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Low Frequency Sound Generation by Loudspeaker Drivers

Robert-H Munnig Schmidt 罗伯特-芒尼格-施密特

(C) 2017 The author, RMS Acoustics & Mechatronics and Grimm Audio. All rights reserved.
(C) 2017 作者、RMS Acoustics & Mechatronics 和 Grimm Audio。保留所有权利。
Copying of the complete document is allowed for personal use only.
The author/publisher is not responsible for any problems that might arise by using the contents of this paper.
Published by RMS Acoustics & Mechatronics
由 RMS Acoustics & Mechatronics 出版
The Netherlands 荷兰

Contents 目录

1 Introduction ..... 3 1 简介 .....3
1.1 Rotary Subwoofers ..... 4
1.1 旋转式低音炮 .....4

2 Sound from a Vibrating Diaphragm ..... 5
2 振动膜片发出的声音 .....5

3 The Lorentz-type actuator ..... 8
3 洛伦兹型推杆 .....8

4 Dynamic Properties of Loudspeaker Drivers ..... 10
4 扬声器驱动器的动态特性 .....10

4.1 Amplifier Voltage Response ..... 11
4.1 放大器电压响应 .....11

4.2 Motion Voltage Response ..... 14
4.2 运动电压响应 .....14

5 Using an Enclosure ..... 15
5 使用机箱 .....15

5.1 Closed-Box Enclosure ..... 16
5.1 封闭式机箱 .....16

5.1.1 Impact of Stiffness of the enclosed Air ..... 17
5.1.1 封闭空气刚度的影响 .....17

5.1.2 Efficiency ..... 19
5.1.2 效率 .....19

5.1.3 Causes of the Low Efficiency ..... 20
5.1.3 低效率的原因 .....20

5.1.4 Horn Shaped Impedance Transformer ..... 21
5.1.4 喇叭形阻抗变压器 .....21

5.1.5 Increase of radiating surface ..... 23
5.1.5 增加辐射表面 .....23

5.1.6 Ultra Low Frequency Efficiency ..... 23
5.1.6 超低频效率 .....23
Figure 1: Cross section of an low frequency loudspeaker.
图 1:低频扬声器的横截面。

1 Introduction 1 引言

A large number of sound generating transducers are developed over time, ranging from small vibrating membranes in a horn to the modulation of plasma by a varying magnetic field for high frequencies and even rotary subwoofers for extremely low frequencies.

With the exception of the mentioned plasma principle most practical loudspeakers always apply an intermediate material, in most cases a flat or conical diaphragm that drives the air molecules based of forces from an actuator The diaphragm can be rigid or non-rigid, depending on the means of actuation. The best example of the latter is the electrostatic loudspeaker where the thin non-rigid membrane is directly driven over its entire surface by electrostatic forces.
除上述等离子体原理外,大多数实用扬声器总是使用一种中间材料,在大多数情况下是一种平面或锥形膜片,它利用来自驱动器的力驱动空气分子 膜片可以是刚性或非刚性的,取决于驱动方式。后者的最佳例子是静电扬声器,在这种扬声器中,薄薄的非刚性膜片通过静电力直接驱动其整个表面。
Most low frequency loudspeakers use a linear Lorentz-type actuator that drives the diaphragm, which is guided in one motion direction (degree of freedom) by an elastic suspension inside a supporting frame (Figure 1).
大多数低频扬声器都使用洛伦兹型线性致动器来驱动振膜,而振膜则由支撑框架内的弹性悬架在一个运动方向(自由度)上引导(图 1)。

The Lorentz-type actuator is also called a moving-coil actuator and consists of a permanent magnet "stator" creating a strong magnetic field inside an air-gap with a moving coil inside that air-gap that transforms the current in the coil into a force on the diaphragm which in its turn "pushes" the air.
洛伦兹型推杆也称为动圈推杆,由一个在气隙内产生强磁场的永磁体 "定子 "和气隙内的动圈组成,动圈将线圈中的电流转化为作用在膜片上的力,进而 "推动 "空气。
Because of the fact that the electrodynamic loudspeaker was one of the first applications of Lorentz actuators, the moving coil is also called a voice coil because that coil gives the "voice" to the loudspeaker.
由于电动扬声器是洛伦兹致动器的首批应用之一,因此动圈也被称为音圈,因为音圈为扬声器提供了 "声音"。
After a short comment on rotary subwoofers, the following sections first describe the physical relations that determine the radiated sound power for a vibrating plate, representative for the diaphragm of a low frequency loudspeaker.

