Low Frequency Sound Generation by Loudspeaker Drivers
扬声器驱动器产生的低频声音

(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.
整个文件的复制仅限于个人使用。

The author/publisher is not responsible for any problems that might arise by using the contents of this paper.
对于因使用本文内容而可能产生的任何问题，作者/出版者概不负责。

Published by RMS Acoustics & Mechatronics
由 RMS Acoustics & Mechatronics 出版

The Netherlands 荷兰

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。

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.
随着时间的推移，人们开发出了大量的发声换能器，从喇叭中的小型振动膜，到通过变化磁场对等离子体进行高频调制，甚至是用于极低频的旋转式低音炮。
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.
在对旋转式超重低音扬声器进行简短评论后，下面的章节首先描述了决定振动板（代表低频扬声器的振膜）辐射声功率的物理关系。
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.
第二部分介绍了放大器、致动器和机械系统的动态特性，因为它们决定了膜片的振动与放大器信号的函数关系。最后介绍外壳的影响。

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.
背面朝向一个较大的箱体容积，而正面则通过覆盖有阻尼材料的第二个腔体传递声压，以降低气流噪音。

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
从这个极端的例子中，我们可以立即看出，这种情况可能会
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.
遗憾的是，这种系统存在一些缺陷，其中最重要的是风扇叶片产生的流动噪音，这就需要使用大量的阻尼结构。
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.
因此，旋转式低音扬声器只适用于专业安装场合，如电影院或大型电子教堂管风琴等大型系统。
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。

A loudspeaker driver diaphragm moves ideally like a piston, creating pressure waves that are propagated through the air.
扬声器驱动膜片像活塞一样理想地运动，产生的压力波在空气中传播。

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.
在计算辐射声功率时，假定扬声器的安装方式是，来自背面的压力永远不会达到来自正面的压力。
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.
这可以通过一个无限大的平板来实现，它将声音限制在半球形范围内辐射；也可以通过一个无限大的管子来实现，它将声音辐射到整个球形空间；还可以通过一个封闭的腔体（外壳）来实现，它的辐射从低频时的球形到高频时的半球形不等。

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:
隔膜上的压力是由隔膜本身的运动对周围空气产生的作用造成的，其关系如下：

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:
声阻在低频时与频率有关，在高频时变为常数。根据经验分析，可以得出以下近似值：

with: 用：

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:
有了这些数值和公式，就可以计算出空气对膜片的作用力：

Using Equation (3) this is equal to:
根据公式 (3)，这等于

The average radiated sound power is equal to:
平均辐射声功率等于：

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:
与频率无关的输出功率水平要求位移振幅与频率平方成反比。这一要求会产生以下重要影响：

This is the reason why powerful low frequency loudspeakers need to be large.
这就是大功率低频扬声器需要很大体积的原因。

Another important aspect is the relation between the sound power and acceleration.
另一个重要方面是声功率与加速度之间的关系。
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.
比气隙更大的（悬空）线圈会增加力的范围，使力更加恒定。

This means that a diaphragm will produce a constant frequency independent sound power when the acceleration is kept at a constant level.
这意味着，当加速度保持在恒定水平时，膜片将产生与频率无关的恒定声功率。

Furthermore the sound pressure is proportional to the square root of the sound power which means:
此外，声压与声功率的平方根成正比，这意味着

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.
声压与低频加速度的这种关系是一个重要现象，因为它意味着可以通过控制振膜的加速度来掌握扬声器的动态特性。

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.
法拉第定律描述了变化的磁场对电荷的影响，从而利用动能发电。
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.
根据能量守恒定律，从运动中产生电能与从电能中产生运动能是完全互补的，因此洛伦兹定律和法拉第定律密切相关。
In vectorial notation the formulation of Lorentz describes the force on a moving charged particle as:
在矢量符号中，洛伦兹的公式描述了运动的带电粒子所受的力：

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.
当开瓶器右旋时，从正电流方向到磁场方向（箭头），开瓶器点的移动决定了力的方向。

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:
如果有多个绕组，洛伦兹力就会变大：

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"）。

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.
不过，在实际例子中我们会发现，相对于其他阻尼原因，声学阻尼非常小，因此可以忽略声学对质量和阻尼的影响。
Also the effect of the surround is small compared with the electromagnetic damping of the actuator in combination with the amplifier.
此外，与执行机构和放大器的电磁阻尼相比，环绕声的影响很小。

