Thiele-Small Parameters

Or T/S Parameters

F

$F_s$ - driver resonance frequency any speaker will have a hard time to play any frequency below the $F_s$

$$f_s=\frac{1}{2\pi \sqrt{C_{ms}\cdot M_{ms}}}$$

$F_0$ - The frequency when frequency response start to drop $F_3$ - frequency where the 3 decibel power drop occurs, less than this you will feel that the sound is quieter $F_b$ - box resonance frequency

Q

Q - Quantity factor, the magnification of resonance factor of any resonant device or circuit. A driver with a high Q is more resonant than one with a low Q.

Q can also be called as “damping” lower Q will have more damping (more damping is more clear more good things)

$Q_{ms}$ - the mechanical Q of the driver. (Spring from surround and spider)

• Higher $Q_{ms}$ drivers displaying a higher impedance peak

$Q_{es}$ - the electrical Q of the driver. This causes by Back EMF (same as motor)

$Q_{ts}$ - the total Q of the driver at Fs.

$\frac{1}{Q_{ts}}=\frac{1}{Q_{ms}}+\frac{1}{Q_{es}}$

$Q_{ts}=\frac{Q_{es}\cdot Q_{ms}}{Q_{es}+Q_{ms}}$

Finding $Q_{ms}$ and $Q_{es}$

$Z_{max}$ = impedance at $F_s$ $Z_0=\sqrt{R_e\times Z_{max}}$ find $F_1$ and $F_2$ in impedance curve ($F_1$ and $F_2$ have difference meaning from $F_3$) $Q_{ms}=\frac{F_s}{F_2-F_1}\times \sqrt{\frac{Z_{max}}{R_e}}$

 Qms = Fs/(F2 - F1)*sqrt(Zmax/Re)

$Q_{es}=\frac{Q_{ms}}{\frac{Z_{max}}{R_e}-1}$ $Q_{es}=\frac{F_s}{F_2-F_1}\times \frac{\sqrt{Z_{max}\cdot R_e}}{Z_{max}-R_e}$

Qes = Qms/(Zmax/Re - 1)

Relation with other variables

$$Q_{es}=\frac{2\pi \cdot f_s\cdot M_{ms}\cdot R_e}{(Bl{)}^2}=\frac{R_e}{(Bl{)}^2}\sqrt{\frac{M_{ms}}{C_{ms}}}$$ $$Q_{ms}=\frac{2\pi \cdot f_s\cdot M_{ms}}{R_{ms}}=\frac{1}{R_{ms}}\sqrt{\frac{M_{ms}}{C_{ms}}}$$

ref. Measuring transducer Qms, Qes, Qts (wavecor.com)

$C_{ms}$ - Compliance Higher (loose suspension) → yield a lower frequency Lower (stiffer suspension) → yield a higher frequency

$R_e$ - DC resistance (normally 30% less than impedance) Impedance, $Z$ - AC resistance

$Bl$ - Magnetic force factor / motor strength ($T\cdot Nm$)

$B\times l$. Which is (the flux density) x (the length of the voice coil)

Or, actually

$$Bl=B\times l\sin \theta$$ $$\theta =90°,\sin (90°)=1$$

So,

$$Bl=B\times l$$

Relative to other parameters

$$Qes=\frac{2\pi \cdot F_s\cdot Mms\cdot R_e}{B{l}^2}$$ $$Bl=\sqrt{\frac{2\pi \cdot F_s\cdot Mms\cdot R_e}{Qes}}$$

Dimensions

V

Vas - air volume equal equal to compliance of driver suspension (springiness of driver) = $C_{ms}$ describe in volume

$$V_{as}=\rho \cdot c^2\cdot S_d^2\cdot C_{ms}$$

Vb - Net enclosure internal volume

$Xmax$ - maximum distance that coin can travel without distortion (they said that travel of 115% Xmax will not introduce audible distortion)

In 5-7” speaker $Xmax$ should be more than 3mm In 8-12” speaker $Xmax$ should be more than 6mm

$Xmech$ - maximum distance a speaker can travel without damaging the driver

$>Xmax$ = Distortion $>Xmech$ = Damage!!

Sd - effective area of the cone ($\pi r^2$) more Sd = more bass less Sd = more spread high frequency

Mmd = moving mass (mass of Coil + Cone + spider/2 + surround/2) Mms = Mmd + mass of air in front of the cone

High Mmd → Low $F_s$ Low Mmd → High $F_s$

SPL

SPL

units

• $dB@W/m$
• $dB@2.83V/m$

Difference driver Impedance will effect $dB@2.83V/m$ unit

$$\begin{array}{rl}W& =V^2/Z\\ & =2.83^2/8\\ & =8/8\\ & =1\end{array}$$ dB\;@\;2.83V/m\;@\;8\ohm &= \pm 0\;dB\;@\;W/m \\ dB\;@\;2.83V/m\;@\;6\ohm &= -1.3\;dB\;@\;W/m \\ dB\;@\;2.83V/m\;@\;4\ohm &= -3\;dB\;@\;W/m \\ dB\;@\;2.83V/m\;@\;2\ohm &= -6\;dB\;@\;W/m \\ \end{split}

twice the power give +3 dB half the power give -3 dB one tenth the power give -10 dB

$\Delta dB=10\log (\frac{P_1}{P_2})$

Ex

$$\begin{array}{rcl}1W& \Rightarrow & 90dB\\ 2W& \Rightarrow & 93dB\\ 4W& \Rightarrow & 96dB\\ 8W& \Rightarrow & 99dB\\ 10W& \Rightarrow & 100dB\\ 16W& \Rightarrow & 102dB\end{array}$$

Max SPL

$MaxSPL=Efficiency@W/m+10\times \log (MaxPower)$

Perceived loudness

• 3 dB difference is difficult to hear
• 10 dB considered to be twice as loud

$\eta (n_0)$ Efficiency

1 Watt of acoustic power = 112 dB SPL 1W/1m

$SPL=112+10\log \eta$ $\eta =10^{\frac{SPL-112}{10}}$  

Efficiency	Percent	Sensitivity
0.2	20 %	105 dB
0.1	10 %	102 dB
0.05	5 %	99 dB
0.02	2 %	95 dB
0.01	1 %	92 dB
0.005	0.5 %	89 dB
0.002	0.2 %	85 dB
0.001	0.1 %	82 dB

$\eta$ relation to other parameters

$$\eta =(\frac{\rho \cdot B^2\cdot l^2\cdot S_d^2}{2\pi c\cdot M_{ms}^2{R}_e})\times 100\%$$ $$\eta =(\frac{4{\pi }^2\cdot f_s^3{V}_{as}}{c^3\cdot Q_{es}})\times 100\%$$

• $\rho /2\pi =5.365\times 10^{-4}{m}^2\cdot s/kg$
• $4{\pi }^2/c^3=9.523\times 10^{-}7s^3/m^3$
• @ 25 °C and 50% humidity

ref. Efficiency and sensitivity conversion - sengpielaudio Sengpiel Berlin

Hofmann’s Iron Law

$\eta =k\cdot V_b\cdot f_3^3$