The second section describes the dynamics of amplifier, actuator and the mechanical system as they determine the vibrations of the diaphragm as function of the signal from the amplifier. Finally the impact of the enclosure is presented.
Figure 2: The Thigpen rotary subwoofer by Eminent Technology uses a fan with controllable pitch of the blades to create pressure variations between the front and the back side of the fan by changing the pitch of the blades.
图 2:Eminent Technology 公司的 Thigpen 旋转式超低音扬声器使用一个叶片间距可控的风扇,通过改变叶片间距在风扇前后两侧产生压力变化。

The back side faces a large enclosure volume, while the front side delivers the sound pressure via a second chamber covered with damping material to reduce the airflow noise.

1.1 Rotary Subwoofers 1.1 旋转式低音炮

To understand physics phenomena in general it is often illustrative to observe extreme situations. With low frequency sound reproduction the static pressure at is such an extreme situation. It refers to a constant pressure that is lower or higher than the average environmental pressure. This phenomenon can only be obtained by pointing a continuous air flow towards an object, like wind in a sail.
要理解一般物理现象,观察极端情况往往很有帮助。在低频声音重现方面, ,静压就是这样一种极端情况。它指的是低于或高于平均环境压力的恒定压力。只有将持续气流对准一个物体(如风帆中的风),才能获得这种现象。

It becomes immediately clear from this extreme example that such a situation can

never be created by a diaphragm that moves over a limited range, as is the case in a "normal" loudspeaker driver.
在 "普通 "扬声器驱动器中,振膜只能在有限的范围内移动,而不能产生任何声音。
It is however possible to generate an artificial wind by means of a fan as shown in Figure 2. By controlling the pitch of the blades the direction and the amount of air that flows through the device can be changed, theoretically to any frequency, even at . It follows the same principle as used in the propeller-drive of an airplane to reverse the thrust when braking.
不过,如图 2 所示,可以通过风扇产生人工风。通过控制叶片的间距,可以改变流经该装置的空气的方向和数量,理论上可以改变任何频率,甚至在 。其原理与飞机螺旋桨驱动装置在制动时反向推力的原理相同。

Unfortunately there are some caveats for this system of which the most important is the flow noise from the fan blades, which demands the use of a voluminous damping structure.

In practice for acceptable noise levels such a subwoofer can only be applied with a large baffle board ("suskast" in Dutch), an anechoic enclosure with an opening to the listening room, internally covered with sufficient damping material.
在实际应用中,为了达到可接受的噪音水平,这种低音炮只能使用大块障板(荷兰语为 "suskast"),这是一个消声箱体,开口通向聆听室,内部覆盖足够的阻尼材料。
For these reasons a rotary subwoofer is only applicable in professional installations, which allow large systems, like with cinema's or large electronic church organs.

A good example of the latter is the Thigpen rotary subwoofers from Eminent technology, which are used in a real church with the OPUS 4 church organ of Marshall and Ogletree.
后者的一个很好的例子是 Eminent Technology 公司的 Thigpen 旋转式低音炮,它与 Marshall 和 Ogletree 公司的 OPUS 4 教堂管风琴一起在一个真实的教堂中使用。
As this paper focuses on installations for music reproduction at home or in recording and mastering rooms in studios with frequencies ranging not lower than , the rotary subwoofer is not considered a viable option and in the following only the more regular driver configuration with a reciprocating diaphragm is presented.
由于本文的重点是家庭或录音室和母带制作室的音乐重放装置,频率范围不低于 ,因此旋转式超低音扬声器不被认为是一种可行的选择,下文将只介绍带有往复振膜的更常规的驱动器配置。

2 Sound from a Vibrating Diaphragm
2 振膜发出的声音

A loudspeaker driver diaphragm moves ideally like a piston, creating pressure waves that are propagated through the air.
To calculate the radiated sound power the loudspeaker is assumed to be mounted in such a way that the pressure from the back side can never reach the pressure from the front side.

