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

(C) 2017 The author, RMS Acoustics & Mechatronics and Grimm Audio. All rights reserved.
(C) 2017 年作者，RMS Acoustics & Mechatronics 和 Grimm Audio。版权所有。

Copying of the complete document is allowed for personal use only.
完整文件的复印仅供个人使用。

The author/publisher is not responsible for any problems that might arise by using the contents of this paper.
作者/出版商对使用本文内容可能出现的任何问题不承担任何责任。

Published by RMS Acoustics & Mechatronics
由 RMS 声学与机电一体化出版

The Netherlands 荷兰人

1 Introduction ..... 3 1 简介 ...... 3
1.1 Rotary Subwoofers ..... 4
1.1 旋转低音炮 ...... 4
2 Sound from a Vibrating Diaphragm ..... 5
2 振动膜发出的声音 ...... 5
3 The Lorentz-type actuator ..... 8
4 Dynamic Properties of Loudspeaker Drivers ..... 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 使用外壳 ...... 15
5.1 Closed-Box Enclosure ..... 16
5.1 封闭式外壳 ...... 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
5.1.3 Causes of the Low Efficiency ..... 20
5.1.3 效率低下的原因...... 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 增加散热面............. 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.

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，人们开发出了大量发声换能器，从喇叭中的小型振动膜到通过变化的高频磁场调制等离子体，甚至用于极低频的旋转低音炮。
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.
除了提到的等离子体原理外，大多数实用的扬声器总是采用中间材料，在大多数情况下是平面或圆锥形隔膜，根据致动器的力驱动空气分子 隔膜可以是刚性的或非刚性的，具体取决于致动方式。后者最好的例子是静电扬声器，其中非刚性薄膜由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).
大多数低频扬声器使用线性洛伦兹型致动器来驱动振膜，振膜由支撑框架内的弹性悬架沿一个运动方向（自由度）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.
洛伦兹型执行器也称为动圈式执行器，由一个永磁体“定子”组成，在气隙内产生强磁场，气隙内有一个动线圈，将线圈中的电流转换为隔膜上的力反过来“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最早应用之一，因此动圈也称为音圈，因为该线圈为扬声器提供“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.
在对旋转低音炮进行简短评论后，以下各节首先描述确定振动板（代表低频扬声器振膜）的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.
第二部分描述了放大器、执行器和机械系统的动力学，因为它们根据来自放大器的信号确定膜片的振动。最后介绍了外壳的影响。

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 的 Thigpen 旋转低音炮使用叶片节距可控的风扇，通过改变叶片节距来在风扇的前侧和后侧之间产生压力变化。
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.
背面面向较大的外壳体积，而正面则通过覆盖有阻尼材料的第二个腔室传递声压，以降低气流噪声。

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.
为了理解一般的物理现象，观察极端情况通常是有说明性的。低频声音再现时的静压 就是这么极端的情况。它是指低于或高于平均环境压力的恒定压力。这种现象只能通过将连续的气流指向某个物体来实现，就像帆中的风一样。
It becomes immediately clear from this extreme example that such a situation can
从这个极端的例子可以立即看出，这种情况可以
never be created by a diaphragm that moves over a limited range, as is the case in a "normal" loudspeaker driver.
永远不会由在有限范围内移动的隔膜产生，就像“正常”扬声器驱动器的情况一样。

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.
然而，可以通过风扇产生人造风，如图 2 所示。通过控制叶片的桨距，可以改变流经该设备的空气的方向和量，理论上可以改变为任何频率，甚至在 。它遵循与飞机螺旋桨驱动装置相同的原理，以在制动时反转推力。
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.
不幸的是，该系统有一些注意事项，其中最重要的是风扇叶片的流动噪音，这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.
在实践中，为了达到可接受的噪声水平，此类低音炮只能使用大型隔音板（荷兰语“suskast”），这是一种消声外壳，具有通向听音室的开口，内部覆盖有足够的阻尼材料。

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.
由于这些原因，旋转低音炮仅适用于专业安装，允许大型系统，例如电影院或大型电子教堂管风琴。
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 的 Thigpen 旋转低音炮，它与 Marshall 和 Ogletree 的 OPUS 4 教堂管风琴一起用于真实的教堂。

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.
由于本文重点关注家庭或工作室录音室和母带室的音乐再现装置，频率范围不低于 ，旋转低音炮不被认为是可行的选择，下面仅介绍带有往复式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.
扬声器驱动器振膜像活塞一样理想地移动，产生通过空气传播的压力波。