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.
扬声器的电路可近似看作是线圈绕组的电阻和自感与运动电压源的串联电路，运动电压源与线圈相对于磁场的速度成正比。

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
将扬声器连接至电压放大器时，放大器的电压源将与运动电压源串联，而总电压源将与运动电压源串联。

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.
实际上，对阻尼影响较大的频率区域非常低，因此在分析时可以忽略自电感，只考虑电阻元件的值 。

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.
首先，由于电压源的阻抗为零，因此可以用短路来代替电机电压，从而确定放大器电压的影响。利用欧姆定律和洛伦兹致动器的力关系，可以求出电机的力。

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.
在这种动态情况下，扬声器致动器的力与音盆运动所产生的力保持平衡，音盆运动所产生的力与扬声器悬挂架的刚度以及箱体的空气刚度保持平衡。

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.
而 以上的响应则与频率无关，没有相位引导。

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:
在 ，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：

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:
根据欧姆定律，产生的电流相等：

As the current will flow in the same magnetic field, a damping Lorentz force will occur:
由于电流将在同一磁场中流动，因此会产生阻尼洛伦兹力：

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。

It is good to notice that the damping effect is in fact identical to velocity feedback as it is a force that proportionally counteracts velocity. This is further explained in the paper "Motional Feedback Theory in a Nutshell".
值得注意的是，阻尼效应实际上与速度反馈相同，因为它是一种按比例抵消速度的力。论文 "运动反馈理论简述 "对此有进一步解释。

The equations of the previous section on acoustic radiation apply to the situation where all energy is radiated from the front and not from the back side.
上一节关于声辐射的公式适用于所有能量都从正面而非背面辐射的情况。
The sound pressure from one side of the diaphragm is radiated in counter phase to the sound pressure from the other side because a rise in pressure at the front surface will correspond with a sink in pressure on the backside and the other way around.
振膜一侧的声压与另一侧的声压反相辐射，因为前表面的压力升高会导致背面的压力下降，反之亦然。
At low frequencies the pressures have ample time to reach the opposite side when propagating at the velocity of sound. As a result destructive interference occurs between both pressures, which means that the resulting pressure difference is reduced by cancellation.
在低频情况下，压力以声速传播时有足够的时间到达对面。因此，两种压力之间会产生破坏性干扰，这意味着所产生的压力差会因抵消而减小。
This phenomenon is called acoustic short-circuiting and at the cancellation is complete. At higher frequencies the cancellation will gradually reduce giving slope in the frequency response until the frequency where the distance from front to back equals half a wavelength , giving phase shift with constructive interference for that frequency. At so double the frequency the distance between front and backside equals a full wavelength with phase and destructive interference again. With higher frequencies this succession of constructive and destructive interference will happen at equal frequency intervals which on the logarithmic scale of a frequency response plot will appear as "hair" at higher frequencies.
这种现象被称为声学短路，在 ，这种抵消是完全的。在较高频率下，抵消将逐渐减小，使 的频率响应出现斜率，直到 前后的距离等于半个波长的频率 ，在该频率下产生 相移和建设性干扰。在 ，频率增加一倍，前后距离等于全波长时， ，再次出现相移和破坏性干扰。随着频率的升高，这种连续的建设性干扰和破坏性干扰将以相等的频率间隔发生，在频率响应图的对数刻度上，频率越高，"发丝 "越明显。
This all is shown in Figure 8 and the characteristic frequency response is called a Comb filter as it would show up in a non-logarithmic frequency plot as equally spaced spikes.
这一切如图 8 所示，其频率响应特征被称为梳状滤波器，因为它在非对数频率图中显示为等距的尖峰。
This basic reasoning neglects the fact that the sound is created over the entire diaphragm surface and not only at the centre but anyway the sound radiation becomes irregular and the low frequency radiation is strongly reduced.
这一基本推理忽略了一个事实，即声音是在整个膜片表面上产生的，而不仅仅是在中心，但无论如何，声音辐射会变得不规则，低频辐射会大大减少。
In fact the radiation pattern of an unmounted loudspeaker will always look like a dipole. On-axis the sound pressure is maximum, while off-axis at both sides when for instance and on the intermediate radiating plane the sound pressure is zero for all frequencies.
事实上，未安装扬声器的辐射模式看起来总是像偶极子。在轴上声压最大，而在两侧离轴处，例如 和中间辐射平面上，所有频率的声压均为零。
This observation matches the situation when measuring the frequency response in an anechoic room without reflections but in a real living room the walls will reflect the sound and in practice some loudspeaker systems are designed such, that they use these reflections to be able to compensate the loss of sound pressure due to acoustic interference.
这一观察结果与在没有反射的消声室中测量频率响应时的情况相吻合，但在真实的起居室中，墙壁会反射声音，而在实际设计中，一些扬声器系统会利用这些反射来补偿声学干扰造成的声压损失。
These so called "open baffle" systems have often a very appreciated sound although the placement in the room is even more critical than when the loudspeaker is built in an enclosure.
这些所谓的 "开放式障板 "系统通常具有非常动听的音效，尽管在室内的摆放位置比安装在箱体内的扬声器更为重要。
To avoid these problems, generally the loudspeaker is mounted in an enclosure, which cancels the sound from the back by blocking (closed-box), dissipates the sound over a long path, which mimics an infinite enclosure (transmission line) or uses an additional resonator (bass-reflex), which reduces the movement of the diaphragm at its resonance frequency and filters the radiation from the back side.
为了避免这些问题，扬声器通常安装在一个箱体中，通过阻挡（封闭箱体）来消除来自背面的声音，或通过长路径消散声音，模拟无限箱体（传输线），或使用额外的谐振器（低音反射），减少振膜在共振频率下的运动，过滤来自背面的辐射。
All these methods have their benefits and drawbacks. The transmission line requires a very large volume due to the need to fit at least one half wavelength of the lowest frequency ( for ) because always an anti-node (high velocity) of the sound must be captured in the damping material, which is very impractical. The closed box has a well controlled
所有这些方法各有利弊。传输线需要非常大的体积，因为至少需要容纳最低频率的一个半波长（ ， ），因为声音的反节点（高速）必须被阻尼材料捕获，这非常不切实际。封闭箱具有良好的控制能力