This can be achieved by means of an infinitely large plate, restricting the sound to be radiated in a hemisphere, by an infinitely large tube, where the sound will be radiated over a full sphere in space or by a closed chamber (enclosure) where the radiation varies from spherical at low frequencies to hemispherical at high frequencies.
The average sound power over one period, radiated from one side of the diaphragm moving with a sinusoidal motion, is equal to the multiplication of the effective (RMS) value of the velocity of the diaphragm with the effective value of the real (in phase) part of the force on the diaphragm caused by the pressure that is exerted on the diaphragm.
从正弦运动的膜片一侧辐射出的一个周期内的平均声功率,等于膜片 速度的有效值(均方根值) 与施加在膜片上的压力所产生的膜片上力的实部(同相)有效值的乘积。
with:  用:
The pressure on the diaphragm is caused by the movement of the diaphragm itself working on the surrounding air with the following relation:
where the acoustic resistance is equal to the real part of the complex, frequency dependent acoustic impedance . Only the real part creates power as it corresponds with the component of the air pressure that is in phase with the velocity 4 .
其中,声阻抗 等于与频率相关的复数声阻抗 的实部。只有实部才会产生功率,因为它对应于与速度同相的气压分量 4 。
The acoustic resistance is frequency dependent at low frequencies and becomes constant at higher frequencies. From empirical analysis it is found that the values can be approximated as follows:
with: 用:
density of air  空气密度
angular frequency
propagation velocity of sound waves
diameter of the diaphragm
The transition frequency between the low and high frequency range is found when both values are equal:
当两个值相等时,低频和高频范围之间的过渡频率 就会出现:
A large loudspeaker with a diaphragm diameter of with the velocity of sound would have a transition frequency of , which indicates that for low frequency sound reproduction with subwoofers it is allowed to use the value from Equation (3).
一个大型扬声器的振膜直径为 ,声速为 ,其过渡频率为 ,这表明在使用超低音重现低频声音时,可以使用公式 (3) 中的值。
With these values and equations the force by the air on the diaphragm can be calculated:
Using Equation (3) this is equal to:
根据公式 (3),这等于
The average radiated sound power is equal to:
With a sinusoidal reciprocating motion of the diaphragm, and the average sound power can be written as:
在振膜做正弦往复运动的情况下, 和平均声功率可写成:
When the diaphragm moves with a constant amplitude for all frequencies, the radiated power increases proportional with frequency to the power 4. In controlengineering terms this represents a slope of +2 in the frequency response plot being decade.
当振膜在所有频率上以恒定振幅运动时,辐射功率随频率的增加而增加,与功率 4 成正比。在控制工程术语中,这表示频率响应图中的斜率为 +2,即 10。
A frequency independent output power level would require that the displacement amplitude is inversely proportional to the frequency squared. This requirement has the following important consequences:
  • A high output power level at low frequencies requires a large displacement amplitude.
  • The displacement amplitude can only be reduced by a larger surface of the diaphragm.
This is the reason why powerful low frequency loudspeakers need to be large.
Another important aspect is the relation between the sound power and acceleration.

Even though the sound power is not generated by the acceleration, the squared relation to the amplitude of the displacement, as found in Equation (9), means that the radiated power is proportional to the acceleration of the diaphragm as also the acceleration increases with the frequency squared with a slope of decade when the amplitude of the displacement is kept constant.
尽管声功率不是由加速度产生的,但等式 (9) 中发现的与位移振幅的平方关系意味着,辐射功率与膜片的加速度成正比,当位移振幅保持不变时,加速度也随频率平方的增加而增加,斜率为 10。
With the effective value of the acceleration , Equation (9) can be written as the following frequency independent relation:
有了加速度的有效值 ,方程 (9) 可以写成以下与频率无关的关系式:
Figure 3: Cross section of a voice coil actuator consisting of a permanent magnetic structure, which generates a strong magnetic field in a circular air-gap, and a moving coil, which is inserted in this air-gap.
图 3:音圈致动器的横截面,包括一个在圆形气隙中产生强磁场的永磁结构和一个插入气隙中的动圈。

A larger (overhung) coil than the air-gap increases the range over which the force is more constant.
This means that a diaphragm will produce a constant frequency independent sound power when the acceleration is kept at a constant level.
Furthermore the sound pressure is proportional to the square root of the sound power which means:
This relation of sound pressure with acceleration at low frequencies is an important phenomenon as it means that the dynamic behaviour of a loudspeaker can be mastered by controlling the acceleration of the diaphragm.

