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6.9. Efficiency

Medium frequencies

The efficiency of a loudspeaker is defined as 100 times the ratio of the acoustic power radiated to the power supplied by the electrical generator. In the medium frequency range between the suspension resonance and the point where the coil inductance starts to contribute to the electrical impedance, the power supplied by the generator is approximately

W_{E} \approx \frac{e_{g (rms)}^{2}}{R_{E}}

$W_{E} \approx \frac{e_{g (rms)}^{2}}{R_{E}}$

(6.46)

where we are assuming that R _g ≪ R _E. Using W from Eq. (6.23), the reference efficiency E _ff is then given by

E_{f f} = 100 \frac{W}{W_{E}} \approx 100 \frac{B^{2} l^{2} S_{D}^{2} ρ_{0}}{π R_{E} M_{M S}^{2} c}, 2 f_{0} < f < \frac{c}{4 π a}

$E_{f f} = 100 \frac{W}{W_{E}} \approx 100 \frac{B^{2} l^{2} S_{D}^{2} ρ_{0}}{π R_{E} M_{M S}^{2} c}, 2 f_{0} < f < \frac{c}{4 π a}$

(6.47)

Not surprisingly, if we compare this with Eq. (6.32), we find that the same parameters which contribute to a high SPL also make for an efficient loudspeaker, namely high field strength, low mass, and a large radiating area. By combining Eqs. (6.8), (6.11), (6.26), and (6.47), we obtain a convenient expression for the reference efficiency in terms of Thiele–Small parameters:

E_{f f} = 100 \frac{W}{W_{E}} \approx 100 \frac{8 π^{2} V_{A S} f_{S}^{3}}{Q_{E S} c^{3}}, 2 f_{0} < f < \frac{c}{4 π a}

$E_{f f} = 100 \frac{W}{W_{E}} \approx 100 \frac{8 π^{2} V_{A S} f_{S}^{3}}{Q_{E S} c^{3}}, 2 f_{0} < f < \frac{c}{4 π a}$

(6.48)

At resonance

When f = f _S, the cone velocity is at a maximum value which is found by letting X _M = 0 in Eq. (6.1) so that

{\tilde{u}}_{c} = \frac{{\tilde{e}}_{g} B l}{(R_{g} + R_{E}) R_{M}}

${\tilde{u}}_{c} = \frac{{\tilde{e}}_{g} B l}{(R_{g} + R_{E}) R_{M}}$

(6.49)

From Eqs. (6.19) and (6.21),

W = \frac{e_{g (rms)}^{2} B^{2} l^{2} ω_{S}^{2} S_{D}^{2} ρ_{0}}{π c {(R_{g} + R_{E})}^{2} R_{M}^{2}}

$W = \frac{e_{g (rms)}^{2} B^{2} l^{2} ω_{S}^{2} S_{D}^{2} ρ_{0}}{π c {(R_{g} + R_{E})}^{2} R_{M}^{2}}$

(6.50)

The input power at resonance, assuming R _MS ≫ 2 R _MR, is given by

W_{E} = \frac{e_{g (rms)}^{2}}{R_{g} + R_{E} + (B^{2} l^{2}) / R_{M S}}

$W_{E} = \frac{e_{g (rms)}^{2}}{R_{g} + R_{E} + (B^{2} l^{2}) / R_{M S}}$

(6.51)

Then the efficiency at resonance, assuming R _E >> R _g, is given by

E_{f f S} = 100 \frac{W}{W_{E}} = 100 \frac{B^{2} l^{2} ω_{S}^{2} S_{D}^{2} ρ_{0}}{π c (B^{2} l^{2} + R_{E} R_{M S}) R_{M S}}

$E_{f f S} = 100 \frac{W}{W_{E}} = 100 \frac{B^{2} l^{2} ω_{S}^{2} S_{D}^{2} ρ_{0}}{π c (B^{2} l^{2} + R_{E} R_{M S}) R_{M S}}$

(6.52)

Noting that Q _MS = ω _S M _MS/R _MS and Q _ES = ω _S M _MS R _E/(Bl)², we obtain

E_{f f S} = 100 \frac{W}{W_{E}} = 100 \frac{Q_{E S} Q_{M S}^{2} B^{2} l^{2} S_{D}^{2} ρ_{0}}{(Q_{E S} + Q_{M S}) π R_{E} M_{M S}^{2} c}

$E_{f f S} = 100 \frac{W}{W_{E}} = 100 \frac{Q_{E S} Q_{M S}^{2} B^{2} l^{2} S_{D}^{2} ρ_{0}}{(Q_{E S} + Q_{M S}) π R_{E} M_{M S}^{2} c}$

(6.53)

Comparing this with Eq. (6.47) and using the relationship of Eq. (6.10) yields the following relationship between the efficiency at resonance E _ffS and the midband reference efficiency E _ff:

\frac{E_{f f S}}{E_{f f}} = Q_{T S} Q_{M S}

$\frac{E_{f f S}}{E_{f f}} = Q_{T S} Q_{M S}$

(6.54)

All frequencies

The power supplied by the generator at all frequencies is

W_{E} = e_{g (rms)}^{2} ℜ (\frac{1}{Z_{E}})

$W_{E} = e_{g (rms)}^{2} ℜ (\frac{1}{Z_{E}})$

(6.55)

where the electrical impedance Z _E is given by Eq. (6.45). The radiated power is given by Eq. (6.19) where the cone velocity is given by Eq. (6.42) and R _MR by

R_{M R} = S_{D} ρ_{0} c (1 - \frac{J_{1} (2 k a)}{k a})

$R_{M R} = S_{D} ρ_{0} c (1 - \frac{J_{1} (2 k a)}{k a})$

(6.56)

from Eq. (13.117). Hence,

E_{f f} = 100 \frac{W}{W_{E}} = 100 {| \frac{B l}{(R_{E} + j ω L_{E}) Z_{M T}} |}^{2} \frac{2 S_{D} ρ_{0} c}{ℜ (1 / Z_{E})} (1 - \frac{J_{1} (2 k a)}{k a})

$E_{f f} = 100 \frac{W}{W_{E}} = 100 {| \frac{B l}{(R_{E} + j ω L_{E}) Z_{M T}} |}^{2} \frac{2 S_{D} ρ_{0} c}{ℜ (1 / Z_{E})} (1 - \frac{J_{1} (2 k a)}{k a})$

(6.57)

where Z _MT is given by Eq. (6.41). The efficiency of a typical 100 mm loudspeaker in an infinite baffle is plotted in Fig. 6.7. Not surprisingly, the loudspeaker is most efficient at the suspension resonance f _S where the input impedance is also at a maximum so that relatively little current is drawn. The efficiency falls off at high frequencies due to diaphragm and coil inertia, where most of the input power is dissipated in heating the voice coil.

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Table of Contents for Acoustics

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6.9. Efficiency

Medium frequencies

At resonance

All frequencies

Table of Contents for
Acoustics