC circuit, you already know the thermal one. Temperature is voltage, heat flow is current, thermal resistance $R_{th}$ is how hard it is to push heat out, thermal capacitance $C_{th}$ is how much heat you must dump in to raise the temperature one degree.

Map it: the coil dumps $P_e$ watts into $C_{th\_v}$ , the small fast can — 10 to 15 J/K in the code. It leaks to the magnet through $R_{th\_v}$ . The magnet is the big slow can, $C_{th\_m} = 250$ J/K, barely warming at first, then storing that heat for minutes. It leaks to ambient through $R_{th\_m}$ . Two cans, two time constants:

\tau_v = R_{th\_v} C_{th\_v}

— coil, seconds

\tau_m = R_{th\_m} C_{th\_m}

— magnet, minutes
e.g. standard preset:

\tau_v \approx 22.5

\tau_m \approx 250

Now add copper's twist. $R_e(T) = R_{e0}(1+\alpha(T-20))$ . As $T_v$ climbs, $R_e$ climbs. The amp is modeled as constant-voltage — $V_{in}^2 = P_{input} R_{e0}$ — so actual electrical power $P_e = V_{in}^2 / R_e(T)$ drops as it heats. That's a stabilizing negative feedback on temperature, but a direct hit to SPL. You trade loudness for survival.

Hot coil $\to$ higher $R_e$ $\to$ lower $I = V_{in}/R_e$ $\to$ lower BlI.
Lower $I$ $\to$ slightly less heating, so temperature asymptotes rather than running away.
But $SPL_{comp} = -20\log_{10}(R_e/R_{e0})$ keeps sliding negative — you hear it as compression.

Short bursts mostly charge $C_{th\_v}$ . Long, heavy drive also charges $C_{th\_m}$ , and that magnet won't cool quickly when you stop. That is why a speaker can sound fine for one song and dull for the next, even at the same meter reading.

Think of it like

Two buckets connected by a slow pipe. You pour water (power) into the small top bucket (coil). It drains through a narrow hose ( $R_{th\_v}$ ) into a huge bottom tank (magnet). The top bucket's water level is temperature. Now imagine the faucet gets stingier as the top bucket fills — higher water level pinches the inlet. Level stabilizes, but flow out the BlI tap drops. That's compression.

Where architecture matters

Standard basket: $R_{th\_v} \approx 1.5$ K/W, sluggish cooling
Vented pole: $R_{th\_v} \approx 0.8$ , much better exhaust
Ferrofluid: $R_{th\_v} \approx 0.4$ , fluid bridges the gap

Same $C_{th\_v}$ range (10–15 J/K) but cutting $R_{th\_v}$ by 3× cuts $\tau_v$ and the final temperature almost proportionally. Check the lower plot — curves separate hard at high power.

The math — from copper physics to code

1. Copper doesn't keep its resistance

The base fact of this whole sim is linear TCR:

R_e(T) = R_{e0}\,[1 + \alpha\,(T - 20)]

\alpha = 0.00393\,K^{-1} \text{ (copper, code }ALPHA\_CU\text{)}

At 150°C, $R_e \approx 1.51 \times R_{e0}$ . A nominal $6.2\,\Omega$ coil is now $9.4\,\Omega$ . The amp hasn't changed. Ohm's law has.

2. Constant-voltage amp model

The slider says "Input power" in RMS watts. That value is interpreted as power into a cold coil. Code stores it as $P_{input}$ and defines the drive voltage once:

V_{in}^2 = P_{input}\,R_{e0}

P_e(T_v) = \frac{V_{in}^2}{R_e(T_v)} = \frac{P_{input}R_{e0}}{R_e(T_v)}

I(T_v) = \frac{V_{in}}{R_e(T_v)}

So $P_e$ falls as the coil heats. If you wanted a constant-power source, you'd hold $P_e$ fixed and temperature would climb faster. Real amps behave closer to constant-voltage into a resistive load, so this is the honest choice for SPL loss.