Figure 8: An electrodynamic loudspeaker radiates sound in two directions with difference. Without an enclosure that stops one of the sound waves they will interfere with each other causing a frequency related spatial dipole radiation pattern and an irregular frequency response on axis, called a "comb filter".
图 8：电动式扬声器向两个方向辐射声音， 不同。如果没有一个能阻挡其中一个声波的外壳，它们就会相互干扰，造成与频率相关的空间偶极子辐射模式和轴上不规则的频率响应，称为 "梳状滤波器"。
The graphs do not include the dynamics of the loudspeaker so a constant acceleration amplitude is assumed over the shown frequency band.
这些图表不包括扬声器的动态效果，因此在所示频段内假定加速度振幅恒定不变。

behaviour but the stiffness of the air spring by the enclosed volume will increase the fundamental resonance frequency with a corresponding decrease of damping (= increase of factor). The bassreflex system allows more acoustic power at the lowest bandwidth frequency, but it introduces a delay in the response due to the resonance and a -order roll off below the lowest badwidth frequency. The bass reflex principle is presented in a separate paper and the conclusion is that it is not useful for low frequency subwoofers.
低音反射系统的共振频率较高，但封闭体积带来的空气弹簧刚度会增加基频共振频率，同时阻尼也会相应减少（= 因子增加）。低音反射系统允许在最低带宽频率上产生更多的声功率，但会因共振而导致响应延迟，并在最低衰减带宽频率以下产生 -order 滚降。低音反射原理在另一篇论文中介绍，结论是低音反射对低频超重低音扬声器并无用处。
While also the transmission line is not useful due to its size, only the closed-box enclosure is further treated in this paper.
由于传输线的尺寸问题，本文只进一步讨论了封闭箱体。

