SLAAE48 Application note

SLAAE48 May 2025 TAS5825M

2.1 Speaker Basics and Models

Typical structure of the speaker can be presented in Figure 2-1. With alternating current at certain frequency applied to the voice coil, magnetic force is generated between the magnet and the voice coil, and drive the attached cone membrane (all the moving parts including the cone, dust cap, surround, and so forth) to move back and forth at the same frequency, leading to sound.

Figure 2-1 Typical Speaker Structure

To better understand and analyze the principles and behaviors of the speaker, mathematical models, including the electromechanical and thermal models of speaker have been developed. Figure 2-2 shows the linearized electromechanical model of typical speakers, and the description of the main parameters has been listed in Table 2-1.

TAS5825M Typical Electromechanical Model of Speakers

Figure 2-2 Typical Electromechanical Model of Speakers

Table 2-1 Parameters of the Electromechanical Model

Parameters	Unit	Description
R_e	Ω	DC resistance of the voice coil
S_d	cm²	Area of the diaphragm
Bl	T∙m	Force factor
R_ms	N∙s/m	Mechanical damping factor
M_ms	g	Mechanical mass
C_ms	m/N	Mechanical compliance
L_e	mH	Leakage inductance of the voice coil
L₂	mH	Inductance of the voice coil
K_e	sH	Semi inductance of the voice coil
u	V	Input voltage
i	A	Input current
v	m/s	Velocity of the membrane
X	m	Membrane excursion

Based on the above electromechanical model, the transfer function of typical speakers can be derived. For simplicity, parasitic parameters with small values, such like L_e, L₂ and K_e can be omitted in further analysis. Therefore, the input electrical impedance of the speaker can be deduced as:

Equation 1.

Z_{i n} (s) = \frac{u (s)}{i (s)} = R_{e} + \frac{{(B l)}^{2}}{s M_{m s} + R_{m s} + 1 / s C_{m s}}

And the transfer function from input voltage to the excursion can be derived as:

Equation 2.

H_{e x c} (s) = \frac{X (s)}{u (s)} = \frac{B l}{s R_{e}} \cdot \frac{1}{s M_{m s} + (R_{m s} + {(B l)}^{2} / R_{e}) + 1 / s C_{m s}}

Furthermore, the equivalent Thiele/Small (T/S) parameters of the electromechanical model of typical speakers can be derived, as listed in Table 2-2.

Table 2-2 T/S Parameters of the Electromechanical Model

Parameters	Unit	Description
F_s	Hz	Resonance frequency of the speaker
Q_es	–	Electrical quality factor at Fs
Q_ts	–	Mechanical quality factor at Fs
Q_ms	–	Total quality factor at Fs
V_as	liter	Equivalent compliance volume

Equation 3.

F_{s} = \frac{1}{2 π \sqrt{M_{m s} C_{m s}}} = \frac{ω_{s}}{2 π}

Equation 4.

Q_{e s} = \frac{R_{e}}{{(B l)}^{2}} \sqrt{\frac{M_{m s}}{C_{m s}}}

Equation 5.

Q_{m s} = \frac{1}{R_{m s}} \sqrt{\frac{M_{m s}}{C_{m s}}}

Equation 6.

Q_{t s} = \frac{Q_{e s} Q_{m s}}{Q_{e s} + Q_{m s}}

Equation 7.

V_{a s} = 1000 \cdot ρ c^{2} {S_{d}}^{2} C_{m s}

In the table, ρ is the density of air (1.184kg/m³ at 25 °C), and c is the speed of sound (346.1m/s at 25 °C). In this case, the transfer function of the electromechanical model can be transformed into the following equations.

Input electrical impedance:

Equation 8.

Z_{i n} (s) = R_{e} + \frac{{(B l)}^{2}}{M_{m s}} \times \frac{s}{s^{2} + s ω_{s} / Q_{m s} + {ω_{s}}^{2}}

Excursion transfer function:

Equation 9.

H_{e x c} (s) = \frac{B l}{M_{m s} R_{e}} \times \frac{1}{s^{2} + s ω_{s} / Q_{t s} + {ω_{s}}^{2}}

Similarly, the thermal behavior of speakers can also be described with the linearized mathematical model, that is, the thermal model, as shown in Figure 2-3. Table 2-3 lists the corresponding parameters of the thermal model of typical speakers.

TAS5825M Typical Thermal Model of Speakers

Figure 2-3 Typical Thermal Model of Speakers

Table 2-3 Parameters of the Thermal Model

Parameters	Unit	Description
R_tv	K/W	Thermal resistance from voice coil to magnet
C_tv	J/K	Thermal capacitance of voice coil
R_tm	K/W	Thermal resistance from magnet to ambient air
C_tm	J/K	Thermal capacitance of the magnet
R_tva	K/W	Thermal resistance from voice coil to air gap
P	W	Power dissipation on voice coil as heat
T_v	K	Voice coil temperature
T_m	K	Magnet temperature
T_a	K	Ambient temperature
∆T_v	K	Temperature difference between voice coil and ambient
∆T_m	K	Temperature difference between magnet and ambient

For better understanding, the relation between the dissipated power and the temperature difference in the thermal model is similar as the relation between current and voltage in electric circuits. Thus, for thermal resistance:

Equation 10.

Δ T (s) = R_{t h e r m a l} \times P_{d i s s i p a t e d} (s)

And for the thermal capacitance:

Equation 11.

Δ T (s) = \frac{1}{s C_{t h e r m a l}} \times P_{d i s s i p a t e d} (s)

Therefore, the transfer function from the dissipated power to the voice coil temperature needs to be:

Equation 12.

H_{c o i l} (s) = \frac{Δ T_{v} (s)}{P (s)} = (R_{t v} + \frac{1}{s C_{t m}} ∥ R_{t m}) ∥ (\frac{1}{s C_{t v}}) ∥ R_{t v a}

Similarly, the transfer function from the dissipated power to the magnet temperature is:

Equation 13.

H_{m a g} (s) = \frac{Δ T_{m} (s)}{P (s)} = H_{c o i l} (s) \times (\frac{1}{s C_{t m}} ∥ R_{t m}) \times {(R_{t v} + (\frac{1}{s C_{t m}}) ∥ R_{t m})}^{- 1}