3. Two coupled thermal ODEs

This is the lumped ladder the code actually integrates with explicit Euler every $PHYSICS\_DT = 0.02$ s:

\begin{aligned} \frac{dT_v}{dt} &= \frac{P_e(T_v) - \frac{T_v - T_m}{R_{th\_v}}}{C_{th\_v}} \\ \frac{dT_m}{dt} &= \frac{\frac{T_v - T_m}{R_{th\_v}} - \frac{T_m - T_{amb}}{R_{th\_m}}}{C_{th\_m}} \end{aligned}

Inside stepPhysics(), heatFlowVC_to_Mag = (T_coil - T_mag)/Rth_v and heatFlowMag_to_Amb = (T_mag - Tamb)/Rth_m. The update is T_coil += dT_coil with dT_coil = (Pe - flow)/Cth_v * dt. That is exactly the math above.

The rolling chart keeps $WINDOW\_SECONDS = 60$ simulated seconds, sampling about 30 Hz, with MAX_BUFFER = 1800 points. Time reads as simulated seconds, not wall-clock — accelerated by the slider up to $40\times$ , safely stepping with an accumulator and a 200-iteration cap per frame.

4. SPL compression from lost current

Force is $Bl \cdot I$ . If $Bl$ held constant, $F \propto I \propto 1/R_e$ . Sound pressure is proportional to force, so:

\Delta L = -20\,\log_{10}\!\left(\frac{R_e(T_v)}{R_{e0}}\right)\;[\text{dB}]

At $1.5\times R_{e0}$ , that's -3.5 dB — you need $2.2\times$ electrical power to make up what heat stole. The top canvas draws this alongside temperature, so you see compression lag temperature slightly — it tracks $R_e$ , which tracks $T_v$ .

5. Where it settles — steady-state solve

Set derivatives to zero. Heat in equals heat out through the series $R_{th}$ :

\frac{V_{in}^2}{R_{e0}[1+\alpha(T_{ss}-20)]} = \frac{T_{ss}-T_{amb}}{R_{th\_v}+R_{th\_m}}

That's implicit in $T_{ss}$ because $R_e$ depends on it. Code function steadyStateTemp(P_input, Re0, Tamb, alpha, Rth_v, Rth_m) Newton-iterates 60 times: residual $r = P_e - (T-T_{amb})/R_{th}$ , derivative $dr/dT = dP_e/dT - 1/R_{th}$ , $T \gets T - r/(dr/dT)$ . Initial guess ignores TCR: $T_{guess} = T_{amb} + P_{input}R_{th}$ . Result shows as the dashed $T_{ss}$ line and the "Steady-state T" badge. The power-sweep canvas uses the same solver at 100 power points up to $2\times P_{input}$ .

State vector

$\mathbf{x} = [T_v, T_m]^{\top}$

Two states, two time scales. Code keeps them as T_coil and T_mag, plus simulatedTime. No hidden integrators pulling tricks.

Variables you actually touch

$R_{th\_v}$ coil-to-magnet K/W — slider $rth\_v$
$C_{th\_v}$ coil J/K — slider $cth\_v$
$R_{th\_m}$ , $C_{th\_m}$ magnet side — preset only
$\alpha$ copper TCR — slider, default 0.00393
$P_{input}$ , $R_{e0}$ , $T_{amb}$ direct controls
$PHYSICS\_DT=0.02$ , $WINDOW\_SECONDS=60$

Time constant decoded

$\tau = R_{th}C_{th}\;[s]$

After one $\tau$ , the step response reaches $63\%$ . Standard basket: $\tau_v=1.5\times15=22.5$ s. You see it on the top plot — knee around 20–30 s. Ferrofluid slashes $R_{th\_v}$ to 0.4, so $\tau_v \approx 4$ s and $T_{ss}$ drops dramatically, but $C_{th\_v}$ also 10 J/K in that preset, keeping it responsive.

Integrator honesty

Forward Euler, fixed 0.02 s, is stable here because thermal times are seconds, not microseconds. The accumulator pattern decouples sim time from frame rate, and time acceleration is a multiplier on elapsed wall time. No implicit solver, no artificial damping — what you see is the ODE, stepped honestly.