When a loudspeaker is mounted in a closed-box enclosure the air inside the box can not escape nor can the sound pressure from the back side of the diaphragm reach the front side.
当扬声器安装在封闭箱体内时，箱内的空气无法排出，振膜背面的声压也无法传到正面。
A drawback of the closed-box configuration is the fact that the sound from the back side of the diaphragm can also not reach our ears.
闭箱结构的一个缺点是，来自振膜背面的声音也无法传入我们的耳朵。
For high frequencies, where the size of the enclosure is large in respect to the wavelength, this sound from the back side should be absorbed by damping material as otherwise that sound would be reflected by the inner wall of the enclosure and return through the diaphragm, which is very light.
对于高频而言，由于箱体尺寸相对于波长较大，来自背面的声音应被阻尼材料吸收，否则声音会被箱体内壁反射，并通过振膜返回，而振膜非常轻。
This mixing of direct sound from the front side of the diaphragm and reflected sound from the back side in an enclosure causes a diffuse sound.
这种来自振膜正面的直达声和来自箱体背面的反射声的混合会产生扩散声。
For this reason (and to reduce the stiffness by isothermal expansion as shown in the following section) a piece of damping material is applied inside the enclosure to absorb the radiated energy.
因此（如下节所示，通过等热膨胀降低刚度），在箱体内部使用一块阻尼材料来吸收辐射能量。
Fortunately, at low frequencies with long wavelengths larger than the size of the enclosure, this is not a problem. This can be understood from the following reasoning, while using the relations given in Section 2. In that section the radiated sound power of a diaphragm in free air was
幸运的是，在波长大于外壳尺寸的低频情况下，这并不是一个问题。利用第 2 节中给出的关系，我们可以从下面的推理中理解这一点。在该章节中，自由空气中振膜的辐射声功率为
derived from the velocity of the diaphragm and the pressure on the diaphragm. This pressure appeared to be related to the velocity, thereby creating power, because the velocity and the pressure are in phase.
从膜片的速度和膜片上的压力得出。这种压力似乎与速度有关，从而产生了动力，因为速度和压力是相位的。

Inside an enclosure with small dimensions relative to the wavelength, the pressure is only related to the position of the diaphragm as the enclosed air acts as a spring. With sinusoidal movements the velocity is 90 degrees out of phase with the position, because when the position , the velocity equals and cos differs 90 degrees from sine. The multiplication of and sine of the same term, which is the case when determining the power, averages to zero as follows from the following trigonometric identity: , which is average zero. As a consequence the acoustic power at the back side is zero for low frequencies.
在相对于波长尺寸较小的外壳内，压力只与膜片的位置有关，因为外壳内的空气就像弹簧一样。正弦运动时，速度与位置相位差 90 度，因为当位置 时，速度等于 ，而 cos 与正弦相差 90 度。 和正弦的同项相乘，即确定功率时的情况，平均为零，如下所示的三角恒等式： 即平均为零。因此，低频时背面的声功率为零。

The enclosed air in the box will act as an air spring that adds its stiffness to the stiffness of the suspension of the loudspeaker diaphragm resulting in a higher fundamental resonance frequency, thereby limiting the low frequency bandwidth.
箱体中的封闭空气将起到空气弹簧的作用，在扬声器振膜悬挂刚度的基础上增加其刚度，导致基频共振频率升高，从而限制了低频带宽。
For that reason it is important to determine this stiffness. To calculate this coupling stiffness by the air in the enclosure, the pressure on the surface area of the diaphragm of a loudspeaker is calculated as function of the displacement. The compression/expansion is somewhere between adiabatic and isothermal.
因此，确定这一刚度非常重要。为了计算箱体内空气的耦合刚度，扬声器振膜 表面积上的压力与位移成函数关系。压缩/膨胀介于绝热和等温线之间。
When the cabinet is filled with fibre wadding like wool or any other material consisting of soft fine fibre like polyester, the temperature will fluctuate less at expansion/compresssion.
如果橱柜内填充的是羊毛等纤维絮状物或任何其他由聚酯等柔软细纤维组成的材料，在膨胀/压缩时，温度波动会更小。
In that case the relation between the pressure and the compressed volume can be derived by applying the simple isothermal gas law . For adiabatic expansion/compression the pressure change is larger due to the temperature rise at compression. For air this is a factor . This factor will linearly increase the stiffness of the air spring with a resulting higher fundamental resonance frequency of the combination of the moving mass of the diaphragm with this stiffness.
在这种情况下，压力和压缩体积之间的关系可以通过应用简单的等温气体定律 得出。对于绝热膨胀/压缩，由于压缩时温度升高，压力变化较大。对于空气来说，这一系数为 。该系数将线性增加空气弹簧的刚度，从而提高膜片运动质量与该刚度组合的基本共振频率。
This resonance frequency should be as low as possible for an extended low frequency bandwidth of the loudspeaker.
为了扩大扬声器的低频带宽，共振频率应尽可能低。
A slight reduction of the stiffness can be achieved by making the expansion/compression more isothermal by filling the cabinet with very fine fibre padding, even when this padding is not required for other reasons like the abovementioned damping of internal reflections and standing waves inside the cabinet at higher frequencies.
通过在箱体中填充极细的纤维衬垫，使膨胀/压缩更加等温，可以略微降低刚度，即使由于其他原因，如上述阻尼箱体内部反射和高频驻波等，并不需要填充衬垫。
In practice a reduction of to approximately 1.2 can be achieved when applying padding fibre giving a reduction of the resonance frequency with approximately .
在实际应用中，当使用填充纤维时，可将 降低到约 1.2，从而将共振频率降低到约 。

