If you haven’t read Part I of this series, it is strongly recommended that the user starts there.

Part II: Mechanical and Electrical Characteristics of Loudspeakers

Transfer Function for Free-Air Loudspeaker
Using Resonance and Mass Loading to Find the Mechanical Compliance and Mass
Equivalent Compliance Volume
Quantifying the Quality Factors
Bl Product
Calculating the Loudspeaker Efficiency Parameter
Conclusion

Transfer Function for Free-Air Loudspeaker

Brüel and Kjær published a quality report a while ago on the transfer function for a loudspeaker in free air:

$\large H(\omega) = \frac{Bl}{M_{mr}}\cdot \frac{\omega}{\omega^2+\frac{R_{mr}}{M_{mr}}\omega+\frac{1}{M_{mr}C_{mr}}}$

where ω is the frequency input to the system, C_mr is the mechanical compliance, M_mr is the mass of the acoustic and mechanical diaphragm and coil loading, R_mr is the mechanical resistance, and Bl is a product of magnetic field strength and coil length which determines the amount of force imposed on the system in accordance with Faraday's law of induction.

The constants above can be found for specific changes in impedance due to frequency, which we will continue to explore in the sections to follow.

Using Resonance and Mass Loading to Find the Mechanical Compliance and Mass

If we take the derivative of the transfer function above:

$\large H^\prime(\omega) = \frac{Bl}{M_{mr}}\cdot \frac{-1+\frac{1}{\omega^2M_{mr}C_{mr}}}{\left(\omega+\frac{R_{mr}}{M_{mr}}+\frac{1}{\omega M_{mr}C_{mr}}\right)^2}$

The frequency at which the derivative is equal to zero is the value which we call the resonance frequency:

$\large \frac{Bl}{M_{mr}}\cdot \frac{-\omega_r^2+\frac{1}{M_{mr}C_{mr}}}{\left(\omega_r^2+\frac{R_{mr}}{M_{mr}}\omega_r+\frac{1}{M_{mr}C_{mr}}\right)^2} = 0$

simplifying and solving for the resonance frequency, ω_r:

$\large \omega_r^2=\frac{1}{M_{mr}C_{mr}}$

and rewriting in a more familiar format:

$\large f_r = \frac{1}{2\pi}\frac{1}{\sqrt{C_{mr}\cdot M_{mr}}}$

with C_mr is the mechanical compliance, and M_mr is the mass of the acoustic and mechanical diaphragm and coil loading.

One method for finding the mechanical compliance and mass is to add known masses to the mechanical mass and observe the change in resonance frequency. This allows us to setup a relationship between resonance frequency and added mass in order to solve for mechanical compliance and mass in a linear function. We can start by writing the added mass and new resonance as a function of one another:

$\large f_{r,i} = \frac{1}{2\pi}\frac{1}{\sqrt{C_{mr}\cdot (M_{mr}+m_i)}}$

where: m_i is the added mass amount and f_r,i is the resonance at that added mass. We can rewrite the above equation in linear form:

$\large \frac{1}{(2\pi f_{r,i})^2} = C_{mr}m_i+ C_{mr} M_{mr}$

which is of the form:

$\large y = nx+b$

where:

$\large \\ y = \frac{1}{(2\pi f_{r,i})^2} \\ \\ n = C_{mr} \\ \\ x = m_i \\ \\ b = C_{mr} M_{mr}$

and so the general method for solving for C_mr and M_mr is we use multiple masses and resonance calculations to fit a line and solve for slope and intercept. I used resonances for added masses at: 0g (actual resonance), 2g, 5g, 10g, 15g, 20g. The linear fit based on the square relationship above is shown below:

The plot above gives us the approximate values for the compliance and mass:

$\large \\ C_{mr} = 6.17 \times 10^{-4} \ \textup{m}/\textup{N}\\ \\ M_{mr} = \frac{b}{C_{mr}} = \frac{3.35 \times 10^{-6}}{6.17 \times 10^{-4}} = 5.43 \ \textup{g}$

The Python code for implementing the added mass and resonance fit is shown below. All the user needs to do is add the mass and measure the resonance frequency, then input the values into the code and it prints out the plot shown above.

## linear fitting the mass loading of loudspeaker
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
plt.rcParams["font.family"] = 'Times New Roman'
plt.rcParams["mathtext.fontset"] = 'stix'
plt.style.use('ggplot')

m = np.array([0.0,2.0,5.0,10.0,15.0,20.0])*0.001
f = np.array([86.4,74.0,62.6,52.1,45.0,40.0])

y = 1.0/((2.0*np.pi*f)**2.0)

slope,intercept,r_val,p_val,std_err = stats.linregress(m,y)

fig = plt.figure(figsize=(12,9))
ax = fig.add_subplot(111)

plt.plot(m,y,label='data',linewidth=3.0)
plt.plot(m,(m*slope)+intercept,label='fit',linewidth=3.0)
plt.legend(fontsize=20)

diff_val = (y-((m*slope)+intercept))
MAPE = 100.0*np.mean(np.abs(diff_val/y))
print('MAPE: {0:2.2f}%'.format(MAPE))