Playbook — what to push, what to watch

Start here

150 W, Standard basket, 20°C. The default. Hit Play — coil climbs toward $\sim 150$ –200°C depending on $R_{e0}$ , SPL sags about -1 to -2 dB over 60 s. Watch $\tau_v$ : time readout shows the early rise is steep, then bends as magnet warms. That's $C_{th\_m}$ waking up.

Crank it

1000 W. Same cooling, $5\times$ heat. $T_{ss}$ in the code's Newton solver jumps beyond 400°C — well past the 200°C adhesive warning. Current drops from $\sim 10$ A to $\sim 6$ A, electrical power actually falls as it heats. You asked for 1000 W, you get $\sim 600$ W hot. The red banner is not theatrical.

Swap to Ferrofluid. Same 1000 W now settles near 180–220°C, SPL sag about half. The power-sweep chart shows why: the orange-red curve (your current architecture) pulls down to green when you click that preset. Ferrofluid bridges the 0.2 mm gap with liquid that conducts heat — $R_{th\_v} = 0.4$ vs 1.5. Physics you can buy.

Find the constants

Measure $\tau$ . Pause at ambient, hit Play, time how long to get $63\%$ of $T_{ss}-T_{amb}$ . That's $\tau$ — compare to $R_{th\_v}C_{th\_v}$ badge. Now lower $C_{th\_v}$ to 0.5 J/K (custom). Rise becomes near-instant — you removed thermal inertia. Raise it to 50 J/K and you get a slow cooker.

Ambient matters. Set $T_{amb} = 40°C$ — a summer stage. $T_v$ baseline lifts by 20 K, $T_{ss}$ lifts by the same plus a little extra because hotter start means higher $R_e$ means a touch less power. Small, but real.

Tweaks that matter

Time acceleration

Physics $dt$ stays $PHYSICS\_DT = 0.02$ s — speed just runs more steps per render frame (capped at 200 per frame). At $40\times$ you watch minutes of thermal drift in seconds. At $1\times$ you can see the sampling — data points every 0.033 s — and feel how $\tau_m$ really dominates long runs.

Power sweep + show all architectures

Top marker shows current live $T_{coil}$ vs steady-state $T_{ss}$ at $P_{input}$ . When they diverge, you're still in transient. Click "Show all architectures" — three dashed curves use PRESETS directly: standard 1.5/1.0, vented 0.8/0.4, ferrofluid 0.4/0.3 for $R_{th\_v}/R_{th\_m}$ . Slope at low power is $R_{th\_v}+R_{th\_m}$ : linear for cold.

200°C warning

At 200°C the SVG coil turns orange-red via coilHexColor() and the top plot shades the danger zone. Not arbitrary — most voice-coil adhesives (polyimide) soften around there. The simulation still runs hotter, but reality would have already distorted or seized.

Where the code lives

steadyStateTemp() Newton solve, stepPhysics() Euler ladder, renderTimeCanvas() dual stacked plot with Tamb and Tss asymptotes, renderSweepCanvas() pre-computed 100-point curves. Sampling interval 0.033 s, $WINDOW_SECONDS=60$ window trim. No magic.

Honesty — what this is and isn't

What is real physics

Lumped two-time-constant thermal network, copper TCR $\alpha=0.00393$ , constant-voltage interpretation $V_{in}^2 = P_{input}R_{e0}$ , coupled ODEs as written, Newton solve for steady-state, SPL compression as $-20\log_{10}(R_e/R_{e0})$ . Time constants $\tau=R_{th}C_{th}$ behave exactly as labeled. Numbers track a real 15" woofer within a factor of two.

What it simplifies

Uniform temperature — real coil has hot spots near the gap exit. Constant $R_{th}$ , $C_{th}$ — in reality convection scales with $\Delta T$ and gap airflow modulates with excursion. No radiation term $\propto T^4$ . No frequency dependence — music is peaky, RMS is a crude average. No eddy-current heating in pole pieces, no $Bl(T)$ drop from magnet demagnetization, no former material or adhesive thermal conductance, no excursion-driven pumping. The model overestimates low-frequency heating and underestimates HF cooling — take it as a clear, teachable bounding case.