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.
为了计算辐射声功率，假设扬声器的安装方式使得来自背面的压力永远无法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.
这可以通过无限大的板来实现，限制声音在半球内辐射，通过无限大的管，声音将在空间中的整个球体上辐射，或者通过封闭的室（外壳），其中辐射从低频的球形辐射到高频的半球形辐射。

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.
从正弦运动的振膜一侧辐射的一个周期内的平均声功率等于有效 (RMS) 值的乘积 隔膜的速度 由施加在隔膜上的压力引起的隔膜上的力的真实（同相）部分的有效值。

with:
和：

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作用于周围空气而引起的，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 等于复数、频率相关声阻抗的实部 。只有实部才能产生功率，因为​​它对应于与速度 4 同相的气压分量。

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相关，在较高频率时变得恒定。通过实证分析发现，其值可近似为：

with: 和：

The transition frequency between the low and high frequency range is found when both values are equal:
转变频率 当两个值相等时，找到低频和高频范围之间的值：

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).
振膜直径为 的大型扬声器 以声速 会有一个过渡频率 ，这表明对于低音炮的低频声音再现，允许使用等式（3）中的值。

With these values and equations the force by the air on the diaphragm can be calculated:
利用这些值和方程，可以计算空气作用在隔膜上的力：

Using Equation (3) this is equal to:
使用等式（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:
随着隔膜的正弦往复运动， 平均声功率可写为：

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.
当膜片在所有频率下以恒定幅度移动时，辐射功率与频率的功率 4 成正比增加。在控制工程术语中，这表示频率响应图中的斜率为 +2 十年。

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电平要求位移幅度与频率的平方成反比。这一要求具有以下重要后果：

This is the reason why powerful low frequency loudspeakers need to be large.
这就是为什么强大的低频扬声器需要很大的原因。

Another important aspect is the relation between the sound power and acceleration.
另一个重要方面是声功率和加速度之间的关系。
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，但与位移幅度的平方关系（如方程（9）所示）意味着辐射功率与膜片的加速度成正比，因为加速度也随着膜片加速度的增加而增加。频率的平方与斜率 当位移幅度保持恒定时。

With the effective value of the acceleration , Equation (9) can be written as the following frequency independent relation:
与加速度的有效值 ，方程（9）可以写成以下与频率无关的关系：

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，该音圈致动器由永磁结构和插入该气隙中的动圈组成，永磁结构在圆形气隙中产生强磁场。
A larger (overhung) coil than the air-gap increases the range over which the force is more constant.
比气隙更大（悬垂）的线圈增加了力更恒定的范围。

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恒定水平时，隔膜将产生恒定频率的独立声功率。

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.
低频声压与加速度的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控制扬声器的动态行为。

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·安东·洛伦兹（Hendrik Antoon Lorentz，1853 - 1928）提出了洛伦兹力作为麦克斯韦方程组的完善。
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完全互补，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，洛伦兹公式将运动带电粒子上的力描述为：

Figure 4: Determining the direction of the Lorentz force with the corkscrew rule.
图 4：用螺旋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.
当开瓶器向右旋转时，从正电流方向转向磁场方向（箭头），开瓶器尖端的运动决定了力的方向。

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:
和 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。第二个术语用于电磁执行器。除了作用在运动粒子上的力之外，它同样表示作用在流经长度为导线的电流上的力 ，插入磁场中。对于这种情况，移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:
对于以一定角度作用在电线上的磁力 相对于磁通密度的磁场方向 ，承载电流 ，这种关系导致电磁执行器洛伦兹力的标量表示法，其大小由下式给出：

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.
由于矢量洛伦兹方程中的“叉积”，该力的方向与由磁场和电流方向确定的平面正交。
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.
该规则可以被记住为右手规则或螺旋规则，即当将螺旋形从正电流方向旋转到磁场方向时找到正力方向，如图 4 所示。
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.
当然，对于真正的机械工程师来说，任何普通的右旋螺钉也都适合该目的，但开瓶器更容易记住。

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 尽可能保持等于一。这意味着简化的方程等于：

Figure 5: A damped mass-spring system with an external force stimulus.
图 5：具有外力刺激的阻尼质量弹簧系统。

And with multiple windings the Lorentz force becomes:
对于多个绕组，洛伦兹力变为：

where equals the total length of the wire inserted in the magnetic field.
在哪里 等于插入磁场中的导线的总长度。