stats_str = '% Error = {0:2.2f}%\n$R^2$ = {1:2.3f}'.format(MAPE,r_val**2.0)
plt.text(0.0, .000012, stats_str, size=20,
         ha="left", va="center",
         bbox=dict(boxstyle="round",
                   ec=(0.2, 0.2, 0.2),
                   fc=(0.95, 0.95, 0.95),
                   )
         )

eqn_str = '$(2\pi f_{r,i})^{-2} = $'+'{0:.2E}'.format(slope)+'$m_i$+'+'{1:.2E}'.format(slope,intercept)
eqn_str = eqn_str.replace('E','e')
plt.text(np.mean(m), .000005, eqn_str, size=20,
         ha="left", va="center",
         )
plt.xlabel('$m_i$ [grams]',fontsize=20)
plt.ylabel('$(2 \pi f_{r,i})^{-2}$ [1/Hz$^2$]',fontsize=20)
plt.savefig('mass_loading.png',dpi=300,facecolor=np.array([252,252,252])/255.0)
plt.show()

Equivalent Compliance Volume, V_ar

The compliance value above allows us to calculate another important parameter called the equivalent compliance volume, V_ar , which is a measure of the amount of air that the suspension system is capable of moving. It is measured in liters or m³, and generally is a measure of what size enclosure the loudspeaker should be housed in. It can be calculated as follows:

$\large V_{ar} = \rho c^2 \cdot S_d^2 \cdot C_{mr}$

where ρ is the density of air, c is the speed of sound in air, S_d is the effective area of the moving speaker cone, and C_mr is the compliance calculated above. For the speaker cone area I measure across the width of the speaker until about halfway past the speaker surround. This gives an effective diameter of about 90-95% of the cited diameter.

For the speaker used in this series, the equivalent compliance volume is calculated as follows:

$\large V_{ar} = \rho c^2 \cdot (\pi (D/2)^2)^2 \cdot C_{mr}$

$\large V_{ar} = \1.225 \cdot 343^2 \cdot \left(\pi \left(\frac{0.146}{2}\right)^2\right)^2\cdot 6.17\times 10^{-6}$

$\large V_{ar} = 0.025 \ \textup{m}^3 = 25 \ \textup{L}$

For comparison with other drivers of similar size and specifications, this value seems reasonable. Now we will continue with the analysis of impedance by investigating the significance of impedance magnitudes in relation to damping and the rest of the parameters associated with loudspeaker characterization.

Quantifying the Quality Factors

The natural succession after finding C_mr and M_mr is to quanitfy the quality factor (inverse of damping), Q , for both mechanical and electrical domains. The quality factor for an oscillating system can be defined as:

$\large Q \equiv \frac{f_r}{f_{2,r}-f_{1,r}}$

where Q is the quality factor, which in this case is defined by two resonance sidelobe frequencies. Typically, the sidelobe frequencies are chosen at -3dB from the peak value (Z_max ) at resonance, which is also one half of the value. Therefore, if we wanted to define the Q _mr, we would have to solve for the frequencies where the impedance is half the maximum. In my case, the sidelobe frequencies are roughly 75 Hz and 99 Hz, resulting in:

$\large Q_{mr} = 3.61$

The electrical quality factor, Q _er, can be approximated using the mechanical with the following relation:

$\large \frac{Z_{max}}{R_e} = \left(\frac{Q_{mr}}{Q_{er}}+1 \right )$

$\large Q_{er} = \frac{Q_{mr}}{\frac{Z_{max}}{R_e}-1}$

$\large Q_{er} = 1.81$

It is customary to combine the quality factors in series to produce a total quality factor:

$\large Q_{tr} = \frac{Q_{er}\cdot Q_{mr}}{Q_{er}+Q_{mr}}$

$\large Q_{tr} = 1.21$

Bl Product

The Bl product, which is a measure of the magnetic field strength with respect to the voice coil, comes quite easily from the quality factor in the electrical domain:

$\large Q_{er} = \frac{\omega_r M_{mr}R_e}{(Bl)^2}$

which gives us the result:

$\large Bl = 2.8 \textup{ T-}\textup{m}$

The Bl product above is in Tesla-meters - a unit of magnetic field strenght and length.

Calculating the Loudspeaker Efficiency Parameter

This parameter is one of the more important and perhaps intuitive of the Thiele-Small parameters - as it approximates the efficiency of the loudspeaker driver with reference to the input power.

$\large \eta_0 = \frac{4\pi^2 f_r^3V_{ar}}{c^3 Q_{er}}$

and now we can see that without some crucial parameters derived using the methods above, we wouldn't be able to calculate the efficiency quite so easily. The approximate efficiency for our loudspeaker is:

$\large \eta_0 = 1.3 \%$

which appears to be quite poor, however, in comparison with other loudspeakers - it’s not far from the average for its size and design.

Conclusion

This concludes the investigation into the Thiele-Small parameters for a loudspeaker. A lot of the analysis and calculation was based on decades-old methods and parameters. In the next entry into this series I will focus on experiments with loudspeakers, such as an application with microphones and even an experiment with a non-newtonian fluid.

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