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# Uncovering the relationship between the main parameters of loudspeakers

2019-11-2358

The relationship between the main parameters of the disclosed loudspeaker is the result of the interaction of the physical parameters such as electricity, mechanics, acoustics and magnetism. The performance of the loudspeaker is determined by the performance of the key components such as drum paper, elastic wave, voice coil and magnetic circuit. Some of the parameters restrict and influence each other, so it must be considered and designed comprehensively.
1. Comprehensive design and analysis of main parameters
The commonly used mechanical and electrical parameters, calculation formulas and measurement methods of loudspeakers are as follows:
DC resistance re
Determined by voice coil, it can be directly measured by DC bridge.
Resonance frequency fo
It is determined by the equivalent vibration mass MS and the equivalent compliance CMS of the loudspeaker. See formula (5). Fo can be measured directly by fo tester or by measuring the impedance curve.
* impedance Zo at resonance frequency
Determined by voice coil, magnetic circuit and vibration system (drum paper and elastic wave), it can be measured by alternative method or by measuring impedance curve.
Zo = Re+[(BL)2/(Rms+Rmr)] (10)
Mechanical resistance RMS
Determined by the internal damping of drum paper and elastic wave and the characteristics of glue used, it can be calculated by the following formula after measuring the mechanical quality factor QMS:
Rms =(1/Qms)*SQR(Mms/Cms) (11)
Here SQR () represents the square root of the value in brackets (), the same below.
It is determined by aperture and frequency, and can be ignored at low frequency.
Rmr = 0.022*(f/Sd)2 (12)
It is only related to the caliber (equivalent radius a).
Sd =π* a2 (13)
Electromechanical coupling factor bl
It is determined by the BG value of the magnetic circuit and the effective length L of the voice coil. It can also be calculated by the following formula after measuring the electrical quality factor QES:
(BL)2 =(Re/Qes)*SQR(Mms/Cms) (14)
Equivalent vibration mass MS
It is determined by the mass of the voice coil MM1, the equivalent mass of the drum paper mm2 and the radiation mass MMR, which can be measured by the additional mass method.
Mms=Mm1+Mm2+2Mmr
It is only related to the caliber (equivalent radius a).
Mmr =2.67*ρo* a3 (16)
Where ρ o = 1.21kg/m3 is the air density and a is the equivalent radius of loudspeaker.
Equivalent compliance CMS
It refers to the flexibility of the supporting parts of the loudspeaker vibration system. The larger the value, the softer the whole loudspeaker vibration system. Unit: mm / N (mm / N)
It is determined by the drum paper's compliance CM1 and elastic wave's compliance cm2. This compliance is what we call deflection, but the unit needs to be converted to the international system of units: M / N, and the deflection can be measured directly by the deflection instrument. CMS can be measured by additional volume method.
Cms=(Cm1*Cm2)/(Cm1+Cm2) (17)
Equivalent volume VAS
It is only related to the equivalent compliance and the equivalent radiation area.
Vas =ρo*c2*Sd2*Cms (18)
Here C is the velocity of sound in the air, C = 344M / S
Mechanical quality factor QMS
It is determined by the equivalent vibration mass MS, the equivalent compliance CMS and the mechanical resistance RMS of the vibration system. The QMS can be obtained by the measurement of the impedance curve.
Qms =(1/Rms)*SQR(Mms/Cms)=(Fo/Δf)*(Zo/Re) (19)
F is the difference between two frequencies corresponding to the impedance equal to SQR (Zo * re) on the impedance curve.
Electrical quality factor QES
It is determined by the equivalent vibration mass MS, the equivalent compliance CMS and the electromechanical coupling factor BL of the vibration system, and is obtained by the measurement of the impedance curve.
Qes =[Re/(BL)2]*SQR(Mms/Cms)=(Fo/Δf)*SQR(Zo*Re)/(Zo-Re) (20)
Total quality factor QTS
It is determined by mechanical quality factor QMS and electrical quality factor QES.
Qts =(Qms*Qes)/(Qms+Qes)=(Fo/Δf)*SQR(Re/Zo) (21)
Reference electroacoustic conversion efficiency η o
It is determined by the electromechanical coupling factor BL, the equivalent radiation area SD and the equivalent vibration mass Ms.
ηo =(ρo/2πc)*(BL*Sd/Mms)2/Re (22)
Reference sensitivity level splo
It is directly related to the reference electroacoustic conversion efficiency η o.
SPLo = 112+10lgηo (23)
Reference amplitude ξ
It is related to the reference electroacoustic conversion efficiency η o, electric power PE, equivalent radius a and frequency f.
ξ = 0.481*SQR(Pe*ηo)/(a*f)2
The above parameters can be measured and calculated by Speaker computer test system. Common test systems include LMS, Clio, mlssa, DAAS, SYSID, laud, imp, etc. In addition, some computer simulation software can also be used for the basic design of speaker parameters, such as leap, calsod, speaker easy, DLC design, audiocad, soundeasy, etc.