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".
除以电流时 该方程给出了执行器的力与电流之比，也称为力常数，这将被证明根本不是常数（请参阅论文“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，动圈扬声器驱动器可以用简单的模型来描述，如图 5 所示。具有质量的运动体 由振膜和线圈组成。机体由支架和橡胶辊环绕悬挂，具有一定的刚度 。最后是阻尼器，有阻尼系数 由声音辐射、橡胶包围和致动器放大器组合引起的阻尼组成。通过等式（31），可以计算由声音引起的阻尼。它是一个与频率相关的值，因此有必要针对特定​​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.
然而，如在实际示例中将变得清楚的，相对于阻尼的其他原因，声阻尼是如此之低，以至于允许忽略对质量和阻尼的声学影响。
Also the effect of the surround is small compared with the electromagnetic damping of the actuator in combination with the amplifier.
此外，与与放大器结合的执行器的电磁阻尼相比，环绕声的影响也很小。

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行为，使用了图 6 的电路。对于扬声器的电信号源，通常选择具有电压源输出的放大器。
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 ， （在实践中 ）。

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.
扬声器的电路可以近似为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
当将扬声器连接到电压放大器时，放大器的电压源与运动电压源和总电压串联。

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：放大器扬声器组合的电气模型。总电流 尽管扬声器由以下部分组成 由放大器和部件的电压源提供 由扬声器线圈的运动感应电压传递。

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。该电路确定电流 这会对扬声器的移动部分产生力。两个电压对总电流的影响可以单独分析，因为由于该近似中电路的线性特性，它们对电路的电流贡献可以叠加。
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.
在实践中，对阻尼影响较大的频率区域是如此之低，以至于分析时可以忽略自感，仅考虑具有值的电阻元件 被认为。

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，放大器电压的影响仅通过用短路代替电机电压来确定，短路是允许的，因为电压源的阻抗为零。电机的力是通过使用欧姆定律和洛伦兹执行器的力关系来找到的。

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 :
在哪里 等于通过线圈的磁通密度，以特斯拉 [T] 为单位， 等于磁场内线圈绕组的长度。从动态分析频率响应 锥体偏移 有质量 到激振力 如下所示为径向频率的函数 :

The defined damping ratio , compliance and resonating eigenfrequency are equal to:
定义的阻尼比 ， 遵守 和共振本征频率 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.
共振本征频率 以赫兹为单位等于 它有不同的名称，例如第一或基本谐振频率，因为在较高频率下，扬声器中会发生许多附加谐振。频率低于 ，方程（18）分母中的前两项将相对于 1 变小，并且频率响应接近常数因子 。这意味着对于给定的激振力，锥体偏移的幅度变得与频率无关。
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.
这是动态情况，其中扬声器致动器的力与由于锥体运动抵抗扬声器悬架的刚度以及外壳的空气刚度而产生的力平衡。

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.
在以上频率 等式（18）分母中的第一项占主导地位，锥体偏移的幅度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锥体质量的加速度平衡的动态情况。在 锥体偏移的幅度可以变得非常大，理论上甚至是无限的，如果 。这是任何电动扬声器的低端谐振频率。

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) 的频率响应必须与相应的 +2 ( 十进制）频率响应中的斜率，与加速度相关，作为位置的二阶导数。该组合是通过将等式（18）与 。与公式 17 一起得出频率响应 从电压到加速度成为径向频率的函数 :

With: 和：

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 显示了幅度和相位与频率响应的关系，归一化为 。很明显，对于以下频率 原来的公寓

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：辐射声音的频率响应作为电动扬声器周期性激振力的函数，标准化为 具有不同的阻尼设置。

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.
现在响应已变成斜率为 +2 的低频滚降，并且 相位领先，而上述响应 已经变得与频率无关，没有相位超前。

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，在该图中，阻尼比 和品质因数 被提到为 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.
下一步是通过电流确定力 它是由运动线圈的运动电压通过放大器引起的，而放大器又可以近似为低阻抗电路。该电流值由运动电压和 并产生抵消运动的力，因此它起到阻尼器的作用。

The motion induced voltage of a moving coil with velocity in a magnetic field equals:
运动线圈随速度变化的运动感应电压 在磁场中等于：

The resulting current equals according to Ohm's law:
根据欧姆定律，所得电流等于：

As the current will flow in the same magnetic field, a damping Lorentz force will occur:
当电流在同一磁场中流动时，会产生阻尼洛伦兹力：

With the previously derived value for this gives:
与之前导出的值 这给出：

So the damping coefficient is:
所以阻尼系数 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，得出电 数据表中给出的因素。

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，因为它是按比例抵消速度的力。这在论文《Motional Feedback Theory in a Nutshell》中得到了进一步的解释。