3 The Lorentz-type actuator
3 洛伦兹型推杆

The Dutch physicist and Nobel prize winner Hendrik Antoon Lorentz (1853 - 1928) formulated the Lorentz force as a completion to the Maxwell equations.
荷兰物理学家、诺贝尔奖获得者亨德里克-安通-洛伦兹(1853 - 1928 年)提出了洛伦兹力,作为麦克斯韦方程的补充。

The law of Faraday describes the effect of a changing magnetic field on electrical charges hence generating electricity from kinetic energy.

Based on energy conservation laws creating electrical energy from motion is fully complementary to creating motion energy from electrical energy so the laws of Lorentz and Faraday are strongly related.

In vectorial notation the formulation of Lorentz describes the force on a moving charged particle as:
Figure 4: Determining the direction of the Lorentz force with the corkscrew rule.
图 4:用开瓶器法则确定洛伦兹力的方向。

When the corkscrew is rotated right handed, from the direction of the positive current to the direction of the magnetic field (arrow), the movement of the point of the corkscrew determines the direction of the force.
with equals the instantaneous velocity of the particle. The first part of the Equation is the electrostatic force and the second part is the electromagnetic force. This second term is used in electromagnetic actuators. Next to the force on a moving particle it equally represents the force on a current flowing through a wire with length , inserted in the magnetic field. For this situation the moving charge equals the current times the length, , and with this relation the electromagnetic Lorentz force is equal to:
等于粒子的瞬时速度。方程 的第一部分是静电力,第二部分是电磁力。第二项用于电磁致动器。除了运动粒子所受的力之外,它还同样代表了流经长度为 、插入磁场中的导线的电流所受的力。在这种情况下,运动电荷等于电流乘以长度 ,根据这种关系,电磁洛伦兹力等于:
For the magnetic force on a wire at an angle relative to the direction of a magnetic field with flux density , carrying a current , this relation leads to the scalar notation of the Lorentz force of electromagnetic actuators of which the magnitude is given by:
对于相对于磁通密度为 的磁场方向成 角的导线上的磁力,其携带的电流为 ,根据这一关系可以得出电磁致动器的洛伦兹力的标量符号,其大小由以下公式给出:
The direction of this force is orthogonal to the plane that is determined by the direction of the magnetic field and the current, due to the "cross product" in the vectorial Lorentz equation.
由于洛伦兹矢量方程中的 "交叉积",这种力的方向与磁场和电流方向所决定的平面正交。

This rule can be remembered as the right hand or corkscrew rule, which states that the positive force direction is found when rotating a corkscrew from the positive current direction onto the direction of the magnetic field as shown in Figure 4. Of course for a real mechanical engineer any normal right turning screw will also suit the purpose, but the corkscrew is more easy to remember.
如图 4 所示,当把螺旋从正电流方向转到磁场方向时,就能找到正力方向。当然,对于真正的机械工程师来说,任何普通的右旋螺钉也能达到目的,但螺旋线更容易记住。
In most practical cases the Lorentz force must be maximised, which means that is kept as much as possible equal to one. This means that the simplified equation becomes equal to:
在大多数实际情况下,洛伦兹力必须最大化,这意味着 要尽可能保持等于 1。这意味着简化方程等于
Figure 5: A damped mass-spring system with an external force stimulus.
图 5:受到外力刺激的阻尼质量弹簧系统。
And with multiple windings the Lorentz force becomes:
where equals the total length of the wire inserted in the magnetic field.
其中 等于插入磁场的导线总长度。
When divided by the current this equation gives the force to current ratio, also called the force constant, of the actuator, which will prove to be not constant at all (see paper "Distortion Sources in Loudspeaker Drivers".
当除以电流 时,这个等式给出了致动器的力与电流比,也称为力常数,但事实将证明它根本不是常数(参见论文 "扬声器驱动器中的失真源")。