If you want to push further: make $R_{th\_v}$ a function of $x_{coil}(t)$ to model pumping, add a third node for pole piece with its own $C$ , let $Bl = Bl_0[1 - \beta(T_m-20)]$ for ferrite tempco, or drive $P_e(t)$ from a real audio envelope instead of continuous sine. All fit inside the same ladder — more nodes, same idea.

Anatomy of the simulator

Every slider, pixel, and readout traces back to the two coupled thermal ODEs. Here is what each piece of the dashboard is doing and why it is built that way.

The time-history canvas

01
Dual stacked plot. Top half: coil temperature $T_v$ in red, with a dashed $T_{amb}$ baseline and a dotted $T_{ss}$ asymptote. Bottom half: SPL compression in blue — same $-20\log_{10}(R_e/R_{e0})$ formula, different axis. The two traces share one x-axis so you see compression lagging temperature directly.
02
Danger zone shading. The region above 200°C is tinted red. It is not cosmetic — the warning banner fires from the same threshold. The shade persists even if the actively controlled slider briefly pushes past.
03
Rolling window. Only the most recent $WINDOW\_SECONDS = 60$ simulated seconds are visible. The buffer trims with a while-loop on each data push — MAX_BUFFER = 1800 points at ~30 Hz. The x-axis always ends at the current time, so you read left-to-right like an oscilloscope.
04
Live dot indicator. The most recent data point is a filled circle on each trace — red on the temperature curve, blue on the SPL curve — so you can tell at a glance where the sim is right now relative to the history.

Controls, readouts, and the motor SVG

01
Seven metric readouts. Coil temp (color-coded: green below 80°C, yellow to 150°C, orange to 200°C, red above), $R_e$ (live from TCR), current, electrical power (drops as resistance climbs), SPL compression in dB, coil thermal time constant $\tau_v = R_{th\_v}C_{th\_v}$ , and the Newton-solved steady-state $T_{ss}$ . They all derive from one state vector $[T_v, T_m]^{\top}$ .
02
Cooling architecture presets. Standard, vented pole-piece, ferrofluid — each swaps $R_{th\_v}, C_{th\_v}, R_{th\_m}, C_{th\_m}$ from the PRESETS map. The custom button unlocks the thermal-parameter sliders for manual tweaking. Toggling back to a preset restores its exact values.
03
Motor cross-section SVG. Voice-coil rectangles are colored by the coilHexColor() function — a three-segment gradient from copper-orange through red to danger-red based on temperature normalized against $T_{amb}$ . Heat-flow arrows fade opacity with the temperature gradient. Not a physics solver, just an honest visual indicator.
04
Power sweep canvas. Pre-computes $T_{ss}$ at 100 power points up to $2\times P_{input}$ using the same Newton solver. The live operating point is a filled circle; a separate hollow circle shows current $T_{coil}$ when it diverges from steady-state (transient). The "Show all architectures" toggle overlays dashed curves for standard, vented, and ferrofluid presets — revealing how $R_{th\_v}$ separates the curves at high power.
05
The render loop. Runs a fixed $PHYSICS\_DT = 0.02$ s Euler step via an accumulator pattern decoupled from frame rate. Time acceleration (up to $40\times$ ) is a multiplier on wall-clock elapsed time, capped at 200 steps per frame to prevent spiral-of-death. All canvases repaint every frame; the sim pauses when off-screen via IntersectionObserver.

Gear behind this build

Akous · DSP audio stack · 33 picks

Audio DSP hardware33

$159.00Headphones

Audio-Technica ATH-M50X Professional Studio Monitor Headphones, Black, Professional Grade, Critically Acclaimed, with Detachable Cable

Reference monitoring headphones used for akous's binaural audio testing.

Speaker Power Compression

Thermal Controls

Motor cross-section & thermal gradient

Your speaker gets quieter the longer you punish it. That's copper doing math.