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方程适用于所有能量从正面而不是从背面辐射的情况。
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.
来自振膜一侧的声压与另一侧的声压反相辐射，因为前表面压力的上升将对应于背面压力的下降，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.
在低频下，压力以声速传播时有足够的时间到达另一侧。结果，两个压力之间发生相消干涉，这意味着所产生的压力差通过抵消而减小。
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.
这种现象称为声短路，在 取消完成。在较高的频率下，取消将逐渐减少给予 频率响应的斜率，直到距离的频率 从前到后等于半个波长 , 给予 该频率具有相长干扰的相移。在 因此，频率的两倍，正面和背面之间的距离等于全波长 再次相位和相消干涉。对于较高的频率，这种连续的相长和相消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，在较高频率下将显示为“头发”。
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，特征频率响应称为梳状滤波器，因为它将在非对数频率图中显示为等间隔的尖峰。
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.
这种基本推理忽略了这样一个事实：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，声音辐射变得不规则，并且低频辐射大大减少。
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，未安装的扬声器的辐射图总是看起来像偶极子。轴上声压最大，而离轴时两侧声压最大 在中间辐射平面上，所有频率的声压为零。
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匹配，但在真实的客厅中，墙壁会反射声音，并且在实践中，一些扬声器系统的设计是这样的，它们使用这些反射来补偿损失由于声学干扰而产生的声压。
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“开放式挡板”系统通常具有非常令人欣赏的声音，尽管在房间中的放置比扬声器内置在外壳中更为重要。
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，通常将扬声器安装在一个外壳中，通过阻挡（封闭箱）来消除来自后面的声音，通过模仿无限外壳（传输线）的长路径消散声音或使用附加的谐振器（低音反射），可减少振膜以其谐振频率的运动并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
所有这些方法都有其优点和缺点。由于需要适应最低频率的至少半波长（ 为了 ）因为声音的波腹（高速）必须始终被阻尼材料捕获，这是非常不切实际的。封闭式箱体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：电动扬声器向两个方向辐射声音 不同之处。如果没有阻止其中一个声波的外壳，它们将相互干扰，从而导致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特性，因此假定在所示频带内加速度幅度恒定。

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.
行为，但空气弹簧的封闭体积的刚度将增加基本共振频率，并相应减少阻尼（=增加 因素）。低音反射系统在最低带宽频率下允许更大的声功率，但由于共振和 -阶数滚降到最低坏宽度频率以下。低音反射原理在另一篇论文中介绍，结论是它对于低频低音炮没有用处。
While also the transmission line is not useful due to its size, only the closed-box enclosure is further treated in this paper.
虽然传输线由于其尺寸而没有用处，但本文仅进一步处理封闭盒外壳。

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的箱体中时，箱体内部的空气无法逸出，振膜背面的声压也无法到达正面。
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.
封闭式结构的一个缺点是来自振膜背面的声音也无法到达我们的耳朵。
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，外壳尺寸相对于波长较大，来自背面的声音应被阻尼材料吸收，否则声音将被外壳内壁反射并通过隔膜返回，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.
来自振膜前侧的直达声音与来自外壳后侧的反射声音的混合会产生扩散声音。
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（并通过等温膨胀来降低刚度，如下节所示），在外壳内部应用了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.
幸运的是，在长波长大于外壳尺寸的低频下，这不是问题。使用第 2 节中给出的关系，可以通过以下推理来理解这一点。
In that section the radiated sound power of a diaphragm in free air was
在该部分中，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.
由隔膜的速度和隔膜上的压力得出。这种压力似乎与速度有关，从而产生能量，因为速度和压力是同相的。

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.
在尺寸相对于波长较小的外壳内，压力仅与隔膜的位置相关，因为封闭的空气充当弹簧。
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.
对于正弦运动，速度与位置异相 90 度，因为当位置 ，速度等于 cos 与正弦相差 90 度。的乘法 和同项的正弦（确定幂时的情况）根据以下三角恒等式平均为零： ，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.
盒子中封闭的空气将充当空气弹簧，将其刚度增加到扬声器振膜悬架的刚度，从而产生更高的基本谐振频率，从而限制低频带宽。
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，确定该刚度非常重要。为了计算外壳中空气的耦合刚度，隔膜表面积上的压力 扬声器的位移被计算为位移的函数。压缩/膨胀介于绝热和等温之间。
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.
当柜体填充纤维填料（如羊毛）或任何其他由柔软细纤维（如聚酯）组成的材料时，膨胀/压缩时温度波动较小。
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，可以通过应用简单的等温气体定律导出压力和压缩体积之间的关系 。对于绝热膨胀/压缩，由于压缩时的温升，压力变化更大。对于空气来说这是一个因素 。该因素将线性增加空气弹簧的刚度，从而导致膜片的移动质量与该刚度的组合的基本共振频率更高。
This resonance frequency should be as low as possible for an extended low frequency bandwidth of the loudspeaker.
对于扬声器的扩展低频带宽，该谐振频率应该尽可能低。
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.
通过用非常细的纤维填充物填充箱体，使膨胀/压缩更加等温，可以稍微降低刚度，即使由于其他原因（例如上述箱体内反射和驻波的阻尼）不需要该填充物在更高的频率下。
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 .
在实践中减少 当应用填充纤维时，可以实现约 1.2，从而使谐振频率降低约 。

