⚙️ ENGINEER LEVEL: Transfer Functions and System Modeling

Closed-Box Transfer Function

The low-frequency response of a sealed system is a second-order high-pass function:

H(s) = s² / (s² + (ωc/Qtc)×s + ωc²)

Where ωc = 2πFc

Magnitude response:

|H(jω)| = (ω/ωc)² / √[(1 − (ω/ωc)²)² + (ω/(ωc×Qtc))²]

Group delay:

τg(ω) = [2ω/ωc² × (1 − (ω/ωc)²) + (ω/ωc)³/Qtc] / [(1 − (ω/ωc)²)² + (ω/(ωc×Qtc))²]

Peak group delay at Fc for Butterworth (Qtc=0.707):

τg_peak = √2 / ωc = √2 / (2πFc)

At Fc = 50 Hz: τg_peak = 4.5 ms — well within inaudibility threshold at bass frequencies.

Power compression modeling:

Voice coil temperature rise:

ΔT = P_input × Re / (thermal_resistance × Rvc_cold)

Resistance rise:

R_hot = R_cold × (1 + α × ΔT)

Where α = 0.00393 /°C for copper

Power compression:

PC_dB = 20 × log₁₀(√(R_cold/R_hot))

At 200°C voice coil temperature (sustained heavy use):

R_hot = R_cold × (1 + 0.00393 × 200) = R_cold × 1.786
PC = 20 × log₁₀(√(1/1.786)) = 20 × log₁₀(0.748) = −2.5 dB

A driver that measured 105 dB at startup may only produce 102.5 dB after thermal equilibrium — significant in competition where every dB matters.