The intuition — heat has mass and insulation

The math — from copper physics to code

1. Copper doesn't keep its resistance

2. Constant-voltage amp model

3. Two coupled thermal ODEs

4. SPL compression from lost current

5. Where it settles — steady-state solve

Playbook — what to push, what to watch

Start here

Crank it

Find the constants

Honesty — what this is and isn't

Anatomy of the simulator

The time-history canvas

Controls, readouts, and the motor SVG

Gear behind this build

Audio DSP hardware33

Audio-Technica ATH-M50X Professional Studio Monitor Headphones, Black, Professional Grade, Critically Acclaimed, with Detachable Cable

Comimark 1Pcs ADMP401 MEMS Microphone Breakout Module Board for Arduino Universal 1.3cm*1cm

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Behringer UMC1820 Audiophile 18x20 USB Audio/MIDI Interface with Midas Mic Preamplifiers and ADAT I/O | For Recording Microphones and Instruments

Bit Trade One BIMIMI - Binaural Stereo Microphone Easy BINAURAL Recording/ASMR/Podcast/Recording/Streaming / W7.2×H1.9×D2.6 inches/ADBMM

Blue Yeti Nano Premium USB Microphone - Shadow Grey (Renewed)

Digital Signal Processing

High Speed Digital Design: A Handbook of Black Magic (Prentice Hall Modern Semiconductor Design)

HiLetgo USB Logic Analyzer Device with EMI Ferrite Ring USB Cable 24MHz 8CH 24MHz 8 Channel UART IIC SPI Debug

IK Multimedia iRig Pro Duo I/O USB audio interface, TRS balanced & headphones outputs, audio mixer to 24-bit, midi interface for music studio, recording, podcasting, streaming & social apps

Loudspeaker Design Cookbook

Microwave Engineering

SEESII Upgraded NanoVNA-H4 Vector Network Analyzer, Latest V4.4 9KHz-1.5GHz HF VHF UHF 4" Touch Screen VNA Antenna Analyzer Measures S Parameters,Voltage Standing Wave Ratio, Phase,Delay, Smith Chart

RF Circuit Design

RØDE NT-USB+ Professional-Grade USB Condenser Microphone For Recording Studio Quality Audio Directly To A Computer Or Mobile Device, Black

RTL-SDR Blog V4 R828D RTL2832U 1PPM TCXO HF Bias Tee SMA Software Defined Radio with Dipole Antenna Kit

The Scientist & Engineer's Guide to Digital Signal Processing

Sennheiser Pro Audio Sennheiser Pro Audio Wireless Microphone System, Black (MKE600)

Teensy 4.0

Shure MV7+ Podcast Dynamic Microphone with Stand – OBS Certified, Enhanced Audio, LED Panel, USB-C & XLR Outputs, Auto Level Mode, Digital Pop Filter – for Podcasting, Streaming, and Recording, Black

Shure MV88+ Video Kit Digital Stereo Condenser Microphone for iPhone, Android, Mac & PC - Portable Recording Mic with DSP Controls, Headphone Monitoring & Tripod, Black

Signals and Systems (Prentice-hall Signal Processing Series)

AURSINC SMA Calibration Kit and SMA Male to Male Jumper for NanoVNA H H4 F V2 V3 SAA Mini1300 PS100 Vector Network Analyzer Antenna Analyzer

Sony MDR7506 Professional Large Diaphragm Headphone

Immersive Sound: The Art and Science of Binaural and Multi-Channel Audio (Audio Engineering Society Presents)

Teensy 4.0 (Headers)

AURSINC TinySA Ultra+ ZS406 Spectrum Analyzer with 50W N-Type 50dB RF Attenuator, 100kHz-5.4GHz, 4.0" Touchscreen HW V0.4.6, 2-in-1 Signal Generator, DC-3GHz 50Ω Ham Radio Test Kit

miniDSP UMIK-1 USB Measurement Calibrated Microphone

Understanding Digital Signal Processing

SABRENT USB External Stereo Sound Adapter for Windows and Mac. Plug and Play No Drivers Needed. (AU-MMSA)

miniDSP UMA-8 USB Microphone Array

Sennheiser HD 650 Open-Back Headphones

SR3D® Dummy Head MKIII

Two gotchas worth knowing

The solver, copyable

Frequently asked questions

The instrument, captured—not illustrated.