Suppose displacement of the diaphragm, the displaced volume equals:
认为 隔膜的位移，位移体积 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
排出的体积将引起环境压力的相对压力变化 可以用绝热气体定律和相对变化来计算
of the volume of the enclosure :
外壳体积 :

With the surface area of the diaphragm , the stiffness of the spring due to the enclosed air is equal to the following expression:
与隔膜的表面积 ，由于封闭空气而产生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 节中提出的关系中计算出的刚度值，可以得出结论，当安装在封闭箱体中时，扬声器的谐振频率会增加，而阻尼比会增加 将减少，给出更大 -价值。在实际设计中，大多数基本谐振频率至少增加两倍，这是很重要的。为了改善这种行为，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.
逆滤波器的使用是最常用的原理，尽管很难准确地做到这一点，特别是在阻尼更低的情况下。因此，动态响应的精确调整（包括过冲）通常不太可接受。
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.
数字电子技术的使用使得通过数字信号处理器 (DSP) 和每个扬声器专用的内置放大器实现更精确的逆传递函数成为可能。
In the first example that will be shown a sophisticated DSP compensator reaches the best one could obtain with this method.
在第一个示例中，将显示复杂的 DSP 补偿器达到了使用此方法可以获得的最佳补偿器。
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.
这种方法的一个剩余问题是，它纯粹基于前馈补偿，同时它会迫使扬声器在非常低的频率下达到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.
这将导致严重的失真，除非使用非常好的（因此昂贵的）扬声器。尽管如此，相关的失真仍然经常被接受，因为人们倾向于喜欢失真。
This is caused by a special property of human ears that distort themselves quite heavily as function of the loudness.
这是由人耳的特殊特性引起的，人耳会根据响度严重扭曲自身。
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.
一定程度的失真实际上会被感知为响亮的信号，而人脑也可以根据其谐波重建基频，从而给出较低信号频率的感知，即使该较低频率的幅度减小。
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然能给整个音乐留下正确的印象，否则较低的音调将被视为一两个
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.
将女高音中的男中音提高八度。但它并不是首选，因为低音缺失，失真和共振会引起头痛。

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用作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也被麦克风延迟和检测到的房间反射的限制。
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.
虽然这一原理可以与足够的滤波器一起使用来抑制房间混响，这通常会降低音质，但这些延迟会导致系统变得不稳定，从而产生啸叫声。
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相干性（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.
最好的方法是使用移动膜片的主动反馈控制，并且在“Motional Feedback in a Nutshell”的单独论文中概括地介绍了这种方法。所有主动反馈控制系统都基于封闭式外壳，因为它的行为最具确定性。

Loudspeakers are normally designed to be operated with a voltage controlled amplifier with a near zero output impedance.
扬声器通常设计为与输出阻抗接近于零的压控放大器一起运行。
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).
通常，多个扬声器一起安装在一个外壳中，以覆盖不同的频率范围，这意味着这些扬声器必须在声音强度（功率）上匹配。
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，需要使用标准信号来测量扬声器的辐射声级。该标准信号是 8 欧姆纯电阻产生 1 瓦电功率的电压水平。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 .
为了了解低音炮的效率，一些平均数字将说明电动扬声器将电能转换为声能的效率极低的事实。大多数低音炮的线圈电阻约为 ，这意味着标准参考电压 表示输入功率电平 。许多扬声器的指定声功率 实际上，半球上的米范围在 80 到 80 之间 对于这个参考输入电压。最佳情况值为 是 以下 ，它代表声功率强度 。 A 较低的功率水平相当于 ， 意思就是 等于声功率强度 。具有半球表面@ 的 总辐射功率变为 。和 输入功率的效率仅为 。

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.
看待这种低效率的另一种方法是辐射声音对基本共振的阻尼效应。高水平的阻尼意味着高水平的辐射能量。

This acoustic damping can be derived from Equation (7):
该声阻尼可以从方程（7）导出：

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.
通过以下具有一些实用价值的示例，可以清楚地看出这种声学阻尼非常有限，这也表明电动扬声器的效率较低。

This results in: 这导致：