Suppose displacement of the diaphragm, the displaced volume equals:
假设 ，膜片的位移量 等于：

The displaced volume will cause a relative pressure change to the environmental pressure that can be calculated with the adiabatic gas law and the relative change
被置换的体积将导致环境压力的相对压力变化 ，该变化可通过绝热气体定律和相对变化计算得出
of the volume of the enclosure :
的体积 ：

With the surface area of the diaphragm , the stiffness of the spring due to the enclosed air is equal to the following expression:
膜片的表面积为 ，封闭空气导致的弹簧刚度等于以下表达式：

By using the calculated stiffness value in the relations that were presented in Section 4 it can be concluded that the resonance frequency of a loudspeaker will increase when mounted in a closed-box enclosure, while the damping ratio will be decreased, giving a larger -value. In practical designs mostly the fundamental resonance frequency is increased with at least a factor two, which is significant. To improve this behaviour there are several possibilities:
通过使用第 4 节所述关系中的计算刚度值，可以得出结论：当扬声器安装在封闭箱体中时，其共振频率将增加，而阻尼比 将减小，从而产生更大的 值。在实际设计中，基本共振频率大多会增加至少 2 倍，这是很重要的。要改善这种特性，有几种可能性：

The use of an inverse filter is the mostly used principle although it is hardly possible to do this accurately especially with situations where the damping is even lower. The exact tuning of the dynamic response including overshoot is for that reason often less than acceptable.
使用反向滤波器是最常用的原理，尽管很难做到精确，尤其是在阻尼更小的情况下。因此，包括过冲在内的动态响应的精确调整往往不那么容易接受。
The use of digital electronics has made it possible to realise a more exact inverse transfer function by means of Digital Signal Processors (DSPs) and built in amplifiers dedicated for each loudspeaker.
通过数字信号处理器（DSP）和每个扬声器专用的内置放大器，数字电子技术的应用使得实现更精确的反向传递函数成为可能。
In the first example that will be shown a sophisticated DSP compensator reaches the best one could obtain with this method.
在将要展示的第一个例子中，一个复杂的 DSP 补偿器达到了这种方法所能达到的最佳效果。
A remaining problem with this approach is that it is purely based on feedforward compensation while it will force the loudspeaker to very high uncontrolled excursions at very low frequencies into the non-linear range of the surround spring and the Lorentz actuator.
这种方法的另一个问题是，它纯粹基于前馈补偿，而会迫使扬声器在非常低的频率下出现非常高的不受控制的偏移，进入环绕弹簧和洛伦兹致动器的非线性范围。
This will cause severe distortion unless a very good (and thus expensive) loudspeaker is used. Still the related distortion is quite often accepted as people tend to like distortion.
这将导致严重失真，除非使用非常好的（因此也是昂贵的）扬声器。尽管如此，由于人们往往喜欢失真，因此相关的失真还是经常被接受。
This is caused by a special property of human ears that distort themselves quite heavily as function of the loudness.
这是由于人耳的一种特殊属性造成的，它随着响度的变化而发生严重的扭曲。
A certain level of distortion will in reality be perceived as a louder signal while also the human brain can reconstruct the fundamental frequency from its harmonics giving the perception of a lower signal frequency even if this lower frequency is reduced in amplitude.
一定程度的失真在现实中会被认为是更大的信号，而人脑也能从谐波中重建基频，使人感觉信号频率更低，即使这个更低的频率在振幅上有所减弱。
This is why a small transistor radio still gives the right impression of the total music as otherwise the lower tones would be perceived as one or two
这就是为什么小型晶体管收音机仍能给人留下完整音乐的正确印象，否则低音就会被认为是一个或两个音调。
octaves higher changing a baritone in a soprano. Yet it is not preferred as the bass is missing and the distortion and resonances will cause headache.
在女高音中，男中音的八度变化较大。然而，这种方法并不可取，因为缺少了低音，失真和共鸣会让人头痛。