The power and distortion index of loudspeakers can not be calculated directly by formula, but can only be analyzed and discussed qualitatively.
The rated sinusoidal power and pure tone listening power of the loudspeaker are basically determined by the low frequency * amplitude ξ o. Generally, the low frequency * amplitude is at the resonance frequency fo. The low frequency * amplitude of the loudspeaker mainly depends on the magnetic circuit structure and the coil width. Of course, it has a great relationship with the vibration system. When the loudspeaker works normally, the voice coil cannot jump out of the magnetic gap, that is to say, ξ o ≤ xmax, otherwise, it will produce great nonlinear distortion (manifested as abnormal amplitude sound), and even cause the voice coil damage (stuck or burnt). The * amplitude ξ o at fo can be calculated by the following formula:
ξo = 1.414*BL*I*Cms*Qts (25)
Where I is the current fed to the loudspeaker, I = SQR (PE / re). It can be seen that if the basic electromechanical parameters (BL, CMS, QTS) of the loudspeaker are determined, the power PE = I2 * re determined by the current I is limited by the low frequency * amplitude ξ o ≤ xmax. On the contrary, if the power of loudspeaker must reach a certain value, the equivalent smoothness of loudspeaker cannot be too large, that is, fo cannot be too small. When there is (BL) 2 / re > > RMS, the formula (25) can be simplified as follows:
ξo = 0.225*V/(BL*Fo) (26)
Where V is the voltage fed to the loudspeaker, v = SQR (PE * re). This formula more intuitively shows the relationship between * amplitude ξ O and voltage V, electromechanical coupling factor BL, resonance frequency fo.  Generally speaking, the control ability of the total quality factor QTS to the low-frequency amplitude is reflected and reflected by the formulas (25) and (26), among which the effect of BL value is more obvious.
The low frequency sound power PA of the loudspeaker is also limited:
Pa= Pe*ηo=4.33*ξ2*a 4*f 4 (27)
It can be seen that the sound power Pa is not only related to the electric power Pe, but also directly related to the electroacoustic conversion efficiency η o, in fact * is ultimately related to the amplitude, diameter and frequency of the loudspeaker. In order to achieve a certain sound power PA, under the same frequency, the smaller the aperture is, the larger its amplitude is, and the amplitude is generally limited, so the aperture cannot be too small. That is to say, it is impossible for small-diameter speakers to produce large sound power, because small-diameter speakers are generally limited by the structure, with small amplitude and low efficiency, and the voice coil is not large, the wire diameter used is limited, and the electric power they can bear is limited.
The rated noise power and long-term power of loudspeakers are not only related to the low frequency * amplitude, but also directly related to the coil diameter, material, heat dissipation conditions of the system, glue used, etc.  High power loudspeakers generally use high-strength and high-temperature resistant voice coil wire, voice coil framework, glue, large stroke and good heat dissipation magnetic circuit structure, voice coil adopts wide coil width and wire diameter, elastic wave adopts materials with good strength and fatigue resistance, of course, large caliber series are also generally used. The rated noise power and long-term * power of loudspeakers can only be obtained and verified through load test.
2. Parameters of horn unit
T / s index (Thiele / small specs)
T / s index is the basic parameter of the mathematical model of loudspeaker system invented by Australian A.N. Thiele and Richard small in the early 1970s. Nowadays, almost all people produce loudspeaker speakers according to this theory. The T / s indicators are as follows:
FS (FO) is the resonance frequency of the horn in the free field.
Vas is the volume of air equivalent to horn smoothness.
QES is the electrical Q value of the horn, which reflects the resonance ability of the unit under the electromagnetic control when fo is used. The lower the value is, the stronger the damping is, and the lower the resonance ability is.
QMS is the mechanical Q value of the horn. It reflects the resonance ability of the element in the mechanical structure when fo is used. The lower the value, the stronger the damping.
QTS is the total Q value of the horn (coupled by QMS and QES in parallel). It reflects the resonance capacity of the element at fo. The lower the value, the stronger the damping
Electrical mechanical parameter
MS: total vibration mass of the horn (including the mass of diaphragm, voice coil, air loaded before and after)
CMS: smoothness of horn unit
RMS: mechanical damping, including friction and radiation resistance of vibration.
RME: electrical damping factor, which reflects the mechanical control and damping of the electromagnetic system of the unit to the diaphragm. It is often used to measure the capability of the electromagnetic system of the unit.
Re: DC resistance of voice coil
BL: magnetic field strength of coil gap
D: diaphragm diameter
Le: voice coil inductance
SD: surface area of diaphragm
Fle: inductance measurement frequency
Large signal parameter
Xmax: * linear displacement, or linear stroke, is calculated as 1 / 2 of the displacement value of the full stroke. Generally, this value is relatively moisture, and some manufacturers will give the physical * displacement of the unit. However, some manufacturers use the whole P-P value (peak to peak) to indicate that at this time, we should pay attention to halving in comparison.