4 Dynamic Properties of Loudspeaker Drivers
4 扬声器驱动单元的动态特性

For low frequencies a moving coil loudspeaker driver can be described in a simple model as shown in Figure 5. The moving body with mass consists of the diaphragm and the coil. The body is suspended by the spider and the rubber roll surround, with a certain stiffness . Finally the damper, with damping coefficient consists of the sound radiation, the rubber surround and the damping caused by actuator-amplifier combination. With Equation (31) the damping due to the sound can be calculated. It is a frequency dependent value so it is necessary to calculate it for a certain frequency.
对于低频而言,动圈扬声器驱动器可以用一个简单的模型来描述,如图 5 所示。质量为 的动圈体由振膜和线圈组成。本体由蜘蛛和橡胶辊环绕悬挂,具有一定的刚度 。最后是阻尼器,其阻尼系数为 ,由声辐射、橡胶环绕和致动器-放大器组合产生的阻尼组成。根据公式 (31) 可以计算出声音产生的阻尼。它是一个与频率相关的数值,因此有必要针对某一频率进行计算。

As will become clear in the actual example however, the acoustic damping is so low in respect to the other causes of damping that it is allowed to neglect the acoustic effects on mass and damping.

Also the effect of the surround is small compared with the electromagnetic damping of the actuator in combination with the amplifier.
To determine the dynamic behaviour of the total system, the electrical circuit of Figure 6 is used. For the electrical signal source to the loudspeaker a usually applied amplifier with a voltage source output is chosen.
为了确定整个系统的动态特性,使用了图 6 中的电路。对于扬声器的电信号源,我们选择了一个通常应用的放大器,其输出为电压源。

A voltage source output means that the amplifier has a very low output impedance approximating , (in practice ).
电压源输出意味着放大器具有非常低的输出阻抗 ,近似于 ,(实际上 )。
The electrical circuit of the loudspeaker can be approximated as a series circuit of the resistance and self inductance of the coil windings with a motion voltage source, which is proportional to the velocity of the coil relative to the magnetic field.
When connecting the loudspeaker to a voltage amplifier, the voltage source of the amplifier becomes connected in series with the motion voltage source and the total
Figure 6: The electrical model of the amplifier loudspeaker combination. The total current though the loudspeaker consists of the part delivered by the voltage source of the amplifier and the part delivered by the motion induced voltage of the loudspeaker coil.
图 6:放大器与扬声器组合的电气模型。扬声器的总电流 包括放大器电压源提供的部分 和扬声器线圈运动感应电压提供的部分
impedance. This circuit determines the current that creates the force to the moving part of the loudspeaker. The effect of both voltages on the total current can be analysed separately as their current contributions to the circuit can be super imposed because of the linear properties of the circuit in this approximation.
阻抗。该电路决定了电流 ,从而对扬声器的移动部分产生作用力。这两个电压对总电流的影响可分别进行分析,因为在这种近似情况下,由于电路的线性特性,它们对电路的电流贡献是超强的。

In practice the frequency area where the effect on damping is large is so low that the self inductance can be neglected for the analysis and only the resistive element with value is considered.

4.1 Amplifier Voltage Response
4.1 放大器电压响应

First the effect of the amplifier voltage is determined only by replacing the motor voltage by a short circuit which is allowed as the impedance of a voltage source is zero. The force of the motor is found by using Ohm's law and the force relation of a Lorentz actuator.
where equals the magnetic flux density in Tesla [T] through the coil and equals the length of the windings of the coil inside the magnetic field. From dynamic analysis the frequency response of the cone excursion with mass to an excitation force is given as follows as function of radial frequency :
其中 等于通过线圈的磁通密度(以特斯拉[T]为单位), 等于线圈在磁场内的绕组长度。通过动态分析,质量为 的锥体偏移 对激振力 的频率响应 与径向频率的函数关系如下:
The defined damping ratio , compliance and resonating eigenfrequency are equal to:
所定义的阻尼比 、顺应性 和共振特征频率 等于:
The resonating eigenfrequency in hertz is equal to and is called by different names like the first or fundamental resonance frequency, because at higher frequencies many additional resonances occur in a loudspeaker. At frequencies below , the first two terms in the denominator of Equation (18) will become small relative to one and the frequency response approaches the constant factor . This means that the magnitude of the cone excursion becomes frequency independent for a given excitation force.
以赫兹为单位的共振特征频率 等于 ,并有不同的名称,如第一共振频率或基频共振频率,因为在更高的频率下,扬声器中会出现许多附加共振。当频率低于 时,等式 (18) 中分母的前两项相对于一项会变得很小,频率响应接近常数系数 。这意味着对于给定的激振力,振膜偏移的大小与频率无关。