The second method is to measure the sound itself and use that as feedback signal for an actively controlled system. This approach is limited by the delay between the loudspeaker and the microphone and the room reflections that are also delayed and detected by the microphone.
第二种方法是测量声音本身，并将其作为主动控制系统的反馈信号。这种方法受到扬声器和麦克风之间的延迟以及室内反射的限制，而室内反射也会延迟并被麦克风检测到。
Although this principle can be used with adequate filters to suppress room reverberations, which often deteriorate the sound quality, these delays would cause the system to become unstable resulting in a howling sound.
虽然这一原理可与适当的滤波器配合使用，以抑制通常会降低音质的室内混响，但这些延迟会导致系统变得不稳定，从而产生嚎叫声。
Furthermore it would be limited to the very low frequencies where the wavelength is long due to the delays and lack of coherence (phase relationship) in the reflected sound waves.
此外，由于反射声波的延迟和缺乏一致性（相位关系），它将仅限于波长较长的极低频率。

The best method is to use active feedback control of the moving diaphragm and is in general terms presented in a separate paper on "Motional Feedback in a Nutshell". All active feedback controlled systems are based on the closed-box enclosure as it behaves the most deterministic.
最好的方法是对运动膜片进行主动反馈控制，具体方法在另一篇题为 "运动反馈简述 "的论文中作了概括介绍。所有主动反馈控制系统都以封闭箱体为基础，因为它的表现最具有确定性。

Loudspeakers are normally designed to be operated with a voltage controlled amplifier with a near zero output impedance.
扬声器通常设计为使用输出阻抗接近于零的电压控制放大器。
Often more loudspeakers are mounted together in one enclosure to cover different frequency ranges which means that these loudspeakers have to match in sound intensity (power).
为了覆盖不同的频率范围，通常会在一个箱体内安装多个扬声器，这意味着这些扬声器的声强（功率）必须匹配。
In order to achieve this matching the radiated sound level of a loudspeaker is measured with a standard signal. This standard signal is the voltage level that would result in an electrical Power of 1 Watt in a 8 Ohm pure resistance. The corresponding voltage has an effective value of .
为了实现这种匹配，扬声器的辐射声级是用标准信号测量的。该标准信号是 在 8 欧姆纯电阻上产生 1 瓦功率的电压电平。相应电压的有效值为 。

To get a feel of the efficiency of subwoofers some average numbers will illustrate the fact that an electrodynamic loudspeaker has an extremely low efficiency in converting electrical power into acoustical power. Most subwoofers have a coil resistance in the order of , which means that the standard reference voltage of represents an input power level of . The specified acoustic power of many loudspeaker metre over a hemisphere ranges in practice between 80 and for this reference input voltage. The best case value of is below , which represents a sound power intensity of . A lower power level is equivalent to , which means that is equal to a sound power intensity of . With a surface of a hemisphere @ of the total radiated power becomes . With input power this is an efficiency of only .
为了了解超低音扬声器的效率，一些平均数字可以说明这样一个事实，即电动扬声器将电能转化为声能的效率极低。大多数超低音扬声器的线圈电阻为 ，这意味着 的标准参考电压代表 的输入功率水平。对于该参考输入电压，许多扬声器 米半球的指定声功率实际上在 80 到 之间。 的最佳值为 ，低于 ，表示声功率强度为 。 较低的功率水平相当于 ，这意味着 相当于 的声功率强度。半球表面 @ 的 ，总辐射功率变为 。输入功率为 时，效率仅为 。

Another way to look at this low efficiency is in the damping effect on the fundamental resonance by the radiated sound. A high level of damping would imply a high level of radiated energy.
另一种看待低效率的方法是辐射声对基频共振的阻尼效应。高阻尼意味着高辐射能量。

This acoustic damping can be derived from Equation (7):
这种声学阻尼可以从公式 (7) 中推导出来：

By the following example with some practical values it will become clear that this acoustic damping is very limited, which also indicates the low efficiency of an electrodynamic loudspeaker.
通过下面的例子和一些实际数值，我们可以清楚地看到，这种声学阻尼非常有限，这也表明电动扬声器的效率很低。

This results in: 其结果是