Xlim: undamaged * displacement. (or other xmec, * mechanical displacement)
HC: coil height
Hg: clearance height
VD: pushing air volume of * when the horn is in linear range
PE: input power * that can work continuously without burning.
Discussion:
◆ in fact, all t / S parameters are measured around the resonance peak of the bass unit, reflecting the characteristics of the resonance peak of the bass unit, and various speaker boxes are designed according to the characteristics. The resonance peak of the treble unit is meaningless for the box making (the amplitude of the treble is also very small), and no special description is needed for application, so we will not make t / S parameters on the treble unit.
◆ FO value refers to the resonance frequency of the unit, i.e. the frequency when the horn amplitude is *. Basically, this is the low-frequency playback limit of the unit, because after the resonance point, the sound pressure of the unit will drop sharply, (generally, the cut-off frequency at - 3dB is expressed as F3)
◆ when we describe the unit, there are many Q values, which is actually a mathematical value describing the sharpness of the impedance peak caused by resonance. The higher the Q value, the smaller the damping, the weaker the control, and the larger the resonance amplitude, so as to produce a stronger low-frequency sound pressure, but this brings about the distortion caused by uncontrolled vibration.
◆ there is a lot of debate on the question of Q value and what box is suitable for. Generally speaking, the horn with low Q value has good damping and high control force, so it is suitable to be used as phase inversion box. The unit with high Q value is suitable for airtight box. In fact, this is a fuzzy boundary choice. Generally, the unit with Q value higher than 0.5 is suitable for airtight box, while the unit with Q value lower than 0.3 needs to be inverted phase box. In the industry, EBP value is usually used to measure which kind of box the unit is suitable for.
3. QTc: total Q value of the whole system of the speaker
Q value of box loss
QL - leakage loss Q value, which is caused by the poor sealing of the box and unit, usually has a great impact on the phase inversion box. Generally, the value is 5-20, which is unpredictable. 5 means the seal is very good! Usually the default value is 10.
QA - absorption loss Q value, which is generated by the absorption of sound wave by the box body. The filling material in the box will greatly enhance the absorption. The inner wall of a dry and smooth rigid box is generally about QA = 30-100, and it will reach 3-5 when it is filled in large quantities.
QP - loss of phase inversion tube, produced by the phase inversion tube, due to the friction of the tube wall when air passes through, the phase inversion tube will have some damping. In fact, if you set the Q value to a very small value (which means that the damping is very large), the inverted phase box will become a closed box.
On the understanding of Q value
Q value is a mathematical quantity describing the resonance situation. It is always introduced with the concept of damping (in resonance system), or it is equivalent to damping value. For a resonant system, the larger the damping is, the more the resonance of the system is clamped, resulting in a low Q-value resonance curve. When the damping is small, the opposite is true, the resonance is violent, forming a high-Q curve.
Generally speaking, for loudspeaker system, the appropriate Q value is between 0.5-1.5. When it is lower than 0.5, the damping is too strong, and there is no resonance. Therefore, some people call 0.5q as critical damping, and even smaller Q as over damping. On the contrary, if q is greater than 1.5, it can be called under damping.
On the frequency amplitude curve of resonance system, we can see the curves represented by different Q values and the significance of different Q values.
4. Q of horn
QES is the electrical Q value of the horn, which reflects the resonance ability of the unit under the electromagnetic control when fo is used. The lower the value, the stronger the damping and the smaller the system resonance.
QMS is the mechanical Q value of the horn. It reflects the resonance ability of the element in the mechanical structure when fo is used. The lower the numerical value is, the stronger the damping is and the smaller the resonance of the system is.
QTS is the total Q value of the horn (coupled by QMS and QES in parallel). It reflects the resonance ability of the element at fo. The lower the value, the stronger the damping.
5. Q value of the system
The whole system includes power amplifier output, speaker line and speaker. This is the actual Q value in operation. Compared with the Q value QTc of the box, the damping factor is added here.
The influence of damping coefficient includes the output damping coefficient of power amplifier, the damping coefficient of horn line, the damping coefficient of serial horn (if any), and the damping coefficient of frequency divider.
Therefore, in order to ensure that the Q-value design of the original box is not affected, the general power amplifier requires that the damping coefficient be as small as possible, * * at least more than 10, but generally more than 100.  The frequency divider is mainly affected by the resistance of the inductance, generally speaking, more than 20. Wire should also be as small as possible.
For the series connected horn, the damping coefficient is more than 1 inevitably, so the general design is parallel horn.
Damping and Q value describe the working condition of the element near the resonance point, that is, the change of the sound generation near the resonance point, which has no effect on the frequency response of other frequency areas.