This is the dynamic situation where the force of the loudspeaker actuator is in balance with the force due to the motion of the cone against the stiffness of the loudspeaker suspension combined with the air stiffness of the enclosure.
At frequencies above the first term in the denominator of Equation (18) becomes dominant and the magnitude of the cone excursion becomes inverse proportional to the frequency squared.
当频率高于 时,等式 (18) 中分母的第一项变得占主导地位,锥体偏移的大小与频率的平方成反比。

This is the dynamic situation where the force of the loudspeaker actuator is in balance with the acceleration of the mass of the cone. At the magnitude of the cone excursion can become very large, in theory even infinite if . This is the low-end resonance frequency of any electrodynamic loudspeaker.
这是扬声器致动器的力与音盆质量的加速度相平衡的动态情况。在 时,音盆偏移的幅度会变得非常大,理论上如果 ,甚至可以达到无限大。这是任何电动扬声器的低端共振频率。
The acoustic response of a loudspeaker is shown to be proportional to acceleration which means that the frequency response of Equation (18) has to be combined with the corresponding +2 ( decade) slope in the frequency response that is related to acceleration as being the second derivative of the position. This combination is obtained by multiplication of Equation (18) with . Together with Equation 17 the frequency response from voltage to acceleration becomes as function of radial frequency :
扬声器的声学响应与加速度成正比,这意味着等式 (18) 的频率响应必须与频率响应中相应的 +2 ( 10)斜率相结合,该斜率与加速度有关,是位置的二阶导数。将公式 (18) 与 相乘即可得到这一组合。连同公式 17,频率响应 从电压到加速度成为径向频率 的函数:
With: 有了
Figure 7 shows this relation in an amplitude and phase to frequency response normalised to . It is clear that for frequencies below the original flat
图 7 显示了以 归一化的振幅和相位与频率响应的关系。很明显,对于低于 的频率,原来的平
Figure 7: Frequency response of the radiated sound as function of a periodic excitation force of an electrodynamic loudspeaker, normalised to with different damping settings.
图 7:在不同阻尼设置下,辐射声的频率响应与电动扬声器的周期激振力的函数关系,归一化为
response now has become a low frequency roll-off with a slope of +2 and phase lead, while the response above has become frequency independent without a phase lead.
At the damping determines the response and in this graph both the damping ratio and the quality factor are mentioned as is mostly used in loudspeaker systems. These terms relate as follows:
,阻尼决定响应,在此图中提到了阻尼比 和品质因数 ,因为 主要用于扬声器系统。这些术语的关系如下:

4.2 Motion Voltage Response
4.2 运动电压响应

The next step is to determine the force by the current that is induced by the motion voltage of the moving coil through the amplifier that in its turn can be approximated as a low impedance circuit. This current value is determined by the motion voltage and and causes a force that counteracts the movement so it acts like a damper.
下一步是通过电流 来确定力的大小,该电流是由运动线圈的运动电压通过放大器产生的,而放大器又可以近似地看作是一个低阻抗电路。该电流值由运动电压和 决定,并产生一个与运动相抵消的力,因此它就像一个阻尼器。
The motion induced voltage of a moving coil with velocity in a magnetic field equals:
速度为 的运动线圈在磁场中的运动感应电压等于:
The resulting current equals according to Ohm's law:
As the current will flow in the same magnetic field, a damping Lorentz force will occur:
With the previously derived value for this gives:
根据之前得出的 值,可以得出
So the damping coefficient is:
因此,阻尼系数 是:
Combining this damping coefficient with the spring-stiffness and mass gives the electrical factor given in the data sheets.
将该阻尼系数与弹簧刚度和质量相结合,就得出了数据表中给出的电气 因子。